Subgroups — C₆₇×#60371 #30739622

Order 17152.

4810 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#585×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#575×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×#537×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#3013	C₂	1	8	yes	maximal, complemented
5	C₆₇×#3012	C₂	1	8	yes	maximal, complemented
6	C₆₇×#3007	C₂	1	4	yes	maximal, complemented
7	C₆₇×#3006	C₂	1	4	yes	maximal, complemented
8	C₂×#417×C₆₇	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2879	C₂	1	1	yes	maximal, characteristic, complemented
10	#585×C₆₇	—	2	16	no	—
11	C₂×2₋¹⁺⁴×C₆₇	—	2	1	no	—
12	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	4	yes	complemented
13	C₂×H(5,𝔽₂)×C₆₇	—	2	1	no	—
14	C₂×H(5,𝔽₂)×C₆₇	—	2	2	no	—
15	C₂×H(5,𝔽₂)×C₆₇	C₂²	1	4	yes	complemented
16	C₂²×#41×C₆₇	—	2	4	no	—
17	C₂²×#41×C₆₇	C₂²	1	4	yes	—
18	C₂²×#41×C₆₇	C₂²	1	2	yes	—
19	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic, complemented
21	#575×C₆₇	—	2	16	no	—
22	C₂×#137×C₆₇	C₂²	1	4	yes	complemented
23	C₂×#137×C₆₇	C₂²	1	4	yes	complemented
24	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	complemented
25	C₂×AGL(1,ℤ/8)×C₆₇	C₂²	1	4	yes	complemented
26	C₂×#135×C₆₇	C₂²	1	8	yes	complemented
27	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	complemented
28	#537×C₆₇	—	2	4	no	—
29	#537×C₆₇	C₂²	1	8	yes	complemented
30	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
31	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
32	C₂×#121×C₆₇	—	2	1	no	—
33	C₂×#121×C₆₇	—	2	2	no	—
34	C₂×#121×C₆₇	C₂²	1	2	yes	complemented
35	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#452×C₆₇	C₂²	1	16	yes	complemented
37	#451×C₆₇	C₂²	1	8	yes	complemented
38	#450×C₆₇	C₂²	1	8	yes	complemented
39	#449×C₆₇	C₂²	1	16	yes	complemented
40	#448×C₆₇	C₂²	1	8	yes	complemented
41	#447×C₆₇	C₂²	1	8	yes	complemented
42	#417×C₆₇	C₂²	1	8	yes	complemented
43	#417×C₆₇	C₂²	1	8	yes	complemented
44	C₂×#103×C₆₇	C₂²	1	4	yes	complemented
45	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
46	#408×C₆₇	C₂²	1	8	yes	complemented
47	#407×C₆₇	C₂²	1	4	yes	complemented
48	2₋¹⁺⁴×C₆₇	—	2	16	no	—
49	2₋¹⁺⁴×C₆₇	—	2	16	no	—
50	2₋¹⁺⁴×C₆₇	—	2	16	no	—
51	H(5,𝔽₂)×C₆₇	—	2	16	no	—
52	H(5,𝔽₂)×C₆₇	—	2	32	no	—
53	H(5,𝔽₂)×C₆₇	—	2	16	no	—
54	H(5,𝔽₂)×C₆₇	—	2	16	no	—
55	C₂×#41×C₆₇	—	2	64	no	—
56	C₂×#41×C₆₇	—	2	32	no	—
57	C₂×#41×C₆₇	—	2	4	no	—
58	C₂×#41×C₆₇	—	2	4	no	—
59	C₂×#41×C₆₇	—	2	8	no	—
60	C₂×#41×C₆₇	—	2	4	no	—
61	C₂×#41×C₆₇	—	2	2	no	—
62	C₂×#41×C₆₇	—	2	4	no	—
63	C₂×#41×C₆₇	—	2	2	no	—
64	C₂×#41×C₆₇	—	2	8	no	—
65	C₂×#41×C₆₇	—	2	8	no	—
66	C₂×#41×C₆₇	—	2	2	no	—
67	C₂×#41×C₆₇	—	2	8	no	—
68	C₂×#41×C₆₇	D₄	1	4	yes	—
69	C₂×#41×C₆₇	D₄	1	16	yes	complemented
70	C₂×#41×C₆₇	C₂³	1	8	yes	—
71	C₂×#41×C₆₇	C₂³	1	4	yes	—
72	C₂×#41×C₆₇	C₂³	1	4	yes	—
73	C₂×#41×C₆₇	C₂³	1	4	yes	—
74	C₂×#41×C₆₇	C₂³	1	4	yes	—
75	C₂²×Q₈×C₆₇	—	2	4	no	—
76	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
77	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
78	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
80	C₂²×D₄×C₆₇	—	2	4	no	—
81	C₂²×D₄×C₆₇	—	2	2	no	—
82	C₂²×D₄×C₆₇	—	2	8	no	—
83	C₂²×D₄×C₆₇	—	2	4	no	—
84	C₂²×D₄×C₆₇	—	2	1	no	—
85	C₂²×D₄×C₆₇	—	2	2	no	—
86	C₂²×D₄×C₆₇	D₄	1	2	yes	—
87	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
88	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
89	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
90	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
91	C₂³×C₄×C₆₇	—	2	4	no	—
92	C₂³×C₄×C₆₇	—	2	2	no	—
93	C₂³×C₄×C₆₇	—	2	1	no	—
94	C₂³×C₄×C₆₇	—	2	1	no	—
95	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
96	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
97	#137×C₆₇	—	2	16	no	—
98	#137×C₆₇	—	2	16	no	—
99	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
100	AGL(1,ℤ/8)×C₆₇	—	2	16	no	—
101	#135×C₆₇	—	2	32	no	—
102	C₂×Q₁₆×C₆₇	C₂³	1	8	yes	—
103	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
104	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
105	C₂×D₈×C₆₇	C₂³	1	8	yes	—
106	C₂×M₁₆×C₆₇	—	2	8	no	—
107	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
108	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
109	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
110	C₂²⋊Q₈×C₆₇	—	2	8	no	—
111	C₂²⋊Q₈×C₆₇	—	2	4	no	—
112	C₂²⋊Q₈×C₆₇	C₂³	1	8	yes	—
113	#121×C₆₇	—	2	8	no	—
114	#121×C₆₇	—	2	16	no	—
115	#121×C₆₇	—	2	4	no	—
116	#121×C₆₇	C₂³	1	8	yes	—
117	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
121	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
122	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
123	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
124	#103×C₆₇	C₂³	1	16	yes	—
125	#102×C₆₇	C₂³	1	16	yes	—
126	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
127	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
128	C₂⁴×C₆₇	—	2	4	no	—
129	C₂⁴×C₆₇	—	2	2	no	—
130	C₂⁴×C₆₇	—	2	1	no	—
131	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
132	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
133	#41×C₆₇	—	2	32	no	—
134	#41×C₆₇	—	2	64	no	—
135	#41×C₆₇	—	2	32	no	—
136	#41×C₆₇	—	2	32	no	—
137	#41×C₆₇	—	2	32	no	—
138	#41×C₆₇	—	2	16	no	—
139	#41×C₆₇	—	2	16	no	—
140	#41×C₆₇	—	2	16	no	—
141	#41×C₆₇	—	2	16	no	—
142	#41×C₆₇	—	2	16	no	—
143	#41×C₆₇	—	2	8	no	—
144	#41×C₆₇	—	2	8	no	—
145	#41×C₆₇	—	2	8	no	—
146	#41×C₆₇	—	2	8	no	—
147	#41×C₆₇	—	2	8	no	—
148	#41×C₆₇	—	2	8	no	—
149	#41×C₆₇	—	4	8	no	—
150	#41×C₆₇	C₂×D₄	1	16	yes	—
151	#41×C₆₇	C₂×D₄	1	16	yes	complemented
152	C₂×Q₈×C₆₇	—	2	32	no	—
153	C₂×Q₈×C₆₇	—	2	16	no	—
154	C₂×Q₈×C₆₇	—	2	4	no	—
155	C₂×Q₈×C₆₇	—	2	4	no	—
156	C₂×Q₈×C₆₇	—	2	2	no	—
157	C₂×Q₈×C₆₇	—	2	4	no	—
158	C₂×Q₈×C₆₇	—	2	1	no	—
159	C₂×Q₈×C₆₇	—	2	4	no	—
160	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
161	C₂×Q₈×C₆₇	C₂×D₄	1	8	yes	complemented
162	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
163	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
164	C₂×D₄×C₆₇	—	2	32	no	—
165	C₂×D₄×C₆₇	—	2	64	no	—
166	C₂×D₄×C₆₇	—	2	32	no	—
167	C₂×D₄×C₆₇	—	2	16	no	—
168	C₂×D₄×C₆₇	—	2	8	no	—
169	C₂×D₄×C₆₇	—	2	8	no	—
170	C₂×D₄×C₆₇	—	2	4	no	—
171	C₂×D₄×C₆₇	—	2	4	no	—
172	C₂×D₄×C₆₇	—	2	16	no	—
173	C₂×D₄×C₆₇	—	2	4	no	—
174	C₂×D₄×C₆₇	—	2	8	no	—
175	C₂×D₄×C₆₇	—	2	4	no	—
176	C₂×D₄×C₆₇	—	2	4	no	—
177	C₂×D₄×C₆₇	—	2	2	no	—
178	C₂×D₄×C₆₇	—	2	4	no	—
179	C₂×D₄×C₆₇	—	2	4	no	—
180	C₂×D₄×C₆₇	—	2	1	no	—
181	C₂×D₄×C₆₇	—	4	4	no	—
182	C₂×D₄×C₆₇	—	8	4	no	—
183	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	—
184	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
185	C₂×D₄×C₆₇	C₂×D₄	1	8	yes	complemented
186	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
187	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
188	C₂²×C₄×C₆₇	—	2	32	no	—
189	C₂²×C₄×C₆₇	—	2	4	no	—
190	C₂²×C₄×C₆₇	—	2	2	no	—
191	C₂²×C₄×C₆₇	—	2	16	no	—
192	C₂²×C₄×C₆₇	—	2	4	no	—
193	C₂²×C₄×C₆₇	—	2	8	no	—
194	C₂²×C₄×C₆₇	—	2	4	no	—
195	C₂²×C₄×C₆₇	—	2	8	no	—
196	C₂²×C₄×C₆₇	—	2	2	no	—
197	C₂²×C₄×C₆₇	—	2	1	no	—
198	C₂²×C₄×C₆₇	—	2	4	no	—
199	C₂²×C₄×C₆₇	—	2	4	no	—
200	C₂²×C₄×C₆₇	—	4	4	no	—
201	C₂²×C₄×C₆₇	—	8	2	no	—
202	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
203	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
204	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
205	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
206	C₂²×C₄×C₆₇	C₂×D₄	1	8	yes	—
207	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
208	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
210	Q₁₆×C₆₇	—	2	16	no	—
211	SD₁₆×C₆₇	—	2	16	no	—
212	SD₁₆×C₆₇	—	2	16	no	—
213	D₈×C₆₇	—	2	16	no	—
214	M₁₆×C₆₇	—	2	8	no	—
215	M₁₆×C₆₇	—	2	8	no	—
216	C₂×C₈×C₆₇	—	2	8	no	—
217	C₂×C₈×C₆₇	C₂⁴	1	8	yes	—
218	C₄⋊C₄×C₆₇	—	2	2	no	—
219	C₄⋊C₄×C₆₇	—	2	4	no	—
220	C₄⋊C₄×C₆₇	—	2	8	no	—
221	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
222	C₂²⋊C₄×C₆₇	—	2	8	no	—
223	C₂²⋊C₄×C₆₇	—	2	8	no	—
224	C₂³×C₆₇	—	2	16	no	—
225	C₂³×C₆₇	—	2	4	no	—
226	C₂³×C₆₇	—	2	4	no	—
227	C₂³×C₆₇	—	2	8	no	—
228	C₂³×C₆₇	—	2	2	no	—
229	C₂³×C₆₇	—	4	1	no	—
230	C₂³×C₆₇	—	4	1	no	—
231	C₂³×C₆₇	—	4	2	no	—
232	C₂³×C₆₇	—	4	2	no	—
233	C₂³×C₆₇	—	4	2	no	—
234	C₂³×C₆₇	—	4	2	no	—
235	C₂³×C₆₇	—	4	2	no	—
236	C₂³×C₆₇	—	8	2	no	—
237	C₂³×C₆₇	—	8	2	no	—
238	C₂³×C₆₇	—	8	4	no	—
239	C₂³×C₆₇	—	8	4	no	—
240	C₂³×C₆₇	#120	1	4	yes	—
241	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
242	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
243	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
244	Q₈×C₆₇	—	2	16	no	—
245	Q₈×C₆₇	—	2	8	no	—
246	Q₈×C₆₇	—	2	8	no	—
247	Q₈×C₆₇	—	2	4	no	—
248	Q₈×C₆₇	—	2	4	no	—
249	Q₈×C₆₇	—	4	2	no	—
250	Q₈×C₆₇	C₂²×D₄	1	8	yes	—
251	D₄×C₆₇	—	2	32	no	—
252	D₄×C₆₇	—	2	16	no	—
253	D₄×C₆₇	—	2	16	no	—
254	D₄×C₆₇	—	2	16	no	—
255	D₄×C₆₇	—	2	16	no	—
256	D₄×C₆₇	—	2	8	no	—
257	D₄×C₆₇	—	2	8	no	—
258	D₄×C₆₇	—	2	4	no	—
259	D₄×C₆₇	—	2	4	no	—
260	D₄×C₆₇	—	4	8	no	—
261	D₄×C₆₇	—	4	2	no	—
262	D₄×C₆₇	—	8	8	no	—
263	D₄×C₆₇	C₂²×D₄	1	8	yes	—
264	C₂×C₄×C₆₇	—	2	16	no	—
265	C₂×C₄×C₆₇	—	2	32	no	—
266	C₂×C₄×C₆₇	—	2	8	no	—
267	C₂×C₄×C₆₇	—	2	4	no	—
268	C₂×C₄×C₆₇	—	2	4	no	—
269	C₂×C₄×C₆₇	—	2	16	no	—
270	C₂×C₄×C₆₇	—	2	8	no	—
271	C₂×C₄×C₆₇	—	2	16	no	—
272	C₂×C₄×C₆₇	—	2	8	no	—
273	C₂×C₄×C₆₇	—	2	4	no	—
274	C₂×C₄×C₆₇	—	2	2	no	—
275	C₂×C₄×C₆₇	—	4	4	no	—
276	C₂×C₄×C₆₇	—	4	2	no	—
277	C₂×C₄×C₆₇	—	4	8	no	—
278	C₂×C₄×C₆₇	—	8	4	no	—
279	C₂×C₄×C₆₇	#120	1	8	yes	—
280	C₂×C₄×C₆₇	#120	1	4	yes	—
281	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
282	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
284	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
285	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
286	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
287	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
288	C₈×C₆₇	—	2	8	no	—
289	C₂²×C₆₇	—	2	1	no	—
290	C₂²×C₆₇	—	2	2	no	—
291	C₂²×C₆₇	—	2	1	no	—
292	C₂²×C₆₇	—	2	2	no	—
293	C₂²×C₆₇	—	2	4	no	—
294	C₂²×C₆₇	—	2	4	no	—
295	C₂²×C₆₇	—	2	4	no	—
296	C₂²×C₆₇	—	2	8	no	—
297	C₂²×C₆₇	—	4	4	no	—
298	C₂²×C₆₇	—	4	4	no	—
299	C₂²×C₆₇	—	4	4	no	—
300	C₂²×C₆₇	—	4	2	no	—
301	C₂²×C₆₇	—	4	2	no	—
302	C₂²×C₆₇	—	4	2	no	—
303	C₂²×C₆₇	—	8	4	no	—
304	C₂²×C₆₇	—	8	8	no	—
305	C₂²×C₆₇	—	8	4	no	—
306	C₂²×C₆₇	—	8	16	no	—
307	C₂²×C₆₇	C₂×#120	1	2	yes	—
308	C₂²×C₆₇	C₂×#120	1	4	yes	—
309	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
310	C₄×C₆₇	—	2	8	no	—
311	C₄×C₆₇	—	2	4	no	—
312	C₄×C₆₇	—	4	4	no	—
313	C₄×C₆₇	—	4	2	no	—
314	C₄×C₆₇	C₂×#120	1	2	yes	—
315	C₄×C₆₇	C₂×#120	1	4	yes	—
316	C₄×C₆₇	C₂³×D₄	1	2	yes	—
317	#60371	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
318	C₂×C₆₇	—	2	2	no	—
319	C₂×C₆₇	—	2	4	no	—
320	C₂×C₆₇	—	4	4	no	—
321	C₂×C₆₇	—	4	8	no	—
322	C₂×C₆₇	—	8	2	no	—
323	C₂×C₆₇	C₂²×#120	1	1	yes	characteristic
324	C₂×C₆₇	#3585	1	1	yes	characteristic
325	C₂×C₆₇	#3586	1	1	yes	characteristic
326	C₂×#585	C₂×C₆₇	1	1	yes	characteristic, complemented
327	C₂×#575	C₂×C₆₇	1	2	yes	complemented
328	C₂×#537	C₂×C₆₇	1	1	yes	characteristic, complemented
329	#3013	C₂×C₆₇	1	8	yes	complemented
330	#3012	C₂×C₆₇	1	8	yes	complemented
331	#3007	C₂×C₆₇	1	4	yes	complemented
332	#3006	C₂×C₆₇	1	4	yes	complemented
333	C₂×#417	C₂×C₆₇	1	2	yes	complemented
334	#2879	C₂×C₆₇	1	1	yes	characteristic, complemented
335	C₆₇	#60371	1	1	yes	characteristic, direct product factor, Sylow
336	#585	—	2	16	no	—
337	C₂×2₋¹⁺⁴	—	2	1	no	—
338	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	4	yes	complemented
339	C₂×H(5,𝔽₂)	—	2	1	no	—
340	C₂×H(5,𝔽₂)	—	2	2	no	—
341	C₂×H(5,𝔽₂)	C₂²×C₆₇	1	4	yes	complemented
342	C₂²×#41	—	2	4	no	—
343	C₂²×#41	C₂²×C₆₇	1	4	yes	—
344	C₂²×#41	C₂²×C₆₇	1	2	yes	—
345	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
346	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic, complemented
347	#575	—	2	16	no	—
348	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
349	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
350	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	complemented
351	C₂×AGL(1,ℤ/8)	C₂²×C₆₇	1	4	yes	complemented
352	C₂×#135	C₂²×C₆₇	1	8	yes	complemented
353	C₂²×M₁₆	C₂²×C₆₇	1	2	yes	complemented
354	#537	—	2	4	no	—
355	#537	C₂²×C₆₇	1	8	yes	complemented
356	C₂×C₂²⋊Q₈	—	2	1	no	—
357	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
358	C₂×#121	—	2	1	no	—
359	C₂×#121	—	2	2	no	—
360	C₂×#121	C₂²×C₆₇	1	2	yes	complemented
361	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
362	#452	C₂²×C₆₇	1	16	yes	complemented
363	#451	C₂²×C₆₇	1	8	yes	complemented
364	#450	C₂²×C₆₇	1	8	yes	complemented
365	#449	C₂²×C₆₇	1	16	yes	complemented
366	#448	C₂²×C₆₇	1	8	yes	complemented
367	#447	C₂²×C₆₇	1	8	yes	complemented
368	#417	C₂²×C₆₇	1	8	yes	complemented
369	#417	C₂²×C₆₇	1	8	yes	complemented
370	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
371	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
372	#408	C₂²×C₆₇	1	8	yes	complemented
373	#407	C₂²×C₆₇	1	4	yes	complemented
374	2₋¹⁺⁴	—	2	16	no	—
375	2₋¹⁺⁴	—	2	16	no	—
376	2₋¹⁺⁴	—	2	16	no	—
377	H(5,𝔽₂)	—	2	16	no	—
378	H(5,𝔽₂)	—	2	32	no	—
379	H(5,𝔽₂)	—	2	16	no	—
380	H(5,𝔽₂)	—	2	16	no	—
381	C₂×#41	—	2	64	no	—
382	C₂×#41	—	2	32	no	—
383	C₂×#41	—	2	4	no	—
384	C₂×#41	—	2	4	no	—
385	C₂×#41	—	2	8	no	—
386	C₂×#41	—	2	4	no	—
387	C₂×#41	—	2	2	no	—
388	C₂×#41	—	2	4	no	—
389	C₂×#41	—	2	2	no	—
390	C₂×#41	—	2	8	no	—
391	C₂×#41	—	2	8	no	—
392	C₂×#41	—	2	2	no	—
393	C₂×#41	—	2	8	no	—
394	C₂×#41	D₄×C₆₇	1	4	yes	—
395	C₂×#41	D₄×C₆₇	1	16	yes	complemented
396	C₂×#41	C₂³×C₆₇	1	4	yes	—
397	C₂×#41	C₂³×C₆₇	1	8	yes	—
398	C₂×#41	C₂³×C₆₇	1	4	yes	—
399	C₂×#41	C₂³×C₆₇	1	4	yes	—
400	C₂×#41	C₂³×C₆₇	1	4	yes	—
401	C₂²×Q₈	—	2	4	no	—
402	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
403	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
404	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
405	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
406	C₂²×D₄	—	2	4	no	—
407	C₂²×D₄	—	2	2	no	—
408	C₂²×D₄	—	2	8	no	—
409	C₂²×D₄	—	2	4	no	—
410	C₂²×D₄	—	2	1	no	—
411	C₂²×D₄	—	2	2	no	—
412	C₂²×D₄	D₄×C₆₇	1	2	yes	—
413	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
414	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
415	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
416	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
417	C₂³×C₄	—	2	4	no	—
418	C₂³×C₄	—	2	2	no	—
419	C₂³×C₄	—	2	1	no	—
420	C₂³×C₄	—	2	1	no	—
421	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
422	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
423	#137	—	2	16	no	—
424	#137	—	2	16	no	—
425	AGL(1,ℤ/8)	—	2	16	no	—
426	AGL(1,ℤ/8)	—	2	16	no	—
427	#135	—	2	32	no	—
428	C₂×Q₁₆	C₂³×C₆₇	1	8	yes	—
429	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
430	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
431	C₂×D₈	C₂³×C₆₇	1	8	yes	—
432	C₂×M₁₆	—	2	8	no	—
433	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
434	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
435	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
436	C₂²⋊Q₈	—	2	8	no	—
437	C₂²⋊Q₈	—	2	4	no	—
438	C₂²⋊Q₈	C₂³×C₆₇	1	8	yes	—
439	#121	—	2	8	no	—
440	#121	—	2	16	no	—
441	#121	—	2	4	no	—
442	#121	C₂³×C₆₇	1	8	yes	—
443	C₂×C₄⋊C₄	—	2	1	no	—
444	C₂×C₄⋊C₄	—	2	2	no	—
445	C₂×C₄⋊C₄	—	2	2	no	—
446	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
447	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
448	C₂×C₂²⋊C₄	—	2	2	no	—
449	C₂×C₂²⋊C₄	—	2	2	no	—
450	#103	C₂³×C₆₇	1	16	yes	—
451	#102	C₂³×C₆₇	1	16	yes	—
452	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
453	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
454	C₂⁴	—	2	4	no	—
455	C₂⁴	—	2	2	no	—
456	C₂⁴	—	2	1	no	—
457	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
458	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
459	#41	—	2	32	no	—
460	#41	—	2	64	no	—
461	#41	—	2	32	no	—
462	#41	—	2	32	no	—
463	#41	—	2	32	no	—
464	#41	—	2	16	no	—
465	#41	—	2	16	no	—
466	#41	—	2	16	no	—
467	#41	—	2	16	no	—
468	#41	—	2	16	no	—
469	#41	—	2	8	no	—
470	#41	—	2	8	no	—
471	#41	—	2	8	no	—
472	#41	—	2	8	no	—
473	#41	—	2	8	no	—
474	#41	—	2	8	no	—
475	#41	—	4	8	no	—
476	#41	C₂×D₄×C₆₇	1	16	yes	—
477	#41	C₂×D₄×C₆₇	1	16	yes	complemented
478	C₂×Q₈	—	2	32	no	—
479	C₂×Q₈	—	2	16	no	—
480	C₂×Q₈	—	2	4	no	—
481	C₂×Q₈	—	2	4	no	—
482	C₂×Q₈	—	2	2	no	—
483	C₂×Q₈	—	2	4	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	4	no	—
486	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
487	C₂×Q₈	C₂×D₄×C₆₇	1	8	yes	complemented
488	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
489	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
490	C₂×D₄	—	2	32	no	—
491	C₂×D₄	—	2	64	no	—
492	C₂×D₄	—	2	32	no	—
493	C₂×D₄	—	2	16	no	—
494	C₂×D₄	—	2	8	no	—
495	C₂×D₄	—	2	8	no	—
496	C₂×D₄	—	2	4	no	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	—	2	16	no	—
499	C₂×D₄	—	2	4	no	—
500	C₂×D₄	—	2	8	no	—
501	C₂×D₄	—	2	4	no	—
502	C₂×D₄	—	2	4	no	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	2	4	no	—
505	C₂×D₄	—	2	4	no	—
506	C₂×D₄	—	2	1	no	—
507	C₂×D₄	—	4	4	no	—
508	C₂×D₄	—	8	4	no	—
509	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	—
510	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
511	C₂×D₄	C₂×D₄×C₆₇	1	8	yes	complemented
512	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
513	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
514	C₂²×C₄	—	2	32	no	—
515	C₂²×C₄	—	2	4	no	—
516	C₂²×C₄	—	2	2	no	—
517	C₂²×C₄	—	2	16	no	—
518	C₂²×C₄	—	2	4	no	—
519	C₂²×C₄	—	2	8	no	—
520	C₂²×C₄	—	2	4	no	—
521	C₂²×C₄	—	2	8	no	—
522	C₂²×C₄	—	2	2	no	—
523	C₂²×C₄	—	2	1	no	—
524	C₂²×C₄	—	2	4	no	—
525	C₂²×C₄	—	2	4	no	—
526	C₂²×C₄	—	4	4	no	—
527	C₂²×C₄	—	8	2	no	—
528	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
529	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
530	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
531	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
532	C₂²×C₄	C₂×D₄×C₆₇	1	8	yes	—
533	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
534	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
535	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
536	Q₁₆	—	2	16	no	—
537	SD₁₆	—	2	16	no	—
538	SD₁₆	—	2	16	no	—
539	D₈	—	2	16	no	—
540	M₁₆	—	2	8	no	—
541	M₁₆	—	2	8	no	—
542	C₂×C₈	—	2	8	no	—
543	C₂×C₈	C₂⁴×C₆₇	1	8	yes	—
544	C₄⋊C₄	—	2	2	no	—
545	C₄⋊C₄	—	2	4	no	—
546	C₄⋊C₄	—	2	8	no	—
547	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
548	C₂²⋊C₄	—	2	8	no	—
549	C₂²⋊C₄	—	2	8	no	—
550	C₂³	—	2	16	no	—
551	C₂³	—	2	4	no	—
552	C₂³	—	2	4	no	—
553	C₂³	—	2	8	no	—
554	C₂³	—	2	2	no	—
555	C₂³	—	4	1	no	—
556	C₂³	—	4	1	no	—
557	C₂³	—	4	2	no	—
558	C₂³	—	4	2	no	—
559	C₂³	—	4	2	no	—
560	C₂³	—	4	2	no	—
561	C₂³	—	4	2	no	—
562	C₂³	—	8	2	no	—
563	C₂³	—	8	2	no	—
564	C₂³	—	8	4	no	—
565	C₂³	—	8	4	no	—
566	C₂³	#120×C₆₇	1	4	yes	—
567	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
568	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
569	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
570	Q₈	—	2	16	no	—
571	Q₈	—	2	8	no	—
572	Q₈	—	2	8	no	—
573	Q₈	—	2	4	no	—
574	Q₈	—	2	4	no	—
575	Q₈	—	4	2	no	—
576	Q₈	C₂²×D₄×C₆₇	1	8	yes	—
577	D₄	—	2	32	no	—
578	D₄	—	2	16	no	—
579	D₄	—	2	16	no	—
580	D₄	—	2	16	no	—
581	D₄	—	2	16	no	—
582	D₄	—	2	8	no	—
583	D₄	—	2	8	no	—
584	D₄	—	2	4	no	—
585	D₄	—	2	4	no	—
586	D₄	—	4	8	no	—
587	D₄	—	4	2	no	—
588	D₄	—	8	8	no	—
589	D₄	C₂²×D₄×C₆₇	1	8	yes	—
590	C₂×C₄	—	2	16	no	—
591	C₂×C₄	—	2	32	no	—
592	C₂×C₄	—	2	8	no	—
593	C₂×C₄	—	2	4	no	—
594	C₂×C₄	—	2	4	no	—
595	C₂×C₄	—	2	16	no	—
596	C₂×C₄	—	2	8	no	—
597	C₂×C₄	—	2	16	no	—
598	C₂×C₄	—	2	8	no	—
599	C₂×C₄	—	2	4	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	4	4	no	—
602	C₂×C₄	—	4	2	no	—
603	C₂×C₄	—	4	8	no	—
604	C₂×C₄	—	8	4	no	—
605	C₂×C₄	#120×C₆₇	1	8	yes	—
606	C₂×C₄	#120×C₆₇	1	4	yes	—
607	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
608	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
609	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
610	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
611	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
612	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
613	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
614	C₈	—	2	8	no	—
615	C₂²	—	2	1	no	—
616	C₂²	—	2	2	no	—
617	C₂²	—	2	1	no	—
618	C₂²	—	2	2	no	—
619	C₂²	—	2	4	no	—
620	C₂²	—	2	4	no	—
621	C₂²	—	2	4	no	—
622	C₂²	—	2	8	no	—
623	C₂²	—	4	4	no	—
624	C₂²	—	4	4	no	—
625	C₂²	—	4	4	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	2	no	—
628	C₂²	—	4	2	no	—
629	C₂²	—	8	4	no	—
630	C₂²	—	8	8	no	—
631	C₂²	—	8	4	no	—
632	C₂²	—	8	16	no	—
633	C₂²	C₂×#120×C₆₇	1	2	yes	—
634	C₂²	C₂×#120×C₆₇	1	4	yes	—
635	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
636	C₄	—	2	8	no	—
637	C₄	—	2	4	no	—
638	C₄	—	4	4	no	—
639	C₄	—	4	2	no	—
640	C₄	C₂×#120×C₆₇	1	2	yes	—
641	C₄	C₂×#120×C₆₇	1	4	yes	—
642	C₄	C₂³×D₄×C₆₇	1	2	yes	—
643	C₂	—	2	2	no	—
644	C₂	—	2	4	no	—
645	C₂	—	4	4	no	—
646	C₂	—	4	8	no	—
647	C₂	—	8	2	no	—
648	C₂	C₂²×#120×C₆₇	1	1	yes	characteristic
649	C₂	C₆₇×#3585	1	1	yes	characteristic
650	C₂	C₆₇×#3586	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 ◁ C₂²×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#585×C₆₇	C₂	semidirect
2. C₂×#575×C₆₇	C₂	semidirect
3. C₂×#537×C₆₇	C₂	semidirect
4. C₆₇×#3013	C₂	semidirect
5. C₆₇×#3012	C₂	semidirect
6. C₆₇×#3007	C₂	semidirect
7. C₆₇×#3006	C₂	semidirect
8. C₂×#417×C₆₇	C₂	semidirect
9. C₆₇×#2879	C₂	semidirect
12. C₂×2₋¹⁺⁴×C₆₇	C₂²	semidirect
15. C₂×H(5,𝔽₂)×C₆₇	C₂²	semidirect
20. C₂²×#41×C₆₇	C₂²	semidirect
22. C₂×#137×C₆₇	C₂²	semidirect
23. C₂×#137×C₆₇	C₂²	semidirect
24. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
25. C₂×AGL(1,ℤ/8)×C₆₇	C₂²	semidirect
26. C₂×#135×C₆₇	C₂²	semidirect
27. C₂²×M₁₆×C₆₇	C₂²	semidirect
29. #537×C₆₇	C₂²	semidirect
31. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
34. C₂×#121×C₆₇	C₂²	semidirect
35. C₂²×C₄⋊C₄×C₆₇	C₂²	semidirect
36. #452×C₆₇	C₂²	semidirect
37. #451×C₆₇	C₂²	semidirect
38. #450×C₆₇	C₂²	semidirect
39. #449×C₆₇	C₂²	semidirect
40. #448×C₆₇	C₂²	semidirect
41. #447×C₆₇	C₂²	semidirect
42. #417×C₆₇	C₂²	semidirect
43. #417×C₆₇	C₂²	semidirect
44. C₂×#103×C₆₇	C₂²	semidirect
45. C₂×#102×C₆₇	C₂²	semidirect
46. #408×C₆₇	C₂²	semidirect
47. #407×C₆₇	C₂²	semidirect
69. C₂×#41×C₆₇	D₄	semidirect
151. #41×C₆₇	C₂×D₄	semidirect
161. C₂×Q₈×C₆₇	C₂×D₄	semidirect
185. C₂×D₄×C₆₇	C₂×D₄	semidirect
317. #60371	C₆₇	direct (N × H)
326. C₂×#585	C₂×C₆₇	semidirect
327. C₂×#575	C₂×C₆₇	semidirect
328. C₂×#537	C₂×C₆₇	semidirect
329. #3013	C₂×C₆₇	semidirect
330. #3012	C₂×C₆₇	semidirect
331. #3007	C₂×C₆₇	semidirect
332. #3006	C₂×C₆₇	semidirect
333. C₂×#417	C₂×C₆₇	semidirect
334. #2879	C₂×C₆₇	semidirect
335. C₆₇	#60371	direct (N × H)
338. C₂×2₋¹⁺⁴	C₂²×C₆₇	semidirect
341. C₂×H(5,𝔽₂)	C₂²×C₆₇	semidirect
346. C₂²×#41	C₂²×C₆₇	semidirect
348. C₂×#137	C₂²×C₆₇	semidirect
349. C₂×#137	C₂²×C₆₇	semidirect
350. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
351. C₂×AGL(1,ℤ/8)	C₂²×C₆₇	semidirect
352. C₂×#135	C₂²×C₆₇	semidirect
353. C₂²×M₁₆	C₂²×C₆₇	semidirect
355. #537	C₂²×C₆₇	semidirect
357. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
360. C₂×#121	C₂²×C₆₇	semidirect
361. C₂²×C₄⋊C₄	C₂²×C₆₇	semidirect
362. #452	C₂²×C₆₇	semidirect
363. #451	C₂²×C₆₇	semidirect
364. #450	C₂²×C₆₇	semidirect
365. #449	C₂²×C₆₇	semidirect
366. #448	C₂²×C₆₇	semidirect
367. #447	C₂²×C₆₇	semidirect
368. #417	C₂²×C₆₇	semidirect
369. #417	C₂²×C₆₇	semidirect
370. C₂×#103	C₂²×C₆₇	semidirect
371. C₂×#102	C₂²×C₆₇	semidirect
372. #408	C₂²×C₆₇	semidirect
373. #407	C₂²×C₆₇	semidirect
395. C₂×#41	D₄×C₆₇	semidirect
477. #41	C₂×D₄×C₆₇	semidirect
487. C₂×Q₈	C₂×D₄×C₆₇	semidirect
511. C₂×D₄	C₂×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×#585×C₆₇	2
C₂×#575×C₆₇	2
C₂×#537×C₆₇	2
C₆₇×#3013	2
C₆₇×#3012	2
C₆₇×#3007	2
C₆₇×#3006	2
C₂×#417×C₆₇	2
C₆₇×#2879	2
#60371	67

Subgroups — C67×#60371 #30739622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60371 #30739622