Subgroups — #36368702

Order 22144.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012396	C₂	1	2	yes	maximal
2	#26011857	C₂	1	1	yes	maximal, characteristic
3	#19149738	—	2	2	no	—
4	#19149676	C₂²	1	1	yes	characteristic
5	#19149599	—	2	2	no	—
6	#19149590	C₄	1	1	yes	characteristic, complemented
7	#125×C₁₇₃	C₄	1	1	yes	characteristic, complemented
8	C₂²⋊#10621924	—	4	2	no	—
9	C₂×D₆₉₂	—	4	1	no	—
10	C₂×D₆₉₂	D₄	1	2	yes	—
11	C₂×#13063436	—	2	1	no	—
12	C₄⋊C₄×C₁₇₃	—	2	1	no	—
13	C₄⋊C₄×C₁₇₃	—	4	1	no	—
14	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
15	C₂×#10621924	—	8	2	no	—
16	C₂²×D₁₇₃	—	2	2	no	—
17	D₆₉₂	—	4	2	no	—
18	D₆₉₂	—	8	2	no	—
19	#13063436	—	4	1	no	—
20	C₂×C₄×C₁₇₃	—	2	1	no	—
21	C₂×C₄×C₁₇₃	—	4	1	no	—
22	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₂²×C₁₇₃	#99	1	1	yes	characteristic
24	C₂×D₁₇₃	—	4	2	no	—
25	C₂×D₁₇₃	—	8	2	no	—
26	#10621924	—	16	2	no	—
27	C₄×C₁₇₃	—	2	1	no	—
28	C₄×C₁₇₃	—	4	1	no	—
29	C₄×C₁₇₃	—	8	1	no	—
30	C₂×C₁₇₃	—	2	1	no	—
31	C₂×C₁₇₃	#354	1	1	yes	characteristic
32	D₁₇₃	—	8	2	no	—
33	C₁₇₃	#1412	1	1	yes	characteristic, complemented, Sylow
34	#1412	—	173	1	no	maximal, Sylow
35	#490	—	173	1	no	—
36	#353	—	173	2	no	—
37	#127	—	173	1	no	—
38	#125	#10621924	1	1	yes	characteristic, complemented
39	#102	—	346	2	no	—
40	#99	—	346	2	no	—
41	#97	—	173	1	no	—
42	C₂×D₄	—	173	2	no	—
43	C₂×D₄	—	692	1	no	—
44	C₂×C₈	—	346	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₂²⋊C₄	—	692	2	no	—
48	C₄²	C₂×#10621924	1	1	yes	characteristic
49	C₂³	—	346	2	no	—
50	D₄	—	1384	2	no	—
51	D₄	—	692	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	1384	2	no	—
55	C₂×C₄	C₂²⋊#10621924	1	1	yes	characteristic
56	C₈	—	692	1	no	—
57	C₂²	—	1384	2	no	—
58	C₂²	—	692	2	no	—
59	C₂²	#19149737	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	8	1	no	—
63	C₄	—	2768	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	1384	2	no	—
66	C₂	#26012397	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26012397
commutator	C₄²×C₁₇₃	C₂×C₄
Frattini	C₄²	C₂×#10621924
Fitting	#125×C₁₇₃	C₄
socle	C₂×C₁₇₃	#354
prime radical, cosocle	#19149676	C₂²
hypercenter	#125	#10621924
hypocenter	C₁₇₃	#1412
LCS	G ▷ C₄²×C₁₇₃ ▷ C₂×C₄×C₁₇₃ ▷ C₂²×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
6. #19149590	C₄	semidirect
7. #125×C₁₇₃	C₄	semidirect
33. C₁₇₃	#1412	semidirect
38. #125	#10621924	semidirect

Maximal subgroups

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

subgroup	index
#26012396	2
#26011857	2
#1412	173