Subgroups — #37559659

Order 23872.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₉₈₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#26045767	C₂	1	2	yes	maximal, complemented
4	#26045712	C₂	1	2	yes	maximal, complemented
5	#107×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₄₉₂	—	2	2	no	—
7	C₂×D₁₄₉₂	C₂²	1	2	yes	—
8	C₂×C₄×D₃₇₃	C₂²	1	1	yes	characteristic
9	D₂₉₈₄	—	2	2	no	—
10	C₂×#15688412	C₂²	1	1	yes	characteristic
11	C₈×D₃₇₃	—	2	2	no	—
12	#20238587	—	2	2	no	—
13	C₂×C₈×C₃₇₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₃₇₃	C₂²	1	2	yes	—
15	C₂²×D₃₇₃	—	2	2	no	—
16	C₂²×D₃₇₃	D₄	1	1	yes	characteristic
17	D₁₄₉₂	—	2	2	no	—
18	D₁₄₉₂	—	2	2	no	—
19	D₁₄₉₂	—	4	4	no	—
20	C₂×Dic₃₇₃	D₄	1	1	yes	characteristic
21	C₄×D₃₇₃	—	2	1	no	—
22	C₄×D₃₇₃	—	2	1	no	—
23	#15688412	—	2	1	no	—
24	C₂×C₄×C₃₇₃	—	2	2	no	—
25	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
26	C₈×C₃₇₃	D₄	1	2	yes	complemented
27	C₂²×C₃₇₃	C₂×D₄	1	1	yes	characteristic
28	C₂×D₃₇₃	—	2	1	no	—
29	C₂×D₃₇₃	—	2	1	no	—
30	C₂×D₃₇₃	—	4	2	no	—
31	C₂×D₃₇₃	—	4	2	no	—
32	C₂×D₃₇₃	—	4	2	no	—
33	C₂×D₃₇₃	D₈	1	2	yes	complemented
34	C₄×C₃₇₃	—	4	2	no	—
35	C₄×C₃₇₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₃₇₃	#41	1	1	yes	characteristic
37	Dic₃₇₃	—	2	1	no	—
38	C₂×C₃₇₃	#121	1	1	yes	characteristic
39	C₂×C₃₇₃	C₂×D₈	1	1	yes	characteristic
40	C₂×C₃₇₃	#135	1	1	yes	characteristic
41	D₃₇₃	—	2	2	no	—
42	D₃₇₃	—	8	2	no	—
43	C₃₇₃	#466	1	1	yes	characteristic, complemented, Sylow
44	#466	—	373	1	no	maximal, Sylow
45	C₂×D₈	—	373	1	no	—
46	C₂²×C₈	—	373	1	no	—
47	#121	—	373	2	no	—
48	#107	D₃₇₃	1	1	yes	characteristic, complemented
49	#102	—	373	2	no	—
50	C₂×D₄	—	373	2	no	—
51	C₂×D₄	—	746	2	no	—
52	C₂²×C₄	—	373	1	no	—
53	D₈	—	746	2	no	—
54	C₂×C₈	—	373	1	no	—
55	C₂×C₈	—	746	2	no	—
56	C₂×C₈	C₂×D₃₇₃	1	1	yes	characteristic
57	C₄⋊C₄	C₂×D₃₇₃	1	2	yes	—
58	C₂²⋊C₄	—	746	2	no	—
59	C₂³	—	373	1	no	—
60	C₂³	—	746	2	no	—
61	D₄	—	1492	4	no	—
62	D₄	—	746	2	no	—
63	D₄	—	746	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	373	1	no	—
66	C₂×C₄	—	746	1	no	—
67	C₂×C₄	—	746	1	no	—
68	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
69	C₈	—	746	1	no	—
70	C₈	D₁₄₉₂	1	2	yes	complemented
71	C₂²	—	373	2	no	—
72	C₂²	—	1492	2	no	—
73	C₂²	—	1492	2	no	—
74	C₂²	—	1492	2	no	—
75	C₂²	—	746	1	no	—
76	C₂²	—	746	1	no	—
77	C₂²	D₄×D₃₇₃	1	1	yes	characteristic
78	C₄	—	4	2	no	—
79	C₄	—	746	1	no	—
80	C₄	C₂×D₁₄₉₂	1	1	yes	characteristic
81	C₄	#20238611	1	1	yes	characteristic
82	C₂	—	2984	2	no	—
83	C₂	—	746	2	no	—
84	C₂	#26045767	1	1	yes	characteristic
85	C₂	D₈×D₃₇₃	1	1	yes	characteristic
86	C₂	#26045815	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₇₃
commutator, cosocle	C₂×C₄×C₃₇₃	C₂³
Frattini	C₂×C₄	C₂²×D₃₇₃
Fitting	#107×C₃₇₃	C₂
socle	C₂²×C₃₇₃	C₂×D₄
hypercenter	#107	D₃₇₃
hypocenter	C₃₇₃	#466
LCS	G ▷ C₂×C₄×C₃₇₃ ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
1. C₂×D₂₉₈₄	C₂	semidirect
2. C₂×C₈×D₃₇₃	C₂	semidirect
3. #26045767	C₂	semidirect
4. #26045712	C₂	semidirect
5. #107×C₃₇₃	C₂	semidirect
26. C₈×C₃₇₃	D₄	semidirect
33. C₂×D₃₇₃	D₈	semidirect
43. C₃₇₃	#466	semidirect
48. #107	D₃₇₃	semidirect
70. C₈	D₁₄₉₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₉₈₄	2
C₂×C₈×D₃₇₃	2
#26045767	2
#26045712	2
#107×C₃₇₃	2
#466	373