Subgroups — #37511315

Order 22912.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26038871	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₂₈₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#26038278	C₂	1	2	yes	maximal, complemented
4	D₅₇₂₈	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19153135	—	2	1	no	—
7	C₂×D₁₄₃₂	—	2	1	no	—
8	Dic₁₄₃₂	C₂²	1	1	yes	characteristic, complemented
9	#19153058	—	2	1	no	—
10	D₂₈₆₄	—	2	1	no	—
11	D₂₈₆₄	C₂²	1	2	yes	—
12	D₂₈₆₄	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
14	C₃₂×C₁₇₉	C₂²	1	2	yes	complemented
15	#15668259	—	4	1	no	—
16	C₂×D₇₁₆	—	4	1	no	—
17	Dic₇₁₆	—	2	1	no	—
18	#15668248	—	4	1	no	—
19	D₁₄₃₂	—	2	2	no	—
20	D₁₄₃₂	—	2	1	no	—
21	D₁₄₃₂	—	4	1	no	—
22	C₂×C₈×C₁₇₉	D₄	1	1	yes	characteristic
23	C₁₆×C₁₇₉	D₄	1	1	yes	characteristic
24	C₁₆×C₁₇₉	C₂³	1	1	yes	characteristic
25	C₂²×D₁₇₉	—	8	1	no	—
26	Dic₃₅₈	—	4	1	no	—
27	D₇₁₆	—	4	2	no	—
28	D₇₁₆	—	4	1	no	—
29	D₇₁₆	—	8	1	no	—
30	C₂²⋊D₁₇₉	—	8	1	no	—
31	C₄×D₁₇₉	—	8	1	no	—
32	C₂×C₄×C₁₇₉	D₈	1	1	yes	characteristic
33	C₈×C₁₇₉	D₈	1	1	yes	characteristic
34	C₈×C₁₇₉	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₁₇₉	D₁₆	1	1	yes	characteristic
36	C₂×D₁₇₉	—	8	1	no	—
37	C₂×D₁₇₉	—	8	2	no	—
38	C₂×D₁₇₉	—	16	2	no	—
39	C₄×C₁₇₉	D₁₆	1	1	yes	characteristic
40	C₄×C₁₇₉	C₂×D₈	1	1	yes	characteristic
41	Dic₁₇₉	—	8	1	no	—
42	C₂×C₁₇₉	—	2	1	no	—
43	C₂×C₁₇₉	C₂×D₁₆	1	1	yes	characteristic
44	D₁₇₉	—	16	1	no	—
45	D₁₇₉	—	16	2	no	—
46	C₁₇₉	#2263	1	1	yes	characteristic, complemented, Sylow
47	#2263	—	179	1	no	maximal, Sylow
48	#508	—	179	1	no	—
49	C₂×D₁₆	—	179	1	no	—
50	SD₆₄	—	179	2	no	—
51	D₃₂	—	179	2	no	—
52	M₆₄	D₁₇₉	1	1	yes	characteristic, complemented
53	#135	—	358	1	no	—
54	C₂×D₈	—	358	1	no	—
55	Q₃₂	—	179	1	no	—
56	SD₃₂	—	358	1	no	—
57	D₁₆	—	358	1	no	—
58	D₁₆	—	179	1	no	—
59	D₁₆	—	179	2	no	—
60	C₂×C₁₆	C₂×D₁₇₉	1	1	yes	characteristic
61	C₃₂	C₂×D₁₇₉	1	2	yes	complemented
62	#41	—	716	1	no	—
63	C₂×D₄	—	716	1	no	—
64	Q₁₆	—	358	1	no	—
65	SD₁₆	—	716	1	no	—
66	D₈	—	358	2	no	—
67	D₈	—	358	1	no	—
68	D₈	—	716	1	no	—
69	C₂×C₈	D₇₁₆	1	1	yes	characteristic
70	C₁₆	D₇₁₆	1	1	yes	characteristic
71	C₁₆	C₂²×D₁₇₉	1	1	yes	characteristic
72	C₂³	—	1432	1	no	—
73	Q₈	—	716	1	no	—
74	D₄	—	1432	1	no	—
75	D₄	—	1432	1	no	—
76	D₄	—	716	2	no	—
77	D₄	—	716	1	no	—
78	C₂×C₄	—	1432	1	no	—
79	C₂×C₄	D₁₄₃₂	1	1	yes	characteristic
80	C₈	D₁₄₃₂	1	1	yes	characteristic
81	C₈	C₂×D₇₁₆	1	1	yes	characteristic
82	C₂²	—	2864	2	no	—
83	C₂²	—	1432	1	no	—
84	C₂²	—	1432	2	no	—
85	C₂²	D₂₈₆₄	1	1	yes	characteristic
86	C₄	—	1432	1	no	—
87	C₄	D₂₈₆₄	1	1	yes	characteristic
88	C₄	C₂×D₁₄₃₂	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	2864	2	no	—
91	C₂	—	2864	1	no	—
92	C₂	C₂×D₂₈₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₂₈₆₄
commutator, cosocle	C₁₆×C₁₇₉	C₂³
Frattini	C₁₆	C₂²×D₁₇₉
Fitting	M₆₄×C₁₇₉	C₂
socle	C₂×C₁₇₉	C₂×D₁₆
hypercenter	M₆₄	D₁₇₉
hypocenter	C₁₇₉	#2263
LCS	G ▷ C₁₆×C₁₇₉ ▷ C₈×C₁₇₉ ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
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
1. #26038871	C₂	semidirect
2. C₂×D₂₈₆₄	C₂	semidirect
3. #26038278	C₂	semidirect
4. D₅₇₂₈	C₂	semidirect
5. M₆₄×C₁₇₉	C₂	semidirect
8. Dic₁₄₃₂	C₂²	semidirect
12. D₂₈₆₄	C₂²	semidirect
14. C₃₂×C₁₇₉	C₂²	semidirect
46. C₁₇₉	#2263	semidirect
52. M₆₄	D₁₇₉	semidirect
61. C₃₂	C₂×D₁₇₉	semidirect

Maximal subgroups

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

subgroup	index
#26038871	2
C₂×D₂₈₆₄	2
#26038278	2
D₅₇₂₈	2
M₆₄×C₁₇₉	2
#2263	179