Subgroups — C₆₇×#32959 #30712210

Order 17152.

1720 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#478×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2851	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1864	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1862	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1853	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1852	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1804	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1786	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1786	C₂	1	1	yes	maximal, characteristic
12	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₆₇	—	2	1	no	—
14	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₄×D₄×C₆₇	—	2	8	no	—
16	C₂×C₄×D₄×C₆₇	—	2	1	no	—
17	C₂×C₄×D₄×C₆₇	—	2	1	no	—
18	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊C₄×C₆₇	—	2	1	no	—
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	#478×C₆₇	—	2	8	no	—
24	C₂×#107×C₆₇	C₂²	1	2	yes	complemented
25	C₂×#106×C₆₇	C₂²	1	2	yes	complemented
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	complemented
27	C₂×#102×C₆₇	C₂²	1	2	yes	—
28	C₂×#102×C₆₇	C₂²	1	2	yes	—
29	C₂×#102×C₆₇	C₂²	1	2	yes	—
30	C₂×#102×C₆₇	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	#389×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	#389×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	#387×C₆₇	—	2	1	no	—
36	#387×C₆₇	—	2	1	no	—
37	#385×C₆₇	—	2	1	no	—
38	#385×C₆₇	—	2	1	no	—
39	#382×C₆₇	C₂²	1	1	yes	characteristic, complemented
40	#380×C₆₇	C₂²	1	1	yes	characteristic
41	#380×C₆₇	C₂²	1	1	yes	characteristic
42	#340×C₆₇	C₂²	1	2	yes	complemented
43	#340×C₆₇	C₂²	1	2	yes	complemented
44	#336×C₆₇	C₂²	1	2	yes	complemented
45	C₂⁵×C₆₇	—	2	1	no	—
46	C₂²×D₄×C₆₇	—	2	4	no	—
47	C₂²×D₄×C₆₇	—	2	4	no	—
48	C₂²×D₄×C₆₇	—	2	1	no	—
49	C₂²×D₄×C₆₇	—	2	2	no	—
50	C₂²×D₄×C₆₇	Q₈	1	4	yes	—
51	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
52	C₂³×C₄×C₆₇	—	2	8	no	—
53	C₂³×C₄×C₆₇	—	2	1	no	—
54	C₂³×C₄×C₆₇	—	2	2	no	—
55	C₂³×C₄×C₆₇	—	2	2	no	—
56	C₂³×C₄×C₆₇	—	4	4	no	—
57	C₂³×C₄×C₆₇	—	4	4	no	—
58	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
62	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
63	#125×C₆₇	—	2	4	no	—
64	C₄×D₄×C₆₇	—	2	8	no	—
65	C₄×D₄×C₆₇	—	2	8	no	—
66	C₄×D₄×C₆₇	—	2	4	no	—
67	C₄×D₄×C₆₇	—	2	4	no	—
68	C₄×D₄×C₆₇	—	2	4	no	—
69	C₄×D₄×C₆₇	D₄	1	8	yes	—
70	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
86	C₂×C₄²×C₆₇	—	2	4	no	—
87	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
89	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
91	#107×C₆₇	—	2	4	no	—
92	#106×C₆₇	—	2	4	no	—
93	C₄⋊C₈×C₆₇	—	2	4	no	—
94	#102×C₆₇	—	2	4	no	—
95	#102×C₆₇	—	2	4	no	—
96	#102×C₆₇	—	2	4	no	—
97	#102×C₆₇	—	2	4	no	—
98	C₂²⋊C₈×C₆₇	—	2	2	no	—
99	C₂²⋊C₈×C₆₇	—	2	2	no	—
100	#95×C₆₇	—	2	1	no	—
101	#95×C₆₇	—	2	1	no	—
102	#95×C₆₇	—	2	1	no	—
103	#95×C₆₇	—	2	1	no	—
104	#95×C₆₇	—	2	1	no	—
105	#95×C₆₇	—	2	1	no	—
106	C₂⁴×C₆₇	—	2	4	no	—
107	C₂⁴×C₆₇	—	2	4	no	—
108	C₂⁴×C₆₇	—	2	4	no	—
109	C₂⁴×C₆₇	—	2	2	no	—
110	C₂⁴×C₆₇	—	4	2	no	—
111	C₂⁴×C₆₇	—	4	2	no	—
112	C₂⁴×C₆₇	—	4	2	no	—
113	C₂⁴×C₆₇	—	4	2	no	—
114	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₂×D₄×C₆₇	—	2	4	no	—
116	C₂×D₄×C₆₇	—	2	4	no	—
117	C₂×D₄×C₆₇	—	2	16	no	—
118	C₂×D₄×C₆₇	—	2	16	no	—
119	C₂×D₄×C₆₇	—	2	4	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	—	2	2	no	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
124	C₂×D₄×C₆₇	C₂×Q₈	1	4	yes	—
125	C₂×D₄×C₆₇	#41	1	4	yes	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	8	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	2	2	no	—
136	C₂²×C₄×C₆₇	—	2	8	no	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	1	no	—
140	C₂²×C₄×C₆₇	—	2	1	no	—
141	C₂²×C₄×C₆₇	—	4	1	no	—
142	C₂²×C₄×C₆₇	—	4	32	no	—
143	C₂²×C₄×C₆₇	—	4	4	no	—
144	C₂²×C₄×C₆₇	—	4	4	no	—
145	C₂²×C₄×C₆₇	—	4	4	no	—
146	C₂²×C₄×C₆₇	—	4	2	no	—
147	C₂²×C₄×C₆₇	—	4	2	no	—
148	C₂²×C₄×C₆₇	—	4	4	no	—
149	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
155	C₂²×C₄×C₆₇	#41	1	2	yes	—
156	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
157	C₂×C₈×C₆₇	—	2	2	no	—
158	C₂×C₈×C₆₇	—	2	2	no	—
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₄⋊C₄×C₆₇	—	2	4	no	—
164	C₄⋊C₄×C₆₇	—	2	2	no	—
165	C₄⋊C₄×C₆₇	—	2	2	no	—
166	C₄⋊C₄×C₆₇	—	2	2	no	—
167	C₄⋊C₄×C₆₇	—	2	2	no	—
168	C₄⋊C₄×C₆₇	—	2	2	no	—
169	C₄⋊C₄×C₆₇	—	4	2	no	—
170	C₄⋊C₄×C₆₇	—	4	2	no	—
171	C₄⋊C₄×C₆₇	—	4	2	no	—
172	C₄⋊C₄×C₆₇	—	4	2	no	—
173	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
174	C₂²⋊C₄×C₆₇	—	2	8	no	—
175	C₂²⋊C₄×C₆₇	—	2	8	no	—
176	C₂²⋊C₄×C₆₇	—	4	2	no	—
177	C₂²⋊C₄×C₆₇	—	4	2	no	—
178	C₄²×C₆₇	—	2	2	no	—
179	C₄²×C₆₇	—	2	2	no	—
180	C₄²×C₆₇	—	2	2	no	—
181	C₄²×C₆₇	C₂×D₄	1	4	yes	—
182	C₂³×C₆₇	—	2	1	no	—
183	C₂³×C₆₇	—	2	1	no	—
184	C₂³×C₆₇	—	2	1	no	—
185	C₂³×C₆₇	—	2	1	no	—
186	C₂³×C₆₇	—	2	16	no	—
187	C₂³×C₆₇	—	2	4	no	—
188	C₂³×C₆₇	—	2	4	no	—
189	C₂³×C₆₇	—	2	4	no	—
190	C₂³×C₆₇	—	4	8	no	—
191	C₂³×C₆₇	—	4	8	no	—
192	C₂³×C₆₇	—	4	8	no	—
193	C₂³×C₆₇	—	4	8	no	—
194	C₂³×C₆₇	—	4	8	no	—
195	C₂³×C₆₇	—	4	2	no	—
196	C₂³×C₆₇	—	4	2	no	—
197	C₂³×C₆₇	—	4	2	no	—
198	C₂³×C₆₇	—	4	2	no	—
199	C₂³×C₆₇	—	4	8	no	—
200	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
201	C₂³×C₆₇	#135	1	2	yes	—
202	C₂³×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
203	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
204	D₄×C₆₇	—	2	16	no	—
205	D₄×C₆₇	—	2	4	no	—
206	D₄×C₆₇	—	2	4	no	—
207	D₄×C₆₇	—	2	4	no	—
208	D₄×C₆₇	C₂²⋊Q₈	1	8	yes	—
209	C₂×C₄×C₆₇	—	2	8	no	—
210	C₂×C₄×C₆₇	—	2	1	no	—
211	C₂×C₄×C₆₇	—	2	1	no	—
212	C₂×C₄×C₆₇	—	2	1	no	—
213	C₂×C₄×C₆₇	—	2	1	no	—
214	C₂×C₄×C₆₇	—	2	1	no	—
215	C₂×C₄×C₆₇	—	2	1	no	—
216	C₂×C₄×C₆₇	—	2	1	no	—
217	C₂×C₄×C₆₇	—	2	1	no	—
218	C₂×C₄×C₆₇	—	2	4	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	4	no	—
222	C₂×C₄×C₆₇	—	2	1	no	—
223	C₂×C₄×C₆₇	—	2	1	no	—
224	C₂×C₄×C₆₇	—	4	4	no	—
225	C₂×C₄×C₆₇	—	4	16	no	—
226	C₂×C₄×C₆₇	—	4	16	no	—
227	C₂×C₄×C₆₇	—	4	8	no	—
228	C₂×C₄×C₆₇	—	4	2	no	—
229	C₂×C₄×C₆₇	—	4	2	no	—
230	C₂×C₄×C₆₇	—	4	1	no	—
231	C₂×C₄×C₆₇	—	4	1	no	—
232	C₂×C₄×C₆₇	—	4	1	no	—
233	C₂×C₄×C₆₇	—	4	1	no	—
234	C₂×C₄×C₆₇	—	4	1	no	—
235	C₂×C₄×C₆₇	—	4	1	no	—
236	C₂×C₄×C₆₇	—	4	1	no	—
237	C₂×C₄×C₆₇	—	4	1	no	—
238	C₂×C₄×C₆₇	—	4	1	no	—
239	C₂×C₄×C₆₇	—	4	1	no	—
240	C₂×C₄×C₆₇	—	4	1	no	—
241	C₂×C₄×C₆₇	—	4	1	no	—
242	C₂×C₄×C₆₇	#120	1	2	yes	—
243	C₂×C₄×C₆₇	#120	1	2	yes	—
244	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
245	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
246	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
247	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
248	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
249	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
250	C₈×C₆₇	—	4	2	no	—
251	C₈×C₆₇	—	4	2	no	—
252	C₂²×C₆₇	—	2	2	no	—
253	C₂²×C₆₇	—	2	2	no	—
254	C₂²×C₆₇	—	2	2	no	—
255	C₂²×C₆₇	—	2	2	no	—
256	C₂²×C₆₇	—	2	2	no	—
257	C₂²×C₆₇	—	2	2	no	—
258	C₂²×C₆₇	—	2	8	no	—
259	C₂²×C₆₇	—	4	32	no	—
260	C₂²×C₆₇	—	4	4	no	—
261	C₂²×C₆₇	—	4	4	no	—
262	C₂²×C₆₇	—	4	4	no	—
263	C₂²×C₆₇	—	4	4	no	—
264	C₂²×C₆₇	—	4	4	no	—
265	C₂²×C₆₇	—	4	4	no	—
266	C₂²×C₆₇	#478	1	4	yes	complemented
267	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
268	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
269	C₂²×C₆₇	#525	1	1	yes	characteristic
270	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
271	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
272	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₂²×C₆₇	#575	1	1	yes	characteristic
274	C₄×C₆₇	—	2	2	no	—
275	C₄×C₆₇	—	2	2	no	—
276	C₄×C₆₇	—	2	2	no	—
277	C₄×C₆₇	—	2	2	no	—
278	C₄×C₆₇	—	4	1	no	—
279	C₄×C₆₇	—	4	4	no	—
280	C₄×C₆₇	—	8	1	no	—
281	C₄×C₆₇	—	8	1	no	—
282	C₄×C₆₇	—	8	1	no	—
283	C₄×C₆₇	—	8	1	no	—
284	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
285	C₄×C₆₇	#525	1	2	yes	—
286	#32959	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
287	C₂×C₆₇	—	2	4	no	—
288	C₂×C₆₇	—	4	8	no	—
289	C₂×C₆₇	#2848	1	1	yes	characteristic
290	C₂×C₆₇	#3002	1	1	yes	characteristic
291	C₂×C₆₇	#3003	1	1	yes	characteristic
292	C₂×C₆₇	C₂×#478	1	1	yes	characteristic
293	C₂×C₆₇	#3077	1	1	yes	characteristic
294	C₂×C₆₇	#3105	1	1	yes	characteristic
295	C₂×C₆₇	#3114	1	1	yes	characteristic
296	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
297	C₂×#478	C₂×C₆₇	1	2	yes	complemented
298	#2851	C₂×C₆₇	1	1	yes	characteristic, complemented
299	#1864	C₂×C₆₇	1	2	yes	complemented
300	#1862	C₂×C₆₇	1	2	yes	complemented
301	#1853	C₂×C₆₇	1	1	yes	characteristic, complemented
302	#1852	C₂×C₆₇	1	1	yes	characteristic, complemented
303	#1804	C₂×C₆₇	1	2	yes	complemented
304	#1797	C₂×C₆₇	1	1	yes	characteristic, complemented
305	#1786	C₂×C₆₇	1	1	yes	characteristic
306	#1786	C₂×C₆₇	1	1	yes	characteristic
307	C₆₇	#32959	1	1	yes	characteristic, direct product factor, Sylow
308	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
309	C₂⁴×C₄	—	2	1	no	—
310	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic, complemented
311	C₂×C₄×D₄	—	2	8	no	—
312	C₂×C₄×D₄	—	2	1	no	—
313	C₂×C₄×D₄	—	2	1	no	—
314	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
315	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
316	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
317	C₂²×C₂²⋊C₄	—	2	1	no	—
318	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
319	#478	—	2	8	no	—
320	C₂×#107	C₂²×C₆₇	1	2	yes	complemented
321	C₂×#106	C₂²×C₆₇	1	2	yes	complemented
322	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	complemented
323	C₂×#102	C₂²×C₆₇	1	2	yes	—
324	C₂×#102	C₂²×C₆₇	1	2	yes	—
325	C₂×#102	C₂²×C₆₇	1	2	yes	—
326	C₂×#102	C₂²×C₆₇	1	2	yes	—
327	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
328	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
329	#389	C₂²×C₆₇	1	1	yes	characteristic, complemented
330	#389	C₂²×C₆₇	1	1	yes	characteristic, complemented
331	#387	—	2	1	no	—
332	#387	—	2	1	no	—
333	#385	—	2	1	no	—
334	#385	—	2	1	no	—
335	#382	C₂²×C₆₇	1	1	yes	characteristic, complemented
336	#380	C₂²×C₆₇	1	1	yes	characteristic
337	#380	C₂²×C₆₇	1	1	yes	characteristic
338	#340	C₂²×C₆₇	1	2	yes	complemented
339	#340	C₂²×C₆₇	1	2	yes	complemented
340	#336	C₂²×C₆₇	1	2	yes	complemented
341	C₂⁵	—	2	1	no	—
342	C₂²×D₄	—	2	4	no	—
343	C₂²×D₄	—	2	4	no	—
344	C₂²×D₄	—	2	1	no	—
345	C₂²×D₄	—	2	2	no	—
346	C₂²×D₄	Q₈×C₆₇	1	4	yes	—
347	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
348	C₂³×C₄	—	2	8	no	—
349	C₂³×C₄	—	2	1	no	—
350	C₂³×C₄	—	2	2	no	—
351	C₂³×C₄	—	2	2	no	—
352	C₂³×C₄	—	4	4	no	—
353	C₂³×C₄	—	4	4	no	—
354	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
355	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
357	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
358	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
359	#125	—	2	4	no	—
360	C₄×D₄	—	2	8	no	—
361	C₄×D₄	—	2	8	no	—
362	C₄×D₄	—	2	4	no	—
363	C₄×D₄	—	2	4	no	—
364	C₄×D₄	—	2	4	no	—
365	C₄×D₄	D₄×C₆₇	1	8	yes	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	4	no	—
368	C₂×C₄⋊C₄	—	2	1	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	1	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
373	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
374	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
375	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
377	C₂×C₂²⋊C₄	—	2	8	no	—
378	C₂×C₂²⋊C₄	—	2	2	no	—
379	C₂×C₂²⋊C₄	—	2	2	no	—
380	C₂×C₂²⋊C₄	—	2	1	no	—
381	C₂×C₂²⋊C₄	—	2	1	no	—
382	C₂×C₄²	—	2	4	no	—
383	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
384	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
385	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
386	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
387	#107	—	2	4	no	—
388	#106	—	2	4	no	—
389	C₄⋊C₈	—	2	4	no	—
390	#102	—	2	4	no	—
391	#102	—	2	4	no	—
392	#102	—	2	4	no	—
393	#102	—	2	4	no	—
394	C₂²⋊C₈	—	2	2	no	—
395	C₂²⋊C₈	—	2	2	no	—
396	#95	—	2	1	no	—
397	#95	—	2	1	no	—
398	#95	—	2	1	no	—
399	#95	—	2	1	no	—
400	#95	—	2	1	no	—
401	#95	—	2	1	no	—
402	C₂⁴	—	2	4	no	—
403	C₂⁴	—	2	4	no	—
404	C₂⁴	—	2	4	no	—
405	C₂⁴	—	2	2	no	—
406	C₂⁴	—	4	2	no	—
407	C₂⁴	—	4	2	no	—
408	C₂⁴	—	4	2	no	—
409	C₂⁴	—	4	2	no	—
410	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	16	no	—
414	C₂×D₄	—	2	16	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
420	C₂×D₄	C₂×Q₈×C₆₇	1	4	yes	—
421	C₂×D₄	#41×C₆₇	1	4	yes	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	8	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	8	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	4	1	no	—
438	C₂²×C₄	—	4	32	no	—
439	C₂²×C₄	—	4	4	no	—
440	C₂²×C₄	—	4	4	no	—
441	C₂²×C₄	—	4	4	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	4	2	no	—
444	C₂²×C₄	—	4	4	no	—
445	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
447	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
450	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
451	C₂²×C₄	#41×C₆₇	1	2	yes	—
452	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
453	C₂×C₈	—	2	2	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	2	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	2	no	—
458	C₂×C₈	—	2	2	no	—
459	C₄⋊C₄	—	2	4	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	4	2	no	—
466	C₄⋊C₄	—	4	2	no	—
467	C₄⋊C₄	—	4	2	no	—
468	C₄⋊C₄	—	4	2	no	—
469	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
470	C₂²⋊C₄	—	2	8	no	—
471	C₂²⋊C₄	—	2	8	no	—
472	C₂²⋊C₄	—	4	2	no	—
473	C₂²⋊C₄	—	4	2	no	—
474	C₄²	—	2	2	no	—
475	C₄²	—	2	2	no	—
476	C₄²	—	2	2	no	—
477	C₄²	C₂×D₄×C₆₇	1	4	yes	—
478	C₂³	—	2	1	no	—
479	C₂³	—	2	1	no	—
480	C₂³	—	2	1	no	—
481	C₂³	—	2	1	no	—
482	C₂³	—	2	16	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	2	4	no	—
485	C₂³	—	2	4	no	—
486	C₂³	—	4	8	no	—
487	C₂³	—	4	8	no	—
488	C₂³	—	4	8	no	—
489	C₂³	—	4	8	no	—
490	C₂³	—	4	8	no	—
491	C₂³	—	4	8	no	—
492	C₂³	—	4	2	no	—
493	C₂³	—	4	2	no	—
494	C₂³	—	4	2	no	—
495	C₂³	—	4	2	no	—
496	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
497	C₂³	#135×C₆₇	1	2	yes	—
498	C₂³	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
499	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
500	D₄	—	2	16	no	—
501	D₄	—	2	4	no	—
502	D₄	—	2	4	no	—
503	D₄	—	2	4	no	—
504	D₄	C₂²⋊Q₈×C₆₇	1	8	yes	—
505	C₂×C₄	—	2	8	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	4	4	no	—
521	C₂×C₄	—	4	16	no	—
522	C₂×C₄	—	4	16	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	—	4	2	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	#120×C₆₇	1	2	yes	—
539	C₂×C₄	#120×C₆₇	1	2	yes	—
540	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
541	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
542	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
543	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
544	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
545	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
546	C₈	—	4	2	no	—
547	C₈	—	4	2	no	—
548	C₂²	—	2	2	no	—
549	C₂²	—	2	2	no	—
550	C₂²	—	2	2	no	—
551	C₂²	—	2	2	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	2	2	no	—
554	C₂²	—	2	8	no	—
555	C₂²	—	4	32	no	—
556	C₂²	—	4	4	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	4	no	—
560	C₂²	—	4	4	no	—
561	C₂²	—	4	4	no	—
562	C₂²	#478×C₆₇	1	4	yes	complemented
563	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
564	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
565	C₂²	#525×C₆₇	1	1	yes	characteristic
566	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
567	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
568	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
569	C₂²	#575×C₆₇	1	1	yes	characteristic
570	C₄	—	2	2	no	—
571	C₄	—	2	2	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	4	1	no	—
575	C₄	—	4	4	no	—
576	C₄	—	8	1	no	—
577	C₄	—	8	1	no	—
578	C₄	—	8	1	no	—
579	C₄	—	8	1	no	—
580	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
581	C₄	#525×C₆₇	1	2	yes	—
582	C₂	—	2	4	no	—
583	C₂	—	4	8	no	—
584	C₂	C₆₇×#2848	1	1	yes	characteristic
585	C₂	C₆₇×#3002	1	1	yes	characteristic
586	C₂	C₆₇×#3003	1	1	yes	characteristic
587	C₂	C₂×#478×C₆₇	1	1	yes	characteristic
588	C₂	C₆₇×#3077	1	1	yes	characteristic
589	C₂	C₆₇×#3105	1	1	yes	characteristic
590	C₂	C₆₇×#3114	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#478×C₆₇	C₂	semidirect
3. C₆₇×#2851	C₂	semidirect
4. C₆₇×#1864	C₂	semidirect
5. C₆₇×#1862	C₂	semidirect
6. C₆₇×#1853	C₂	semidirect
7. C₆₇×#1852	C₂	semidirect
8. C₆₇×#1804	C₂	semidirect
9. C₆₇×#1797	C₂	semidirect
14. C₂×#125×C₆₇	C₂²	semidirect
24. C₂×#107×C₆₇	C₂²	semidirect
25. C₂×#106×C₆₇	C₂²	semidirect
26. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
33. #389×C₆₇	C₂²	semidirect
34. #389×C₆₇	C₂²	semidirect
39. #382×C₆₇	C₂²	semidirect
42. #340×C₆₇	C₂²	semidirect
43. #340×C₆₇	C₂²	semidirect
44. #336×C₆₇	C₂²	semidirect
266. C₂²×C₆₇	#478	semidirect
286. #32959	C₆₇	direct (N × H)
297. C₂×#478	C₂×C₆₇	semidirect
298. #2851	C₂×C₆₇	semidirect
299. #1864	C₂×C₆₇	semidirect
300. #1862	C₂×C₆₇	semidirect
301. #1853	C₂×C₆₇	semidirect
302. #1852	C₂×C₆₇	semidirect
303. #1804	C₂×C₆₇	semidirect
304. #1797	C₂×C₆₇	semidirect
307. C₆₇	#32959	direct (N × H)
310. C₂×#125	C₂²×C₆₇	semidirect
320. C₂×#107	C₂²×C₆₇	semidirect
321. C₂×#106	C₂²×C₆₇	semidirect
322. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
329. #389	C₂²×C₆₇	semidirect
330. #389	C₂²×C₆₇	semidirect
335. #382	C₂²×C₆₇	semidirect
338. #340	C₂²×C₆₇	semidirect
339. #340	C₂²×C₆₇	semidirect
340. #336	C₂²×C₆₇	semidirect
562. C₂²	#478×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×D₄×C₆₇	2
C₂×#478×C₆₇	2
C₆₇×#2851	2
C₆₇×#1864	2
C₆₇×#1862	2
C₆₇×#1853	2
C₆₇×#1852	2
C₆₇×#1804	2
C₆₇×#1797	2
C₆₇×#1786	2
#32959	67

Subgroups — C67×#32959 #30712210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32959 #30712210