Subgroups — C₆₇×#19926 #30699177

Order 17152.

626 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3384	C₂	1	2	yes	maximal
2	C₆₇×#3338	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1692	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1669	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1644	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1637	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₆₇	—	2	1	no	—
13	#554×C₆₇	C₂²	1	2	yes	—
14	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
15	#501×C₆₇	—	2	2	no	—
16	#500×C₆₇	C₂²	1	1	yes	characteristic
17	#498×C₆₇	C₂²	1	2	yes	—
18	#494×C₆₇	C₂²	1	1	yes	characteristic
19	#487×C₆₇	C₂²	1	2	yes	—
20	#484×C₆₇	—	2	2	no	—
21	#478×C₆₇	—	2	2	no	—
22	#477×C₆₇	C₂²	1	2	yes	—
23	#476×C₆₇	—	2	2	no	—
24	#476×C₆₇	C₂²	1	2	yes	—
25	#474×C₆₇	C₂²	1	2	yes	—
26	#467×C₆₇	—	2	1	no	—
27	#467×C₆₇	—	2	1	no	—
28	#462×C₆₇	C₂²	1	2	yes	—
29	#451×C₆₇	—	2	2	no	—
30	#446×C₆₇	C₂²	1	2	yes	—
31	#442×C₆₇	—	2	2	no	—
32	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#107×C₆₇	—	2	1	no	—
37	#335×C₆₇	C₂²	1	1	yes	characteristic
38	#335×C₆₇	C₂²	1	1	yes	characteristic
39	#333×C₆₇	C₂²	1	2	yes	—
40	#332×C₆₇	C₂²	1	2	yes	—
41	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
42	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂²×Q₈×C₆₇	—	2	2	no	—
46	C₂×Q₁₆×C₆₇	—	2	2	no	—
47	C₂×Q₁₆×C₆₇	—	2	1	no	—
48	C₂×Q₁₆×C₆₇	—	2	1	no	—
49	C₂×Q₁₆×C₆₇	—	4	4	no	—
50	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
51	C₂×D₈×C₆₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	—	2	1	no	—
53	C₂²×C₈×C₆₇	—	2	1	no	—
54	C₄⋊Q₈×C₆₇	—	2	2	no	—
55	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
56	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
57	#125×C₆₇	—	2	2	no	—
58	C₂²⋊Q₈×C₆₇	—	2	2	no	—
59	C₂²⋊Q₈×C₆₇	—	2	2	no	—
60	C₂²⋊Q₈×C₆₇	—	4	2	no	—
61	C₂²⋊Q₈×C₆₇	—	4	2	no	—
62	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
63	C₄×D₄×C₆₇	—	2	2	no	—
64	C₄×D₄×C₆₇	—	2	2	no	—
65	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
68	#107×C₆₇	—	2	1	no	—
69	#107×C₆₇	—	2	1	no	—
70	#107×C₆₇	—	2	2	no	—
71	#107×C₆₇	—	2	2	no	—
72	#107×C₆₇	D₄	1	1	yes	characteristic
73	#107×C₆₇	D₄	1	1	yes	characteristic
74	#106×C₆₇	—	2	2	no	—
75	#106×C₆₇	—	2	2	no	—
76	#106×C₆₇	—	2	2	no	—
77	C₄⋊C₈×C₆₇	—	2	2	no	—
78	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
80	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
81	#103×C₆₇	—	2	2	no	—
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	4	2	no	—
84	#103×C₆₇	D₄	1	2	yes	—
85	#102×C₆₇	—	2	2	no	—
86	#102×C₆₇	—	2	2	no	—
87	#102×C₆₇	D₄	1	1	yes	characteristic
88	#102×C₆₇	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₆₇	—	2	1	no	—
90	C₂²⋊C₈×C₆₇	—	2	1	no	—
91	#97×C₆₇	—	2	2	no	—
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₆₇	—	2	2	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	4	2	no	—
98	C₂×Q₈×C₆₇	—	4	4	no	—
99	C₂×Q₈×C₆₇	—	8	2	no	—
100	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
101	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	—	2	1	no	—
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	4	2	no	—
106	C₂²×C₄×C₆₇	—	4	2	no	—
107	Q₁₆×C₆₇	—	2	2	no	—
108	Q₁₆×C₆₇	—	4	4	no	—
109	Q₁₆×C₆₇	—	4	2	no	—
110	Q₁₆×C₆₇	—	8	1	no	—
111	D₈×C₆₇	Q₁₆	1	4	yes	complemented
112	C₂×C₈×C₆₇	—	2	2	no	—
113	C₂×C₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	1	no	—
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₂×C₈×C₆₇	—	4	2	no	—
117	C₂×C₈×C₆₇	—	4	2	no	—
118	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	2	2	no	—
127	C₄⋊C₄×C₆₇	—	4	2	no	—
128	C₄⋊C₄×C₆₇	—	4	2	no	—
129	C₄⋊C₄×C₆₇	—	4	2	no	—
130	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
133	C₂²⋊C₄×C₆₇	—	2	1	no	—
134	C₂²⋊C₄×C₆₇	—	2	1	no	—
135	C₂²⋊C₄×C₆₇	—	4	2	no	—
136	C₂²⋊C₄×C₆₇	—	4	2	no	—
137	C₄²×C₆₇	—	2	2	no	—
138	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
139	C₂³×C₆₇	—	2	1	no	—
140	C₂³×C₆₇	—	2	1	no	—
141	Q₈×C₆₇	—	2	2	no	—
142	Q₈×C₆₇	—	2	2	no	—
143	Q₈×C₆₇	—	4	2	no	—
144	Q₈×C₆₇	—	4	2	no	—
145	Q₈×C₆₇	—	8	2	no	—
146	Q₈×C₆₇	—	8	4	no	—
147	D₄×C₆₇	—	2	1	no	—
148	D₄×C₆₇	—	2	1	no	—
149	D₄×C₆₇	C₂×Q₁₆	1	2	yes	—
150	D₄×C₆₇	C₂×Q₁₆	1	2	yes	complemented
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	4	1	no	—
156	C₂×C₄×C₆₇	—	4	1	no	—
157	C₂×C₄×C₆₇	—	4	1	no	—
158	C₂×C₄×C₆₇	—	4	1	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	8	4	no	—
162	C₂×C₄×C₆₇	—	16	2	no	—
163	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
166	C₈×C₆₇	—	2	1	no	—
167	C₈×C₆₇	—	2	1	no	—
168	C₈×C₆₇	—	4	1	no	—
169	C₈×C₆₇	—	4	1	no	—
170	C₈×C₆₇	—	4	1	no	—
171	C₈×C₆₇	—	4	2	no	—
172	C₈×C₆₇	C₂×Q₁₆	1	2	yes	complemented
173	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
174	C₂²×C₆₇	—	2	2	no	—
175	C₂²×C₆₇	—	2	2	no	—
176	C₂²×C₆₇	—	4	1	no	—
177	C₂²×C₆₇	—	4	1	no	—
178	C₂²×C₆₇	—	4	1	no	—
179	C₂²×C₆₇	—	4	1	no	—
180	C₂²×C₆₇	D₄²	1	1	yes	characteristic
181	C₄×C₆₇	—	2	1	no	—
182	C₄×C₆₇	—	4	1	no	—
183	C₄×C₆₇	—	4	1	no	—
184	C₄×C₆₇	—	4	2	no	—
185	C₄×C₆₇	—	4	2	no	—
186	C₄×C₆₇	—	8	2	no	—
187	C₄×C₆₇	—	8	2	no	—
188	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
189	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₄×C₆₇	#577	1	1	yes	characteristic
191	C₄×C₆₇	#578	1	1	yes	characteristic
192	#19926	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂×C₆₇	—	4	2	no	—
194	C₂×C₆₇	—	4	2	no	—
195	C₂×C₆₇	#3272	1	1	yes	characteristic
196	C₂×C₆₇	D₄×Q₁₆	1	1	yes	characteristic
197	C₂×C₆₇	#3290	1	1	yes	characteristic
198	#3384	C₂×C₆₇	1	2	yes	—
199	#3338	C₂×C₆₇	1	1	yes	characteristic, complemented
200	D₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
201	#1707	C₂×C₆₇	1	1	yes	characteristic, complemented
202	#1692	C₂×C₆₇	1	2	yes	complemented
203	#1675	C₂×C₆₇	1	1	yes	characteristic, complemented
204	#1669	C₂×C₆₇	1	1	yes	characteristic, complemented
205	#1644	C₂×C₆₇	1	2	yes	complemented
206	#1637	C₂×C₆₇	1	2	yes	complemented
207	#1591	C₂×C₆₇	1	1	yes	characteristic, complemented
208	C₈×D₈	C₂×C₆₇	1	1	yes	characteristic
209	C₆₇	#19926	1	1	yes	characteristic, direct product factor, Sylow
210	C₂²×Q₁₆	—	2	1	no	—
211	#554	C₂²×C₆₇	1	2	yes	—
212	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
213	#501	—	2	2	no	—
214	#500	C₂²×C₆₇	1	1	yes	characteristic
215	#498	C₂²×C₆₇	1	2	yes	—
216	#494	C₂²×C₆₇	1	1	yes	characteristic
217	#487	C₂²×C₆₇	1	2	yes	—
218	#484	—	2	2	no	—
219	#478	—	2	2	no	—
220	#477	C₂²×C₆₇	1	2	yes	—
221	#476	—	2	2	no	—
222	#476	C₂²×C₆₇	1	2	yes	—
223	#474	C₂²×C₆₇	1	2	yes	—
224	#467	—	2	1	no	—
225	#467	—	2	1	no	—
226	#462	C₂²×C₆₇	1	2	yes	—
227	#451	—	2	2	no	—
228	#446	C₂²×C₆₇	1	2	yes	—
229	#442	—	2	2	no	—
230	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
231	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic
232	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
233	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
234	C₂×#107	—	2	1	no	—
235	#335	C₂²×C₆₇	1	1	yes	characteristic
236	#335	C₂²×C₆₇	1	1	yes	characteristic
237	#333	C₂²×C₆₇	1	2	yes	—
238	#332	C₂²×C₆₇	1	2	yes	—
239	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
240	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
241	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
242	C₈²	C₂²×C₆₇	1	1	yes	characteristic
243	C₂²×Q₈	—	2	2	no	—
244	C₂×Q₁₆	—	2	2	no	—
245	C₂×Q₁₆	—	2	1	no	—
246	C₂×Q₁₆	—	2	1	no	—
247	C₂×Q₁₆	—	4	4	no	—
248	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
249	C₂×D₈	D₄×C₆₇	1	1	yes	characteristic
250	C₂²×C₈	—	2	1	no	—
251	C₂²×C₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	2	no	—
253	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
254	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
255	#125	—	2	2	no	—
256	C₂²⋊Q₈	—	2	2	no	—
257	C₂²⋊Q₈	—	2	2	no	—
258	C₂²⋊Q₈	—	4	2	no	—
259	C₂²⋊Q₈	—	4	2	no	—
260	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
261	C₄×D₄	—	2	2	no	—
262	C₄×D₄	—	2	2	no	—
263	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
264	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
265	C₂×C₄⋊C₄	—	2	2	no	—
266	#107	—	2	1	no	—
267	#107	—	2	1	no	—
268	#107	—	2	2	no	—
269	#107	—	2	2	no	—
270	#107	D₄×C₆₇	1	1	yes	characteristic
271	#107	D₄×C₆₇	1	1	yes	characteristic
272	#106	—	2	2	no	—
273	#106	—	2	2	no	—
274	#106	—	2	2	no	—
275	C₄⋊C₈	—	2	2	no	—
276	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
277	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
278	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
279	#103	—	2	2	no	—
280	#103	—	2	2	no	—
281	#103	—	4	2	no	—
282	#103	D₄×C₆₇	1	2	yes	—
283	#102	—	2	2	no	—
284	#102	—	2	2	no	—
285	#102	D₄×C₆₇	1	1	yes	characteristic
286	#102	D₄×C₆₇	1	1	yes	characteristic
287	C₂²⋊C₈	—	2	1	no	—
288	C₂²⋊C₈	—	2	1	no	—
289	#97	—	2	2	no	—
290	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
291	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
292	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	4	2	no	—
296	C₂×Q₈	—	4	4	no	—
297	C₂×Q₈	—	8	2	no	—
298	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
299	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
300	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
301	C₂²×C₄	—	2	1	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	—	4	2	no	—
305	Q₁₆	—	2	2	no	—
306	Q₁₆	—	4	4	no	—
307	Q₁₆	—	4	2	no	—
308	Q₁₆	—	8	1	no	—
309	D₈	Q₁₆×C₆₇	1	4	yes	complemented
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	4	2	no	—
315	C₂×C₈	—	4	2	no	—
316	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
317	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
318	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
319	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	4	2	no	—
326	C₄⋊C₄	—	4	2	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
329	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
330	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
331	C₂²⋊C₄	—	2	1	no	—
332	C₂²⋊C₄	—	2	1	no	—
333	C₂²⋊C₄	—	4	2	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₄²	—	2	2	no	—
336	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
337	C₂³	—	2	1	no	—
338	C₂³	—	2	1	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	2	2	no	—
341	Q₈	—	4	2	no	—
342	Q₈	—	4	2	no	—
343	Q₈	—	8	2	no	—
344	Q₈	—	8	4	no	—
345	D₄	—	2	1	no	—
346	D₄	—	2	1	no	—
347	D₄	C₂×Q₁₆×C₆₇	1	2	yes	—
348	D₄	C₂×Q₁₆×C₆₇	1	2	yes	complemented
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	8	4	no	—
360	C₂×C₄	—	16	2	no	—
361	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
362	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
363	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
364	C₈	—	2	1	no	—
365	C₈	—	2	1	no	—
366	C₈	—	4	1	no	—
367	C₈	—	4	1	no	—
368	C₈	—	4	1	no	—
369	C₈	—	4	2	no	—
370	C₈	C₂×Q₁₆×C₆₇	1	2	yes	complemented
371	C₈	AGL(1,ℤ/8)×C₆₇	1	2	yes	—
372	C₂²	—	2	2	no	—
373	C₂²	—	2	2	no	—
374	C₂²	—	4	1	no	—
375	C₂²	—	4	1	no	—
376	C₂²	—	4	1	no	—
377	C₂²	—	4	1	no	—
378	C₂²	D₄²×C₆₇	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	4	1	no	—
381	C₄	—	4	1	no	—
382	C₄	—	4	2	no	—
383	C₄	—	4	2	no	—
384	C₄	—	8	2	no	—
385	C₄	—	8	2	no	—
386	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
387	C₄	C₂×AGL(1,ℤ/8)×C₆₇	1	1	yes	characteristic
388	C₄	#577×C₆₇	1	1	yes	characteristic
389	C₄	#578×C₆₇	1	1	yes	characteristic
390	C₂	—	4	2	no	—
391	C₂	—	4	2	no	—
392	C₂	C₆₇×#3272	1	1	yes	characteristic
393	C₂	D₄×Q₁₆×C₆₇	1	1	yes	characteristic
394	C₂	C₆₇×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂×D₈×C₆₇	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3338	C₂	semidirect
3. D₄×Q₁₆×C₆₇	C₂	semidirect
4. C₆₇×#1707	C₂	semidirect
5. C₆₇×#1692	C₂	semidirect
6. C₆₇×#1675	C₂	semidirect
7. C₆₇×#1669	C₂	semidirect
8. C₆₇×#1644	C₂	semidirect
9. C₆₇×#1637	C₂	semidirect
10. C₆₇×#1591	C₂	semidirect
111. D₈×C₆₇	Q₁₆	semidirect
150. D₄×C₆₇	C₂×Q₁₆	semidirect
172. C₈×C₆₇	C₂×Q₁₆	semidirect
192. #19926	C₆₇	direct (N × H)
199. #3338	C₂×C₆₇	semidirect
200. D₄×Q₁₆	C₂×C₆₇	semidirect
201. #1707	C₂×C₆₇	semidirect
202. #1692	C₂×C₆₇	semidirect
203. #1675	C₂×C₆₇	semidirect
204. #1669	C₂×C₆₇	semidirect
205. #1644	C₂×C₆₇	semidirect
206. #1637	C₂×C₆₇	semidirect
207. #1591	C₂×C₆₇	semidirect
209. C₆₇	#19926	direct (N × H)
309. D₈	Q₁₆×C₆₇	semidirect
348. D₄	C₂×Q₁₆×C₆₇	semidirect
370. C₈	C₂×Q₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3384	2
C₆₇×#3338	2
D₄×Q₁₆×C₆₇	2
C₆₇×#1707	2
C₆₇×#1692	2
C₆₇×#1675	2
C₆₇×#1669	2
C₆₇×#1644	2
C₆₇×#1637	2
C₆₇×#1591	2
C₈×D₈×C₆₇	2
#19926	67

Subgroups — C67×#19926 #30699177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#19926 #30699177