Subgroups — #30864635

Order 17152.

750 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125×Dic₆₇	C₂	1	2	yes	maximal
2	#22561175	C₂	1	2	yes	maximal
3	#22560953	C₂	1	2	yes	maximal
4	#22560948	C₂	1	2	yes	maximal
5	#22560373	C₂	1	1	yes	maximal, characteristic
6	#22560109	C₂	1	2	yes	maximal
7	#22560108	C₂	1	2	yes	maximal
8	C₄×#14523953	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×#14523975	C₂²	1	2	yes	—
12	C₂×#14523953	C₂²	1	1	yes	characteristic
13	C₂×#14523953	C₂²	1	1	yes	characteristic
14	C₂×#14523953	C₂²	1	1	yes	characteristic
15	C₂×#14523953	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	#16863295	C₂²	1	4	yes	—
18	#16863282	—	2	4	no	—
19	#16863262	C₂²	1	2	yes	—
20	C₄⋊C₄×Dic₆₇	—	2	4	no	—
21	C₄⋊C₄×Dic₆₇	C₂²	1	4	yes	—
22	C₄²×Dic₆₇	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	C₂³	1	4	yes	—
34	C₂×C₄×Dic₆₇	—	2	4	no	—
35	C₂×C₄×Dic₆₇	D₄	1	4	yes	—
36	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
38	#14523977	—	2	4	no	—
39	#14523975	—	2	4	no	—
40	#14523953	—	2	2	no	—
41	#14523953	—	2	2	no	—
42	#14523953	Q₈	1	4	yes	—
43	#14523953	Q₈	1	4	yes	—
44	C₄×C₆₇⋊C₈	—	2	2	no	—
45	C₆₇⋊#95	—	2	4	no	—
46	C₆₇⋊#95	—	2	4	no	—
47	C₆₇⋊#95	—	2	4	no	—
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	#125×C₆₇	—	2	4	no	—
51	#125×C₆₇	C₂×C₄	1	8	yes	—
52	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
53	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
55	#107×C₆₇	C₂×C₄	1	8	yes	—
56	#106×C₆₇	C₂×C₄	1	8	yes	—
57	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
58	C₂²×Dic₆₇	—	2	4	no	—
59	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
60	C₂²×Dic₆₇	C₂×D₄	1	2	yes	—
61	C₂×C₆₇⋊C₈	—	2	1	no	—
62	C₂×C₆₇⋊C₈	—	2	1	no	—
63	C₂×C₆₇⋊C₈	—	2	1	no	—
64	C₂×C₆₇⋊C₈	—	2	1	no	—
65	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
66	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
67	C₄⋊Dic₆₇	—	2	4	no	—
68	C₄⋊Dic₆₇	—	2	4	no	—
69	C₄⋊Dic₆₇	—	4	8	no	—
70	C₄×Dic₆₇	—	2	1	no	—
71	C₄×Dic₆₇	—	2	1	no	—
72	C₄×Dic₆₇	—	4	4	no	—
73	C₄×Dic₆₇	—	4	8	no	—
74	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
75	C₄×Dic₆₇	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₆₇	—	2	4	no	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₆₇	#41	1	2	yes	—
84	C₂×C₈×C₆₇	#41	1	2	yes	—
85	C₄⋊C₄×C₆₇	—	2	16	no	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
88	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
89	C₄²×C₆₇	#41	1	2	yes	—
90	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
91	C₂×C₄×C₆₇	—	2	4	no	—
92	C₂×C₄×C₆₇	—	2	8	no	—
93	C₂×C₄×C₆₇	—	4	4	no	—
94	C₂×C₄×C₆₇	—	4	4	no	—
95	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
96	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
101	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
102	C₂×Dic₆₇	—	2	4	no	—
103	C₂×Dic₆₇	—	2	2	no	—
104	C₂×Dic₆₇	—	2	4	no	—
105	C₂×Dic₆₇	—	4	4	no	—
106	C₂×Dic₆₇	—	4	4	no	—
107	C₂×Dic₆₇	—	4	4	no	—
108	C₂×Dic₆₇	C₄⋊Q₈	1	4	yes	—
109	C₈×C₆₇	C₄×Q₈	1	4	yes	—
110	C₈×C₆₇	C₄×Q₈	1	4	yes	—
111	C₆₇⋊C₈	—	2	2	no	—
112	C₆₇⋊C₈	—	2	2	no	—
113	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
115	C₂²×C₆₇	#532	1	1	yes	characteristic
116	C₂²×C₆₇	#575	1	4	yes	—
117	C₄×C₆₇	—	2	4	no	—
118	C₄×C₆₇	—	4	8	no	—
119	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
121	C₄×C₆₇	#533	1	1	yes	characteristic
122	C₄×C₆₇	#533	1	1	yes	characteristic
123	Dic₆₇	—	2	4	no	—
124	Dic₆₇	—	4	2	no	—
125	Dic₆₇	—	8	4	no	—
126	#21728	—	67	1	no	maximal, Sylow
127	C₂×C₆₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₆₇	#2943	1	2	yes	—
129	C₂×C₆₇	#3166	1	2	yes	—
130	C₂×C₆₇	#3183	1	2	yes	—
131	C₂×#499	D₆₇	1	1	yes	characteristic
132	C₄×#125	—	67	2	no	—
133	#1991	—	67	2	no	—
134	#1944	—	67	2	no	—
135	#1943	—	67	2	no	—
136	#1846	—	67	1	no	—
137	#1776	—	67	2	no	—
138	#1776	—	67	2	no	—
139	C₄×#97	—	67	1	no	—
140	C₆₇	#21728	1	1	yes	characteristic, complemented, Sylow
141	C₂×#125	C₂×D₆₇	1	2	yes	—
142	#499	Dic₆₇	1	8	yes	complemented
143	C₂×#107	—	67	2	no	—
144	C₂×#107	C₂×D₆₇	1	2	yes	—
145	C₂×#106	—	67	2	no	—
146	C₂×#106	C₂×D₆₇	1	2	yes	—
147	C₂×#97	—	67	1	no	—
148	C₂×#97	—	67	1	no	—
149	C₂×#97	—	67	1	no	—
150	C₂×#97	—	67	1	no	—
151	C₂×C₄×C₈	—	67	1	no	—
152	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
153	#389	—	67	4	no	—
154	#387	—	134	4	no	—
155	#382	—	67	2	no	—
156	C₄×C₄⋊C₄	—	67	4	no	—
157	C₄×C₄⋊C₄	—	134	4	no	—
158	C₄³	—	67	1	no	—
159	#340	—	67	4	no	—
160	#340	—	67	4	no	—
161	C₂²×C₈	—	67	1	no	—
162	C₂²×C₈	—	67	1	no	—
163	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
165	#125	—	2	4	no	—
166	#125	C₂×Dic₆₇	1	8	yes	—
167	C₂×C₄⋊C₄	—	2	4	no	—
168	C₂×C₄⋊C₄	—	67	4	no	—
169	C₂×C₄⋊C₄	—	134	4	no	—
170	C₂×C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
171	C₂×C₄²	—	67	4	no	—
172	C₂×C₄²	—	67	1	no	—
173	C₂×C₄²	—	67	1	no	—
174	C₂×C₄²	—	134	4	no	—
175	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
176	#107	—	134	4	no	—
177	#107	C₂×Dic₆₇	1	8	yes	—
178	#106	—	134	4	no	—
179	#106	C₂×Dic₆₇	1	8	yes	—
180	#97	—	67	4	no	—
181	#97	—	67	4	no	—
182	#97	—	134	2	no	—
183	#97	—	134	2	no	—
184	C₄×C₈	—	134	2	no	—
185	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
186	#95	—	134	4	no	—
187	#95	—	134	4	no	—
188	#95	—	134	4	no	—
189	C₂²×C₄	—	2	4	no	—
190	C₂²×C₄	—	67	2	no	—
191	C₂²×C₄	—	67	2	no	—
192	C₂²×C₄	—	134	4	no	—
193	C₂²×C₄	D₄×D₆₇	1	2	yes	—
194	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
195	C₂×C₈	—	67	2	no	—
196	C₂×C₈	—	67	2	no	—
197	C₂×C₈	—	134	1	no	—
198	C₂×C₈	—	134	1	no	—
199	C₂×C₈	—	134	1	no	—
200	C₂×C₈	—	134	1	no	—
201	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
202	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
203	C₂×C₈	Q₈×D₆₇	1	2	yes	—
204	C₂×C₈	Q₈×D₆₇	1	2	yes	—
205	C₂×C₈	#11761629	1	2	yes	—
206	C₂×C₈	#11761629	1	2	yes	—
207	C₄⋊C₄	—	2	16	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	268	8	no	—
210	C₄⋊C₄	—	134	4	no	—
211	C₄⋊C₄	—	134	4	no	—
212	C₄⋊C₄	C₂²×Dic₆₇	1	8	yes	—
213	C₄²	—	268	4	no	—
214	C₄²	—	268	8	no	—
215	C₄²	—	67	2	no	—
216	C₄²	—	67	2	no	—
217	C₄²	—	134	1	no	—
218	C₄²	—	134	1	no	—
219	C₄²	C₂²×Dic₆₇	1	2	yes	—
220	C₄²	#11761627	1	2	yes	—
221	C₂³	C₂×D₄×D₆₇	1	1	yes	characteristic
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	8	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	268	4	no	—
227	C₂×C₄	—	268	4	no	—
228	C₂×C₄	—	268	4	no	—
229	C₂×C₄	—	67	4	no	—
230	C₂×C₄	—	134	4	no	—
231	C₂×C₄	—	134	2	no	—
232	C₂×C₄	—	134	4	no	—
233	C₂×C₄	D₄×Dic₆₇	1	4	yes	—
234	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
235	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
236	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
237	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
238	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
239	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
240	C₈	—	134	2	no	—
241	C₈	—	134	2	no	—
242	C₈	Q₈×Dic₆₇	1	4	yes	—
243	C₈	Q₈×Dic₆₇	1	4	yes	—
244	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
245	C₂²	C₄⋊Q₈×D₆₇	1	1	yes	characteristic
246	C₂²	#16863947	1	1	yes	characteristic
247	C₂²	#16864193	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	4	8	no	—
250	C₄	—	268	2	no	—
251	C₄	—	536	4	no	—
252	C₄	—	134	4	no	—
253	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
254	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
255	C₄	#16863951	1	1	yes	characteristic
256	C₄	#16863951	1	1	yes	characteristic
257	C₂	C₄⋊Q₈×Dic₆₇	1	1	yes	characteristic
258	C₂	#22569010	1	2	yes	—
259	C₂	#22570757	1	2	yes	—
260	C₂	#22570940	1	2	yes	—

	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	#21728
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
140. C₆₇	#21728	semidirect
142. #499	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#125×Dic₆₇	2
#22561175	2
#22560953	2
#22560948	2
#22560373	2
#22560109	2
#22560108	2
C₄×#14523953	2
C₂×#499×C₆₇	2
#21728	67