Subgroups — C₆₇×#61084 #30740335

Order 17152.

2834 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3496	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3283	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3280	C₂	1	4	yes	maximal, complemented
5	D₄×D₈×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3273	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3272	C₂	1	4	yes	maximal, complemented
8	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3061	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3036	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3012	C₂	1	4	yes	maximal, complemented
12	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	4	yes	complemented
15	#576×C₆₇	—	2	16	no	—
16	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	complemented
17	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	8	yes	complemented
18	C₂×#135×C₆₇	C₂²	1	4	yes	complemented
19	C₂×#135×C₆₇	C₂²	1	2	yes	complemented
20	C₂×#135×C₆₇	C₂²	1	2	yes	complemented
21	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×D₈×C₆₇	C₂²	1	4	yes	complemented
23	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
24	#547×C₆₇	—	2	2	no	—
25	#547×C₆₇	C₂²	1	4	yes	complemented
26	#546×C₆₇	—	2	4	no	—
27	#546×C₆₇	C₂²	1	8	yes	complemented
28	D₄²×C₆₇	—	2	2	no	—
29	D₄²×C₆₇	C₂²	1	4	yes	complemented
30	#537×C₆₇	C₂²	1	4	yes	complemented
31	#534×C₆₇	—	2	4	no	—
32	#532×C₆₇	C₂²	1	1	yes	characteristic
33	#532×C₆₇	C₂²	1	2	yes	complemented
34	C₂×#121×C₆₇	—	2	4	no	—
35	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	complemented
36	#495×C₆₇	C₂²	1	2	yes	complemented
37	#495×C₆₇	C₂²	1	2	yes	complemented
38	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	#493×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	#492×C₆₇	C₂²	1	2	yes	complemented
41	#468×C₆₇	C₂²	1	8	yes	complemented
42	#467×C₆₇	C₂²	1	4	yes	complemented
43	#466×C₆₇	C₂²	1	4	yes	complemented
44	#463×C₆₇	C₂²	1	8	yes	complemented
45	#461×C₆₇	C₂²	1	4	yes	complemented
46	#459×C₆₇	C₂²	1	4	yes	complemented
47	#452×C₆₇	C₂²	1	8	yes	complemented
48	#450×C₆₇	C₂²	1	8	yes	complemented
49	#449×C₆₇	C₂²	1	8	yes	complemented
50	#447×C₆₇	C₂²	1	8	yes	complemented
51	C₄×D₈×C₆₇	C₂²	1	2	yes	complemented
52	C₄×D₈×C₆₇	C₂²	1	2	yes	complemented
53	C₄×D₈×C₆₇	C₂²	1	2	yes	complemented
54	C₄×D₈×C₆₇	C₂²	1	2	yes	complemented
55	#436×C₆₇	C₂²	1	2	yes	complemented
56	#436×C₆₇	C₂²	1	2	yes	complemented
57	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
58	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
59	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic, complemented
60	#423×C₆₇	C₂²	1	1	yes	characteristic, complemented
61	#417×C₆₇	C₂²	1	4	yes	complemented
62	C₂×#102×C₆₇	C₂²	1	2	yes	complemented
63	#408×C₆₇	C₂²	1	2	yes	complemented
64	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
65	H(5,𝔽₂)×C₆₇	—	2	16	no	—
66	H(5,𝔽₂)×C₆₇	—	4	8	no	—
67	C₂×#41×C₆₇	—	2	4	no	—
68	C₂×#41×C₆₇	C₂³	1	4	yes	—
69	C₂×#41×C₆₇	C₂³	1	8	yes	—
70	C₂×#41×C₆₇	C₂³	1	2	yes	—
71	C₂²×D₄×C₆₇	—	2	8	no	—
72	C₂²×D₄×C₆₇	—	2	4	no	—
73	C₂²×D₄×C₆₇	—	2	4	no	—
74	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
75	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
76	C₂³×C₄×C₆₇	—	2	2	no	—
77	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
78	AGL(1,ℤ/8)×C₆₇	—	2	32	no	—
79	#135×C₆₇	—	2	16	no	—
80	#135×C₆₇	—	2	8	no	—
81	#135×C₆₇	—	4	4	no	—
82	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
83	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
84	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
85	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
86	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
87	C₂×D₈×C₆₇	—	2	16	no	—
88	C₂×D₈×C₆₇	D₄	1	8	yes	complemented
89	C₂×D₈×C₆₇	C₂³	1	8	yes	—
90	C₂×D₈×C₆₇	C₂³	1	2	yes	—
91	C₂×D₈×C₆₇	C₂³	1	2	yes	—
92	C₂×D₈×C₆₇	C₂³	1	2	yes	—
93	C₂×D₈×C₆₇	C₂³	1	2	yes	—
94	#131×C₆₇	—	2	8	no	—
95	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
96	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
97	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
98	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
100	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
101	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
102	#127×C₆₇	D₄	1	1	yes	characteristic
103	#127×C₆₇	C₂³	1	2	yes	—
104	#124×C₆₇	D₄	1	2	yes	—
105	#124×C₆₇	C₂³	1	4	yes	—
106	#123×C₆₇	—	2	8	no	—
107	#123×C₆₇	—	2	8	no	—
108	C₂²⋊Q₈×C₆₇	—	2	4	no	—
109	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
110	#121×C₆₇	—	2	8	no	—
111	#121×C₆₇	—	2	4	no	—
112	#121×C₆₇	—	2	4	no	—
113	#121×C₆₇	—	2	2	no	—
114	#121×C₆₇	—	2	2	no	—
115	#121×C₆₇	—	2	4	no	—
116	#121×C₆₇	—	2	8	no	—
117	#121×C₆₇	D₄	1	2	yes	—
118	#121×C₆₇	D₄	1	2	yes	—
119	#121×C₆₇	C₂³	1	8	yes	—
120	#120×C₆₇	—	2	8	no	—
121	#120×C₆₇	—	2	8	no	—
122	C₄×D₄×C₆₇	—	2	2	no	—
123	C₄×D₄×C₆₇	—	2	2	no	—
124	C₄×D₄×C₆₇	—	2	2	no	—
125	C₄×D₄×C₆₇	—	2	2	no	—
126	C₄×D₄×C₆₇	C₂³	1	4	yes	—
127	C₄×D₄×C₆₇	C₂³	1	4	yes	—
128	C₄×D₄×C₆₇	C₂³	1	2	yes	—
129	C₄×D₄×C₆₇	C₂³	1	2	yes	—
130	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
131	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
132	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
133	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
134	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
135	#107×C₆₇	C₂³	1	2	yes	—
136	#107×C₆₇	C₂³	1	2	yes	—
137	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
138	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
139	#103×C₆₇	C₂³	1	8	yes	—
140	#102×C₆₇	C₂³	1	4	yes	—
141	#102×C₆₇	C₂³	1	8	yes	—
142	#102×C₆₇	C₂³	1	4	yes	—
143	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
144	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
145	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
146	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
147	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
148	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
149	C₂⁴×C₆₇	—	2	2	no	—
150	C₂⁴×C₆₇	—	2	8	no	—
151	#41×C₆₇	—	2	4	no	—
152	#41×C₆₇	—	2	4	no	—
153	#41×C₆₇	—	2	16	no	—
154	#41×C₆₇	—	2	8	no	—
155	#41×C₆₇	—	2	8	no	—
156	#41×C₆₇	—	4	8	no	—
157	#41×C₆₇	—	4	16	no	—
158	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
159	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
160	C₂×D₄×C₆₇	—	2	16	no	—
161	C₂×D₄×C₆₇	—	2	8	no	—
162	C₂×D₄×C₆₇	—	2	4	no	—
163	C₂×D₄×C₆₇	—	2	8	no	—
164	C₂×D₄×C₆₇	—	2	2	no	—
165	C₂×D₄×C₆₇	—	2	2	no	—
166	C₂×D₄×C₆₇	—	2	8	no	—
167	C₂×D₄×C₆₇	—	2	2	no	—
168	C₂×D₄×C₆₇	—	2	4	no	—
169	C₂×D₄×C₆₇	—	4	8	no	—
170	C₂×D₄×C₆₇	—	4	16	no	—
171	C₂×D₄×C₆₇	—	4	32	no	—
172	C₂×D₄×C₆₇	—	8	8	no	—
173	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
174	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
175	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
176	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
177	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	complemented
178	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
179	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₆₇	—	2	2	no	—
181	C₂²×C₄×C₆₇	—	2	8	no	—
182	C₂²×C₄×C₆₇	—	2	4	no	—
183	C₂²×C₄×C₆₇	—	2	4	no	—
184	C₂²×C₄×C₆₇	—	2	4	no	—
185	C₂²×C₄×C₆₇	—	8	4	no	—
186	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
190	Q₁₆×C₆₇	—	2	4	no	—
191	Q₁₆×C₆₇	—	4	2	no	—
192	SD₁₆×C₆₇	—	2	16	no	—
193	SD₁₆×C₆₇	—	2	8	no	—
194	SD₁₆×C₆₇	—	4	4	no	—
195	D₈×C₆₇	—	2	16	no	—
196	D₈×C₆₇	—	2	4	no	—
197	D₈×C₆₇	—	4	2	no	—
198	D₈×C₆₇	C₂×D₄	1	8	yes	complemented
199	D₈×C₆₇	C₂×D₄	1	8	yes	complemented
200	M₁₆×C₆₇	—	2	4	no	—
201	M₁₆×C₆₇	—	2	8	no	—
202	C₂×C₈×C₆₇	—	2	2	no	—
203	C₂×C₈×C₆₇	—	2	8	no	—
204	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	complemented
205	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
206	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
207	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
208	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
209	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
210	C₄⋊C₄×C₆₇	—	2	2	no	—
211	C₄⋊C₄×C₆₇	—	2	8	no	—
212	C₄⋊C₄×C₆₇	—	2	2	no	—
213	C₄⋊C₄×C₆₇	—	2	2	no	—
214	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
215	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
216	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
217	C₂²⋊C₄×C₆₇	—	2	8	no	—
218	C₂²⋊C₄×C₆₇	—	2	4	no	—
219	C₂²⋊C₄×C₆₇	—	2	4	no	—
220	C₂²⋊C₄×C₆₇	—	2	4	no	—
221	C₂²⋊C₄×C₆₇	—	2	8	no	—
222	C₂²⋊C₄×C₆₇	—	2	4	no	—
223	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
224	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
225	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
227	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
228	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
229	C₂³×C₆₇	—	2	4	no	—
230	C₂³×C₆₇	—	2	8	no	—
231	C₂³×C₆₇	—	2	4	no	—
232	C₂³×C₆₇	—	4	16	no	—
233	C₂³×C₆₇	—	8	2	no	—
234	C₂³×C₆₇	—	8	2	no	—
235	C₂³×C₆₇	—	8	8	no	—
236	C₂³×C₆₇	—	8	8	no	—
237	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
238	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
239	Q₈×C₆₇	—	2	4	no	—
240	Q₈×C₆₇	—	2	4	no	—
241	Q₈×C₆₇	—	2	4	no	—
242	D₄×C₆₇	—	2	8	no	—
243	D₄×C₆₇	—	2	8	no	—
244	D₄×C₆₇	—	2	4	no	—
245	D₄×C₆₇	—	2	4	no	—
246	D₄×C₆₇	—	2	4	no	—
247	D₄×C₆₇	—	2	4	no	—
248	D₄×C₆₇	—	2	4	no	—
249	D₄×C₆₇	—	4	2	no	—
250	D₄×C₆₇	—	4	16	no	—
251	D₄×C₆₇	—	4	16	no	—
252	D₄×C₆₇	—	4	8	no	—
253	D₄×C₆₇	—	4	4	no	—
254	D₄×C₆₇	—	8	2	no	—
255	D₄×C₆₇	—	8	8	no	—
256	D₄×C₆₇	C₂²×D₄	1	8	yes	—
257	C₂×C₄×C₆₇	—	2	2	no	—
258	C₂×C₄×C₆₇	—	2	4	no	—
259	C₂×C₄×C₆₇	—	2	4	no	—
260	C₂×C₄×C₆₇	—	2	4	no	—
261	C₂×C₄×C₆₇	—	2	4	no	—
262	C₂×C₄×C₆₇	—	4	1	no	—
263	C₂×C₄×C₆₇	—	4	16	no	—
264	C₂×C₄×C₆₇	—	4	8	no	—
265	C₂×C₄×C₆₇	—	4	8	no	—
266	C₂×C₄×C₆₇	—	8	4	no	—
267	C₂×C₄×C₆₇	—	8	2	no	—
268	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
272	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
276	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
277	C₈×C₆₇	—	2	4	no	—
278	C₈×C₆₇	—	2	1	no	—
279	C₈×C₆₇	—	2	1	no	—
280	C₈×C₆₇	C₂²×D₄	1	2	yes	—
281	C₈×C₆₇	C₂²×D₄	1	2	yes	—
282	C₂²×C₆₇	—	2	1	no	—
283	C₂²×C₆₇	—	2	1	no	—
284	C₂²×C₆₇	—	2	4	no	—
285	C₂²×C₆₇	—	2	4	no	—
286	C₂²×C₆₇	—	2	8	no	—
287	C₂²×C₆₇	—	2	4	no	—
288	C₂²×C₆₇	—	4	4	no	—
289	C₂²×C₆₇	—	4	4	no	—
290	C₂²×C₆₇	—	4	4	no	—
291	C₂²×C₆₇	—	4	4	no	—
292	C₂²×C₆₇	—	4	4	no	—
293	C₂²×C₆₇	—	8	8	no	—
294	C₂²×C₆₇	—	16	16	no	—
295	C₂²×C₆₇	D₄²	1	2	yes	—
296	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
297	C₄×C₆₇	—	2	1	no	—
298	C₄×C₆₇	—	2	1	no	—
299	C₄×C₆₇	—	2	4	no	—
300	C₄×C₆₇	—	2	1	no	—
301	C₄×C₆₇	—	4	4	no	—
302	C₄×C₆₇	—	4	4	no	—
303	C₄×C₆₇	D₄²	1	2	yes	—
304	C₄×C₆₇	#576	1	2	yes	complemented
305	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
306	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
307	#61084	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
308	C₂×C₆₇	—	2	2	no	—
309	C₂×C₆₇	—	4	4	no	—
310	C₂×C₆₇	—	4	8	no	—
311	C₂×C₆₇	—	8	4	no	—
312	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
313	C₂×C₆₇	C₂×#576	1	1	yes	characteristic
314	C₂×C₆₇	#3584	1	1	yes	characteristic
315	C₂×#576	C₂×C₆₇	1	2	yes	complemented
316	#3496	C₂×C₆₇	1	2	yes	complemented
317	#3283	C₂×C₆₇	1	2	yes	complemented
318	#3280	C₂×C₆₇	1	4	yes	complemented
319	D₄×D₈	C₂×C₆₇	1	2	yes	complemented
320	#3273	C₂×C₆₇	1	4	yes	complemented
321	#3272	C₂×C₆₇	1	4	yes	complemented
322	#3147	C₂×C₆₇	1	1	yes	characteristic, complemented
323	#3061	C₂×C₆₇	1	2	yes	complemented
324	#3036	C₂×C₆₇	1	2	yes	complemented
325	#3012	C₂×C₆₇	1	4	yes	complemented
326	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
327	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
328	C₆₇	#61084	1	1	yes	characteristic, direct product factor, Sylow
329	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	4	yes	complemented
330	#576	—	2	16	no	—
331	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	complemented
332	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	8	yes	complemented
333	C₂×#135	C₂²×C₆₇	1	2	yes	complemented
334	C₂×#135	C₂²×C₆₇	1	4	yes	complemented
335	C₂×#135	C₂²×C₆₇	1	2	yes	complemented
336	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
337	C₂²×D₈	C₂²×C₆₇	1	4	yes	complemented
338	C₂×#131	C₂²×C₆₇	1	2	yes	complemented
339	#547	—	2	2	no	—
340	#547	C₂²×C₆₇	1	4	yes	complemented
341	#546	—	2	4	no	—
342	#546	C₂²×C₆₇	1	8	yes	complemented
343	D₄²	—	2	2	no	—
344	D₄²	C₂²×C₆₇	1	4	yes	complemented
345	#537	C₂²×C₆₇	1	4	yes	complemented
346	#534	—	2	4	no	—
347	#532	C₂²×C₆₇	1	1	yes	characteristic
348	#532	C₂²×C₆₇	1	2	yes	complemented
349	C₂×#121	—	2	4	no	—
350	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	complemented
351	#495	C₂²×C₆₇	1	2	yes	complemented
352	#495	C₂²×C₆₇	1	2	yes	complemented
353	#494	C₂²×C₆₇	1	1	yes	characteristic, complemented
354	#493	C₂²×C₆₇	1	1	yes	characteristic, complemented
355	#492	C₂²×C₆₇	1	2	yes	complemented
356	#468	C₂²×C₆₇	1	8	yes	complemented
357	#467	C₂²×C₆₇	1	4	yes	complemented
358	#466	C₂²×C₆₇	1	4	yes	complemented
359	#463	C₂²×C₆₇	1	8	yes	complemented
360	#461	C₂²×C₆₇	1	4	yes	complemented
361	#459	C₂²×C₆₇	1	4	yes	complemented
362	#452	C₂²×C₆₇	1	8	yes	complemented
363	#450	C₂²×C₆₇	1	8	yes	complemented
364	#449	C₂²×C₆₇	1	8	yes	complemented
365	#447	C₂²×C₆₇	1	8	yes	complemented
366	C₄×D₈	C₂²×C₆₇	1	2	yes	complemented
367	C₄×D₈	C₂²×C₆₇	1	2	yes	complemented
368	C₄×D₈	C₂²×C₆₇	1	2	yes	complemented
369	C₄×D₈	C₂²×C₆₇	1	2	yes	complemented
370	#436	C₂²×C₆₇	1	2	yes	complemented
371	#436	C₂²×C₆₇	1	2	yes	complemented
372	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
373	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
374	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic, complemented
375	#423	C₂²×C₆₇	1	1	yes	characteristic, complemented
376	#417	C₂²×C₆₇	1	4	yes	complemented
377	C₂×#102	C₂²×C₆₇	1	2	yes	complemented
378	#408	C₂²×C₆₇	1	2	yes	complemented
379	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
380	H(5,𝔽₂)	—	2	16	no	—
381	H(5,𝔽₂)	—	4	8	no	—
382	C₂×#41	—	2	4	no	—
383	C₂×#41	C₂³×C₆₇	1	4	yes	—
384	C₂×#41	C₂³×C₆₇	1	8	yes	—
385	C₂×#41	C₂³×C₆₇	1	2	yes	—
386	C₂²×D₄	—	2	8	no	—
387	C₂²×D₄	—	2	4	no	—
388	C₂²×D₄	—	2	4	no	—
389	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
390	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
391	C₂³×C₄	—	2	2	no	—
392	AGL(1,ℤ/8)	—	2	16	no	—
393	AGL(1,ℤ/8)	—	2	32	no	—
394	#135	—	2	16	no	—
395	#135	—	2	8	no	—
396	#135	—	4	4	no	—
397	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
398	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
399	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
400	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
401	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
402	C₂×D₈	—	2	16	no	—
403	C₂×D₈	D₄×C₆₇	1	8	yes	complemented
404	C₂×D₈	C₂³×C₆₇	1	2	yes	—
405	C₂×D₈	C₂³×C₆₇	1	8	yes	—
406	C₂×D₈	C₂³×C₆₇	1	2	yes	—
407	C₂×D₈	C₂³×C₆₇	1	2	yes	—
408	C₂×D₈	C₂³×C₆₇	1	2	yes	—
409	#131	—	2	8	no	—
410	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
411	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
412	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
413	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
414	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
415	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
416	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
417	#127	D₄×C₆₇	1	1	yes	characteristic
418	#127	C₂³×C₆₇	1	2	yes	—
419	#124	D₄×C₆₇	1	2	yes	—
420	#124	C₂³×C₆₇	1	4	yes	—
421	#123	—	2	8	no	—
422	#123	—	2	8	no	—
423	C₂²⋊Q₈	—	2	4	no	—
424	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
425	#121	—	2	8	no	—
426	#121	—	2	4	no	—
427	#121	—	2	4	no	—
428	#121	—	2	2	no	—
429	#121	—	2	2	no	—
430	#121	—	2	4	no	—
431	#121	—	2	8	no	—
432	#121	D₄×C₆₇	1	2	yes	—
433	#121	D₄×C₆₇	1	2	yes	—
434	#121	C₂³×C₆₇	1	8	yes	—
435	#120	—	2	8	no	—
436	#120	—	2	8	no	—
437	C₄×D₄	—	2	2	no	—
438	C₄×D₄	—	2	2	no	—
439	C₄×D₄	—	2	2	no	—
440	C₄×D₄	—	2	2	no	—
441	C₄×D₄	C₂³×C₆₇	1	4	yes	—
442	C₄×D₄	C₂³×C₆₇	1	4	yes	—
443	C₄×D₄	C₂³×C₆₇	1	2	yes	—
444	C₄×D₄	C₂³×C₆₇	1	2	yes	—
445	C₂×C₄⋊C₄	—	2	4	no	—
446	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
447	C₂×C₂²⋊C₄	—	2	2	no	—
448	C₂×C₂²⋊C₄	—	2	8	no	—
449	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
450	#107	C₂³×C₆₇	1	2	yes	—
451	#107	C₂³×C₆₇	1	2	yes	—
452	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
453	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
454	#103	C₂³×C₆₇	1	8	yes	—
455	#102	C₂³×C₆₇	1	4	yes	—
456	#102	C₂³×C₆₇	1	4	yes	—
457	#102	C₂³×C₆₇	1	8	yes	—
458	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
459	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
460	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
461	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
462	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
463	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
464	C₂⁴	—	2	2	no	—
465	C₂⁴	—	2	8	no	—
466	#41	—	2	4	no	—
467	#41	—	2	4	no	—
468	#41	—	2	16	no	—
469	#41	—	2	8	no	—
470	#41	—	2	8	no	—
471	#41	—	4	8	no	—
472	#41	—	4	16	no	—
473	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
474	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
475	C₂×D₄	—	2	16	no	—
476	C₂×D₄	—	2	8	no	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	8	no	—
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	8	no	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	4	8	no	—
485	C₂×D₄	—	4	16	no	—
486	C₂×D₄	—	4	32	no	—
487	C₂×D₄	—	8	8	no	—
488	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
489	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
490	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
491	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
492	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	complemented
493	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
494	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
495	C₂²×C₄	—	2	2	no	—
496	C₂²×C₄	—	2	8	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	2	4	no	—
499	C₂²×C₄	—	2	4	no	—
500	C₂²×C₄	—	8	4	no	—
501	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
502	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
503	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
504	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
505	Q₁₆	—	2	4	no	—
506	Q₁₆	—	4	2	no	—
507	SD₁₆	—	2	16	no	—
508	SD₁₆	—	2	8	no	—
509	SD₁₆	—	4	4	no	—
510	D₈	—	2	16	no	—
511	D₈	—	2	4	no	—
512	D₈	—	4	2	no	—
513	D₈	C₂×D₄×C₆₇	1	8	yes	complemented
514	D₈	C₂×D₄×C₆₇	1	8	yes	complemented
515	M₁₆	—	2	4	no	—
516	M₁₆	—	2	8	no	—
517	C₂×C₈	—	2	2	no	—
518	C₂×C₈	—	2	8	no	—
519	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	complemented
520	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
521	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
522	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
523	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
524	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	8	no	—
527	C₄⋊C₄	—	2	2	no	—
528	C₄⋊C₄	—	2	2	no	—
529	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
530	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
531	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
532	C₂²⋊C₄	—	2	8	no	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	8	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
539	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
540	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
541	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
542	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
543	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
544	C₂³	—	2	4	no	—
545	C₂³	—	2	8	no	—
546	C₂³	—	2	4	no	—
547	C₂³	—	4	16	no	—
548	C₂³	—	8	2	no	—
549	C₂³	—	8	2	no	—
550	C₂³	—	8	8	no	—
551	C₂³	—	8	8	no	—
552	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
553	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
554	Q₈	—	2	4	no	—
555	Q₈	—	2	4	no	—
556	Q₈	—	2	4	no	—
557	D₄	—	2	8	no	—
558	D₄	—	2	8	no	—
559	D₄	—	2	4	no	—
560	D₄	—	2	4	no	—
561	D₄	—	2	4	no	—
562	D₄	—	2	4	no	—
563	D₄	—	2	4	no	—
564	D₄	—	4	2	no	—
565	D₄	—	4	16	no	—
566	D₄	—	4	16	no	—
567	D₄	—	4	8	no	—
568	D₄	—	4	4	no	—
569	D₄	—	8	2	no	—
570	D₄	—	8	8	no	—
571	D₄	C₂²×D₄×C₆₇	1	8	yes	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	4	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	16	no	—
579	C₂×C₄	—	4	8	no	—
580	C₂×C₄	—	4	8	no	—
581	C₂×C₄	—	8	4	no	—
582	C₂×C₄	—	8	2	no	—
583	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
584	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
585	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
586	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
587	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
588	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
589	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
590	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
591	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
592	C₈	—	2	4	no	—
593	C₈	—	2	1	no	—
594	C₈	—	2	1	no	—
595	C₈	C₂²×D₄×C₆₇	1	2	yes	—
596	C₈	C₂²×D₄×C₆₇	1	2	yes	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	2	4	no	—
600	C₂²	—	2	4	no	—
601	C₂²	—	2	4	no	—
602	C₂²	—	2	8	no	—
603	C₂²	—	4	4	no	—
604	C₂²	—	4	4	no	—
605	C₂²	—	4	4	no	—
606	C₂²	—	4	4	no	—
607	C₂²	—	4	4	no	—
608	C₂²	—	8	8	no	—
609	C₂²	—	16	16	no	—
610	C₂²	D₄²×C₆₇	1	2	yes	—
611	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
612	C₄	—	2	1	no	—
613	C₄	—	2	1	no	—
614	C₄	—	2	1	no	—
615	C₄	—	2	4	no	—
616	C₄	—	4	4	no	—
617	C₄	—	4	4	no	—
618	C₄	D₄²×C₆₇	1	2	yes	—
619	C₄	#576×C₆₇	1	2	yes	complemented
620	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
621	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
622	C₂	—	2	2	no	—
623	C₂	—	4	4	no	—
624	C₂	—	4	8	no	—
625	C₂	—	8	4	no	—
626	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
627	C₂	C₂×#576×C₆₇	1	1	yes	characteristic
628	C₂	C₆₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #532×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#576×C₆₇	C₂	semidirect
2. C₆₇×#3496	C₂	semidirect
3. C₆₇×#3283	C₂	semidirect
4. C₆₇×#3280	C₂	semidirect
5. D₄×D₈×C₆₇	C₂	semidirect
6. C₆₇×#3273	C₂	semidirect
7. C₆₇×#3272	C₂	semidirect
8. C₆₇×#3147	C₂	semidirect
9. C₆₇×#3061	C₂	semidirect
10. C₆₇×#3036	C₂	semidirect
11. C₆₇×#3012	C₂	semidirect
12. C₂×C₄×D₈×C₆₇	C₂	semidirect
13. C₆₇×#2931	C₂	semidirect
14. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
16. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
17. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
18. C₂×#135×C₆₇	C₂²	semidirect
19. C₂×#135×C₆₇	C₂²	semidirect
20. C₂×#135×C₆₇	C₂²	semidirect
22. C₂²×D₈×C₆₇	C₂²	semidirect
23. C₂×#131×C₆₇	C₂²	semidirect
25. #547×C₆₇	C₂²	semidirect
27. #546×C₆₇	C₂²	semidirect
29. D₄²×C₆₇	C₂²	semidirect
30. #537×C₆₇	C₂²	semidirect
33. #532×C₆₇	C₂²	semidirect
35. C₂×C₄×D₄×C₆₇	C₂²	semidirect
36. #495×C₆₇	C₂²	semidirect
37. #495×C₆₇	C₂²	semidirect
38. #494×C₆₇	C₂²	semidirect
39. #493×C₆₇	C₂²	semidirect
40. #492×C₆₇	C₂²	semidirect
41. #468×C₆₇	C₂²	semidirect
42. #467×C₆₇	C₂²	semidirect
43. #466×C₆₇	C₂²	semidirect
44. #463×C₆₇	C₂²	semidirect
45. #461×C₆₇	C₂²	semidirect
46. #459×C₆₇	C₂²	semidirect
47. #452×C₆₇	C₂²	semidirect
48. #450×C₆₇	C₂²	semidirect
49. #449×C₆₇	C₂²	semidirect
50. #447×C₆₇	C₂²	semidirect
51. C₄×D₈×C₆₇	C₂²	semidirect
52. C₄×D₈×C₆₇	C₂²	semidirect
53. C₄×D₈×C₆₇	C₂²	semidirect
54. C₄×D₈×C₆₇	C₂²	semidirect
55. #436×C₆₇	C₂²	semidirect
56. #436×C₆₇	C₂²	semidirect
57. C₈×D₄×C₆₇	C₂²	semidirect
58. C₈×D₄×C₆₇	C₂²	semidirect
59. C₂×#107×C₆₇	C₂²	semidirect
60. #423×C₆₇	C₂²	semidirect
61. #417×C₆₇	C₂²	semidirect
62. C₂×#102×C₆₇	C₂²	semidirect
63. #408×C₆₇	C₂²	semidirect
64. C₂×C₄×C₈×C₆₇	C₂²	semidirect
88. C₂×D₈×C₆₇	D₄	semidirect
177. C₂×D₄×C₆₇	C₂×D₄	semidirect
198. D₈×C₆₇	C₂×D₄	semidirect
199. D₈×C₆₇	C₂×D₄	semidirect
204. C₂×C₈×C₆₇	C₂×D₄	semidirect
304. C₄×C₆₇	#576	semidirect
307. #61084	C₆₇	direct (N × H)
315. C₂×#576	C₂×C₆₇	semidirect
316. #3496	C₂×C₆₇	semidirect
317. #3283	C₂×C₆₇	semidirect
318. #3280	C₂×C₆₇	semidirect
319. D₄×D₈	C₂×C₆₇	semidirect
320. #3273	C₂×C₆₇	semidirect
321. #3272	C₂×C₆₇	semidirect
322. #3147	C₂×C₆₇	semidirect
323. #3061	C₂×C₆₇	semidirect
324. #3036	C₂×C₆₇	semidirect
325. #3012	C₂×C₆₇	semidirect
326. C₂×C₄×D₈	C₂×C₆₇	semidirect
327. #2931	C₂×C₆₇	semidirect
328. C₆₇	#61084	direct (N × H)
329. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
331. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
332. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
333. C₂×#135	C₂²×C₆₇	semidirect
334. C₂×#135	C₂²×C₆₇	semidirect
335. C₂×#135	C₂²×C₆₇	semidirect
337. C₂²×D₈	C₂²×C₆₇	semidirect
338. C₂×#131	C₂²×C₆₇	semidirect
340. #547	C₂²×C₆₇	semidirect
342. #546	C₂²×C₆₇	semidirect
344. D₄²	C₂²×C₆₇	semidirect
345. #537	C₂²×C₆₇	semidirect
348. #532	C₂²×C₆₇	semidirect
350. C₂×C₄×D₄	C₂²×C₆₇	semidirect
351. #495	C₂²×C₆₇	semidirect
352. #495	C₂²×C₆₇	semidirect
353. #494	C₂²×C₆₇	semidirect
354. #493	C₂²×C₆₇	semidirect
355. #492	C₂²×C₆₇	semidirect
356. #468	C₂²×C₆₇	semidirect
357. #467	C₂²×C₆₇	semidirect
358. #466	C₂²×C₆₇	semidirect
359. #463	C₂²×C₆₇	semidirect
360. #461	C₂²×C₆₇	semidirect
361. #459	C₂²×C₆₇	semidirect
362. #452	C₂²×C₆₇	semidirect
363. #450	C₂²×C₆₇	semidirect
364. #449	C₂²×C₆₇	semidirect
365. #447	C₂²×C₆₇	semidirect
366. C₄×D₈	C₂²×C₆₇	semidirect
367. C₄×D₈	C₂²×C₆₇	semidirect
368. C₄×D₈	C₂²×C₆₇	semidirect
369. C₄×D₈	C₂²×C₆₇	semidirect
370. #436	C₂²×C₆₇	semidirect
371. #436	C₂²×C₆₇	semidirect
372. C₈×D₄	C₂²×C₆₇	semidirect
373. C₈×D₄	C₂²×C₆₇	semidirect
374. C₂×#107	C₂²×C₆₇	semidirect
375. #423	C₂²×C₆₇	semidirect
376. #417	C₂²×C₆₇	semidirect
377. C₂×#102	C₂²×C₆₇	semidirect
378. #408	C₂²×C₆₇	semidirect
379. C₂×C₄×C₈	C₂²×C₆₇	semidirect
403. C₂×D₈	D₄×C₆₇	semidirect
492. C₂×D₄	C₂×D₄×C₆₇	semidirect
513. D₈	C₂×D₄×C₆₇	semidirect
514. D₈	C₂×D₄×C₆₇	semidirect
519. C₂×C₈	C₂×D₄×C₆₇	semidirect
619. C₄	#576×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#576×C₆₇	2
C₆₇×#3496	2
C₆₇×#3283	2
C₆₇×#3280	2
D₄×D₈×C₆₇	2
C₆₇×#3273	2
C₆₇×#3272	2
C₆₇×#3147	2
C₆₇×#3061	2
C₆₇×#3036	2
C₆₇×#3012	2
C₂×C₄×D₈×C₆₇	2
C₆₇×#2931	2
#61084	67

Subgroups — C67×#61084 #30740335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61084 #30740335