Subgroups — #30864594

Order 17152.

734 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562951	C₂	1	2	yes	maximal
2	#22561179	C₂	1	2	yes	maximal
3	#22560953	C₂	1	2	yes	maximal
4	#22560943	C₂	1	2	yes	maximal
5	#22560369	C₂	1	1	yes	maximal, characteristic
6	#22560109	C₂	1	2	yes	maximal
7	#107×Dic₆₇	C₂	1	2	yes	maximal
8	#22559895	C₂	1	1	yes	maximal, characteristic
9	C₂×#499×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×#14523977	C₂²	1	2	yes	—
11	C₂×#14523977	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	#16863295	—	2	4	no	—
18	#16863295	C₂²	1	4	yes	—
19	#16863282	—	2	4	no	—
20	#16863266	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×#125×C₆₇	C₂²	1	2	yes	—
26	#499×C₆₇	C₄	1	8	yes	complemented
27	C₂×#107×C₆₇	C₂²	1	2	yes	—
28	C₂×#106×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	—	2	4	no	—
34	C₂×C₄⋊Dic₆₇	C₂³	1	4	yes	—
35	C₂×C₄×Dic₆₇	—	2	4	no	—
36	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
38	#14523977	—	2	4	no	—
39	#14523977	—	2	4	no	—
40	#14523954	—	2	2	no	—
41	#14523953	—	2	2	no	—
42	#14523953	Q₈	1	4	yes	—
43	C₈×Dic₆₇	—	2	2	no	—
44	C₈×Dic₆₇	Q₈	1	4	yes	—
45	C₆₇⋊#95	—	2	4	no	—
46	C₆₇⋊#95	—	2	4	no	—
47	C₆₇⋊#95	D₄	1	2	yes	—
48	C₆₇⋊#95	D₄	1	2	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	#125×C₆₇	—	2	4	no	—
52	#125×C₆₇	C₂×C₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
54	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
56	#107×C₆₇	C₂×C₄	1	8	yes	—
57	#106×C₆₇	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₂²×Dic₆₇	—	2	4	no	—
60	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×Dic₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
63	C₂×C₆₇⋊C₈	—	2	1	no	—
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	C₂×Q₈	1	2	yes	—
68	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
69	C₄⋊Dic₆₇	—	2	4	no	—
70	C₄⋊Dic₆₇	—	2	4	no	—
71	C₄⋊Dic₆₇	—	8	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	C₂²×C₄×C₆₇	—	2	4	no	—
79	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₆₇	#41	1	2	yes	—
86	C₂×C₈×C₆₇	#41	1	2	yes	—
87	C₄⋊C₄×C₆₇	—	2	16	no	—
88	C₄⋊C₄×C₆₇	—	2	4	no	—
89	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
90	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
91	C₄²×C₆₇	#41	1	2	yes	—
92	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	8	no	—
96	C₂×C₄×C₆₇	—	4	4	no	—
97	C₂×C₄×C₆₇	—	4	4	no	—
98	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
99	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
104	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
105	C₂×Dic₆₇	—	2	2	no	—
106	C₂×Dic₆₇	—	2	2	no	—
107	C₂×Dic₆₇	—	2	1	no	—
108	C₂×Dic₆₇	—	2	1	no	—
109	C₂×Dic₆₇	—	2	2	no	—
110	C₂×Dic₆₇	—	2	2	no	—
111	C₂×Dic₆₇	—	4	4	no	—
112	C₂×Dic₆₇	—	4	4	no	—
113	C₂×Dic₆₇	—	4	4	no	—
114	C₂×Dic₆₇	C₄⋊Q₈	1	4	yes	—
115	C₈×C₆₇	C₄×Q₈	1	4	yes	—
116	C₈×C₆₇	C₄×Q₈	1	4	yes	—
117	C₆₇⋊C₈	—	2	2	no	—
118	C₆₇⋊C₈	—	2	2	no	—
119	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
120	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂²×C₆₇	#532	1	1	yes	characteristic
122	C₂²×C₆₇	#576	1	2	yes	—
123	C₂²×C₆₇	#578	1	2	yes	—
124	C₄×C₆₇	—	2	4	no	—
125	C₄×C₆₇	—	4	8	no	—
126	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
127	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
128	C₄×C₆₇	#533	1	1	yes	characteristic
129	C₄×C₆₇	#533	1	1	yes	characteristic
130	Dic₆₇	—	2	4	no	—
131	Dic₆₇	—	4	1	no	—
132	Dic₆₇	—	4	1	no	—
133	Dic₆₇	—	8	4	no	—
134	#21723	—	67	1	no	maximal, Sylow
135	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×C₆₇	#2951	1	2	yes	—
137	C₂×C₆₇	#3164	1	2	yes	—
138	C₂×C₆₇	#3176	1	1	yes	characteristic
139	C₂×C₆₇	#3177	1	1	yes	characteristic
140	C₂×#499	D₆₇	1	1	yes	characteristic
141	#2330	—	67	2	no	—
142	#1992	—	67	2	no	—
143	#1944	—	67	2	no	—
144	#1942	—	67	2	no	—
145	#1845	—	67	1	no	—
146	#1776	—	67	2	no	—
147	C₄×#107	—	67	2	no	—
148	#1726	—	67	1	no	—
149	C₆₇	#21723	1	1	yes	characteristic, complemented, Sylow
150	C₂×#125	C₂×D₆₇	1	2	yes	—
151	#499	Dic₆₇	1	8	yes	complemented
152	C₂×#107	—	67	2	no	—
153	C₂×#107	—	67	2	no	—
154	C₂×#107	C₂×D₆₇	1	2	yes	—
155	C₂×#106	C₂×D₆₇	1	2	yes	—
156	C₂×#97	—	67	1	no	—
157	C₂×#97	—	67	1	no	—
158	C₂×#97	—	67	1	no	—
159	C₂×C₄×C₈	—	67	1	no	—
160	C₂×C₄×C₈	—	67	1	no	—
161	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
162	#389	—	67	4	no	—
163	#389	—	134	4	no	—
164	#387	—	134	4	no	—
165	#384	—	67	2	no	—
166	C₄×C₄⋊C₄	—	67	4	no	—
167	#376	—	67	1	no	—
168	#340	—	67	4	no	—
169	#340	—	67	4	no	—
170	C₂²×C₈	—	67	1	no	—
171	C₂²×C₈	—	67	1	no	—
172	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
174	#125	—	2	4	no	—
175	#125	C₂×Dic₆₇	1	8	yes	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	—	67	4	no	—
178	C₂×C₄⋊C₄	—	134	4	no	—
179	C₂×C₄⋊C₄	—	134	4	no	—
180	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
181	C₂×C₄²	—	67	1	no	—
182	C₂×C₄²	—	67	1	no	—
183	C₂×C₄²	—	134	4	no	—
184	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
185	#107	—	134	4	no	—
186	#107	—	134	4	no	—
187	#107	C₂×Dic₆₇	1	8	yes	—
188	#106	C₂×Dic₆₇	1	8	yes	—
189	#97	—	67	4	no	—
190	#97	—	134	2	no	—
191	#97	—	134	2	no	—
192	C₄×C₈	—	67	4	no	—
193	C₄×C₈	—	134	2	no	—
194	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
195	#95	—	67	2	no	—
196	#95	—	67	2	no	—
197	#95	—	134	4	no	—
198	#95	—	134	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	67	2	no	—
201	C₂²×C₄	—	67	1	no	—
202	C₂²×C₄	—	67	1	no	—
203	C₂²×C₄	—	134	4	no	—
204	C₂²×C₄	D₄×D₆₇	1	2	yes	—
205	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
206	C₂×C₈	—	67	2	no	—
207	C₂×C₈	—	67	2	no	—
208	C₂×C₈	—	134	1	no	—
209	C₂×C₈	—	134	1	no	—
210	C₂×C₈	—	134	1	no	—
211	C₂×C₈	—	134	1	no	—
212	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
213	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
214	C₂×C₈	Q₈×D₆₇	1	2	yes	—
215	C₂×C₈	Q₈×D₆₇	1	2	yes	—
216	C₂×C₈	#11761629	1	2	yes	—
217	C₂×C₈	#11761629	1	2	yes	—
218	C₄⋊C₄	—	2	16	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	536	4	no	—
221	C₄⋊C₄	—	536	4	no	—
222	C₄⋊C₄	—	134	4	no	—
223	C₄⋊C₄	—	134	4	no	—
224	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
225	C₄²	—	536	4	no	—
226	C₄²	—	67	2	no	—
227	C₄²	—	67	2	no	—
228	C₄²	—	134	1	no	—
229	C₄²	—	134	1	no	—
230	C₄²	C₂²×Dic₆₇	1	2	yes	—
231	C₄²	#11761627	1	2	yes	—
232	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	268	4	no	—
239	C₂×C₄	—	268	4	no	—
240	C₂×C₄	—	268	4	no	—
241	C₂×C₄	—	67	4	no	—
242	C₂×C₄	—	134	2	no	—
243	C₂×C₄	—	134	1	no	—
244	C₂×C₄	—	134	1	no	—
245	C₂×C₄	—	134	2	no	—
246	C₂×C₄	—	134	2	no	—
247	C₂×C₄	—	134	2	no	—
248	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
249	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
250	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
251	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
252	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
253	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
254	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
255	C₈	—	134	2	no	—
256	C₈	—	134	2	no	—
257	C₈	Q₈×Dic₆₇	1	4	yes	—
258	C₈	Q₈×Dic₆₇	1	4	yes	—
259	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
260	C₂²	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
261	C₂²	#16863947	1	1	yes	characteristic
262	C₂²	#16864198	1	2	yes	—
263	C₂²	#16864210	1	2	yes	—
264	C₄	—	2	4	no	—
265	C₄	—	4	8	no	—
266	C₄	—	268	1	no	—
267	C₄	—	268	1	no	—
268	C₄	—	536	4	no	—
269	C₄	—	134	4	no	—
270	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
271	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
272	C₄	#16863951	1	1	yes	characteristic
273	C₄	#16863951	1	1	yes	characteristic
274	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
275	C₂	#22569085	1	2	yes	—
276	C₂	#22570746	1	2	yes	—
277	C₂	#22570860	1	1	yes	characteristic
278	C₂	#22570869	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	C₂×#499×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×D₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₂×#499	D₆₇
hypocenter	C₆₇	#21723
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
26. #499×C₆₇	C₄	semidirect
149. C₆₇	#21723	semidirect
151. #499	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22562951	2
#22561179	2
#22560953	2
#22560943	2
#22560369	2
#22560109	2
#107×Dic₆₇	2
#22559895	2
C₂×#499×C₆₇	2
#21723	67