Subgroups — C₆₇×#23831 #30703082

Order 17152.

682 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2330	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2327	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2101	C₂	1	2	yes	maximal
4	C₆₇×#2072	C₂	1	2	yes	maximal
5	C₆₇×#2043	C₂	1	2	yes	maximal
6	C₆₇×#2026	C₂	1	2	yes	maximal
7	C₆₇×#1763	C₂	1	2	yes	maximal
8	C₄×#103×C₆₇	C₂	1	2	yes	maximal
9	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
13	C₂×#103×C₆₇	C₂²	1	2	yes	—
14	C₂×#103×C₆₇	C₂²	1	2	yes	—
15	C₂×#103×C₆₇	C₂²	1	2	yes	—
16	C₂×#103×C₆₇	C₂²	1	2	yes	—
17	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#391×C₆₇	C₂²	1	2	yes	—
20	#389×C₆₇	—	2	2	no	—
21	#389×C₆₇	C₂²	1	2	yes	—
22	#387×C₆₇	—	2	2	no	—
23	#387×C₆₇	—	2	2	no	—
24	#387×C₆₇	—	2	2	no	—
25	#384×C₆₇	C₂²	1	1	yes	characteristic
26	#382×C₆₇	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	#376×C₆₇	C₂²	1	1	yes	characteristic
29	#340×C₆₇	C₂²	1	2	yes	—
30	#340×C₆₇	C₂²	1	2	yes	—
31	#340×C₆₇	C₂²	1	2	yes	—
32	#340×C₆₇	C₂²	1	2	yes	—
33	#336×C₆₇	C₂²	1	2	yes	—
34	#336×C₆₇	C₂²	1	2	yes	—
35	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	C₄⋊Q₈×C₆₇	—	2	4	no	—
39	#125×C₆₇	—	2	4	no	—
40	C₄×Q₈×C₆₇	—	2	4	no	—
41	C₄×Q₈×C₆₇	—	2	4	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₆₇	—	2	2	no	—
52	C₂×C₄²×C₆₇	—	2	2	no	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	#103×C₆₇	—	2	4	no	—
57	#103×C₆₇	—	2	4	no	—
58	#103×C₆₇	—	2	4	no	—
59	#103×C₆₇	—	2	4	no	—
60	#97×C₆₇	—	2	2	no	—
61	C₄×C₈×C₆₇	—	2	2	no	—
62	#95×C₆₇	—	2	2	no	—
63	#95×C₆₇	—	2	1	no	—
64	#95×C₆₇	—	2	1	no	—
65	#95×C₆₇	—	2	2	no	—
66	#95×C₆₇	—	2	4	no	—
67	#95×C₆₇	D₄	1	2	yes	—
68	#95×C₆₇	D₄	1	2	yes	—
69	C₂×Q₈×C₆₇	—	2	2	no	—
70	C₂×Q₈×C₆₇	—	2	2	no	—
71	C₂×Q₈×C₆₇	—	2	2	no	—
72	C₂×Q₈×C₆₇	—	2	2	no	—
73	C₂×Q₈×C₆₇	—	2	2	no	—
74	C₂×Q₈×C₆₇	#41	1	4	yes	—
75	C₂²×C₄×C₆₇	—	2	2	no	—
76	C₂²×C₄×C₆₇	—	2	2	no	—
77	C₂²×C₄×C₆₇	—	2	2	no	—
78	C₂²×C₄×C₆₇	—	2	2	no	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	2	no	—
86	C₂×C₈×C₆₇	—	2	2	no	—
87	C₂×C₈×C₆₇	—	2	2	no	—
88	C₂×C₈×C₆₇	—	2	2	no	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	—	2	8	no	—
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	—	2	4	no	—
93	C₄⋊C₄×C₆₇	—	2	2	no	—
94	C₄⋊C₄×C₆₇	—	2	2	no	—
95	C₄⋊C₄×C₆₇	—	2	2	no	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	4	4	no	—
99	C₄⋊C₄×C₆₇	—	8	2	no	—
100	C₄⋊C₄×C₆₇	—	8	2	no	—
101	C₄⋊C₄×C₆₇	#41	1	4	yes	—
102	C₄²×C₆₇	—	2	2	no	—
103	C₄²×C₆₇	—	2	2	no	—
104	C₄²×C₆₇	—	4	4	no	—
105	C₄²×C₆₇	—	8	2	no	—
106	C₄²×C₆₇	#41	1	2	yes	—
107	C₄²×C₆₇	#41	1	2	yes	—
108	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₆₇	—	2	4	no	—
110	Q₈×C₆₇	—	2	4	no	—
111	Q₈×C₆₇	—	4	2	no	—
112	Q₈×C₆₇	—	4	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	2	2	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	4	no	—
119	C₂×C₄×C₆₇	—	2	4	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	4	2	no	—
129	C₂×C₄×C₆₇	—	4	2	no	—
130	C₂×C₄×C₆₇	—	4	2	no	—
131	C₂×C₄×C₆₇	—	4	2	no	—
132	C₂×C₄×C₆₇	—	4	2	no	—
133	C₂×C₄×C₆₇	—	4	2	no	—
134	C₂×C₄×C₆₇	#124	1	4	yes	—
135	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
136	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
137	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
139	C₈×C₆₇	—	4	2	no	—
140	C₈×C₆₇	—	4	2	no	—
141	C₂²×C₆₇	#525	1	1	yes	characteristic
142	C₂²×C₆₇	#525	1	1	yes	characteristic
143	C₂²×C₆₇	C₂×#124	1	1	yes	characteristic
144	C₂²×C₆₇	#577	1	2	yes	—
145	C₂²×C₆₇	#578	1	2	yes	—
146	C₄×C₆₇	—	2	4	no	—
147	C₄×C₆₇	—	4	2	no	—
148	C₄×C₆₇	—	4	4	no	—
149	C₄×C₆₇	—	4	1	no	—
150	C₄×C₆₇	—	4	1	no	—
151	C₄×C₆₇	—	4	4	no	—
152	C₄×C₆₇	—	8	2	no	—
153	C₄×C₆₇	—	8	2	no	—
154	C₄×C₆₇	#529	1	2	yes	—
155	C₄×C₆₇	#529	1	2	yes	—
156	#23831	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₆₇	#2398	1	1	yes	characteristic
158	C₂×C₆₇	#3120	1	2	yes	—
159	C₂×C₆₇	#3126	1	1	yes	characteristic
160	C₂×C₆₇	#3127	1	1	yes	characteristic
161	C₂×C₆₇	#3137	1	2	yes	—
162	#2330	C₂×C₆₇	1	1	yes	characteristic
163	#2327	C₂×C₆₇	1	1	yes	characteristic
164	#2101	C₂×C₆₇	1	2	yes	—
165	#2072	C₂×C₆₇	1	2	yes	—
166	#2043	C₂×C₆₇	1	2	yes	—
167	#2026	C₂×C₆₇	1	2	yes	—
168	#1763	C₂×C₆₇	1	2	yes	—
169	C₄×#103	C₂×C₆₇	1	2	yes	—
170	#1746	C₂×C₆₇	1	1	yes	characteristic
171	C₆₇	#23831	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
173	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
174	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
175	C₂×#103	C₂²×C₆₇	1	2	yes	—
176	C₂×#103	C₂²×C₆₇	1	2	yes	—
177	C₂×#103	C₂²×C₆₇	1	2	yes	—
178	C₂×#103	C₂²×C₆₇	1	2	yes	—
179	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
180	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
181	#391	C₂²×C₆₇	1	2	yes	—
182	#389	—	2	2	no	—
183	#389	C₂²×C₆₇	1	2	yes	—
184	#387	—	2	2	no	—
185	#387	—	2	2	no	—
186	#387	—	2	2	no	—
187	#384	C₂²×C₆₇	1	1	yes	characteristic
188	#382	C₂²×C₆₇	1	1	yes	characteristic
189	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
190	#376	C₂²×C₆₇	1	1	yes	characteristic
191	#340	C₂²×C₆₇	1	2	yes	—
192	#340	C₂²×C₆₇	1	2	yes	—
193	#340	C₂²×C₆₇	1	2	yes	—
194	#340	C₂²×C₆₇	1	2	yes	—
195	#336	C₂²×C₆₇	1	2	yes	—
196	#336	C₂²×C₆₇	1	2	yes	—
197	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
198	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
199	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
200	C₄⋊Q₈	—	2	4	no	—
201	#125	—	2	4	no	—
202	C₄×Q₈	—	2	4	no	—
203	C₄×Q₈	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
216	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
217	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
218	#103	—	2	4	no	—
219	#103	—	2	4	no	—
220	#103	—	2	4	no	—
221	#103	—	2	4	no	—
222	#97	—	2	2	no	—
223	C₄×C₈	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	1	no	—
226	#95	—	2	1	no	—
227	#95	—	2	2	no	—
228	#95	—	2	4	no	—
229	#95	D₄×C₆₇	1	2	yes	—
230	#95	D₄×C₆₇	1	2	yes	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	#41×C₆₇	1	4	yes	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
245	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₄⋊C₄	—	2	8	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	8	2	no	—
262	C₄⋊C₄	—	8	2	no	—
263	C₄⋊C₄	#41×C₆₇	1	4	yes	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	4	4	no	—
267	C₄²	—	8	2	no	—
268	C₄²	#41×C₆₇	1	2	yes	—
269	C₄²	#41×C₆₇	1	2	yes	—
270	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
271	Q₈	—	2	4	no	—
272	Q₈	—	2	4	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	#124×C₆₇	1	4	yes	—
297	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
298	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
299	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
300	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
301	C₈	—	4	2	no	—
302	C₈	—	4	2	no	—
303	C₂²	#525×C₆₇	1	1	yes	characteristic
304	C₂²	#525×C₆₇	1	1	yes	characteristic
305	C₂²	C₂×#124×C₆₇	1	1	yes	characteristic
306	C₂²	#577×C₆₇	1	2	yes	—
307	C₂²	#578×C₆₇	1	2	yes	—
308	C₄	—	2	4	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	4	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	4	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	#529×C₆₇	1	2	yes	—
317	C₄	#529×C₆₇	1	2	yes	—
318	C₂	C₆₇×#2398	1	1	yes	characteristic
319	C₂	C₆₇×#3120	1	2	yes	—
320	C₂	C₆₇×#3126	1	1	yes	characteristic
321	C₂	C₆₇×#3127	1	1	yes	characteristic
322	C₂	C₆₇×#3137	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ #376×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
156. #23831	C₆₇	direct (N × H)
171. C₆₇	#23831	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2330	2
C₆₇×#2327	2
C₆₇×#2101	2
C₆₇×#2072	2
C₆₇×#2043	2
C₆₇×#2026	2
C₆₇×#1763	2
C₄×#103×C₆₇	2
C₆₇×#1746	2
#23831	67

Subgroups — C67×#23831 #30703082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#23831 #30703082