Subgroups — #30764658

Order 17152.

606 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22559942	C₂	1	2	yes	maximal
2	#22559926	C₂	1	1	yes	maximal, characteristic
3	#22559918	C₂	1	2	yes	maximal
4	#22559905	C₂	1	1	yes	maximal, characteristic
5	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761612	C₂²	1	1	yes	characteristic
7	C₂×#14524002	C₂²	1	2	yes	—
8	C₂×#14524002	C₂²	1	1	yes	characteristic
9	#16863163	—	2	4	no	—
10	#16863159	—	2	2	no	—
11	#16863155	—	2	4	no	—
12	#16863150	—	2	2	no	—
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#117×C₆₇	C₂²	1	2	yes	—
15	#427×C₆₇	—	2	2	no	—
16	#427×C₆₇	—	2	2	no	—
17	C₂×#107×C₆₇	C₄	1	2	yes	complemented
18	C₂×#106×C₆₇	C₄	1	2	yes	complemented
19	C₂×#11761612	—	2	1	no	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	D₄	1	2	yes	—
23	C₂²×C₆₇⋊C₈	—	2	1	no	—
24	#14524002	—	2	2	no	—
25	#14524002	—	2	2	no	—
26	#14524002	—	2	2	no	—
27	#14524002	—	2	2	no	—
28	#14524002	—	2	2	no	—
29	#14524002	Q₈	1	4	yes	—
30	C₂×C₄⋊Dic₆₇	—	2	2	no	—
31	C₂×C₄⋊Dic₆₇	—	2	1	no	—
32	C₂×C₂²⋊Dic₆₇	—	2	2	no	—
33	C₂×C₂²⋊Dic₆₇	D₄	1	2	yes	—
34	C₂×C₄×Dic₆₇	—	2	2	no	—
35	C₂×C₄×Dic₆₇	—	2	1	no	—
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	—	2	2	no	—
38	C₂²×C₈×C₆₇	—	2	2	no	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	D₄	1	2	yes	—
42	C₂²×C₈×C₆₇	D₄	1	2	yes	—
43	#117×C₆₇	—	2	2	no	—
44	#117×C₆₇	—	2	2	no	—
45	#117×C₆₇	—	2	2	no	—
46	#117×C₆₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
48	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
49	C₂×C₄²×C₆₇	—	2	2	no	—
50	#107×C₆₇	—	2	4	no	—
51	#106×C₆₇	—	2	4	no	—
52	C₂²×Dic₆₇	—	2	2	no	—
53	C₂²×Dic₆₇	—	4	2	no	—
54	#11761612	—	2	4	no	—
55	#11761612	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	2	2	no	—
57	C₂×C₆₇⋊C₈	—	4	1	no	—
58	C₂×C₆₇⋊C₈	—	4	1	no	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	—	4	2	no	—
61	C₄⋊Dic₆₇	—	4	2	no	—
62	C₂²⋊Dic₆₇	—	2	4	no	—
63	C₂²⋊Dic₆₇	—	2	4	no	—
64	C₂²⋊Dic₆₇	—	4	2	no	—
65	C₄×Dic₆₇	—	2	4	no	—
66	C₄×Dic₆₇	—	4	1	no	—
67	C₄×Dic₆₇	—	4	2	no	—
68	C₄×Dic₆₇	—	4	1	no	—
69	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	—	2	4	no	—
71	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
77	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
78	C₂²×C₄×C₆₇	#41	1	2	yes	—
79	C₂²×C₄×C₆₇	#41	1	2	yes	—
80	C₂²×C₄×C₆₇	#41	1	2	yes	—
81	C₂×C₈×C₆₇	—	2	2	no	—
82	C₂×C₈×C₆₇	—	2	2	no	—
83	C₂×C₈×C₆₇	—	2	2	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	4	no	—
86	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
87	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₄⋊C₄×C₆₇	—	2	4	no	—
90	C₂²⋊C₄×C₆₇	—	4	2	no	—
91	C₂²⋊C₄×C₆₇	—	4	2	no	—
92	C₄²×C₆₇	—	4	2	no	—
93	C₄²×C₆₇	—	4	2	no	—
94	C₂³×C₆₇	—	4	2	no	—
95	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
96	C₂³×C₆₇	#121	1	1	yes	characteristic
97	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₆₇	#123	1	2	yes	—
99	C₂³×C₆₇	#127	1	1	yes	characteristic
100	C₂×C₄×C₆₇	—	4	4	no	—
101	C₂×C₄×C₆₇	—	8	4	no	—
102	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₆₇	#117	1	2	yes	—
104	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
110	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
111	C₂×C₄×C₆₇	#123	1	2	yes	—
112	C₂×C₄×C₆₇	#123	1	2	yes	—
113	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
114	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
115	C₂×C₄×C₆₇	#125	1	2	yes	—
116	C₂×C₄×C₆₇	#125	1	2	yes	—
117	C₂×C₄×C₆₇	#126	1	2	yes	—
118	C₂×C₄×C₆₇	#126	1	2	yes	—
119	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×Dic₆₇	—	4	4	no	—
121	C₂×Dic₆₇	—	4	2	no	—
122	C₂×Dic₆₇	—	4	4	no	—
123	C₂×Dic₆₇	—	8	4	no	—
124	C₈×C₆₇	—	2	2	no	—
125	C₈×C₆₇	—	2	2	no	—
126	C₆₇⋊C₈	—	4	2	no	—
127	C₂²×C₆₇	—	2	2	no	—
128	C₂²×C₆₇	—	2	2	no	—
129	C₂²×C₆₇	—	2	2	no	—
130	C₂²×C₆₇	—	4	2	no	—
131	C₂²×C₆₇	—	4	2	no	—
132	C₂²×C₆₇	#387	1	2	yes	—
133	C₂²×C₆₇	#390	1	1	yes	characteristic
134	C₂²×C₆₇	#397	1	1	yes	characteristic
135	C₂²×C₆₇	#398	1	1	yes	characteristic
136	C₂²×C₆₇	#399	1	2	yes	—
137	C₄×C₆₇	—	8	4	no	—
138	C₄×C₆₇	#381	1	1	yes	characteristic
139	C₄×C₆₇	#387	1	2	yes	—
140	C₄×C₆₇	#391	1	1	yes	characteristic
141	C₄×C₆₇	#397	1	1	yes	characteristic
142	C₄×C₆₇	#398	1	1	yes	characteristic
143	C₄×C₆₇	#400	1	2	yes	—
144	Dic₆₇	—	8	2	no	—
145	Dic₆₇	—	8	4	no	—
146	#11364	—	67	1	no	maximal, Sylow
147	C₂×C₆₇	—	2	1	no	—
148	C₂×C₆₇	—	2	2	no	—
149	C₂×C₆₇	—	2	2	no	—
150	C₂×C₆₇	—	2	1	no	—
151	C₂×C₆₇	#1445	1	1	yes	characteristic
152	C₂×C₆₇	#1976	1	2	yes	—
153	C₂×#427	D₆₇	1	1	yes	characteristic
154	#1740	—	67	2	no	—
155	#1735	—	67	1	no	—
156	#1733	—	67	2	no	—
157	#1729	—	67	1	no	—
158	C₆₇	#11364	1	1	yes	characteristic, complemented, Sylow
159	C₂²×M₁₆	—	67	1	no	—
160	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
161	C₂×#117	—	67	1	no	—
162	C₂×#117	—	67	2	no	—
163	C₂×#117	C₂×D₆₇	1	2	yes	—
164	#427	—	2	2	no	—
165	#427	—	2	2	no	—
166	C₂×#107	Dic₆₇	1	2	yes	complemented
167	C₂×#106	Dic₆₇	1	2	yes	complemented
168	#342	—	134	4	no	—
169	#340	—	134	2	no	—
170	#339	—	134	4	no	—
171	#336	—	134	2	no	—
172	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
173	C₂×M₁₆	—	67	2	no	—
174	C₂×M₁₆	—	134	2	no	—
175	C₂×M₁₆	—	134	2	no	—
176	C₂×M₁₆	—	134	1	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	134	1	no	—
180	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
181	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
182	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
183	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	67	4	no	—
189	#117	—	134	2	no	—
190	#117	—	134	2	no	—
191	#117	—	134	2	no	—
192	#117	—	134	2	no	—
193	#117	—	134	2	no	—
194	C₂×C₄⋊C₄	—	134	1	no	—
195	C₂×C₄⋊C₄	—	134	2	no	—
196	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	67	2	no	—
199	C₂×C₂²⋊C₄	—	134	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	134	1	no	—
202	C₂×C₄²	—	134	2	no	—
203	#107	—	2	4	no	—
204	#106	—	2	4	no	—
205	C₂⁴	D₄×D₆₇	1	1	yes	characteristic
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	268	2	no	—
208	C₂²×C₄	—	134	2	no	—
209	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
213	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
214	C₂²×C₄	#11761627	1	1	yes	characteristic
215	C₂²×C₄	#11761627	1	1	yes	characteristic
216	C₂²×C₄	#11761628	1	2	yes	—
217	C₂²×C₄	#11761628	1	2	yes	—
218	C₂²×C₄	#11761629	1	2	yes	—
219	M₁₆	—	268	2	no	—
220	M₁₆	—	134	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	268	1	no	—
227	C₂×C₈	—	268	1	no	—
228	C₂×C₈	—	134	2	no	—
229	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
230	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	268	2	no	—
234	C₄⋊C₄	—	268	2	no	—
235	C₄⋊C₄	—	134	4	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	268	2	no	—
239	C₂²⋊C₄	—	134	4	no	—
240	C₂²⋊C₄	—	134	4	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	268	1	no	—
244	C₄²	—	268	2	no	—
245	C₄²	—	268	1	no	—
246	C₄²	—	134	4	no	—
247	C₂³	—	4	2	no	—
248	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
249	C₂³	#14524014	1	1	yes	characteristic
250	C₂³	#14524021	1	2	yes	—
251	C₂³	#14524028	1	2	yes	—
252	C₂³	#14524039	1	1	yes	characteristic
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	—	268	4	no	—
256	C₂×C₄	—	268	2	no	—
257	C₂×C₄	—	268	4	no	—
258	C₂×C₄	—	536	4	no	—
259	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
260	C₂×C₄	#14524002	1	2	yes	—
261	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
262	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
263	C₂×C₄	#14524014	1	1	yes	characteristic
264	C₂×C₄	#14524019	1	2	yes	—
265	C₂×C₄	#14524021	1	2	yes	—
266	C₂×C₄	#14524026	1	1	yes	characteristic
267	C₂×C₄	#14524026	1	1	yes	characteristic
268	C₂×C₄	#14524026	1	2	yes	—
269	C₂×C₄	#14524028	1	2	yes	—
270	C₂×C₄	#14524030	1	1	yes	characteristic
271	C₂×C₄	#14524032	1	1	yes	characteristic
272	C₂×C₄	#14524033	1	2	yes	—
273	C₂×C₄	#14524033	1	2	yes	—
274	C₂×C₄	#14524038	1	2	yes	—
275	C₂×C₄	#14524038	1	2	yes	—
276	C₂×C₄	#14524041	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	2	2	no	—
279	C₈	—	268	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	#16863282	1	2	yes	—
286	C₂²	#16863299	1	1	yes	characteristic
287	C₂²	#16863322	1	1	yes	characteristic
288	C₂²	#16863327	1	1	yes	characteristic
289	C₂²	#16863331	1	2	yes	—
290	C₄	—	8	4	no	—
291	C₄	—	536	2	no	—
292	C₄	—	536	4	no	—
293	C₄	#16863259	1	1	yes	characteristic
294	C₄	#16863282	1	2	yes	—
295	C₄	#16863303	1	1	yes	characteristic
296	C₄	#16863322	1	1	yes	characteristic
297	C₄	#16863327	1	1	yes	characteristic
298	C₄	#16863334	1	2	yes	—
299	C₂	—	2	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	2	no	—
302	C₂	—	2	1	no	—
303	C₂	#22558477	1	1	yes	characteristic
304	C₂	#22561108	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524014
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#427×C₆₇	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#427	D₆₇
hypocenter	C₆₇	#11364
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
17. C₂×#107×C₆₇	C₄	semidirect
18. C₂×#106×C₆₇	C₄	semidirect
158. C₆₇	#11364	semidirect
166. C₂×#107	Dic₆₇	semidirect
167. C₂×#106	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
#22559926	2
#22559918	2
#22559905	2
C₂×#427×C₆₇	2
#11364	67