Subgroups — #30764792

Order 17152.

850 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22559942	C₂	1	1	yes	maximal, characteristic
2	#22559925	C₂	1	2	yes	maximal
3	#22559918	C₂	1	2	yes	maximal
4	#22559906	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#3458	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	1	yes	characteristic
7	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#14524002	C₂²	1	2	yes	—
9	#16863163	—	2	2	no	—
10	#16863158	—	2	4	no	—
11	#16863155	—	2	4	no	—
12	#16863151	—	2	2	no	—
13	#544×C₆₇	—	2	4	no	—
14	#544×C₆₇	C₄	1	4	yes	complemented
15	#543×C₆₇	—	2	2	no	—
16	#533×C₆₇	—	2	4	no	—
17	#533×C₆₇	—	2	2	no	—
18	C₂×#117×C₆₇	C₂²	1	2	yes	—
19	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	1	no	—
22	C₂×#11761612	—	2	2	no	—
23	C₂×#11761612	Q₈	1	2	yes	complemented
24	C₂²×C₆₇⋊C₈	—	2	1	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₆₇⋊C₈	—	2	1	no	—
27	C₂²×C₆₇⋊C₈	—	2	1	no	—
28	C₂²×C₆₇⋊C₈	—	2	2	no	—
29	C₂²×C₆₇⋊C₈	Q₈	1	2	yes	complemented
30	#14524002	—	2	2	no	—
31	#14524002	—	2	2	no	—
32	#14524002	—	2	2	no	—
33	#14524002	Q₈	1	4	yes	complemented
34	C₂×C₄⋊Dic₆₇	—	2	2	no	—
35	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
36	C₂×C₄×Dic₆₇	—	2	2	no	—
37	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
38	C₄⋊Q₈×C₆₇	—	2	4	no	—
39	C₄⋊Q₈×C₆₇	—	2	8	no	—
40	#125×C₆₇	—	2	4	no	—
41	#125×C₆₇	—	2	8	no	—
42	C₂²⋊Q₈×C₆₇	—	2	16	no	—
43	C₂²⋊Q₈×C₆₇	—	2	8	no	—
44	C₄×Q₈×C₆₇	—	2	8	no	—
45	#117×C₆₇	—	2	2	no	—
46	#117×C₆₇	—	2	2	no	—
47	#117×C₆₇	—	2	2	no	—
48	#117×C₆₇	—	2	2	no	—
49	#117×C₆₇	C₂×C₄	1	4	yes	—
50	#117×C₆₇	D₄	1	4	yes	—
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
53	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
54	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
55	C₂×C₄²×C₆₇	—	2	2	no	—
56	C₂×C₄²×C₆₇	D₄	1	2	yes	—
57	C₂²×Dic₆₇	—	4	2	no	—
58	#11761612	—	2	4	no	—
59	#11761612	—	4	2	no	—
60	C₂×C₆₇⋊C₈	—	2	4	no	—
61	C₂×C₆₇⋊C₈	—	2	2	no	—
62	C₂×C₆₇⋊C₈	—	2	2	no	—
63	C₂×C₆₇⋊C₈	—	4	1	no	—
64	C₂×C₆₇⋊C₈	—	4	1	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	C₂×C₆₇⋊C₈	—	4	1	no	—
67	C₂×C₆₇⋊C₈	—	4	1	no	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	C₄⋊Dic₆₇	—	4	2	no	—
70	C₂²⋊Dic₆₇	—	2	4	no	—
71	C₂²⋊Dic₆₇	—	4	2	no	—
72	C₄×Dic₆₇	—	2	4	no	—
73	C₄×Dic₆₇	—	4	2	no	—
74	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂×Q₈×C₆₇	—	2	8	no	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
86	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	#41	1	2	yes	—
88	C₄⋊C₄×C₆₇	—	2	16	no	—
89	C₄⋊C₄×C₆₇	—	2	8	no	—
90	C₄⋊C₄×C₆₇	—	2	4	no	—
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
93	C₂²⋊C₄×C₆₇	—	2	4	no	—
94	C₂²⋊C₄×C₆₇	—	2	4	no	—
95	C₂²⋊C₄×C₆₇	—	2	4	no	—
96	C₄²×C₆₇	—	2	4	no	—
97	C₄²×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	—	2	2	no	—
100	C₂³×C₆₇	—	4	2	no	—
101	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂³×C₆₇	#120	1	2	yes	—
103	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
104	C₂³×C₆₇	#124	1	1	yes	characteristic
105	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
106	Q₈×C₆₇	—	8	8	no	—
107	C₂×C₄×C₆₇	—	2	4	no	—
108	C₂×C₄×C₆₇	—	2	8	no	—
109	C₂×C₄×C₆₇	—	4	4	no	—
110	C₂×C₄×C₆₇	—	8	4	no	—
111	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₆₇	C₄×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
116	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
117	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₆₇	#124	1	2	yes	—
124	C₂×C₄×C₆₇	#124	1	2	yes	—
125	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
126	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
127	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
129	C₂×Dic₆₇	—	4	4	no	—
130	C₂×Dic₆₇	—	8	4	no	—
131	C₆₇⋊C₈	—	4	2	no	—
132	C₆₇⋊C₈	—	4	2	no	—
133	C₂²×C₆₇	—	2	4	no	—
134	C₂²×C₆₇	—	2	2	no	—
135	C₂²×C₆₇	—	4	2	no	—
136	C₂²×C₆₇	—	4	1	no	—
137	C₂²×C₆₇	—	4	1	no	—
138	C₂²×C₆₇	#379	1	1	yes	characteristic
139	C₂²×C₆₇	#391	1	1	yes	characteristic
140	C₂²×C₆₇	#391	1	1	yes	characteristic
141	C₂²×C₆₇	#393	1	2	yes	—
142	C₂²×C₆₇	#395	1	2	yes	—
143	C₄×C₆₇	—	4	4	no	—
144	C₄×C₆₇	—	4	8	no	—
145	C₄×C₆₇	#391	1	2	yes	—
146	C₄×C₆₇	#391	1	2	yes	—
147	C₄×C₆₇	#393	1	1	yes	characteristic
148	C₄×C₆₇	#393	1	1	yes	characteristic
149	C₄×C₆₇	#395	1	1	yes	characteristic
150	C₄×C₆₇	#395	1	1	yes	characteristic
151	Dic₆₇	—	8	4	no	—
152	#11393	—	67	1	no	maximal, Sylow
153	C₂×C₆₇	—	2	2	no	—
154	C₂×C₆₇	—	2	1	no	—
155	C₂×C₆₇	—	2	1	no	—
156	C₂×C₆₇	—	2	2	no	—
157	C₂×C₆₇	#1446	1	1	yes	characteristic
158	C₂×C₆₇	#1794	1	2	yes	—
159	#3458	D₆₇	1	1	yes	characteristic
160	#1740	—	67	1	no	—
161	#1735	—	67	2	no	—
162	#1733	—	67	2	no	—
163	#1729	—	67	1	no	—
164	C₆₇	#11393	1	1	yes	characteristic, complemented, Sylow
165	C₂²×M₁₆	—	67	1	no	—
166	C₂³×C₈	—	67	1	no	—
167	#544	—	2	4	no	—
168	#544	Dic₆₇	1	4	yes	complemented
169	#543	—	2	2	no	—
170	#533	—	2	4	no	—
171	#533	—	2	2	no	—
172	C₂×#117	—	67	2	no	—
173	C₂×#117	C₂×D₆₇	1	2	yes	—
174	C₂×#117	C₂×D₆₇	1	1	yes	characteristic
175	#342	—	134	2	no	—
176	#340	—	134	4	no	—
177	#339	—	134	4	no	—
178	#336	—	134	2	no	—
179	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
180	C₂×M₁₆	—	67	2	no	—
181	C₂×M₁₆	—	134	2	no	—
182	C₂×M₁₆	—	134	2	no	—
183	C₂×M₁₆	—	134	1	no	—
184	C₂²×C₈	—	67	2	no	—
185	C₂²×C₈	—	134	1	no	—
186	C₂²×C₈	—	134	2	no	—
187	C₂²×C₈	—	134	1	no	—
188	C₂²×C₈	—	134	1	no	—
189	C₂²×C₈	—	134	2	no	—
190	C₄⋊Q₈	—	2	4	no	—
191	C₄⋊Q₈	—	2	8	no	—
192	#125	—	2	4	no	—
193	#125	—	2	8	no	—
194	C₂²⋊Q₈	—	2	16	no	—
195	C₂²⋊Q₈	—	2	8	no	—
196	C₄×Q₈	—	2	8	no	—
197	#117	—	2	2	no	—
198	#117	—	2	2	no	—
199	#117	—	2	2	no	—
200	#117	—	2	2	no	—
201	#117	—	67	4	no	—
202	#117	—	134	2	no	—
203	#117	—	134	2	no	—
204	#117	—	134	2	no	—
205	#117	C₂×Dic₆₇	1	4	yes	—
206	#117	C₂²⋊D₆₇	1	4	yes	—
207	C₂×C₄⋊C₄	—	134	2	no	—
208	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
209	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	4	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	—	134	2	no	—
212	C₂×C₂²⋊C₄	C₂²⋊D₆₇	1	2	yes	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	134	2	no	—
215	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
216	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
217	C₂×Q₈	—	2	8	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	268	2	no	—
220	C₂²×C₄	C₂²⋊Dic₆₇	1	4	yes	—
221	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
223	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
224	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
225	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
226	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
227	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
228	C₂²×C₄	#11761629	1	1	yes	characteristic
229	C₂²×C₄	#11761629	1	1	yes	characteristic
230	C₂²×C₄	#11761629	1	2	yes	—
231	M₁₆	—	268	2	no	—
232	M₁₆	—	134	4	no	—
233	C₂×C₈	—	268	1	no	—
234	C₂×C₈	—	268	1	no	—
235	C₂×C₈	—	268	2	no	—
236	C₂×C₈	—	268	1	no	—
237	C₂×C₈	—	268	1	no	—
238	C₂×C₈	—	134	4	no	—
239	C₂×C₈	—	134	2	no	—
240	C₂×C₈	—	134	2	no	—
241	C₄⋊C₄	—	2	16	no	—
242	C₄⋊C₄	—	2	8	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	268	2	no	—
246	C₄⋊C₄	—	134	4	no	—
247	C₄⋊C₄	C₂²⋊Dic₆₇	1	8	yes	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	268	2	no	—
252	C₂²⋊C₄	—	134	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	268	2	no	—
258	C₄²	—	134	4	no	—
259	C₂³	—	4	2	no	—
260	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
261	C₂³	#14524012	1	2	yes	—
262	C₂³	#14524019	1	2	yes	—
263	C₂³	#14524032	1	1	yes	characteristic
264	C₂³	#14524041	1	1	yes	characteristic
265	Q₈	—	8	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	—	268	4	no	—
271	C₂×C₄	—	536	4	no	—
272	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	2	yes	—
273	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
274	C₂×C₄	Q₈×Dic₆₇	1	1	yes	characteristic
275	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
276	C₂×C₄	#14524012	1	1	yes	characteristic
277	C₂×C₄	#14524012	1	1	yes	characteristic
278	C₂×C₄	#14524019	1	2	yes	—
279	C₂×C₄	#14524019	1	2	yes	—
280	C₂×C₄	#14524019	1	2	yes	—
281	C₂×C₄	#14524019	1	2	yes	—
282	C₂×C₄	#14524019	1	1	yes	characteristic
283	C₂×C₄	#14524019	1	1	yes	characteristic
284	C₂×C₄	#14524032	1	2	yes	—
285	C₂×C₄	#14524032	1	2	yes	—
286	C₂×C₄	#14524032	1	1	yes	characteristic
287	C₂×C₄	#14524041	1	2	yes	—
288	C₂×C₄	#14524041	1	2	yes	—
289	C₂×C₄	#14524041	1	1	yes	characteristic
290	C₈	—	268	2	no	—
291	C₈	—	268	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	#16863252	1	1	yes	characteristic
298	C₂²	#16863303	1	1	yes	characteristic
299	C₂²	#16863303	1	1	yes	characteristic
300	C₂²	#16863307	1	2	yes	—
301	C₂²	#16863315	1	2	yes	—
302	C₄	—	4	4	no	—
303	C₄	—	4	8	no	—
304	C₄	—	536	4	no	—
305	C₄	#16863303	1	2	yes	—
306	C₄	#16863303	1	2	yes	—
307	C₄	#16863307	1	1	yes	characteristic
308	C₄	#16863307	1	1	yes	characteristic
309	C₄	#16863315	1	1	yes	characteristic
310	C₄	#16863315	1	1	yes	characteristic
311	C₂	—	2	2	no	—
312	C₂	—	2	1	no	—
313	C₂	—	2	1	no	—
314	C₂	—	2	2	no	—
315	C₂	#22558480	1	1	yes	characteristic
316	C₂	#22560173	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524012
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₆₇×#3458	C₂
socle	C₂²×C₆₇	#379
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	#3458	D₆₇
hypocenter	C₆₇	#11393
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3458

Complements

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

N	H ≅ G/N	type
14. #544×C₆₇	C₄	semidirect
23. C₂×#11761612	Q₈	semidirect
29. C₂²×C₆₇⋊C₈	Q₈	semidirect
33. #14524002	Q₈	semidirect
164. C₆₇	#11393	semidirect
168. #544	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22559942	2
#22559925	2
#22559918	2
#22559906	2
C₆₇×#3458	2
#11393	67