Subgroups — #36368709

Order 22144.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26012400	C₂	1	1	yes	maximal, characteristic
2	#26012398	C₂	1	1	yes	maximal, characteristic
3	#26011848	C₂	1	1	yes	maximal, characteristic
4	#19149740	—	2	1	no	—
5	#19149738	—	2	1	no	—
6	#19149667	C₂²	1	1	yes	characteristic
7	#19149602	—	2	1	no	—
8	#19149599	—	2	1	no	—
9	C₄×#13063436	C₄	1	1	yes	characteristic, complemented
10	#125×C₁₇₃	C₄	1	1	yes	characteristic, complemented
11	#14583835	—	4	1	no	—
12	C₂²⋊#10621924	—	4	1	no	—
13	C₂×Dic₃₄₆	D₄	1	1	yes	characteristic
14	C₂×D₆₉₂	D₄	1	1	yes	characteristic
15	C₂×#13063436	—	2	1	no	—
16	#14583803	—	4	1	no	—
17	C₄⋊C₄×C₁₇₃	—	2	1	no	—
18	C₄⋊C₄×C₁₇₃	—	4	1	no	—
19	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×#10621924	—	8	1	no	—
21	#13063443	—	8	1	no	—
22	C₂²×D₁₇₃	—	2	1	no	—
23	Dic₃₄₆	—	4	1	no	—
24	D₆₉₂	—	4	1	no	—
25	C₂×Dic₁₇₃	—	2	1	no	—
26	#13063436	—	4	1	no	—
27	C₂×C₄×C₁₇₃	—	2	1	no	—
28	C₂×C₄×C₁₇₃	—	4	1	no	—
29	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₁₇₃	#99	1	1	yes	characteristic
31	C₂×D₁₇₃	—	4	1	no	—
32	C₂×D₁₇₃	—	8	1	no	—
33	#10621924	—	16	1	no	—
34	C₄×C₁₇₃	—	2	1	no	—
35	C₄×C₁₇₃	—	4	1	no	—
36	C₄×C₁₇₃	—	8	1	no	—
37	Dic₁₇₃	—	4	1	no	—
38	C₂×C₁₇₃	—	2	1	no	—
39	C₂×C₁₇₃	#354	1	1	yes	characteristic
40	D₁₇₃	—	8	1	no	—
41	C₁₇₃	#1414	1	1	yes	characteristic, complemented, Sylow
42	#1414	—	173	1	no	maximal, Sylow
43	#488	—	173	1	no	—
44	#355	—	173	1	no	—
45	#354	—	173	1	no	—
46	#125	#10621924	1	1	yes	characteristic, complemented
47	#124	—	173	1	no	—
48	#103	—	346	1	no	—
49	#102	—	346	1	no	—
50	#101	—	346	1	no	—
51	#99	—	346	1	no	—
52	C₄×C₈	—	173	1	no	—
53	C₂×Q₈	—	173	1	no	—
54	C₂×D₄	—	173	1	no	—
55	M₁₆	—	692	1	no	—
56	C₂×C₈	—	346	1	no	—
57	C₄⋊C₄	—	2	1	no	—
58	C₄⋊C₄	—	4	1	no	—
59	C₂²⋊C₄	—	692	1	no	—
60	C₂²⋊C₄	—	692	1	no	—
61	C₄²	C₂×#10621924	1	1	yes	characteristic
62	C₂³	—	346	1	no	—
63	Q₈	—	692	1	no	—
64	D₄	—	692	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	346	1	no	—
68	C₂×C₄	—	1384	1	no	—
69	C₂×C₄	C₂²⋊#10621924	1	1	yes	characteristic
70	C₈	—	1384	1	no	—
71	C₈	—	692	1	no	—
72	C₂²	—	1384	1	no	—
73	C₂²	—	692	1	no	—
74	C₂²	#19149737	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	8	1	no	—
78	C₄	—	692	1	no	—
79	C₄	—	2768	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	1384	1	no	—
82	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	C₂×D₆₉₂	D₄
cosocle	#19149667	C₂²
hypercenter	#125	#10621924
hypocenter	C₁₇₃	#1414
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
9. C₄×#13063436	C₄	semidirect
10. #125×C₁₇₃	C₄	semidirect
41. C₁₇₃	#1414	semidirect
46. #125	#10621924	semidirect

Maximal subgroups

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

subgroup	index
#26012400	2
#26012398	2
#26011848	2
#1414	173