Subgroups — #26012374

Order 11072.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583836	C₂	1	1	yes	maximal, characteristic
2	#19149758	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₁₇₃	C₂	1	1	yes	maximal, characteristic
4	#14583836	—	2	1	no	—
5	#14583836	—	2	1	no	—
6	C₂×#13063443	C₄	1	2	yes	complemented
7	C₄⋊#10621924	—	2	1	no	—
8	C₂²⋊#10621924	C₄	1	2	yes	—
9	C₄×#10621924	—	2	1	no	—
10	C₂×C₄×D₁₇₃	C₂²	1	1	yes	characteristic
11	#14583810	C₄	1	1	yes	characteristic, complemented
12	#14583809	—	2	1	no	—
13	C₈×D₁₇₃	—	2	1	no	—
14	M₁₆×C₁₇₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#10621924	—	2	2	no	—
16	#13063443	—	2	2	no	—
17	C₂²×D₁₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
19	C₄×D₁₇₃	D₄	1	1	yes	characteristic
20	C₄×D₁₇₃	D₄	1	1	yes	characteristic
21	C₄×D₁₇₃	D₄	1	1	yes	characteristic
22	C₄×D₁₇₃	Q₈	1	1	yes	characteristic
23	#13063436	—	2	1	no	—
24	C₂×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
25	C₈×C₁₇₃	—	2	1	no	—
26	C₂²×C₁₇₃	C₄²	1	1	yes	characteristic
27	C₂×D₁₇₃	—	4	1	no	—
28	C₂×D₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₁₇₃	C₄⋊C₄	1	1	yes	characteristic
30	#10621924	—	4	2	no	—
31	C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₁₇₃	C₄⋊C₄	1	1	yes	characteristic
33	Dic₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₁₇₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₁₇₃	—	2	1	no	—
36	C₂×C₁₇₃	#95	1	1	yes	characteristic
37	D₁₇₃	—	2	1	no	—
38	D₁₇₃	—	2	1	no	—
39	C₁₇₃	#344	1	1	yes	characteristic, complemented, Sylow
40	#344	—	173	1	no	maximal, Sylow
41	C₂×M₁₆	—	173	1	no	—
42	C₂×M₁₆	—	173	1	no	—
43	#117	—	173	1	no	—
44	C₂²×C₄	—	173	1	no	—
45	M₁₆	—	346	1	no	—
46	M₁₆	—	346	1	no	—
47	M₁₆	—	346	1	no	—
48	M₁₆	—	173	1	no	—
49	M₁₆	#10621924	1	1	yes	characteristic, complemented
50	C₂×C₈	—	346	1	no	—
51	C₂×C₈	—	173	2	no	—
52	C₄⋊C₄	—	346	1	no	—
53	C₂²⋊C₄	—	173	2	no	—
54	C₄²	—	346	1	no	—
55	C₂³	—	173	1	no	—
56	C₂×C₄	—	346	2	no	—
57	C₂×C₄	—	173	1	no	—
58	C₂×C₄	—	173	1	no	—
59	C₂×C₄	—	173	1	no	—
60	C₂×C₄	—	173	1	no	—
61	C₂×C₄	—	173	1	no	—
62	C₂×C₄	C₂×#10621924	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	346	2	no	—
65	C₈	—	346	1	no	—
66	C₂²	—	173	1	no	—
67	C₂²	—	173	1	no	—
68	C₂²	—	692	1	no	—
69	C₂²	C₄×#10621924	1	1	yes	characteristic
70	C₄	—	173	1	no	—
71	C₄	—	173	1	no	—
72	C₄	—	692	2	no	—
73	C₄	C₂²⋊#10621924	1	1	yes	characteristic
74	C₄	C₄⋊#10621924	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	346	1	no	—
77	C₂	—	346	1	no	—
78	C₂	(C₂×#10621924)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#10621924)⋊C₄
commutator	C₂²×C₁₇₃	C₄²
Frattini	C₂×C₄	C₂×#10621924
Fitting	M₁₆×C₁₇₃	C₄
socle	C₂×C₁₇₃	#95
prime radical	C₂²×D₁₇₃	C₂×C₄
cosocle	C₂×C₄×D₁₇₃	C₂²
hypercenter	M₁₆	#10621924
hypocenter	C₁₇₃	#344
LCS	G ▷ C₂²×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
6. C₂×#13063443	C₄	semidirect
11. #14583810	C₄	semidirect
14. M₁₆×C₁₇₃	C₄	semidirect
39. C₁₇₃	#344	semidirect
49. M₁₆	#10621924	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583836	2
#19149758	2
M₁₆×D₁₇₃	2
#344	173