Subgroups — C₆₇×#23843 #30703094

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 312 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	2	yes	maximal
2	C₆₇×#2084	C₂	1	2	yes	maximal
3	C₆₇×#2083	C₂	1	2	yes	maximal
4	C₆₇×#2059	C₂	1	2	yes	maximal
5	C₆₇×#2059	C₂	1	2	yes	maximal
6	C₆₇×#1776	C₂	1	2	yes	maximal
7	C₄×#107×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₄×#106×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×#125×C₆₇	C₂²	1	2	yes	—
11	C₂×#107×C₆₇	C₂²	1	2	yes	—
12	C₂×#107×C₆₇	C₂²	1	2	yes	—
13	C₂×#106×C₆₇	C₂²	1	2	yes	—
14	C₂×#106×C₆₇	C₂²	1	2	yes	—
15	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#389×C₆₇	—	2	2	no	—
18	#389×C₆₇	—	2	2	no	—
19	#389×C₆₇	C₂²	1	2	yes	—
20	#389×C₆₇	C₂²	1	2	yes	—
21	#387×C₆₇	—	2	2	no	—
22	#387×C₆₇	—	2	2	no	—
23	#384×C₆₇	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
26	#376×C₆₇	C₂²	1	1	yes	characteristic
27	#340×C₆₇	C₂²	1	1	yes	characteristic
28	#340×C₆₇	C₂²	1	1	yes	characteristic
29	#340×C₆₇	C₂²	1	1	yes	characteristic
30	#340×C₆₇	C₂²	1	1	yes	characteristic
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×C₈×C₆₇	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
38	#125×C₆₇	—	2	4	no	—
39	#125×C₆₇	—	2	4	no	—
40	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
46	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄²×C₆₇	—	2	2	no	—
51	C₂×C₄²×C₆₇	—	2	2	no	—
52	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	#107×C₆₇	—	2	4	no	—
55	#107×C₆₇	—	2	4	no	—
56	#106×C₆₇	—	2	4	no	—
57	#106×C₆₇	—	2	4	no	—
58	#97×C₆₇	Q₈	1	4	yes	—
59	C₄×C₈×C₆₇	Q₈	1	4	yes	—
60	#95×C₆₇	—	2	2	no	—
61	#95×C₆₇	—	2	2	no	—
62	#95×C₆₇	—	2	2	no	—
63	#95×C₆₇	—	2	2	no	—
64	#95×C₆₇	D₄	1	1	yes	characteristic
65	#95×C₆₇	D₄	1	2	yes	—
66	#95×C₆₇	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	—	2	2	no	—
68	C₂²×C₄×C₆₇	—	2	2	no	—
69	C₂²×C₄×C₆₇	—	2	2	no	—
70	C₂²×C₄×C₆₇	—	2	2	no	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
76	C₂×C₈×C₆₇	—	2	2	no	—
77	C₂×C₈×C₆₇	—	2	2	no	—
78	C₂×C₈×C₆₇	—	2	2	no	—
79	C₂×C₈×C₆₇	—	2	2	no	—
80	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₆₇	C₂×Q₈	1	4	yes	—
83	C₄⋊C₄×C₆₇	—	2	8	no	—
84	C₄⋊C₄×C₆₇	—	2	8	no	—
85	C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₄⋊C₄×C₆₇	—	2	2	no	—
87	C₄⋊C₄×C₆₇	—	2	2	no	—
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	—	2	2	no	—
91	C₄⋊C₄×C₆₇	—	8	2	no	—
92	C₄⋊C₄×C₆₇	—	8	2	no	—
93	C₄⋊C₄×C₆₇	—	8	2	no	—
94	C₄⋊C₄×C₆₇	—	8	2	no	—
95	C₄⋊C₄×C₆₇	#41	1	4	yes	—
96	C₄⋊C₄×C₆₇	#41	1	4	yes	—
97	C₄²×C₆₇	—	2	2	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	—	8	2	no	—
100	C₄²×C₆₇	—	8	2	no	—
101	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
102	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
103	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₇	—	2	1	no	—
105	C₂×C₄×C₆₇	—	2	2	no	—
106	C₂×C₄×C₆₇	—	2	1	no	—
107	C₂×C₄×C₆₇	—	2	2	no	—
108	C₂×C₄×C₆₇	—	2	2	no	—
109	C₂×C₄×C₆₇	—	2	4	no	—
110	C₂×C₄×C₆₇	—	2	4	no	—
111	C₂×C₄×C₆₇	—	2	2	no	—
112	C₂×C₄×C₆₇	—	2	2	no	—
113	C₂×C₄×C₆₇	—	2	2	no	—
114	C₂×C₄×C₆₇	—	2	2	no	—
115	C₂×C₄×C₆₇	—	4	2	no	—
116	C₂×C₄×C₆₇	—	4	2	no	—
117	C₂×C₄×C₆₇	—	4	2	no	—
118	C₂×C₄×C₆₇	—	4	2	no	—
119	C₂×C₄×C₆₇	—	4	2	no	—
120	C₂×C₄×C₆₇	—	4	2	no	—
121	C₂×C₄×C₆₇	—	4	2	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	2	no	—
124	C₂×C₄×C₆₇	—	4	2	no	—
125	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
126	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
130	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
131	C₈×C₆₇	—	2	2	no	—
132	C₈×C₆₇	—	2	4	no	—
133	C₈×C₆₇	—	2	2	no	—
134	C₂²×C₆₇	#525	1	2	yes	—
135	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₂²×C₆₇	#576	1	1	yes	characteristic
137	C₂²×C₆₇	#577	1	2	yes	—
138	C₂²×C₆₇	#578	1	1	yes	characteristic
139	C₄×C₆₇	—	2	2	no	—
140	C₄×C₆₇	—	2	2	no	—
141	C₄×C₆₇	—	4	2	no	—
142	C₄×C₆₇	—	4	4	no	—
143	C₄×C₆₇	—	4	2	no	—
144	C₄×C₆₇	—	4	4	no	—
145	C₄×C₆₇	—	8	2	no	—
146	C₄×C₆₇	—	8	2	no	—
147	C₄×C₆₇	#533	1	1	yes	characteristic
148	C₄×C₆₇	#533	1	2	yes	—
149	C₄×C₆₇	#533	1	1	yes	characteristic
150	#23843	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₆₇	#2399	1	1	yes	characteristic
152	C₂×C₆₇	#3119	1	2	yes	—
153	C₂×C₆₇	#3120	1	2	yes	—
154	C₂×C₆₇	#3163	1	1	yes	characteristic
155	C₂×C₆₇	#3164	1	1	yes	characteristic
156	#2330	C₂×C₆₇	1	2	yes	—
157	#2084	C₂×C₆₇	1	2	yes	—
158	#2083	C₂×C₆₇	1	2	yes	—
159	#2059	C₂×C₆₇	1	2	yes	—
160	#2059	C₂×C₆₇	1	2	yes	—
161	#1776	C₂×C₆₇	1	2	yes	—
162	C₄×#107	C₂×C₆₇	1	1	yes	characteristic
163	C₄×#106	C₂×C₆₇	1	1	yes	characteristic
164	#1746	C₂×C₆₇	1	1	yes	characteristic
165	C₆₇	#23843	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×#125	C₂²×C₆₇	1	2	yes	—
167	C₂×#107	C₂²×C₆₇	1	2	yes	—
168	C₂×#107	C₂²×C₆₇	1	2	yes	—
169	C₂×#106	C₂²×C₆₇	1	2	yes	—
170	C₂×#106	C₂²×C₆₇	1	2	yes	—
171	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
173	#389	—	2	2	no	—
174	#389	—	2	2	no	—
175	#389	C₂²×C₆₇	1	2	yes	—
176	#389	C₂²×C₆₇	1	2	yes	—
177	#387	—	2	2	no	—
178	#387	—	2	2	no	—
179	#384	C₂²×C₆₇	1	2	yes	—
180	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
181	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
182	#376	C₂²×C₆₇	1	1	yes	characteristic
183	#340	C₂²×C₆₇	1	1	yes	characteristic
184	#340	C₂²×C₆₇	1	1	yes	characteristic
185	#340	C₂²×C₆₇	1	2	yes	—
186	#340	C₂²×C₆₇	1	1	yes	characteristic
187	#340	C₂²×C₆₇	1	2	yes	—
188	#340	C₂²×C₆₇	1	1	yes	characteristic
189	#336	C₂²×C₆₇	1	2	yes	—
190	#336	C₂²×C₆₇	1	2	yes	—
191	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
194	#125	—	2	4	no	—
195	#125	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
209	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
210	#107	—	2	4	no	—
211	#107	—	2	4	no	—
212	#106	—	2	4	no	—
213	#106	—	2	4	no	—
214	#97	Q₈×C₆₇	1	4	yes	—
215	C₄×C₈	Q₈×C₆₇	1	4	yes	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	D₄×C₆₇	1	1	yes	characteristic
221	#95	D₄×C₆₇	1	2	yes	—
222	#95	D₄×C₆₇	1	1	yes	characteristic
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
231	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
237	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
238	C₂×C₈	C₂×Q₈×C₆₇	1	4	yes	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	8	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	8	2	no	—
248	C₄⋊C₄	—	8	2	no	—
249	C₄⋊C₄	—	8	2	no	—
250	C₄⋊C₄	—	8	2	no	—
251	C₄⋊C₄	#41×C₆₇	1	4	yes	—
252	C₄⋊C₄	#41×C₆₇	1	4	yes	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	8	2	no	—
256	C₄²	—	8	2	no	—
257	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
258	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
259	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
282	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
283	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
284	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
285	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
286	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
287	C₈	—	2	2	no	—
288	C₈	—	2	4	no	—
289	C₈	—	2	2	no	—
290	C₂²	#525×C₆₇	1	2	yes	—
291	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
292	C₂²	#576×C₆₇	1	1	yes	characteristic
293	C₂²	#577×C₆₇	1	2	yes	—
294	C₂²	#578×C₆₇	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	4	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	#533×C₆₇	1	1	yes	characteristic
304	C₄	#533×C₆₇	1	2	yes	—
305	C₄	#533×C₆₇	1	1	yes	characteristic
306	C₂	C₆₇×#2399	1	1	yes	characteristic
307	C₂	C₆₇×#3119	1	2	yes	—
308	C₂	C₆₇×#3120	1	2	yes	—
309	C₂	C₆₇×#3163	1	1	yes	characteristic
310	C₂	C₆₇×#3164	1	1	yes	characteristic

	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
150. #23843	C₆₇	direct (N × H)
165. C₆₇	#23843	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×#2084	2
C₆₇×#2083	2
C₆₇×#2059	2
C₆₇×#1776	2
C₄×#107×C₆₇	2
C₄×#106×C₆₇	2
C₆₇×#1746	2
#23843	67

Subgroups — C67×#23843 #30703094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#23843 #30703094