Subgroups — #30595956

Order 15616.

606 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21461532	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×#6923	C₂	1	1	yes	maximal, characteristic
4	#21461015	C₂	1	1	yes	maximal, characteristic
5	C₈×D₆₁⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₈×D₆₁⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#322×D₆₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#14519831	C₄	1	4	yes	—
9	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×D₆₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈×#6923	C₄	1	4	yes	—
13	C₄²×#6923	C₂²	1	1	yes	characteristic
14	(C₂×#6923)⋊C₈	C₄	1	4	yes	—
15	#92420⋊C₈	C₄	1	4	yes	complemented
16	C₈×#92420	C₄	1	4	yes	complemented
17	C₈×#92420	C₄	1	4	yes	complemented
18	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
20	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
21	#16839484	C₄	1	1	yes	characteristic, complemented
22	#16839484	C₄	1	1	yes	characteristic, complemented
23	#16839483	C₄	1	1	yes	characteristic, complemented
24	C₆₁×#322	C₄	1	1	yes	characteristic, complemented
25	C₂×D₆₁⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×D₆₁⋊C₈	C₂×C₄	1	2	yes	—
27	C₂×D₆₁⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄×#6923	C₂×C₄	1	2	yes	—
29	C₂×C₄×#6923	C₂×C₄	1	2	yes	—
30	C₂×C₄×#6923	C₂×C₄	1	2	yes	—
31	#14519831	C₈	1	16	yes	complemented
32	C₄×#92420	C₂×C₄	1	2	yes	—
33	C₄×#92420	C₂×C₄	1	2	yes	—
34	C₄×#92420	C₂×C₄	1	2	yes	—
35	C₄×#92420	C₂×C₄	1	2	yes	—
36	C₄×#92420	C₂×C₄	1	2	yes	—
37	C₄×#92420	C₂×C₄	1	2	yes	—
38	C₈×#6923	—	2	8	no	—
39	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
42	C₄²×D₆₁	C₂³	1	1	yes	characteristic
43	C₄×#92418	C₂×C₄	1	1	yes	characteristic
44	C₄×#92418	C₂×C₄	1	1	yes	characteristic
45	C₄×#92418	C₂×C₄	1	1	yes	characteristic
46	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
47	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
48	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×#6923	C₄²	1	4	yes	—
53	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×#92420	C₄²	1	4	yes	—
57	C₂×#92420	C₄²	1	4	yes	—
58	C₂×#92420	C₄²	1	4	yes	—
59	C₄×#6923	—	2	4	no	—
60	C₄×#6923	—	2	4	no	—
61	C₄×#6923	C₂×C₈	1	8	yes	—
62	D₆₁⋊C₈	—	2	4	no	—
63	D₆₁⋊C₈	C₂×C₈	1	8	yes	—
64	D₆₁⋊C₈	M₁₆	1	8	yes	—
65	C₂×C₈×C₆₁	C₄²	1	2	yes	—
66	C₂×C₈×C₆₁	C₄²	1	2	yes	—
67	C₂×C₈×C₆₁	C₄²	1	2	yes	—
68	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
71	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×#92418	C₄²	1	2	yes	—
73	C₂×#92418	C₄²	1	2	yes	—
74	C₂×#92418	C₄²	1	2	yes	—
75	C₈×D₆₁	—	2	4	no	—
76	C₈×D₆₁	—	2	4	no	—
77	C₈×D₆₁	C₂×C₈	1	8	yes	—
78	C₂²×D₆₁	C₂×C₄²	1	1	yes	characteristic
79	C₂×#6923	—	2	4	no	—
80	C₂×#6923	—	2	4	no	—
81	C₂×#6923	C₄×C₈	1	8	yes	—
82	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
83	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
85	C₂×Dic₆₁	C₂×C₄²	1	1	yes	characteristic
86	C₂×Dic₆₁	C₂×C₄²	1	1	yes	characteristic
87	C₂×Dic₆₁	C₂×C₄²	1	1	yes	characteristic
88	C₄×D₆₁	C₂²×C₈	1	4	yes	—
89	C₄×D₆₁	C₂×M₁₆	1	4	yes	—
90	C₄×D₆₁	#131	1	4	yes	—
91	#92420	—	2	4	no	—
92	#92420	C₄×C₈	1	8	yes	complemented
93	#92420	#97	1	8	yes	complemented
94	C₈×C₆₁	—	2	2	no	—
95	C₈×C₆₁	—	2	2	no	—
96	C₈×C₆₁	C₄×C₈	1	4	yes	complemented
97	#92418	—	2	2	no	—
98	#92418	—	2	2	no	—
99	#92418	C₄×C₈	1	4	yes	complemented
100	C₈×#97	—	61	1	no	maximal, Sylow
101	C₂²×C₆₁	C₄³	1	1	yes	characteristic
102	C₂×D₆₁	C₂×C₄×C₈	1	2	yes	—
103	C₂×D₆₁	C₄×M₁₆	1	2	yes	—
104	C₂×D₆₁	#405	1	2	yes	—
105	#6923	—	2	8	no	—
106	C₄×C₆₁	C₂×C₄×C₈	1	2	yes	—
107	C₄×C₆₁	C₄×M₁₆	1	2	yes	—
108	C₄×C₆₁	#405	1	2	yes	—
109	Dic₆₁	C₂×C₄×C₈	1	2	yes	—
110	Dic₆₁	C₂×#97	1	2	yes	—
111	Dic₆₁	#405	1	2	yes	—
112	#1744	—	61	1	no	—
113	C₄×#97	—	61	1	no	—
114	C₄²×C₈	—	61	1	no	—
115	C₂×#322	—	61	1	no	—
116	C₂×#322	—	61	1	no	—
117	C₂×C₈²	—	61	1	no	—
118	C₂×C₈²	—	61	1	no	—
119	C₂×C₆₁	C₄²×C₈	1	1	yes	characteristic
120	C₂×C₆₁	C₄×#97	1	1	yes	characteristic
121	C₂×C₆₁	#1726	1	1	yes	characteristic
122	D₆₁	C₈×M₁₆	1	4	yes	—
123	C₂×#97	—	61	4	no	—
124	C₂×C₄×C₈	—	61	1	no	—
125	C₂×C₄×C₈	—	61	1	no	—
126	C₂×C₄×C₈	—	61	1	no	—
127	C₂×C₄×C₈	—	61	1	no	—
128	C₂×C₄×C₈	—	61	1	no	—
129	C₂×C₄×C₈	—	61	1	no	—
130	C₂×C₄×C₈	—	61	4	no	—
131	C₄³	—	61	1	no	—
132	#336	—	61	4	no	—
133	#322	—	61	4	no	—
134	#322	—	61	1	no	—
135	#322	—	61	1	no	—
136	#322	—	61	1	no	—
137	#322	#6923	1	1	yes	characteristic, complemented
138	C₈²	—	61	4	no	—
139	C₈²	—	61	4	no	—
140	C₆₁	C₈×#97	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₈	—	61	2	no	—
142	C₂²×C₈	—	61	2	no	—
143	C₂²×C₈	—	61	2	no	—
144	C₂²×C₈	—	61	2	no	—
145	C₂²×C₈	—	61	2	no	—
146	C₂²×C₈	—	61	2	no	—
147	C₂×C₄²	—	61	2	no	—
148	C₂×C₄²	—	61	1	no	—
149	C₂×C₄²	—	61	2	no	—
150	C₂×C₄²	—	61	2	no	—
151	#97	—	61	16	no	—
152	C₄×C₈	—	61	2	no	—
153	C₄×C₈	—	61	2	no	—
154	C₄×C₈	—	61	2	no	—
155	C₄×C₈	—	61	2	no	—
156	C₄×C₈	—	61	2	no	—
157	C₄×C₈	—	61	2	no	—
158	C₄×C₈	—	61	2	no	—
159	C₄×C₈	—	61	2	no	—
160	C₄×C₈	—	61	2	no	—
161	C₄×C₈	—	61	1	no	—
162	C₄×C₈	—	61	1	no	—
163	C₄×C₈	—	61	1	no	—
164	C₄×C₈	—	122	8	no	—
165	C₄×C₈	C₂×#6923	1	1	yes	characteristic
166	C₄×C₈	C₂×#6923	1	1	yes	characteristic
167	C₄×C₈	C₂×#6923	1	1	yes	characteristic
168	C₂²×C₄	—	61	1	no	—
169	C₂²×C₄	—	61	1	no	—
170	C₂²×C₄	—	61	1	no	—
171	C₂²×C₄	—	61	4	no	—
172	C₂×C₈	—	61	4	no	—
173	C₂×C₈	—	61	4	no	—
174	C₂×C₈	—	61	8	no	—
175	C₂×C₈	—	61	8	no	—
176	C₂×C₈	—	61	2	no	—
177	C₂×C₈	—	61	8	no	—
178	C₂×C₈	—	61	4	no	—
179	C₂×C₈	—	61	2	no	—
180	C₂×C₈	—	61	2	no	—
181	C₂×C₈	—	122	4	no	—
182	C₂×C₈	—	122	4	no	—
183	C₂×C₈	—	122	4	no	—
184	C₂×C₈	C₄×#6923	1	2	yes	—
185	C₂×C₈	C₄×#6923	1	2	yes	—
186	C₂×C₈	C₄×#6923	1	2	yes	—
187	C₄²	—	61	8	no	—
188	C₄²	—	61	1	no	—
189	C₄²	—	61	1	no	—
190	C₄²	—	61	1	no	—
191	C₄²	—	122	4	no	—
192	C₄²	—	122	4	no	—
193	C₄²	C₂²×#6923	1	1	yes	characteristic
194	C₂³	—	61	1	no	—
195	C₂×C₄	—	61	1	no	—
196	C₂×C₄	—	61	1	no	—
197	C₂×C₄	—	61	1	no	—
198	C₂×C₄	—	61	4	no	—
199	C₂×C₄	—	61	4	no	—
200	C₂×C₄	—	61	4	no	—
201	C₂×C₄	—	61	8	no	—
202	C₂×C₄	—	122	4	no	—
203	C₂×C₄	—	122	4	no	—
204	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	2	no	—
209	C₈	—	61	8	no	—
210	C₈	—	61	8	no	—
211	C₈	—	61	4	no	—
212	C₈	—	122	4	no	—
213	C₈	—	122	2	no	—
214	C₈	—	122	2	no	—
215	C₈	C₈×#6923	1	4	yes	complemented
216	C₂²	—	61	2	no	—
217	C₂²	—	61	2	no	—
218	C₂²	—	61	2	no	—
219	C₂²	C₄²×#6923	1	1	yes	characteristic
220	C₄	—	61	2	no	—
221	C₄	—	61	2	no	—
222	C₄	—	61	2	no	—
223	C₄	—	122	8	no	—
224	C₄	C₂×C₈×#6923	1	2	yes	—
225	C₄	M₁₆×#6923	1	2	yes	—
226	C₄	#16840728	1	2	yes	—
227	C₂	—	61	4	no	—
228	C₂	C₄×C₈×#6923	1	1	yes	characteristic
229	C₂	#97×#6923	1	1	yes	characteristic
230	C₂	#21461477	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×#6923
commutator	C₂×C₆₁	C₄²×C₈
Fitting	C₆₁×#322	C₄
socle	C₂²×C₆₁	C₄³
prime radical	C₂²×D₆₁	C₂×C₄²
cosocle	C₄²×D₆₁	C₂³
hypercenter	#322	#6923
hypocenter	C₆₁	C₈×#97
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
15. #92420⋊C₈	C₄	semidirect
16. C₈×#92420	C₄	semidirect
17. C₈×#92420	C₄	semidirect
21. #16839484	C₄	semidirect
22. #16839484	C₄	semidirect
23. #16839483	C₄	semidirect
24. C₆₁×#322	C₄	semidirect
31. #14519831	C₈	semidirect
92. #92420	C₄×C₈	semidirect
93. #92420	#97	semidirect
96. C₈×C₆₁	C₄×C₈	semidirect
99. #92418	C₄×C₈	semidirect
137. #322	#6923	semidirect
140. C₆₁	C₈×#97	semidirect
215. C₈	C₈×#6923	semidirect

Maximal subgroups

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

subgroup	index
#21461532	2
C₄×#14519831	2
C₄×C₈×#6923	2
#21461015	2
C₈×D₆₁⋊C₈	2
#322×D₆₁	2
C₈×#97	61