Subgroups — C₆₇×#61088 #30740339

Order 17152.

2714 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#577×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3500	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3282	C₂	1	8	yes	maximal, complemented
4	C₆₇×#3277	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3273	C₂	1	4	yes	maximal, complemented
6	C₂×#495×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#3057	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3040	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3011	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3008	C₂	1	2	yes	maximal, complemented
11	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2931	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	2	yes	—
14	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	2	yes	complemented
15	#577×C₆₇	—	2	16	no	—
16	C₂×#137×C₆₇	C₂²	1	2	yes	complemented
17	C₂×#137×C₆₇	C₂²	1	4	yes	complemented
18	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	2	yes	—
19	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	complemented
20	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#135×C₆₇	C₂²	1	4	yes	complemented
22	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
23	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
24	C₂²×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
25	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#131×C₆₇	C₂²	1	2	yes	complemented
27	#548×C₆₇	C₂²	1	8	yes	complemented
28	#546×C₆₇	—	2	8	no	—
29	#546×C₆₇	C₂²	1	8	yes	complemented
30	#536×C₆₇	—	2	2	no	—
31	#536×C₆₇	C₂²	1	2	yes	complemented
32	#535×C₆₇	—	2	2	no	—
33	#535×C₆₇	C₂²	1	2	yes	—
34	#532×C₆₇	—	2	2	no	—
35	#532×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#124×C₆₇	—	2	2	no	—
37	C₂×#124×C₆₇	C₂²	1	2	yes	complemented
38	C₄×#41×C₆₇	C₂²	1	2	yes	complemented
39	#495×C₆₇	C₂²	1	4	yes	complemented
40	#495×C₆₇	C₂²	1	4	yes	complemented
41	#468×C₆₇	C₂²	1	8	yes	complemented
42	#465×C₆₇	C₂²	1	8	yes	complemented
43	#464×C₆₇	C₂²	1	8	yes	complemented
44	#463×C₆₇	C₂²	1	8	yes	complemented
45	#452×C₆₇	C₂²	1	8	yes	complemented
46	#451×C₆₇	C₂²	1	8	yes	complemented
47	#449×C₆₇	C₂²	1	8	yes	complemented
48	#447×C₆₇	C₂²	1	8	yes	complemented
49	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	complemented
50	C₄×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
51	C₄×D₈×C₆₇	C₂²	1	2	yes	complemented
52	#436×C₆₇	C₂²	1	4	yes	complemented
53	C₈×D₄×C₆₇	C₂²	1	4	yes	complemented
54	#427×C₆₇	C₂²	1	1	yes	characteristic, complemented
55	#423×C₆₇	C₂²	1	1	yes	characteristic, complemented
56	#419×C₆₇	C₂²	1	2	yes	complemented
57	#418×C₆₇	C₂²	1	2	yes	complemented
58	#416×C₆₇	C₂²	1	2	yes	complemented
59	#408×C₆₇	C₂²	1	2	yes	complemented
60	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
61	2₋¹⁺⁴×C₆₇	—	2	8	no	—
62	2₋¹⁺⁴×C₆₇	—	4	4	no	—
63	H(5,𝔽₂)×C₆₇	—	2	8	no	—
64	H(5,𝔽₂)×C₆₇	—	4	4	no	—
65	C₂×#41×C₆₇	—	2	2	no	—
66	C₂×#41×C₆₇	—	2	2	no	—
67	C₂×#41×C₆₇	—	2	4	no	—
68	C₂×#41×C₆₇	C₂³	1	4	yes	—
69	C₂×#41×C₆₇	C₂³	1	4	yes	—
70	C₂×#41×C₆₇	C₂³	1	2	yes	—
71	C₂×#41×C₆₇	C₂³	1	2	yes	—
72	C₂²×Q₈×C₆₇	—	2	2	no	—
73	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
74	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
75	C₂²×D₄×C₆₇	—	2	2	no	—
76	C₂²×D₄×C₆₇	—	2	4	no	—
77	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
78	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
79	#137×C₆₇	—	2	8	no	—
80	#137×C₆₇	—	2	16	no	—
81	AGL(1,ℤ/8)×C₆₇	—	2	8	no	—
82	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
83	#135×C₆₇	—	2	16	no	—
84	#135×C₆₇	D₄	1	8	yes	complemented
85	C₂×Q₁₆×C₆₇	—	4	2	no	—
86	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
87	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
88	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×SD₁₆×C₆₇	—	2	16	no	—
90	C₂×SD₁₆×C₆₇	—	2	8	no	—
91	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
92	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
93	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
94	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
95	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
96	C₂×D₈×C₆₇	—	4	2	no	—
97	C₂×D₈×C₆₇	C₂³	1	4	yes	—
98	C₂×D₈×C₆₇	C₂³	1	4	yes	—
99	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
100	#131×C₆₇	—	2	8	no	—
101	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
102	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
103	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
104	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
105	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
106	C₄⋊Q₈×C₆₇	—	2	2	no	—
107	C₄⋊Q₈×C₆₇	D₄	1	1	yes	characteristic
108	#127×C₆₇	—	2	2	no	—
109	#127×C₆₇	D₄	1	1	yes	characteristic
110	#124×C₆₇	—	2	2	no	—
111	#124×C₆₇	—	2	2	no	—
112	#124×C₆₇	—	2	8	no	—
113	#124×C₆₇	—	2	2	no	—
114	#124×C₆₇	—	2	2	no	—
115	#124×C₆₇	D₄	1	2	yes	—
116	#124×C₆₇	C₂³	1	8	yes	—
117	#123×C₆₇	—	2	4	no	—
118	#123×C₆₇	—	2	8	no	—
119	C₂²⋊Q₈×C₆₇	—	2	8	no	—
120	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
121	#121×C₆₇	—	2	8	no	—
122	#121×C₆₇	—	2	8	no	—
123	#121×C₆₇	D₄	1	4	yes	—
124	#121×C₆₇	C₂³	1	8	yes	—
125	#120×C₆₇	—	2	8	no	—
126	#120×C₆₇	—	2	4	no	—
127	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
128	C₄×D₄×C₆₇	—	2	4	no	—
129	C₄×D₄×C₆₇	—	2	8	no	—
130	C₄×D₄×C₆₇	C₂³	1	4	yes	—
131	C₄×D₄×C₆₇	C₂³	1	4	yes	—
132	#117×C₆₇	—	2	2	no	—
133	#117×C₆₇	C₂³	1	2	yes	—
134	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
135	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
136	C₂×C₄²×C₆₇	—	2	2	no	—
137	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
138	#107×C₆₇	C₂³	1	2	yes	—
139	#106×C₆₇	C₂³	1	2	yes	—
140	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
141	#103×C₆₇	C₂³	1	4	yes	—
142	#103×C₆₇	C₂³	1	8	yes	—
143	#102×C₆₇	C₂³	1	8	yes	—
144	#102×C₆₇	C₂³	1	4	yes	—
145	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
146	C₄×C₈×C₆₇	C₂³	1	2	yes	—
147	C₄×C₈×C₆₇	C₂³	1	2	yes	—
148	C₂⁴×C₆₇	—	2	4	no	—
149	C₂⁴×C₆₇	—	2	2	no	—
150	#41×C₆₇	—	2	4	no	—
151	#41×C₆₇	—	2	4	no	—
152	#41×C₆₇	—	2	4	no	—
153	#41×C₆₇	—	2	8	no	—
154	#41×C₆₇	—	2	8	no	—
155	#41×C₆₇	—	4	4	no	—
156	#41×C₆₇	—	4	4	no	—
157	#41×C₆₇	—	4	8	no	—
158	#41×C₆₇	—	4	8	no	—
159	#41×C₆₇	—	4	16	no	—
160	#41×C₆₇	C₂×D₄	1	8	yes	—
161	C₂×Q₈×C₆₇	—	2	4	no	—
162	C₂×Q₈×C₆₇	—	2	2	no	—
163	C₂×Q₈×C₆₇	—	2	8	no	—
164	C₂×Q₈×C₆₇	—	2	4	no	—
165	C₂×Q₈×C₆₇	—	2	4	no	—
166	C₂×Q₈×C₆₇	—	4	4	no	—
167	C₂×Q₈×C₆₇	—	4	8	no	—
168	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
169	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
170	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
171	C₂×D₄×C₆₇	—	2	2	no	—
172	C₂×D₄×C₆₇	—	2	8	no	—
173	C₂×D₄×C₆₇	—	2	4	no	—
174	C₂×D₄×C₆₇	—	2	4	no	—
175	C₂×D₄×C₆₇	—	2	8	no	—
176	C₂×D₄×C₆₇	—	2	4	no	—
177	C₂×D₄×C₆₇	—	2	4	no	—
178	C₂×D₄×C₆₇	—	2	2	no	—
179	C₂×D₄×C₆₇	—	2	4	no	—
180	C₂×D₄×C₆₇	—	4	4	no	—
181	C₂×D₄×C₆₇	—	4	8	no	—
182	C₂×D₄×C₆₇	—	4	16	no	—
183	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
184	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
185	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
186	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
187	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₆₇	—	2	2	no	—
189	C₂²×C₄×C₆₇	—	2	2	no	—
190	C₂²×C₄×C₆₇	—	2	4	no	—
191	C₂²×C₄×C₆₇	—	2	4	no	—
192	C₂²×C₄×C₆₇	—	2	4	no	—
193	C₂²×C₄×C₆₇	—	2	4	no	—
194	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
196	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
197	Q₁₆×C₆₇	—	2	8	no	—
198	Q₁₆×C₆₇	—	4	4	no	—
199	Q₁₆×C₆₇	C₂×D₄	1	4	yes	—
200	SD₁₆×C₆₇	—	2	8	no	—
201	SD₁₆×C₆₇	—	2	8	no	—
202	SD₁₆×C₆₇	—	2	8	no	—
203	SD₁₆×C₆₇	—	2	8	no	—
204	SD₁₆×C₆₇	C₂×D₄	1	8	yes	—
205	D₈×C₆₇	—	2	8	no	—
206	D₈×C₆₇	—	4	4	no	—
207	D₈×C₆₇	C₂×D₄	1	4	yes	—
208	M₁₆×C₆₇	—	2	4	no	—
209	M₁₆×C₆₇	—	2	8	no	—
210	C₂×C₈×C₆₇	—	2	2	no	—
211	C₂×C₈×C₆₇	—	2	8	no	—
212	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	—
213	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
214	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
215	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
216	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
217	C₄⋊C₄×C₆₇	—	2	4	no	—
218	C₄⋊C₄×C₆₇	—	2	8	no	—
219	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
220	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
221	C₂²⋊C₄×C₆₇	—	2	8	no	—
222	C₂²⋊C₄×C₆₇	—	2	2	no	—
223	C₂²⋊C₄×C₆₇	—	2	8	no	—
224	C₂²⋊C₄×C₆₇	—	2	8	no	—
225	C₂²⋊C₄×C₆₇	—	2	8	no	—
226	C₂²⋊C₄×C₆₇	—	2	4	no	—
227	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
228	C₄²×C₆₇	—	2	2	no	—
229	C₄²×C₆₇	—	2	4	no	—
230	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
231	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
232	C₄²×C₆₇	C₂⁴	1	2	yes	—
233	C₂³×C₆₇	—	2	2	no	—
234	C₂³×C₆₇	—	2	4	no	—
235	C₂³×C₆₇	—	4	8	no	—
236	C₂³×C₆₇	—	4	4	no	—
237	C₂³×C₆₇	—	8	4	no	—
238	C₂³×C₆₇	—	8	8	no	—
239	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
240	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	Q₈×C₆₇	—	2	4	no	—
242	Q₈×C₆₇	—	2	2	no	—
243	Q₈×C₆₇	—	2	4	no	—
244	Q₈×C₆₇	—	2	4	no	—
245	Q₈×C₆₇	—	4	8	no	—
246	Q₈×C₆₇	—	4	4	no	—
247	Q₈×C₆₇	C₂²×D₄	1	4	yes	—
248	D₄×C₆₇	—	2	8	no	—
249	D₄×C₆₇	—	2	8	no	—
250	D₄×C₆₇	—	2	4	no	—
251	D₄×C₆₇	—	2	4	no	—
252	D₄×C₆₇	—	2	2	no	—
253	D₄×C₆₇	—	2	4	no	—
254	D₄×C₆₇	—	4	4	no	—
255	D₄×C₆₇	—	4	8	no	—
256	D₄×C₆₇	—	4	8	no	—
257	D₄×C₆₇	—	4	8	no	—
258	D₄×C₆₇	—	4	4	no	—
259	D₄×C₆₇	—	8	8	no	—
260	D₄×C₆₇	C₂²×D₄	1	4	yes	—
261	C₂×C₄×C₆₇	—	2	2	no	—
262	C₂×C₄×C₆₇	—	2	8	no	—
263	C₂×C₄×C₆₇	—	2	4	no	—
264	C₂×C₄×C₆₇	—	2	4	no	—
265	C₂×C₄×C₆₇	—	2	4	no	—
266	C₂×C₄×C₆₇	—	2	4	no	—
267	C₂×C₄×C₆₇	—	4	2	no	—
268	C₂×C₄×C₆₇	—	4	8	no	—
269	C₂×C₄×C₆₇	—	4	8	no	—
270	C₂×C₄×C₆₇	—	4	8	no	—
271	C₂×C₄×C₆₇	—	4	4	no	—
272	C₂×C₄×C₆₇	—	8	4	no	—
273	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
276	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
277	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
278	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
279	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
280	C₈×C₆₇	—	2	4	no	—
281	C₈×C₆₇	—	2	2	no	—
282	C₈×C₆₇	C₂²×D₄	1	2	yes	—
283	C₈×C₆₇	C₂²×D₄	1	2	yes	—
284	C₂²×C₆₇	—	2	2	no	—
285	C₂²×C₆₇	—	2	8	no	—
286	C₂²×C₆₇	—	2	4	no	—
287	C₂²×C₆₇	—	2	4	no	—
288	C₂²×C₆₇	—	4	4	no	—
289	C₂²×C₆₇	—	4	8	no	—
290	C₂²×C₆₇	—	4	4	no	—
291	C₂²×C₆₇	—	16	4	no	—
292	C₂²×C₆₇	—	16	8	no	—
293	C₂²×C₆₇	D₄²	1	2	yes	—
294	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
295	C₄×C₆₇	—	2	2	no	—
296	C₄×C₆₇	—	2	2	no	—
297	C₄×C₆₇	—	2	4	no	—
298	C₄×C₆₇	—	2	4	no	—
299	C₄×C₆₇	—	4	4	no	—
300	C₄×C₆₇	—	4	4	no	—
301	C₄×C₆₇	D₄²	1	2	yes	—
302	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
303	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
304	#61088	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
305	C₂×C₆₇	—	2	2	no	—
306	C₂×C₆₇	—	4	4	no	—
307	C₂×C₆₇	—	4	4	no	—
308	C₂×C₆₇	—	8	4	no	—
309	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
310	C₂×C₆₇	#3584	1	2	yes	—
311	C₂×#577	C₂×C₆₇	1	2	yes	complemented
312	#3500	C₂×C₆₇	1	2	yes	complemented
313	#3282	C₂×C₆₇	1	8	yes	complemented
314	#3277	C₂×C₆₇	1	4	yes	complemented
315	#3273	C₂×C₆₇	1	4	yes	complemented
316	C₂×#495	C₂×C₆₇	1	1	yes	characteristic, complemented
317	#3057	C₂×C₆₇	1	2	yes	complemented
318	#3040	C₂×C₆₇	1	2	yes	complemented
319	#3011	C₂×C₆₇	1	2	yes	complemented
320	#3008	C₂×C₆₇	1	2	yes	complemented
321	C₄×#135	C₂×C₆₇	1	1	yes	characteristic, complemented
322	#2931	C₂×C₆₇	1	1	yes	characteristic, complemented
323	C₆₇	#61088	1	1	yes	characteristic, direct product factor, Sylow
324	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	2	yes	—
325	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	2	yes	complemented
326	#577	—	2	16	no	—
327	C₂×#137	C₂²×C₆₇	1	2	yes	complemented
328	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
329	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	2	yes	—
330	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	complemented
331	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×#135	C₂²×C₆₇	1	4	yes	complemented
333	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
334	C₂²×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
335	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
336	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
337	C₂×#131	C₂²×C₆₇	1	2	yes	complemented
338	#548	C₂²×C₆₇	1	8	yes	complemented
339	#546	—	2	8	no	—
340	#546	C₂²×C₆₇	1	8	yes	complemented
341	#536	—	2	2	no	—
342	#536	C₂²×C₆₇	1	2	yes	complemented
343	#535	—	2	2	no	—
344	#535	C₂²×C₆₇	1	2	yes	—
345	#532	—	2	2	no	—
346	#532	C₂²×C₆₇	1	1	yes	characteristic
347	C₂×#124	—	2	2	no	—
348	C₂×#124	C₂²×C₆₇	1	2	yes	complemented
349	C₄×#41	C₂²×C₆₇	1	2	yes	complemented
350	#495	C₂²×C₆₇	1	4	yes	complemented
351	#495	C₂²×C₆₇	1	4	yes	complemented
352	#468	C₂²×C₆₇	1	8	yes	complemented
353	#465	C₂²×C₆₇	1	8	yes	complemented
354	#464	C₂²×C₆₇	1	8	yes	complemented
355	#463	C₂²×C₆₇	1	8	yes	complemented
356	#452	C₂²×C₆₇	1	8	yes	complemented
357	#451	C₂²×C₆₇	1	8	yes	complemented
358	#449	C₂²×C₆₇	1	8	yes	complemented
359	#447	C₂²×C₆₇	1	8	yes	complemented
360	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	complemented
361	C₄×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
362	C₄×D₈	C₂²×C₆₇	1	2	yes	complemented
363	#436	C₂²×C₆₇	1	4	yes	complemented
364	C₈×D₄	C₂²×C₆₇	1	4	yes	complemented
365	#427	C₂²×C₆₇	1	1	yes	characteristic, complemented
366	#423	C₂²×C₆₇	1	1	yes	characteristic, complemented
367	#419	C₂²×C₆₇	1	2	yes	complemented
368	#418	C₂²×C₆₇	1	2	yes	complemented
369	#416	C₂²×C₆₇	1	2	yes	complemented
370	#408	C₂²×C₆₇	1	2	yes	complemented
371	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
372	2₋¹⁺⁴	—	2	8	no	—
373	2₋¹⁺⁴	—	4	4	no	—
374	H(5,𝔽₂)	—	2	8	no	—
375	H(5,𝔽₂)	—	4	4	no	—
376	C₂×#41	—	2	2	no	—
377	C₂×#41	—	2	2	no	—
378	C₂×#41	—	2	4	no	—
379	C₂×#41	C₂³×C₆₇	1	4	yes	—
380	C₂×#41	C₂³×C₆₇	1	4	yes	—
381	C₂×#41	C₂³×C₆₇	1	2	yes	—
382	C₂×#41	C₂³×C₆₇	1	2	yes	—
383	C₂²×Q₈	—	2	2	no	—
384	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
385	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
386	C₂²×D₄	—	2	2	no	—
387	C₂²×D₄	—	2	4	no	—
388	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
389	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
390	#137	—	2	8	no	—
391	#137	—	2	16	no	—
392	AGL(1,ℤ/8)	—	2	8	no	—
393	AGL(1,ℤ/8)	—	2	16	no	—
394	#135	—	2	16	no	—
395	#135	D₄×C₆₇	1	8	yes	complemented
396	C₂×Q₁₆	—	4	2	no	—
397	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
398	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
399	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
400	C₂×SD₁₆	—	2	16	no	—
401	C₂×SD₁₆	—	2	8	no	—
402	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
403	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
404	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
405	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
406	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
407	C₂×D₈	—	4	2	no	—
408	C₂×D₈	C₂³×C₆₇	1	4	yes	—
409	C₂×D₈	C₂³×C₆₇	1	4	yes	—
410	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
411	#131	—	2	8	no	—
412	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
413	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
414	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
415	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
416	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
417	C₄⋊Q₈	—	2	2	no	—
418	C₄⋊Q₈	D₄×C₆₇	1	1	yes	characteristic
419	#127	—	2	2	no	—
420	#127	D₄×C₆₇	1	1	yes	characteristic
421	#124	—	2	2	no	—
422	#124	—	2	2	no	—
423	#124	—	2	8	no	—
424	#124	—	2	2	no	—
425	#124	—	2	2	no	—
426	#124	D₄×C₆₇	1	2	yes	—
427	#124	C₂³×C₆₇	1	8	yes	—
428	#123	—	2	4	no	—
429	#123	—	2	8	no	—
430	C₂²⋊Q₈	—	2	8	no	—
431	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
432	#121	—	2	8	no	—
433	#121	—	2	8	no	—
434	#121	D₄×C₆₇	1	4	yes	—
435	#121	C₂³×C₆₇	1	8	yes	—
436	#120	—	2	8	no	—
437	#120	—	2	4	no	—
438	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
439	C₄×D₄	—	2	4	no	—
440	C₄×D₄	—	2	8	no	—
441	C₄×D₄	C₂³×C₆₇	1	4	yes	—
442	C₄×D₄	C₂³×C₆₇	1	4	yes	—
443	#117	—	2	2	no	—
444	#117	C₂³×C₆₇	1	2	yes	—
445	C₂×C₂²⋊C₄	—	2	4	no	—
446	C₂×C₂²⋊C₄	—	2	8	no	—
447	C₂×C₄²	—	2	2	no	—
448	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
449	#107	C₂³×C₆₇	1	2	yes	—
450	#106	C₂³×C₆₇	1	2	yes	—
451	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
452	#103	C₂³×C₆₇	1	4	yes	—
453	#103	C₂³×C₆₇	1	8	yes	—
454	#102	C₂³×C₆₇	1	4	yes	—
455	#102	C₂³×C₆₇	1	8	yes	—
456	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
457	C₄×C₈	C₂³×C₆₇	1	2	yes	—
458	C₄×C₈	C₂³×C₆₇	1	2	yes	—
459	C₂⁴	—	2	4	no	—
460	C₂⁴	—	2	2	no	—
461	#41	—	2	4	no	—
462	#41	—	2	4	no	—
463	#41	—	2	4	no	—
464	#41	—	2	8	no	—
465	#41	—	2	8	no	—
466	#41	—	4	4	no	—
467	#41	—	4	4	no	—
468	#41	—	4	8	no	—
469	#41	—	4	8	no	—
470	#41	—	4	16	no	—
471	#41	C₂×D₄×C₆₇	1	8	yes	—
472	C₂×Q₈	—	2	4	no	—
473	C₂×Q₈	—	2	2	no	—
474	C₂×Q₈	—	2	8	no	—
475	C₂×Q₈	—	2	4	no	—
476	C₂×Q₈	—	2	4	no	—
477	C₂×Q₈	—	4	4	no	—
478	C₂×Q₈	—	4	8	no	—
479	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
480	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
481	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	—	2	8	no	—
484	C₂×D₄	—	2	4	no	—
485	C₂×D₄	—	2	4	no	—
486	C₂×D₄	—	2	8	no	—
487	C₂×D₄	—	2	4	no	—
488	C₂×D₄	—	2	4	no	—
489	C₂×D₄	—	2	2	no	—
490	C₂×D₄	—	2	4	no	—
491	C₂×D₄	—	4	4	no	—
492	C₂×D₄	—	4	8	no	—
493	C₂×D₄	—	4	16	no	—
494	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
495	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
496	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
497	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
498	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	2	4	no	—
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	2	4	no	—
505	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
506	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
507	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
508	Q₁₆	—	2	8	no	—
509	Q₁₆	—	4	4	no	—
510	Q₁₆	C₂×D₄×C₆₇	1	4	yes	—
511	SD₁₆	—	2	8	no	—
512	SD₁₆	—	2	8	no	—
513	SD₁₆	—	2	8	no	—
514	SD₁₆	—	2	8	no	—
515	SD₁₆	C₂×D₄×C₆₇	1	8	yes	—
516	D₈	—	2	8	no	—
517	D₈	—	4	4	no	—
518	D₈	C₂×D₄×C₆₇	1	4	yes	—
519	M₁₆	—	2	4	no	—
520	M₁₆	—	2	8	no	—
521	C₂×C₈	—	2	2	no	—
522	C₂×C₈	—	2	8	no	—
523	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	—
524	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
525	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
526	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
527	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
528	C₄⋊C₄	—	2	4	no	—
529	C₄⋊C₄	—	2	8	no	—
530	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
531	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
532	C₂²⋊C₄	—	2	8	no	—
533	C₂²⋊C₄	—	2	2	no	—
534	C₂²⋊C₄	—	2	8	no	—
535	C₂²⋊C₄	—	2	8	no	—
536	C₂²⋊C₄	—	2	8	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
539	C₄²	—	2	2	no	—
540	C₄²	—	2	4	no	—
541	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
542	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
543	C₄²	C₂⁴×C₆₇	1	2	yes	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	4	no	—
546	C₂³	—	4	8	no	—
547	C₂³	—	4	4	no	—
548	C₂³	—	8	4	no	—
549	C₂³	—	8	8	no	—
550	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
551	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
552	Q₈	—	2	4	no	—
553	Q₈	—	2	2	no	—
554	Q₈	—	2	4	no	—
555	Q₈	—	2	4	no	—
556	Q₈	—	4	8	no	—
557	Q₈	—	4	4	no	—
558	Q₈	C₂²×D₄×C₆₇	1	4	yes	—
559	D₄	—	2	8	no	—
560	D₄	—	2	8	no	—
561	D₄	—	2	4	no	—
562	D₄	—	2	4	no	—
563	D₄	—	2	2	no	—
564	D₄	—	2	4	no	—
565	D₄	—	4	4	no	—
566	D₄	—	4	8	no	—
567	D₄	—	4	8	no	—
568	D₄	—	4	8	no	—
569	D₄	—	4	4	no	—
570	D₄	—	8	8	no	—
571	D₄	C₂²×D₄×C₆₇	1	4	yes	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	4	2	no	—
579	C₂×C₄	—	4	8	no	—
580	C₂×C₄	—	4	8	no	—
581	C₂×C₄	—	4	8	no	—
582	C₂×C₄	—	4	4	no	—
583	C₂×C₄	—	8	4	no	—
584	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
585	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
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	H(5,𝔽₂)×C₆₇	1	2	yes	—
590	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
591	C₈	—	2	4	no	—
592	C₈	—	2	2	no	—
593	C₈	C₂²×D₄×C₆₇	1	2	yes	—
594	C₈	C₂²×D₄×C₆₇	1	2	yes	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	8	no	—
597	C₂²	—	2	4	no	—
598	C₂²	—	2	4	no	—
599	C₂²	—	4	4	no	—
600	C₂²	—	4	8	no	—
601	C₂²	—	4	4	no	—
602	C₂²	—	16	4	no	—
603	C₂²	—	16	8	no	—
604	C₂²	D₄²×C₆₇	1	2	yes	—
605	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	4	no	—
609	C₄	—	2	4	no	—
610	C₄	—	4	4	no	—
611	C₄	—	4	4	no	—
612	C₄	D₄²×C₆₇	1	2	yes	—
613	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
614	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
615	C₂	—	2	2	no	—
616	C₂	—	4	4	no	—
617	C₂	—	4	4	no	—
618	C₂	—	8	4	no	—
619	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
620	C₂	C₆₇×#3584	1	2	yes	—

	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×#577×C₆₇	C₂	semidirect
2. C₆₇×#3500	C₂	semidirect
3. C₆₇×#3282	C₂	semidirect
4. C₆₇×#3277	C₂	semidirect
5. C₆₇×#3273	C₂	semidirect
6. C₂×#495×C₆₇	C₂	semidirect
7. C₆₇×#3057	C₂	semidirect
8. C₆₇×#3040	C₂	semidirect
9. C₆₇×#3011	C₂	semidirect
10. C₆₇×#3008	C₂	semidirect
11. C₄×#135×C₆₇	C₂	semidirect
12. C₆₇×#2931	C₂	semidirect
14. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
16. C₂×#137×C₆₇	C₂²	semidirect
17. C₂×#137×C₆₇	C₂²	semidirect
19. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
21. C₂×#135×C₆₇	C₂²	semidirect
22. C₂²×Q₁₆×C₆₇	C₂²	semidirect
23. C₂²×SD₁₆×C₆₇	C₂²	semidirect
24. C₂²×SD₁₆×C₆₇	C₂²	semidirect
26. C₂×#131×C₆₇	C₂²	semidirect
27. #548×C₆₇	C₂²	semidirect
29. #546×C₆₇	C₂²	semidirect
31. #536×C₆₇	C₂²	semidirect
37. C₂×#124×C₆₇	C₂²	semidirect
38. C₄×#41×C₆₇	C₂²	semidirect
39. #495×C₆₇	C₂²	semidirect
40. #495×C₆₇	C₂²	semidirect
41. #468×C₆₇	C₂²	semidirect
42. #465×C₆₇	C₂²	semidirect
43. #464×C₆₇	C₂²	semidirect
44. #463×C₆₇	C₂²	semidirect
45. #452×C₆₇	C₂²	semidirect
46. #451×C₆₇	C₂²	semidirect
47. #449×C₆₇	C₂²	semidirect
48. #447×C₆₇	C₂²	semidirect
49. C₄×Q₁₆×C₆₇	C₂²	semidirect
50. C₄×SD₁₆×C₆₇	C₂²	semidirect
51. C₄×D₈×C₆₇	C₂²	semidirect
52. #436×C₆₇	C₂²	semidirect
53. C₈×D₄×C₆₇	C₂²	semidirect
54. #427×C₆₇	C₂²	semidirect
55. #423×C₆₇	C₂²	semidirect
56. #419×C₆₇	C₂²	semidirect
57. #418×C₆₇	C₂²	semidirect
58. #416×C₆₇	C₂²	semidirect
59. #408×C₆₇	C₂²	semidirect
60. C₂×C₄×C₈×C₆₇	C₂²	semidirect
84. #135×C₆₇	D₄	semidirect
304. #61088	C₆₇	direct (N × H)
311. C₂×#577	C₂×C₆₇	semidirect
312. #3500	C₂×C₆₇	semidirect
313. #3282	C₂×C₆₇	semidirect
314. #3277	C₂×C₆₇	semidirect
315. #3273	C₂×C₆₇	semidirect
316. C₂×#495	C₂×C₆₇	semidirect
317. #3057	C₂×C₆₇	semidirect
318. #3040	C₂×C₆₇	semidirect
319. #3011	C₂×C₆₇	semidirect
320. #3008	C₂×C₆₇	semidirect
321. C₄×#135	C₂×C₆₇	semidirect
322. #2931	C₂×C₆₇	semidirect
323. C₆₇	#61088	direct (N × H)
325. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
327. C₂×#137	C₂²×C₆₇	semidirect
328. C₂×#137	C₂²×C₆₇	semidirect
330. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
332. C₂×#135	C₂²×C₆₇	semidirect
333. C₂²×Q₁₆	C₂²×C₆₇	semidirect
334. C₂²×SD₁₆	C₂²×C₆₇	semidirect
335. C₂²×SD₁₆	C₂²×C₆₇	semidirect
337. C₂×#131	C₂²×C₆₇	semidirect
338. #548	C₂²×C₆₇	semidirect
340. #546	C₂²×C₆₇	semidirect
342. #536	C₂²×C₆₇	semidirect
348. C₂×#124	C₂²×C₆₇	semidirect
349. C₄×#41	C₂²×C₆₇	semidirect
350. #495	C₂²×C₆₇	semidirect
351. #495	C₂²×C₆₇	semidirect
352. #468	C₂²×C₆₇	semidirect
353. #465	C₂²×C₆₇	semidirect
354. #464	C₂²×C₆₇	semidirect
355. #463	C₂²×C₆₇	semidirect
356. #452	C₂²×C₆₇	semidirect
357. #451	C₂²×C₆₇	semidirect
358. #449	C₂²×C₆₇	semidirect
359. #447	C₂²×C₆₇	semidirect
360. C₄×Q₁₆	C₂²×C₆₇	semidirect
361. C₄×SD₁₆	C₂²×C₆₇	semidirect
362. C₄×D₈	C₂²×C₆₇	semidirect
363. #436	C₂²×C₆₇	semidirect
364. C₈×D₄	C₂²×C₆₇	semidirect
365. #427	C₂²×C₆₇	semidirect
366. #423	C₂²×C₆₇	semidirect
367. #419	C₂²×C₆₇	semidirect
368. #418	C₂²×C₆₇	semidirect
369. #416	C₂²×C₆₇	semidirect
370. #408	C₂²×C₆₇	semidirect
371. C₂×C₄×C₈	C₂²×C₆₇	semidirect
395. #135	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#577×C₆₇	2
C₆₇×#3500	2
C₆₇×#3282	2
C₆₇×#3277	2
C₆₇×#3273	2
C₂×#495×C₆₇	2
C₆₇×#3057	2
C₆₇×#3040	2
C₆₇×#3011	2
C₆₇×#3008	2
C₄×#135×C₆₇	2
C₆₇×#2931	2
#61088	67

Subgroups — C67×#61088 #30740339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61088 #30740339