Subgroups — #30864612

Order 17152.

750 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562958	C₂	1	2	yes	maximal
2	#22561183	C₂	1	2	yes	maximal
3	#22560948	C₂	1	2	yes	maximal
4	#22560943	C₂	1	2	yes	maximal
5	#22560373	C₂	1	1	yes	maximal, characteristic
6	#22560108	C₂	1	2	yes	maximal
7	#107×Dic₆₇	C₂	1	2	yes	maximal
8	#22559895	C₂	1	1	yes	maximal, characteristic
9	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#14523977	C₂²	1	2	yes	—
11	C₂×#14523975	C₂²	1	2	yes	—
12	C₂×#14523954	C₂²	1	1	yes	characteristic
13	C₂×#14523953	C₂²	1	1	yes	characteristic
14	C₂×#14523953	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
17	#16863303	—	2	4	no	—
18	#16863295	C₂²	1	4	yes	—
19	#16863282	—	2	4	no	—
20	#16863262	C₂²	1	2	yes	—
21	C₄⋊C₄×Dic₆₇	C₂²	1	4	yes	—
22	#16863242	C₂²	1	1	yes	characteristic
23	#16863158	C₂²	1	4	yes	—
24	#16863158	C₂²	1	4	yes	—
25	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
26	#500×C₆₇	C₄	1	8	yes	complemented
27	C₂×#107×C₆₇	C₂²	1	2	yes	—
28	C₂×#107×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
31	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₆₇	—	2	4	no	—
33	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
34	C₂×C₄×Dic₆₇	—	2	4	no	—
35	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
37	#14523977	—	2	4	no	—
38	#14523975	—	2	4	no	—
39	#14523954	—	2	2	no	—
40	#14523953	—	2	2	no	—
41	#14523953	Q₈	1	4	yes	complemented
42	C₈×Dic₆₇	—	2	2	no	—
43	C₈×Dic₆₇	Q₈	1	4	yes	complemented
44	C₆₇⋊#95	—	2	4	no	—
45	C₆₇⋊#95	—	2	4	no	—
46	C₆₇⋊#95	—	2	2	no	—
47	C₆₇⋊#95	—	2	2	no	—
48	C₆₇⋊#95	D₄	1	2	yes	—
49	C₆₇⋊#95	D₄	1	2	yes	—
50	C₂²×Q₈×C₆₇	—	2	2	no	—
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₄⋊Q₈×C₆₇	—	2	4	no	—
54	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
57	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
58	#107×C₆₇	C₂×C₄	1	8	yes	—
59	#107×C₆₇	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
61	C₂²×Dic₆₇	—	2	4	no	—
62	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
63	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
64	C₂×C₆₇⋊C₈	—	2	1	no	—
65	C₂×C₆₇⋊C₈	—	2	1	no	—
66	C₂×C₆₇⋊C₈	—	2	1	no	—
67	C₂×C₆₇⋊C₈	—	2	1	no	—
68	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
69	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
70	C₄⋊Dic₆₇	—	2	4	no	—
71	C₄⋊Dic₆₇	—	2	4	no	—
72	C₄⋊Dic₆₇	—	8	4	no	—
73	C₄×Dic₆₇	—	2	1	no	—
74	C₄×Dic₆₇	—	2	1	no	—
75	C₄×Dic₆₇	—	8	4	no	—
76	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
77	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
78	C₂×Q₈×C₆₇	—	2	8	no	—
79	C₂×Q₈×C₆₇	—	8	4	no	—
80	C₂²×C₄×C₆₇	—	2	4	no	—
81	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
87	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₆₇	#41	1	2	yes	—
89	C₂×C₈×C₆₇	#41	1	2	yes	—
90	C₄⋊C₄×C₆₇	—	2	8	no	—
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
93	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
94	C₄²×C₆₇	#41	1	2	yes	—
95	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₆₇	—	8	8	no	—
97	C₂×C₄×C₆₇	—	2	2	no	—
98	C₂×C₄×C₆₇	—	2	2	no	—
99	C₂×C₄×C₆₇	—	2	8	no	—
100	C₂×C₄×C₆₇	—	4	4	no	—
101	C₂×C₄×C₆₇	—	4	4	no	—
102	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
107	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
108	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
110	C₂×Dic₆₇	—	2	2	no	—
111	C₂×Dic₆₇	—	2	2	no	—
112	C₂×Dic₆₇	—	2	2	no	—
113	C₂×Dic₆₇	—	2	2	no	—
114	C₂×Dic₆₇	—	2	2	no	—
115	C₂×Dic₆₇	—	4	4	no	—
116	C₂×Dic₆₇	—	4	4	no	—
117	C₂×Dic₆₇	—	4	4	no	—
118	C₂×Dic₆₇	C₄⋊Q₈	1	4	yes	—
119	C₈×C₆₇	C₄×Q₈	1	4	yes	—
120	C₈×C₆₇	C₄×Q₈	1	4	yes	—
121	C₆₇⋊C₈	—	2	2	no	—
122	C₆₇⋊C₈	—	2	2	no	—
123	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
124	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
125	C₂²×C₆₇	#532	1	1	yes	characteristic
126	C₂²×C₆₇	#576	1	2	yes	—
127	C₂²×C₆₇	#578	1	2	yes	—
128	C₄×C₆₇	—	2	2	no	—
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	—	4	8	no	—
131	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
132	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₄×C₆₇	#533	1	1	yes	characteristic
134	C₄×C₆₇	#533	1	1	yes	characteristic
135	Dic₆₇	—	2	4	no	—
136	Dic₆₇	—	4	2	no	—
137	Dic₆₇	—	8	4	no	—
138	#21725	—	67	1	no	maximal, Sylow
139	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
140	C₂×C₆₇	#2948	1	1	yes	characteristic
141	C₂×C₆₇	#2950	1	1	yes	characteristic
142	C₂×C₆₇	#3164	1	2	yes	—
143	C₂×C₆₇	#3175	1	2	yes	—
144	C₂×#500	D₆₇	1	1	yes	characteristic
145	#2331	—	67	2	no	—
146	#1993	—	67	2	no	—
147	#1943	—	67	2	no	—
148	#1942	—	67	2	no	—
149	#1846	—	67	1	no	—
150	#1776	—	67	2	no	—
151	C₄×#107	—	67	2	no	—
152	#1726	—	67	1	no	—
153	C₆₇	#21725	1	1	yes	characteristic, complemented, Sylow
154	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
155	#500	Dic₆₇	1	8	yes	complemented
156	C₂×#107	—	67	2	no	—
157	C₂×#107	C₂×D₆₇	1	2	yes	—
158	C₂×#107	C₂×D₆₇	1	2	yes	—
159	C₂×#106	—	67	2	no	—
160	C₂×#97	—	67	1	no	—
161	C₂×#97	—	67	1	no	—
162	C₂×#97	—	67	1	no	—
163	C₂×C₄×C₈	—	67	1	no	—
164	C₂×C₄×C₈	—	67	1	no	—
165	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
166	#391	—	134	4	no	—
167	#389	—	67	4	no	—
168	#387	—	134	4	no	—
169	#382	—	67	2	no	—
170	C₄×C₄⋊C₄	—	67	4	no	—
171	#376	—	67	1	no	—
172	#340	—	67	4	no	—
173	#340	—	67	4	no	—
174	C₂²×Q₈	—	2	2	no	—
175	C₂²×C₈	—	67	1	no	—
176	C₂²×C₈	—	67	1	no	—
177	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
179	C₄⋊Q₈	—	2	4	no	—
180	C₄⋊Q₈	C₂×Dic₆₇	1	8	yes	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	67	4	no	—
183	C₂×C₄⋊C₄	—	134	4	no	—
184	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
185	C₂×C₄²	—	67	1	no	—
186	C₂×C₄²	—	67	1	no	—
187	C₂×C₄²	—	134	4	no	—
188	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
189	#107	—	134	4	no	—
190	#107	C₂×Dic₆₇	1	8	yes	—
191	#107	C₂×Dic₆₇	1	8	yes	—
192	#106	—	134	4	no	—
193	#97	—	67	4	no	—
194	#97	—	134	2	no	—
195	#97	—	134	2	no	—
196	C₄×C₈	—	67	4	no	—
197	C₄×C₈	—	134	2	no	—
198	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
199	#95	—	67	2	no	—
200	#95	—	67	2	no	—
201	#95	—	134	4	no	—
202	#95	—	134	4	no	—
203	#95	—	134	2	no	—
204	#95	—	134	2	no	—
205	C₂×Q₈	—	2	8	no	—
206	C₂×Q₈	—	8	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	67	2	no	—
209	C₂²×C₄	—	67	2	no	—
210	C₂²×C₄	—	134	4	no	—
211	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
212	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
213	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
214	C₂×C₈	—	67	2	no	—
215	C₂×C₈	—	67	2	no	—
216	C₂×C₈	—	134	1	no	—
217	C₂×C₈	—	134	1	no	—
218	C₂×C₈	—	134	1	no	—
219	C₂×C₈	—	134	1	no	—
220	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
221	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
222	C₂×C₈	Q₈×D₆₇	1	2	yes	—
223	C₂×C₈	Q₈×D₆₇	1	2	yes	—
224	C₂×C₈	#11761629	1	2	yes	—
225	C₂×C₈	#11761629	1	2	yes	—
226	C₄⋊C₄	—	2	8	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	536	4	no	—
229	C₄⋊C₄	—	134	4	no	—
230	C₄⋊C₄	—	134	4	no	—
231	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
232	C₄²	—	536	4	no	—
233	C₄²	—	67	2	no	—
234	C₄²	—	67	2	no	—
235	C₄²	—	134	1	no	—
236	C₄²	—	134	1	no	—
237	C₄²	C₂²×Dic₆₇	1	2	yes	—
238	C₄²	#11761627	1	2	yes	—
239	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
240	Q₈	—	8	8	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	268	4	no	—
247	C₂×C₄	—	268	4	no	—
248	C₂×C₄	—	268	4	no	—
249	C₂×C₄	—	67	4	no	—
250	C₂×C₄	—	134	2	no	—
251	C₂×C₄	—	134	2	no	—
252	C₂×C₄	—	134	2	no	—
253	C₂×C₄	—	134	2	no	—
254	C₂×C₄	—	134	2	no	—
255	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
256	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
257	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
258	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
259	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
260	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
261	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
262	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
263	C₈	—	134	2	no	—
264	C₈	—	134	2	no	—
265	C₈	Q₈×Dic₆₇	1	4	yes	—
266	C₈	Q₈×Dic₆₇	1	4	yes	—
267	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
268	C₂²	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
269	C₂²	#16863947	1	1	yes	characteristic
270	C₂²	#16864197	1	2	yes	—
271	C₂²	#16864209	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	2	2	no	—
274	C₄	—	4	8	no	—
275	C₄	—	268	2	no	—
276	C₄	—	536	4	no	—
277	C₄	—	134	4	no	—
278	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
279	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
280	C₄	#16863951	1	1	yes	characteristic
281	C₄	#16863951	1	1	yes	characteristic
282	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
283	C₂	#22569061	1	1	yes	characteristic
284	C₂	#22569077	1	1	yes	characteristic
285	C₂	#22570742	1	2	yes	—
286	C₂	#22570851	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#500×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#500	D₆₇
hypocenter	C₆₇	#21725
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
26. #500×C₆₇	C₄	semidirect
41. #14523953	Q₈	semidirect
43. C₈×Dic₆₇	Q₈	semidirect
153. C₆₇	#21725	semidirect
155. #500	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22562958	2
#22561183	2
#22560948	2
#22560943	2
#22560373	2
#22560108	2
#107×Dic₆₇	2
#22559895	2
C₂×#500×C₆₇	2
#21725	67