Subgroups — #37560464

Order 23872.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26045879	C₂	1	2	yes	maximal, complemented
2	#26045863	C₂	1	1	yes	maximal, characteristic, complemented
3	#26045856	C₂	1	2	yes	maximal, complemented
4	C₈×#13093637	C₂	1	1	yes	maximal, characteristic, complemented
5	#26045811	C₂	1	1	yes	maximal, characteristic
6	#20238620	C₂²	1	2	yes	—
7	#20238619	—	2	2	no	—
8	D₃₇₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#15688419	—	2	2	no	—
10	C₄×#13093637	C₂²	1	1	yes	characteristic
11	#20238609	C₂²	1	2	yes	—
12	Dic₁₄₉₂	C₄	1	1	yes	characteristic, complemented
13	#20238598	C₄	1	2	yes	complemented
14	C₈×D₃₇₃	C₂²	1	1	yes	characteristic
15	D₈×C₃₇₃	C₄	1	1	yes	characteristic, complemented
16	C₂×#13093637	—	2	1	no	—
17	#15688419	—	2	2	no	—
18	#15688419	D₄	1	1	yes	characteristic
19	#15688419	D₄	1	1	yes	characteristic
20	Dic₇₄₆	C₂×C₄	1	2	yes	—
21	C₂²⋊D₃₇₃	—	2	2	no	—
22	C₂×Dic₃₇₃	—	2	2	no	—
23	C₄×D₃₇₃	C₂³	1	1	yes	characteristic
24	#15688412	C₂×C₄	1	1	yes	characteristic
25	D₄×C₃₇₃	C₂×C₄	1	2	yes	—
26	C₈×C₃₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃₇₃	—	2	2	no	—
28	C₂×D₃₇₃	#41	1	1	yes	characteristic
29	#13093637	—	2	2	no	—
30	C₄×C₃₇₃	C₂²×C₄	1	1	yes	characteristic
31	Dic₃₇₃	—	2	2	no	—
32	Dic₃₇₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₃₇₃	—	4	2	no	—
34	C₂×C₃₇₃	C₄×D₄	1	1	yes	characteristic
35	D₃₇₃	—	2	1	no	—
36	C₃₇₃	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	373	1	no	maximal, Sylow
38	#135	—	373	1	no	—
39	#131	—	373	2	no	—
40	#108	—	373	1	no	—
41	#104	—	373	2	no	—
42	C₄×C₈	—	373	1	no	—
43	#41	—	373	2	no	—
44	Q₁₆	—	373	1	no	—
45	SD₁₆	—	373	2	no	—
46	D₈	#13093637	1	1	yes	characteristic, complemented
47	M₁₆	—	373	2	no	—
48	M₁₆	—	746	2	no	—
49	C₂×C₈	—	373	1	no	—
50	C₂×C₈	—	373	1	no	—
51	C₂×C₈	—	746	2	no	—
52	C₄²	—	373	1	no	—
53	Q₈	—	373	2	no	—
54	D₄	—	746	2	no	—
55	D₄	C₂×#13093637	1	2	yes	—
56	C₂×C₄	—	373	1	no	—
57	C₂×C₄	—	746	2	no	—
58	C₂×C₄	—	746	1	no	—
59	C₈	—	373	1	no	—
60	C₈	—	373	1	no	—
61	C₈	—	373	1	no	—
62	C₈	—	746	2	no	—
63	C₈	C₂×#13093637	1	1	yes	characteristic
64	C₂²	—	2	2	no	—
65	C₂²	—	373	1	no	—
66	C₄	—	373	1	no	—
67	C₄	—	746	2	no	—
68	C₄	—	746	2	no	—
69	C₄	C₂²×#13093637	1	1	yes	characteristic
70	C₂	—	4	2	no	—
71	C₂	—	746	1	no	—
72	C₂	D₄×#13093637	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#13093637
commutator	C₄×C₃₇₃	C₂²×C₄
Frattini	C₄	C₂²×#13093637
Fitting	D₈×C₃₇₃	C₄
socle	C₂×C₃₇₃	C₄×D₄
prime radical	#26045811	C₂
cosocle	C₄×D₃₇₃	C₂³
hypercenter	D₈	#13093637
hypocenter	C₃₇₃	#443
LCS	G ▷ C₄×C₃₇₃ ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #26045879	C₂	semidirect
2. #26045863	C₂	semidirect
3. #26045856	C₂	semidirect
4. C₈×#13093637	C₂	semidirect
12. Dic₁₄₉₂	C₄	semidirect
13. #20238598	C₄	semidirect
15. D₈×C₃₇₃	C₄	semidirect
36. C₃₇₃	#443	semidirect
46. D₈	#13093637	semidirect

Maximal subgroups

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

subgroup	index
#26045879	2
#26045863	2
#26045856	2
C₈×#13093637	2
#26045811	2
#443	373