Subgroups — #36368525

Order 22144.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012413	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#13063443	C₂	1	1	yes	maximal, characteristic, complemented
3	#26011608	C₂	1	1	yes	maximal, characteristic
4	#19149753	—	2	1	no	—
5	C₄×#13063443	—	2	1	no	—
6	#19149686	—	2	1	no	—
7	#19149615	C₄	1	1	yes	characteristic
8	#19149606	—	2	1	no	—
9	C₈×Dic₁₇₃	C₂²	1	1	yes	characteristic
10	#135×C₁₇₃	C₄	1	1	yes	characteristic
11	C₂×#13063443	—	2	2	no	—
12	#14583829	—	4	1	no	—
13	#14583810	—	2	1	no	—
14	#14583810	—	4	1	no	—
15	C₂×#13063436	—	4	1	no	—
16	C₂×#13063436	D₄	1	1	yes	characteristic
17	C₄×Dic₁₇₃	D₄	1	1	yes	characteristic
18	#41×C₁₇₃	—	2	1	no	—
19	Q₁₆×C₁₇₃	C₈	1	1	yes	characteristic, complemented
20	SD₁₆×C₁₇₃	—	2	1	no	—
21	D₈×C₁₇₃	C₈	1	1	yes	characteristic, complemented
22	C₂×C₈×C₁₇₃	C₂×C₄	1	1	yes	characteristic
23	#13063443	—	2	4	no	—
24	C₂×Dic₁₇₃	—	2	1	no	—
25	#13063436	—	4	1	no	—
26	#13063436	D₈	1	1	yes	characteristic
27	#13063436	SD₁₆	1	1	yes	characteristic
28	Q₈×C₁₇₃	—	2	1	no	—
29	D₄×C₁₇₃	—	2	1	no	—
30	D₄×C₁₇₃	—	4	1	no	—
31	C₂×C₄×C₁₇₃	—	4	1	no	—
32	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₈×C₁₇₃	C₂×C₈	1	1	yes	characteristic
34	C₈×C₁₇₃	M₁₆	1	1	yes	characteristic
35	C₂²×C₁₇₃	—	4	1	no	—
36	C₂²×C₁₇₃	#104	1	1	yes	characteristic
37	C₄×C₁₇₃	—	4	1	no	—
38	C₄×C₁₇₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₇₃	#102	1	1	yes	characteristic
40	Dic₁₇₃	—	2	2	no	—
41	C₂×C₁₇₃	—	2	1	no	—
42	C₂×C₁₇₃	—	8	1	no	—
43	C₂×C₁₇₃	D₄⋊C₈	1	1	yes	characteristic
44	C₁₇₃	#1335	1	1	yes	characteristic, complemented, Sylow
45	#1335	—	173	1	no	maximal, Sylow
46	#443	—	173	1	no	—
47	#364	—	173	1	no	—
48	C₈²	—	173	1	no	—
49	#135	#10621924	1	1	yes	characteristic
50	#131	—	346	1	no	—
51	M₃₂	—	346	1	no	—
52	#108	—	173	1	no	—
53	#104	—	346	1	no	—
54	C₄×C₈	—	346	1	no	—
55	C₄×C₈	—	173	1	no	—
56	#41	—	2	1	no	—
57	Q₁₆	#13063443	1	1	yes	characteristic, complemented
58	SD₁₆	—	2	1	no	—
59	D₈	#13063443	1	1	yes	characteristic, complemented
60	M₁₆	—	346	1	no	—
61	M₁₆	—	692	1	no	—
62	C₂×C₈	—	346	2	no	—
63	C₂×C₈	—	173	1	no	—
64	C₂×C₈	—	692	1	no	—
65	C₂×C₈	C₂×#10621924	1	1	yes	characteristic
66	C₄²	—	173	1	no	—
67	C₁₆	—	692	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	4	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	346	1	no	—
73	C₂×C₄	C₂²⋊#10621924	1	1	yes	characteristic
74	C₈	—	346	4	no	—
75	C₈	—	173	1	no	—
76	C₈	—	173	1	no	—
77	C₈	—	692	1	no	—
78	C₈	C₂×#13063443	1	1	yes	characteristic
79	C₈	#14583835	1	1	yes	characteristic
80	C₂²	—	4	1	no	—
81	C₂²	#19149743	1	1	yes	characteristic
82	C₄	—	4	1	no	—
83	C₄	—	346	2	no	—
84	C₄	C₂²⋊#13063443	1	1	yes	characteristic
85	C₄	#19149741	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	D₄⋊#13063443	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19149741
commutator	C₈×C₁₇₃	C₂×C₈
Frattini	C₂×C₈	C₂×#10621924
Fitting, prime radical	#135×C₁₇₃	C₄
socle	C₂×C₁₇₃	D₄⋊C₈
cosocle	C₈×Dic₁₇₃	C₂²
hypercenter	#135	#10621924
hypocenter	C₁₇₃	#1335
LCS	G ▷ C₈×C₁₇₃ ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #26012413	C₂	semidirect
2. C₈×#13063443	C₂	semidirect
19. Q₁₆×C₁₇₃	C₈	semidirect
21. D₈×C₁₇₃	C₈	semidirect
44. C₁₇₃	#1335	semidirect
57. Q₁₆	#13063443	semidirect
59. D₈	#13063443	semidirect

Maximal subgroups

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

subgroup	index
#26012413	2
C₈×#13063443	2
#26011608	2
#1335	173