Subgroups — D₄×#15688412 #37559488

Order 23872.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15688412	C₂	1	2	yes	maximal, complemented
2	C₄⋊#15688412	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#15688412	C₂	1	2	yes	maximal, complemented
4	C₄×#15688412	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₄×C₃₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#15688412	—	2	4	no	—
7	C₂×#15688412	C₂²	1	1	yes	characteristic
8	C₂×#15688412	C₂²	1	2	yes	—
9	C₂×#15688412	C₂²	1	1	yes	characteristic
10	C₂×D₄×C₃₇₃	C₄	1	1	yes	characteristic
11	C₂²×C₄×C₃₇₃	C₂²	1	2	yes	—
12	C₄⋊C₄×C₃₇₃	C₄	1	1	yes	characteristic
13	C₂²⋊C₄×C₃₇₃	C₄	1	2	yes	—
14	C₄²×C₃₇₃	C₂²	1	1	yes	characteristic
15	#15688412	—	2	2	no	—
16	#15688412	—	2	1	no	—
17	#15688412	D₄	1	2	yes	direct product factor
18	C₂³×C₃₇₃	C₂×C₄	1	2	yes	—
19	D₄×C₃₇₃	C₈	1	4	yes	complemented
20	C₂×C₄×C₃₇₃	—	2	2	no	—
21	C₂×C₄×C₃₇₃	—	2	2	no	—
22	C₂×C₄×C₃₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₃₇₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₃₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
26	C₂²×C₃₇₃	—	2	2	no	—
27	C₂²×C₃₇₃	—	2	2	no	—
28	C₂²×C₃₇₃	C₂×C₈	1	4	yes	complemented
29	C₂²×C₃₇₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₃₇₃	—	2	2	no	—
31	C₄×C₃₇₃	—	2	1	no	—
32	C₄×C₃₇₃	C₂×C₈	1	2	yes	complemented
33	C₄×C₃₇₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₃₇₃	#41	1	1	yes	characteristic
35	C₂×C₃₇₃	—	2	4	no	—
36	C₂×C₃₇₃	C₄×D₄	1	1	yes	characteristic
37	C₂×C₃₇₃	C₂²×C₈	1	1	yes	characteristic
38	C₂×C₃₇₃	#131	1	1	yes	characteristic
39	C₃₇₃	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
40	C₈×D₄	—	373	1	no	maximal, Sylow
41	C₂²×C₈	—	373	2	no	—
42	C₄×D₄	D₃₇₃	1	1	yes	characteristic
43	C₄⋊C₈	—	373	1	no	—
44	C₂²⋊C₈	—	373	2	no	—
45	C₄×C₈	—	373	1	no	—
46	C₂×D₄	Dic₃₇₃	1	1	yes	characteristic
47	C₂²×C₄	C₂×D₃₇₃	1	2	yes	—
48	C₂×C₈	—	373	1	no	—
49	C₂×C₈	—	373	1	no	—
50	C₂×C₈	—	373	2	no	—
51	C₂×C₈	—	746	4	no	—
52	C₄⋊C₄	Dic₃₇₃	1	1	yes	characteristic
53	C₂²⋊C₄	Dic₃₇₃	1	2	yes	—
54	C₄²	C₂×D₃₇₃	1	1	yes	characteristic
55	C₂³	C₂×Dic₃₇₃	1	2	yes	—
56	D₄	#15688412	1	4	yes	direct product factor
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	C₂×Dic₃₇₃	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₃₇₃	1	2	yes	—
61	C₂×C₄	C₂×Dic₃₇₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
63	C₈	—	373	2	no	—
64	C₈	—	746	2	no	—
65	C₈	—	746	1	no	—
66	C₂²	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂×#15688412	1	4	yes	complemented
69	C₂²	C₂²×Dic₃₇₃	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	C₂×#15688412	1	2	yes	complemented
73	C₄	D₄×D₃₇₃	1	1	yes	characteristic
74	C₄	#20238609	1	1	yes	characteristic
75	C₂	—	2	4	no	—
76	C₂	D₄×Dic₃₇₃	1	1	yes	characteristic
77	C₂	C₂²×#15688412	1	1	yes	characteristic
78	C₂	#26045799	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₇₃
commutator	C₂×C₃₇₃	C₂²×C₈
Fitting	C₄×D₄×C₃₇₃	C₂
socle	C₂²×C₃₇₃	C₂²×C₄
prime radical	C₂×D₄×C₃₇₃	C₄
cosocle	C₂×C₄×C₃₇₃	C₂³
hypercenter	C₄×D₄	D₃₇₃
hypocenter	C₃₇₃	C₈×D₄
LCS	G ▷ C₂×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#15688412	C₂	semidirect
2. C₄⋊#15688412	C₂	semidirect
3. C₂²⋊#15688412	C₂	semidirect
4. C₄×#15688412	C₂	semidirect
17. #15688412	D₄	direct (N × H)
19. D₄×C₃₇₃	C₈	semidirect
28. C₂²×C₃₇₃	C₂×C₈	semidirect
32. C₄×C₃₇₃	C₂×C₈	semidirect
39. C₃₇₃	C₈×D₄	semidirect
56. D₄	#15688412	direct (N × H)
68. C₂²	C₂×#15688412	semidirect
72. C₄	C₂×#15688412	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15688412	2
C₄⋊#15688412	2
C₂²⋊#15688412	2
C₄×#15688412	2
C₄×D₄×C₃₇₃	2
C₈×D₄	373

Subgroups — D4×#15688412 #37559488

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#15688412 #37559488