Subgroups — #37560403

Order 23872.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26045871	C₂	1	2	yes	maximal, complemented
2	#26045851	C₂	1	4	yes	maximal, complemented
3	C₂×#20238611	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#13093637	—	2	2	no	—
5	C₂²⋊#13093637	C₂²	1	4	yes	—
6	C₄×#13093637	—	2	2	no	—
7	#20238611	—	2	2	no	—
8	#20238611	—	2	2	no	—
9	C₂×D₁₄₉₂	C₄	1	1	yes	characteristic, complemented
10	C₂×D₁₄₉₂	C₂²	1	2	yes	—
11	C₂×C₄×D₃₇₃	C₄	1	2	yes	complemented
12	C₂×C₄×D₃₇₃	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₃₇₃	C₄	1	1	yes	characteristic, complemented
14	C₂×#13093637	—	2	4	no	—
15	C₂²×D₃₇₃	C₂×C₄	1	2	yes	—
16	C₂²×D₃₇₃	C₂³	1	1	yes	characteristic
17	D₁₄₉₂	—	2	4	no	—
18	D₁₄₉₂	—	2	2	no	—
19	C₂×Dic₃₇₃	C₂×C₄	1	1	yes	characteristic
20	C₄×D₃₇₃	—	2	2	no	—
21	C₄×D₃₇₃	—	2	2	no	—
22	C₄×D₃₇₃	D₄	1	2	yes	—
23	C₄×D₃₇₃	D₄	1	2	yes	—
24	Q₈×C₃₇₃	—	2	2	no	—
25	C₂×C₄×C₃₇₃	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₃₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃₇₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×D₃₇₃	—	2	2	no	—
29	C₂×D₃₇₃	—	4	1	no	—
30	C₂×D₃₇₃	—	4	2	no	—
31	C₂×D₃₇₃	C₂×D₄	1	2	yes	—
32	#13093637	—	4	4	no	—
33	C₄×C₃₇₃	—	2	2	no	—
34	C₄×C₃₇₃	C₂²⋊C₄	1	2	yes	—
35	Dic₃₇₃	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₃₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₃₇₃	—	2	1	no	—
38	C₂×C₃₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	D₃₇₃	—	2	2	no	—
40	D₃₇₃	—	4	2	no	—
41	C₃₇₃	#410	1	1	yes	characteristic, complemented, Sylow
42	#410	—	373	1	no	maximal, Sylow
43	C₂×#41	—	373	1	no	—
44	#117	—	373	2	no	—
45	#99	—	373	4	no	—
46	#41	—	746	2	no	—
47	#41	—	746	2	no	—
48	C₂×Q₈	#13093637	1	1	yes	characteristic, complemented
49	C₂×D₄	—	373	2	no	—
50	C₂×D₄	—	373	1	no	—
51	C₂²×C₄	—	373	1	no	—
52	C₂²×C₄	—	373	2	no	—
53	C₄⋊C₄	—	746	2	no	—
54	C₂²⋊C₄	—	373	4	no	—
55	C₄²	—	746	2	no	—
56	C₂³	—	373	1	no	—
57	C₂³	—	373	2	no	—
58	Q₈	—	2	2	no	—
59	D₄	—	746	4	no	—
60	D₄	—	746	2	no	—
61	C₂×C₄	—	373	1	no	—
62	C₂×C₄	—	373	2	no	—
63	C₂×C₄	—	373	2	no	—
64	C₂×C₄	—	746	2	no	—
65	C₂×C₄	—	746	2	no	—
66	C₂×C₄	—	746	4	no	—
67	C₂×C₄	C₂×#13093637	1	2	yes	—
68	C₂×C₄	C₂×#13093637	1	1	yes	characteristic
69	C₂²	—	373	2	no	—
70	C₂²	—	1492	2	no	—
71	C₂²	—	1492	1	no	—
72	C₂²	—	746	2	no	—
73	C₂²	C₂²×#13093637	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	373	1	no	—
76	C₄	—	373	1	no	—
77	C₄	—	1492	4	no	—
78	C₄	C₂²⋊#13093637	1	2	yes	—
79	C₂	—	2	1	no	—
80	C₂	—	1492	2	no	—
81	C₂	—	746	2	no	—
82	C₂	C₂×C₂²⋊#13093637	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#13093637
commutator	C₂²×C₃₇₃	C₂²×C₄
Frattini	C₂²	C₂²×#13093637
Fitting	C₂×Q₈×C₃₇₃	C₄
socle	C₂×C₃₇₃	C₂×C₂²⋊C₄
prime radical	C₂×#20238611	C₂
cosocle	C₂²×D₃₇₃	C₂³
hypercenter	C₂×Q₈	#13093637
hypocenter	C₃₇₃	#410
LCS	G ▷ C₂²×C₃₇₃ ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #26045871	C₂	semidirect
2. #26045851	C₂	semidirect
9. C₂×D₁₄₉₂	C₄	semidirect
11. C₂×C₄×D₃₇₃	C₄	semidirect
13. C₂×Q₈×C₃₇₃	C₄	semidirect
41. C₃₇₃	#410	semidirect
48. C₂×Q₈	#13093637	semidirect

Maximal subgroups

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

subgroup	index
#26045871	2
#26045851	2
C₂×#20238611	2
#410	373