Group #37560087 — subgroups

Subgroups — #37560087

Order 23872.

280 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₃₇₃	C₂	1	2	yes	maximal, complemented
2	#26045791	C₂	1	2	yes	maximal, complemented
3	#26045770	C₂	1	4	yes	maximal, complemented
4	#26045768	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₇₄₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₁₄₉₂	C₂	1	1	yes	maximal, characteristic
7	#26045754	C₂	1	2	yes	maximal
8	C₄⋊Q₈×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₇₄₆	C₂²	1	4	yes	—
10	C₂×Dic₇₄₆	C₂²	1	1	yes	characteristic
11	Q₈×D₃₇₃	—	2	8	no	—
12	C₂×D₁₄₉₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₃₇₃	C₂²	1	4	yes	—
14	C₂×C₄×D₃₇₃	C₂²	1	2	yes	—
15	C₄⋊Dic₃₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₃₇₃	C₂²	1	1	yes	characteristic
17	#20238589	C₂²	1	2	yes	—
18	#20238589	C₂²	1	4	yes	—
19	#20238587	C₂²	1	4	yes	—
20	#20238587	C₂²	1	2	yes	—
21	C₄×Dic₃₇₃	C₂²	1	2	yes	—
22	C₂×Q₈×C₃₇₃	C₂²	1	2	yes	—
23	C₄⋊C₄×C₃₇₃	C₂²	1	2	yes	—
24	C₄⋊C₄×C₃₇₃	C₂²	1	2	yes	—
25	C₄²×C₃₇₃	C₂²	1	1	yes	characteristic
26	C₂²×D₃₇₃	C₂³	1	2	yes	—
27	Dic₇₄₆	—	2	8	no	—
28	Dic₇₄₆	D₄	1	4	yes	complemented
29	D₁₄₉₂	Q₈	1	4	yes	complemented
30	C₂×Dic₃₇₃	C₂³	1	2	yes	—
31	C₂×Dic₃₇₃	C₂³	1	2	yes	—
32	C₂×Dic₃₇₃	C₂³	1	2	yes	—
33	C₄×D₃₇₃	—	2	4	no	—
34	C₄×D₃₇₃	—	2	8	no	—
35	Q₈×C₃₇₃	—	2	4	no	—
36	C₂×C₄×C₃₇₃	C₂³	1	4	yes	—
37	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
40	C₂²×C₃₇₃	C₂⁴	1	1	yes	characteristic
41	C₂×D₃₇₃	—	2	2	no	—
42	C₂×D₃₇₃	—	2	2	no	—
43	C₂×D₃₇₃	C₂×Q₈	1	4	yes	complemented
44	C₄×C₃₇₃	—	2	4	no	—
45	C₄×C₃₇₃	—	2	1	no	—
46	C₄×C₃₇₃	C₂×D₄	1	2	yes	—
47	C₄×C₃₇₃	C₂×Q₈	1	2	yes	complemented
48	Dic₃₇₃	—	2	2	no	—
49	Dic₃₇₃	—	2	2	no	—
50	Dic₃₇₃	C₂×D₄	1	4	yes	—
51	C₂×C₃₇₃	C₂²×D₄	1	1	yes	characteristic
52	C₂×C₃₇₃	C₂²×Q₈	1	1	yes	characteristic
53	C₂×C₃₇₃	2₋¹⁺⁴	1	1	yes	characteristic
54	D₃₇₃	—	2	4	no	—
55	C₃₇₃	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
56	D₄×Q₈	—	373	1	no	maximal, Sylow
57	C₂²×Q₈	—	373	2	no	—
58	C₄⋊Q₈	—	373	2	no	—
59	C₄⋊Q₈	D₃₇₃	1	1	yes	characteristic, complemented
60	C₂²⋊Q₈	—	373	2	no	—
61	C₂²⋊Q₈	—	373	4	no	—
62	C₄×Q₈	—	373	1	no	—
63	C₄×D₄	—	373	2	no	—
64	C₄×D₄	—	373	1	no	—
65	C₂×Q₈	—	373	1	no	—
66	C₂×Q₈	—	373	4	no	—
67	C₂×Q₈	—	746	8	no	—
68	C₂×Q₈	C₂×D₃₇₃	1	2	yes	—
69	C₂×D₄	—	373	1	no	—
70	C₂²×C₄	—	373	4	no	—
71	C₂²×C₄	—	373	2	no	—
72	C₄⋊C₄	—	373	1	no	—
73	C₄⋊C₄	—	373	2	no	—
74	C₄⋊C₄	—	373	1	no	—
75	C₄⋊C₄	—	373	4	no	—
76	C₄⋊C₄	C₂×D₃₇₃	1	2	yes	—
77	C₄⋊C₄	C₂×D₃₇₃	1	2	yes	—
78	C₂²⋊C₄	—	373	4	no	—
79	C₂²⋊C₄	—	373	2	no	—
80	C₄²	—	373	2	no	—
81	C₄²	C₂×D₃₇₃	1	1	yes	characteristic
82	C₂³	—	373	2	no	—
83	Q₈	—	2	4	no	—
84	Q₈	—	373	4	no	—
85	Q₈	—	746	8	no	—
86	D₄	—	373	4	no	—
87	C₂×C₄	—	373	2	no	—
88	C₂×C₄	—	373	2	no	—
89	C₂×C₄	—	373	2	no	—
90	C₂×C₄	—	746	8	no	—
91	C₂×C₄	—	746	4	no	—
92	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₃₇₃	1	4	yes	—
94	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
96	C₂²	—	373	4	no	—
97	C₂²	—	746	2	no	—
98	C₂²	—	746	2	no	—
99	C₂²	C₂³×D₃₇₃	1	1	yes	characteristic
100	C₄	—	2	4	no	—
101	C₄	—	2	1	no	—
102	C₄	—	373	4	no	—
103	C₄	—	746	2	no	—
104	C₄	—	746	2	no	—
105	C₄	D₄×D₃₇₃	1	2	yes	—
106	C₄	Q₈×D₃₇₃	1	2	yes	complemented
107	C₂	—	746	4	no	—
108	C₂	C₂×D₄×D₃₇₃	1	1	yes	characteristic
109	C₂	C₂×Q₈×D₃₇₃	1	1	yes	characteristic
110	C₂	#26045840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₃₇₃

C₂

yes

maximal, complemented

#26045791

C₂

yes

maximal, complemented

#26045770

C₂

yes

maximal, complemented

#26045768

C₂

yes

maximal, complemented

C₄×Dic₇₄₆

C₂

yes

maximal, characteristic, complemented

C₄×D₁₄₉₂

C₂

yes

maximal, characteristic

#26045754

C₂

yes

maximal

C₄⋊Q₈×C₃₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₄₆

C₂²

yes

—

C₂×Dic₇₄₆

C₂²

yes

characteristic

Q₈×D₃₇₃

—

C₂×D₁₄₉₂

C₂²

yes

characteristic

C₂×C₄×D₃₇₃

C₂²

yes

—

C₂×C₄×D₃₇₃

C₂²

yes

—

C₄⋊Dic₃₇₃

C₂²

yes

characteristic

C₄⋊Dic₃₇₃

C₂²

yes

characteristic

#20238589

C₂²

yes

—

#20238589

C₂²

yes

—

#20238587

C₂²

yes

—

#20238587

C₂²

yes

—

C₄×Dic₃₇₃

C₂²

yes

—

C₂×Q₈×C₃₇₃

C₂²

yes

—

C₄⋊C₄×C₃₇₃

C₂²

yes

—

C₄⋊C₄×C₃₇₃

C₂²

yes

—

C₄²×C₃₇₃

C₂²

yes

characteristic

C₂²×D₃₇₃

C₂³

yes

—

Dic₇₄₆

—

Dic₇₄₆

D₄

yes

complemented

D₁₄₉₂

Q₈

yes

complemented

C₂×Dic₃₇₃

C₂³

yes

—

C₂×Dic₃₇₃

C₂³

yes

—

C₂×Dic₃₇₃

C₂³

yes

—

C₄×D₃₇₃

—

C₄×D₃₇₃

—

Q₈×C₃₇₃

—

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂×C₄×C₃₇₃

C₂³

yes

characteristic

C₂×C₄×C₃₇₃

C₂³

yes

characteristic

C₂×C₄×C₃₇₃

C₂³

yes

characteristic

C₂²×C₃₇₃

C₂⁴

yes

characteristic

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

C₂×Q₈

yes

complemented

C₄×C₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

C₂×D₄

yes

—

C₄×C₃₇₃

C₂×Q₈

yes

complemented

Dic₃₇₃

—

Dic₃₇₃

—

Dic₃₇₃

C₂×D₄

yes

—

C₂×C₃₇₃

C₂²×D₄

yes

characteristic

C₂×C₃₇₃

C₂²×Q₈

yes

characteristic

C₂×C₃₇₃

2₋¹⁺⁴

yes

characteristic

D₃₇₃

—

C₃₇₃

D₄×Q₈

yes

characteristic, complemented, Sylow

D₄×Q₈

—

373

maximal, Sylow

C₂²×Q₈

—

373

—

C₄⋊Q₈

—

373

—

C₄⋊Q₈

D₃₇₃

yes

characteristic, complemented

C₂²⋊Q₈

—

373

—

C₂²⋊Q₈

—

373

—

C₄×Q₈

—

373

—

C₄×D₄

—

373

—

C₄×D₄

—

373

—

C₂×Q₈

—

373

—

C₂×Q₈

—

373

—

C₂×Q₈

—

746

—

C₂×Q₈

C₂×D₃₇₃

yes

—

C₂×D₄

—

373

—

C₂²×C₄

—

373

—

C₂²×C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

C₂×D₃₇₃

yes

—

C₄⋊C₄

C₂×D₃₇₃

yes

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₄²

—

373

—

C₄²

C₂×D₃₇₃

yes

characteristic

C₂³

—

373

—

Q₈

—

Q₈

—

373

—

Q₈

—

746

—

D₄

—

373

—

C₂×C₄

—

373

—

C₂×C₄

—

373

—

C₂×C₄

—

373

—

C₂×C₄

—

746

—

C₂×C₄

—

746

—

C₂×C₄

C₂²×D₃₇₃

yes

characteristic

C₂×C₄

C₂²×D₃₇₃

yes

—

C₂×C₄

C₂²×D₃₇₃

yes

characteristic

C₂×C₄

C₂²×D₃₇₃

yes

characteristic

C₂²

—

373

—

C₂²

—

746

—

C₂²

—

746

—

C₂²

C₂³×D₃₇₃

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

373

—

103

C₄

—

746

—

104

C₄

—

746

—

105

C₄

D₄×D₃₇₃

yes

—

106

C₄

Q₈×D₃₇₃

yes

complemented

107

C₂

—

746

—

108

C₂

C₂×D₄×D₃₇₃

yes

characteristic

109

C₂

C₂×Q₈×D₃₇₃

yes

characteristic

110

C₂

#26045840

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇₃
commutator, socle, cosocle	C₂²×C₃₇₃	C₂⁴
Fitting	C₄⋊Q₈×C₃₇₃	C₂
prime radical	C₂×D₁₄₉₂	C₂²
hypercenter	C₄⋊Q₈	D₃₇₃
hypocenter	C₃₇₃	D₄×Q₈
LCS	G ▷ C₂²×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇₃

commutator, socle, cosocle

C₂²×C₃₇₃

C₂⁴

Fitting

C₄⋊Q₈×C₃₇₃

C₂

prime radical

C₂×D₁₄₉₂

C₂²

hypercenter

C₄⋊Q₈

D₃₇₃

hypocenter

C₃₇₃

D₄×Q₈

LCS

G ▷ C₂²×C₃₇₃ ▷ C₃₇₃

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₃₇₃	C₂	semidirect
2. #26045791	C₂	semidirect
3. #26045770	C₂	semidirect
4. #26045768	C₂	semidirect
5. C₄×Dic₇₄₆	C₂	semidirect
8. C₄⋊Q₈×C₃₇₃	C₂	semidirect
28. Dic₇₄₆	D₄	semidirect
29. D₁₄₉₂	Q₈	semidirect
43. C₂×D₃₇₃	C₂×Q₈	semidirect
47. C₄×C₃₇₃	C₂×Q₈	semidirect
55. C₃₇₃	D₄×Q₈	semidirect
59. C₄⋊Q₈	D₃₇₃	semidirect
106. C₄	Q₈×D₃₇₃	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₃₇₃

C₂

semidirect

2. #26045791

C₂

semidirect

3. #26045770

C₂

semidirect

4. #26045768

C₂

semidirect

5. C₄×Dic₇₄₆

C₂

semidirect

8. C₄⋊Q₈×C₃₇₃

C₂

semidirect

28. Dic₇₄₆

D₄

semidirect

29. D₁₄₉₂

Q₈

semidirect

43. C₂×D₃₇₃

C₂×Q₈

semidirect

47. C₄×C₃₇₃

C₂×Q₈

semidirect

55. C₃₇₃

D₄×Q₈

semidirect

59. C₄⋊Q₈

D₃₇₃

semidirect

106. C₄

Q₈×D₃₇₃

semidirect

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

subgroup	index
C₂×Q₈×D₃₇₃	2
#26045791	2
#26045770	2
#26045768	2
C₄×Dic₇₄₆	2
C₄×D₁₄₉₂	2
#26045754	2
C₄⋊Q₈×C₃₇₃	2
D₄×Q₈	373

subgroup

index

C₂×Q₈×D₃₇₃

#26045791

#26045770

#26045768

C₄×Dic₇₄₆

C₄×D₁₄₉₂

#26045754

C₄⋊Q₈×C₃₇₃

D₄×Q₈

373

Subgroups — #37560087

Conjugacy classes of subgroups

Complements

Maximal subgroups