Subgroups — #37536041

Order 23168.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26041711	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₄₄₈	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₈₉₆	C₂	1	2	yes	maximal, complemented
4	#26041118	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₈₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#19153838	—	2	1	no	—
7	C₂×Dic₇₂₄	—	2	1	no	—
8	Dic₁₄₄₈	—	2	1	no	—
9	Dic₁₄₄₈	C₂²	1	2	yes	—
10	Dic₁₄₄₈	C₂²	1	1	yes	characteristic
11	#19153761	—	2	1	no	—
12	D₂₈₉₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₈₁	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₈₁	C₂²	1	2	yes	—
15	#15668449	—	4	1	no	—
16	C₂×Dic₃₆₂	—	4	1	no	—
17	Dic₇₂₄	—	2	1	no	—
18	Dic₇₂₄	—	2	2	no	—
19	Dic₇₂₄	—	4	1	no	—
20	#15668438	—	4	1	no	—
21	D₁₄₄₈	—	2	1	no	—
22	C₂×C₈×C₁₈₁	D₄	1	1	yes	characteristic
23	C₁₆×C₁₈₁	D₄	1	1	yes	characteristic
24	C₁₆×C₁₈₁	C₂³	1	1	yes	characteristic
25	Dic₃₆₂	—	4	1	no	—
26	Dic₃₆₂	—	4	2	no	—
27	Dic₃₆₂	—	8	1	no	—
28	D₇₂₄	—	4	1	no	—
29	C₂²⋊D₁₈₁	—	8	1	no	—
30	C₂×Dic₁₈₁	—	8	1	no	—
31	C₄×D₁₈₁	—	8	1	no	—
32	C₂×C₄×C₁₈₁	D₈	1	1	yes	characteristic
33	C₈×C₁₈₁	D₈	1	1	yes	characteristic
34	C₈×C₁₈₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₈₁	D₁₆	1	1	yes	characteristic
36	C₂×D₁₈₁	—	8	1	no	—
37	C₄×C₁₈₁	D₁₆	1	1	yes	characteristic
38	C₄×C₁₈₁	C₂×D₈	1	1	yes	characteristic
39	Dic₁₈₁	—	8	1	no	—
40	Dic₁₈₁	—	8	2	no	—
41	C₂×C₁₈₁	—	2	1	no	—
42	C₂×C₁₈₁	C₂×D₁₆	1	1	yes	characteristic
43	D₁₈₁	—	16	1	no	—
44	C₁₈₁	#2264	1	1	yes	characteristic, complemented, Sylow
45	#2264	—	181	1	no	maximal, Sylow
46	#508	—	181	1	no	—
47	C₂×Q₃₂	—	181	1	no	—
48	Q₆₄	—	181	2	no	—
49	SD₆₄	—	181	2	no	—
50	M₆₄	D₁₈₁	1	1	yes	characteristic, complemented
51	#135	—	362	1	no	—
52	C₂×Q₁₆	—	362	1	no	—
53	Q₃₂	—	362	1	no	—
54	Q₃₂	—	181	2	no	—
55	Q₃₂	—	181	1	no	—
56	SD₃₂	—	362	1	no	—
57	D₁₆	—	181	1	no	—
58	C₂×C₁₆	C₂×D₁₈₁	1	1	yes	characteristic
59	C₃₂	C₂×D₁₈₁	1	2	yes	—
60	#41	—	724	1	no	—
61	C₂×Q₈	—	724	1	no	—
62	Q₁₆	—	362	1	no	—
63	Q₁₆	—	362	2	no	—
64	Q₁₆	—	724	1	no	—
65	SD₁₆	—	724	1	no	—
66	D₈	—	362	1	no	—
67	C₂×C₈	D₇₂₄	1	1	yes	characteristic
68	C₁₆	D₇₂₄	1	1	yes	characteristic
69	C₁₆	C₂²×D₁₈₁	1	1	yes	characteristic
70	Q₈	—	1448	1	no	—
71	Q₈	—	724	1	no	—
72	Q₈	—	724	2	no	—
73	D₄	—	1448	1	no	—
74	D₄	—	724	1	no	—
75	C₂×C₄	—	1448	1	no	—
76	C₂×C₄	—	1448	1	no	—
77	C₂×C₄	D₁₄₄₈	1	1	yes	characteristic
78	C₈	D₁₄₄₈	1	1	yes	characteristic
79	C₈	C₂×D₇₂₄	1	1	yes	characteristic
80	C₂²	—	1448	1	no	—
81	C₂²	D₂₈₉₆	1	1	yes	characteristic
82	C₄	—	1448	1	no	—
83	C₄	—	1448	2	no	—
84	C₄	D₂₈₉₆	1	1	yes	characteristic
85	C₄	C₂×D₁₄₄₈	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2896	1	no	—
88	C₂	C₂×D₂₈₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₈₉₆
commutator, cosocle	C₁₆×C₁₈₁	C₂³
Frattini	C₁₆	C₂²×D₁₈₁
Fitting	M₆₄×C₁₈₁	C₂
socle	C₂×C₁₈₁	C₂×D₁₆
prime radical	#26041711	C₂
hypercenter	M₆₄	D₁₈₁
hypocenter	C₁₈₁	#2264
LCS	G ▷ C₁₆×C₁₈₁ ▷ C₈×C₁₈₁ ▷ C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₄₄₈	C₂	semidirect
3. Dic₂₈₉₆	C₂	semidirect
4. #26041118	C₂	semidirect
5. M₆₄×C₁₈₁	C₂	semidirect
44. C₁₈₁	#2264	semidirect
50. M₆₄	D₁₈₁	semidirect

Maximal subgroups

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

subgroup	index
#26041711	2
C₂×Dic₁₄₄₈	2
Dic₂₈₉₆	2
#26041118	2
M₆₄×C₁₈₁	2
#2264	181