Subgroups — #36351599

Order 22144.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26011868	C₂	1	1	yes	maximal, characteristic
2	#26011312	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₃₈₄	—	2	1	no	—
5	C₂×D₁₃₈₄	—	2	1	no	—
6	#19149676	C₄	1	2	yes	—
7	#19149619	—	2	1	no	—
8	M₃₂×C₁₇₃	—	2	1	no	—
9	C₄×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
10	C₂×D₆₉₂	—	2	2	no	—
11	C₂×D₆₉₂	—	4	2	no	—
12	D₁₃₈₄	—	4	1	no	—
13	D₁₃₈₄	—	4	1	no	—
14	D₁₃₈₄	—	4	1	no	—
15	D₁₃₈₄	—	4	1	no	—
16	#14583801	—	4	1	no	—
17	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
18	C₂×C₈×C₁₇₃	D₄	1	1	yes	characteristic
19	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₇₃	—	4	1	no	—
21	C₂²×D₁₇₃	—	4	2	no	—
22	D₆₉₂	—	4	2	no	—
23	D₆₉₂	—	4	2	no	—
24	D₆₉₂	—	8	4	no	—
25	C₂×C₄×C₁₇₃	—	2	1	no	—
26	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₈×C₁₇₃	D₈	1	1	yes	characteristic
28	C₈×C₁₇₃	D₈	1	1	yes	characteristic
29	C₈×C₁₇₃	SD₁₆	1	1	yes	characteristic
30	C₈×C₁₇₃	SD₁₆	1	1	yes	characteristic
31	C₂²×C₁₇₃	#100	1	1	yes	characteristic
32	C₂×D₁₇₃	—	8	4	no	—
33	C₂×D₁₇₃	—	8	2	no	—
34	C₄×C₁₇₃	—	2	2	no	—
35	C₄×C₁₇₃	#102	1	1	yes	characteristic
36	C₄×C₁₇₃	#102	1	1	yes	characteristic
37	C₂×C₁₇₃	—	2	1	no	—
38	C₂×C₁₇₃	#331	1	1	yes	characteristic
39	D₁₇₃	—	16	2	no	—
40	C₁₇₃	#1357	1	1	yes	characteristic, complemented, Sylow
41	#1357	—	173	1	no	maximal, Sylow
42	#493	—	173	1	no	—
43	#364	—	173	1	no	—
44	#364	D₁₇₃	1	1	yes	characteristic, complemented
45	C₂×D₈	—	346	1	no	—
46	C₂×D₈	—	346	1	no	—
47	#127	—	173	2	no	—
48	M₃₂	—	2	1	no	—
49	M₃₂	—	346	1	no	—
50	C₄×C₈	C₂×D₁₇₃	1	1	yes	characteristic
51	C₂×D₄	—	346	2	no	—
52	C₂×D₄	—	692	2	no	—
53	D₈	—	692	1	no	—
54	D₈	—	692	1	no	—
55	D₈	—	692	1	no	—
56	D₈	—	692	1	no	—
57	C₂×C₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
58	C₂×C₈	D₆₉₂	1	1	yes	characteristic
59	C₄²	C₄×D₁₇₃	1	1	yes	characteristic
60	C₁₆	—	4	1	no	—
61	C₁₆	—	692	1	no	—
62	C₂³	—	692	2	no	—
63	D₄	—	1384	4	no	—
64	D₄	—	692	2	no	—
65	D₄	—	692	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	#14583803	1	1	yes	characteristic
68	C₈	#14583811	1	1	yes	characteristic
69	C₈	D₁₃₈₄	1	1	yes	characteristic
70	C₈	#14583813	1	1	yes	characteristic
71	C₈	#14583815	1	1	yes	characteristic
72	C₂²	—	1384	4	no	—
73	C₂²	—	1384	2	no	—
74	C₂²	#19149595	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	#19149599	1	1	yes	characteristic
77	C₄	#19149601	1	1	yes	characteristic
78	C₂	—	2	1	no	—
79	C₂	—	2768	2	no	—
80	C₂	#26011234	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26011234
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₁₇₃
Fitting	#364×C₁₇₃	C₂
socle	C₂×C₁₇₃	#331
prime radical	#26011868	C₂
cosocle	C₄×C₈×C₁₇₃	C₂²
hypercenter	#364	D₁₇₃
hypocenter	C₁₇₃	#1357
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #26011312	C₂	semidirect
3. #364×C₁₇₃	C₂	semidirect
40. C₁₇₃	#1357	semidirect
44. #364	D₁₇₃	semidirect

Maximal subgroups

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

subgroup	index
#26011868	2
#26011312	2
#364×C₁₇₃	2
#1357	173