Subgroups — #30864129

Order 17152.

742 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14524035	C₂	1	1	yes	maximal, characteristic
2	#22560925	C₂	1	2	yes	maximal
3	#22560764	C₂	1	4	yes	maximal
4	#22560074	C₂	1	2	yes	maximal
5	#22560070	C₂	1	4	yes	maximal
6	C₄²×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1831	C₂	1	1	yes	maximal, characteristic
8	C₂×#14524035	C₄	1	1	yes	characteristic
9	C₂×Dic₆₇⋊C₈	C₂²	1	4	yes	—
10	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₆₇	C₂²	1	4	yes	—
15	#16863294	C₄	1	2	yes	—
16	#16863280	C₄	1	4	yes	—
17	#16863262	C₄	1	1	yes	characteristic
18	C₄×Dic₆₇⋊C₄	C₂²	1	4	yes	—
19	C₄×C₄⋊Dic₆₇	C₂²	1	2	yes	—
20	#16863150	C₂²	1	4	yes	—
21	#16863150	C₂²	1	4	yes	—
22	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	#336×C₆₇	C₂²	1	4	yes	—
25	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
27	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
28	#14524035	C₈	1	8	yes	complemented
29	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
31	C₂×Dic₆₇⋊C₄	C₂×C₄	1	4	yes	—
32	C₂×Dic₆₇⋊C₄	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₆₇	C₂³	1	4	yes	—
34	Dic₆₇⋊C₈	—	2	8	no	—
35	C₄×C₆₇⋊C₈	—	2	4	no	—
36	C₄×C₆₇⋊C₈	—	2	4	no	—
37	C₄×C₆₇⋊C₈	—	2	2	no	—
38	C₄×C₆₇⋊C₈	Q₈	1	4	yes	—
39	C₈×Dic₆₇	—	2	8	no	—
40	C₆₇⋊#95	C₂×C₄	1	4	yes	—
41	C₆₇⋊#95	C₂×C₄	1	4	yes	—
42	C₆₇⋊#95	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
44	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
48	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₆₇	—	2	4	no	—
50	C₂²×Dic₆₇	C₂²×C₄	1	4	yes	—
51	C₂²×Dic₆₇	C₂²×C₄	1	4	yes	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	2	2	no	—
54	C₂×C₆₇⋊C₈	—	2	2	no	—
55	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
56	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
57	C₂×C₆₇⋊C₈	#41	1	4	yes	—
58	C₂×C₆₇⋊C₈	#41	1	4	yes	—
59	C₄⋊Dic₆₇	—	2	4	no	—
60	C₄⋊Dic₆₇	C₂×C₈	1	8	yes	—
61	Dic₆₇⋊C₄	—	2	8	no	—
62	Dic₆₇⋊C₄	C₂×C₈	1	16	yes	—
63	C₄×Dic₆₇	—	2	4	no	—
64	C₄×Dic₆₇	—	2	4	no	—
65	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	2	4	no	—
72	C₂×C₈×C₆₇	—	2	4	no	—
73	C₄²×C₆₇	—	2	4	no	—
74	C₄²×C₆₇	—	2	2	no	—
75	C₄²×C₆₇	—	2	2	no	—
76	C₄²×C₆₇	—	2	2	no	—
77	C₄²×C₆₇	C₂×C₈	1	4	yes	—
78	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
79	C₄²×C₆₇	#41	1	2	yes	—
80	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
87	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
88	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
89	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₆₇	#131	1	4	yes	—
91	C₂×C₄×C₆₇	#131	1	2	yes	—
92	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
94	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
96	C₂×Dic₆₇	—	2	4	no	—
97	C₂×Dic₆₇	—	2	4	no	—
98	C₂×Dic₆₇	—	2	4	no	—
99	C₂×Dic₆₇	—	2	4	no	—
100	C₂×Dic₆₇	—	2	4	no	—
101	C₂×Dic₆₇	C₂²×C₈	1	8	yes	—
102	C₈×C₆₇	—	4	4	no	—
103	C₆₇⋊C₈	—	2	4	no	—
104	C₆₇⋊C₈	—	2	2	no	—
105	C₆₇⋊C₈	#125	1	4	yes	complemented
106	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
107	C₂²×C₆₇	C₄×#41	1	2	yes	—
108	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
109	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
110	C₂²×C₆₇	C₂×#131	1	2	yes	—
111	C₄×C₆₇	—	2	2	no	—
112	C₄×C₆₇	—	2	4	no	—
113	C₄×C₆₇	—	2	4	no	—
114	C₄×C₆₇	C₈×Q₈	1	4	yes	—
115	C₄×C₆₇	C₂×#125	1	1	yes	characteristic
116	C₄×C₆₇	#529	1	2	yes	—
117	C₄×C₆₇	#533	1	1	yes	characteristic
118	Dic₆₇	—	2	8	no	—
119	Dic₆₇	—	4	4	no	—
120	C₈×#125	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	C₄×#125	1	1	yes	characteristic
122	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
123	C₂×C₆₇	#2960	1	1	yes	characteristic
124	C₂×C₆₇	C₈×#41	1	2	yes	—
125	C₂×C₆₇	#2966	1	2	yes	—
126	C₄×#125	—	67	1	no	—
127	#1939	—	67	2	no	—
128	#1916	—	67	4	no	—
129	#1831	D₆₇	1	1	yes	characteristic
130	C₈×C₄⋊C₄	—	67	2	no	—
131	C₈×C₄⋊C₄	—	67	4	no	—
132	C₄²×C₈	—	67	1	no	—
133	C₆₇	C₈×#125	1	1	yes	characteristic, complemented, Sylow
134	C₂×#125	—	67	1	no	—
135	C₂×C₄⋊C₈	—	67	4	no	—
136	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
137	C₂×C₄×C₈	—	67	1	no	—
138	C₂×C₄×C₈	—	67	4	no	—
139	C₂×C₄×C₈	—	67	2	no	—
140	C₂×C₄×C₈	—	67	1	no	—
141	C₂×C₄×C₈	—	67	2	no	—
142	#389	—	67	2	no	—
143	#387	—	67	4	no	—
144	#382	—	67	1	no	—
145	C₄×C₄⋊C₄	—	67	2	no	—
146	C₄×C₄⋊C₄	—	67	4	no	—
147	C₄³	C₂×D₆₇	1	1	yes	characteristic
148	#336	—	67	4	no	—
149	#336	—	67	4	no	—
150	#336	C₂×D₆₇	1	4	yes	—
151	C₂²×C₈	—	67	1	no	—
152	C₂²×C₈	—	67	1	no	—
153	C₂²×C₈	—	67	2	no	—
154	C₂²×C₈	C₂²×D₆₇	1	4	yes	—
155	#125	—	67	8	no	—
156	C₂×C₄⋊C₄	—	67	4	no	—
157	C₂×C₄⋊C₄	—	67	2	no	—
158	C₂×C₄⋊C₄	—	67	4	no	—
159	C₂×C₄⋊C₄	—	67	2	no	—
160	C₂×C₄²	—	67	4	no	—
161	C₂×C₄²	C₄×D₆₇	1	2	yes	—
162	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
163	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
164	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
165	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
166	C₄⋊C₈	—	2	4	no	—
167	C₄⋊C₈	—	134	8	no	—
168	C₄×C₈	—	67	4	no	—
169	C₄×C₈	—	134	4	no	—
170	C₄×C₈	—	134	4	no	—
171	C₄×C₈	—	134	8	no	—
172	C₄×C₈	—	134	2	no	—
173	#95	—	67	4	no	—
174	#95	—	67	4	no	—
175	#95	—	67	4	no	—
176	C₂²×C₄	—	67	4	no	—
177	C₂²×C₄	—	67	4	no	—
178	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
180	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
181	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
182	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	67	2	no	—
187	C₂×C₈	—	67	4	no	—
188	C₂×C₈	—	67	2	no	—
189	C₂×C₈	—	67	4	no	—
190	C₂×C₈	—	134	2	no	—
191	C₂×C₈	—	134	2	no	—
192	C₂×C₈	—	134	2	no	—
193	C₄⋊C₄	—	67	16	no	—
194	C₄⋊C₄	—	67	8	no	—
195	C₄⋊C₄	—	134	8	no	—
196	C₄⋊C₄	—	134	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	134	4	no	—
202	C₄²	—	134	4	no	—
203	C₄²	C₈×D₆₇	1	4	yes	—
204	C₄²	Q₈×D₆₇	1	2	yes	—
205	C₄²	#11761627	1	2	yes	—
206	C₂³	C₂²×C₄×D₆₇	1	1	yes	characteristic
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	67	8	no	—
213	C₂×C₄	—	134	4	no	—
214	C₂×C₄	—	134	4	no	—
215	C₂×C₄	—	134	4	no	—
216	C₂×C₄	—	134	4	no	—
217	C₂×C₄	—	134	4	no	—
218	C₂×C₄	#14524008	1	2	yes	—
219	C₂×C₄	#14524008	1	2	yes	—
220	C₂×C₄	C₂×C₈×D₆₇	1	4	yes	—
221	C₂×C₄	C₂×C₈×D₆₇	1	2	yes	—
222	C₂×C₄	#14524051	1	4	yes	—
223	C₂×C₄	#14524051	1	2	yes	—
224	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
225	C₂×C₄	#41×D₆₇	1	2	yes	—
226	C₂×C₄	#41×D₆₇	1	2	yes	—
227	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
228	C₈	—	4	4	no	—
229	C₈	—	67	4	no	—
230	C₈	—	134	4	no	—
231	C₈	—	134	2	no	—
232	C₂²	C₂×#14524008	1	1	yes	characteristic
233	C₂²	#16863864	1	2	yes	—
234	C₂²	C₂²×C₈×D₆₇	1	1	yes	characteristic
235	C₂²	C₂×#14524051	1	1	yes	characteristic
236	C₂²	C₂×#14524051	1	2	yes	—
237	C₄	—	2	2	no	—
238	C₄	—	2	4	no	—
239	C₄	—	2	4	no	—
240	C₄	—	268	4	no	—
241	C₄	—	134	8	no	—
242	C₄	#16863520	1	4	yes	—
243	C₄	#125×D₆₇	1	1	yes	characteristic
244	C₄	#16863925	1	2	yes	—
245	C₄	#16863952	1	1	yes	characteristic
246	C₂	#22562764	1	1	yes	characteristic
247	C₂	C₂×#16863520	1	1	yes	characteristic
248	C₂	#22569139	1	1	yes	characteristic
249	C₂	#22569165	1	2	yes	—
250	C₂	#22569178	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Fitting	C₆₇×#1831	C₂
socle, prime radical	C₂³×C₆₇	C₂³×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#1831	D₆₇
hypocenter	C₆₇	C₈×#125
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #1831

Complements

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

N	H ≅ G/N	type
28. #14524035	C₈	semidirect
105. C₆₇⋊C₈	#125	semidirect
133. C₆₇	C₈×#125	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14524035	2
#22560925	2
#22560764	2
#22560074	2
#22560070	2
C₄²×C₆₇⋊C₈	2
C₆₇×#1831	2
C₈×#125	67