Subgroups — #37560048

Order 23872.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26045771	C₂	1	4	yes	maximal, complemented
2	#26045769	C₂	1	8	yes	maximal, complemented
3	C₂×C₂²⋊Dic₃₇₃	C₂	1	2	yes	maximal
4	C₂×C₂²⋊C₄×C₃₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₇₄₆	C₂²	1	4	yes	complemented
6	C₂²×Dic₃₇₃	C₂²	1	4	yes	—
7	C₄⋊Dic₃₇₃	C₂²	1	4	yes	complemented
8	#20238589	C₂²	1	8	yes	complemented
9	C₂²⋊Dic₃₇₃	C₂²	1	4	yes	—
10	C₂²⋊Dic₃₇₃	C₂²	1	4	yes	—
11	C₂⁴×C₃₇₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₃₇₃	C₂²	1	2	yes	—
13	C₂²⋊C₄×C₃₇₃	C₂²	1	4	yes	—
14	Dic₇₄₆	—	4	4	no	—
15	C₂×Dic₃₇₃	—	4	4	no	—
16	C₂×Dic₃₇₃	C₂³	1	8	yes	—
17	C₂³×C₃₇₃	—	4	2	no	—
18	C₂³×C₃₇₃	Q₈	1	4	yes	complemented
19	C₂³×C₃₇₃	C₂³	1	2	yes	—
20	C₂³×C₃₇₃	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₃₇₃	—	4	2	no	—
22	C₂×C₄×C₃₇₃	C₂³	1	4	yes	—
23	C₂²×C₃₇₃	—	2	2	no	—
24	C₂²×C₃₇₃	—	2	4	no	—
25	C₂²×C₃₇₃	—	4	4	no	—
26	C₂²×C₃₇₃	C₂×Q₈	1	4	yes	—
27	C₂²×C₃₇₃	C₂×Q₈	1	2	yes	—
28	C₂²×C₃₇₃	C₂⁴	1	1	yes	characteristic
29	C₄×C₃₇₃	—	2	4	no	—
30	Dic₃₇₃	—	2	8	no	—
31	C₂×C₃₇₃	—	2	2	no	—
32	C₂×C₃₇₃	—	2	4	no	—
33	C₂×C₃₇₃	C₂²×Q₈	1	1	yes	characteristic
34	C₂×C₃₇₃	H(5,𝔽₂)	1	2	yes	—
35	C₃₇₃	#543	1	1	yes	characteristic, complemented, Sylow
36	#543	—	373	1	no	maximal, Sylow
37	C₂²⋊Q₈	—	373	4	no	—
38	C₂²⋊Q₈	—	373	8	no	—
39	C₂×C₂²⋊C₄	—	373	2	no	—
40	C₂×C₂²⋊C₄	D₃₇₃	1	1	yes	characteristic
41	C₂⁴	C₂×D₃₇₃	1	1	yes	characteristic
42	C₂×Q₈	—	373	4	no	—
43	C₂²×C₄	—	373	4	no	—
44	C₂²×C₄	C₂×D₃₇₃	1	2	yes	—
45	C₄⋊C₄	—	373	8	no	—
46	C₄⋊C₄	—	373	4	no	—
47	C₂²⋊C₄	—	373	4	no	—
48	C₂²⋊C₄	—	373	4	no	—
49	C₂²⋊C₄	C₂×D₃₇₃	1	4	yes	—
50	C₂³	—	4	2	no	—
51	C₂³	Dic₇₄₆	1	4	yes	complemented
52	C₂³	C₂²×D₃₇₃	1	1	yes	characteristic
53	C₂³	C₂²×D₃₇₃	1	2	yes	—
54	Q₈	—	1492	4	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	373	8	no	—
57	C₂×C₄	—	1492	4	no	—
58	C₂×C₄	C₂²×D₃₇₃	1	4	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	—	2	4	no	—
61	C₂²	—	4	4	no	—
62	C₂²	C₂×Dic₇₄₆	1	2	yes	—
63	C₂²	C₂×Dic₇₄₆	1	4	yes	—
64	C₂²	C₂³×D₃₇₃	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	746	8	no	—
67	C₂	—	2	2	no	—
68	C₂	—	2	4	no	—
69	C₂	C₂²×Dic₇₄₆	1	1	yes	characteristic
70	C₂	#26045837	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇₃
commutator, socle, cosocle	C₂²×C₃₇₃	C₂⁴
Fitting	C₂×C₂²⋊C₄×C₃₇₃	C₂
prime radical	C₂⁴×C₃₇₃	C₂²
hypercenter	C₂×C₂²⋊C₄	D₃₇₃
hypocenter	C₃₇₃	#543
LCS	G ▷ C₂²×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. #26045771	C₂	semidirect
2. #26045769	C₂	semidirect
5. C₂×Dic₇₄₆	C₂²	semidirect
7. C₄⋊Dic₃₇₃	C₂²	semidirect
8. #20238589	C₂²	semidirect
18. C₂³×C₃₇₃	Q₈	semidirect
35. C₃₇₃	#543	semidirect
51. C₂³	Dic₇₄₆	semidirect

Maximal subgroups

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

subgroup	index
#26045771	2
#26045769	2
C₂×C₂²⋊Dic₃₇₃	2
C₂×C₂²⋊C₄×C₃₇₃	2
#543	373