Group #37560176 — subgroups

Subgroups — #37560176

Order 23872.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#26045791	C₂	1	2	yes	maximal, complemented
2	#26045787	C₂	1	2	yes	maximal, complemented
3	#26045787	C₂	1	2	yes	maximal, complemented
4	#26045784	C₂	1	1	yes	maximal, characteristic, complemented
5	#26045780	C₂	1	1	yes	maximal, characteristic
6	#26045777	C₂	1	2	yes	maximal
7	#26045772	C₂	1	2	yes	maximal, complemented
8	#26045770	C₂	1	2	yes	maximal, complemented
9	#125×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₇₄₆	C₂²	1	2	yes	—
11	C₂×Dic₇₄₆	C₂²	1	1	yes	characteristic
12	C₂×D₁₄₉₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₃₇₃	C₂²	1	2	yes	—
14	C₄⋊Dic₃₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₃₇₃	C₂²	1	2	yes	—
16	C₄⋊Dic₃₇₃	C₂²	1	1	yes	characteristic
17	#20238589	C₂²	1	2	yes	—
18	#20238589	C₂²	1	2	yes	—
19	#20238589	C₂²	1	2	yes	—
20	#20238587	C₂²	1	2	yes	—
21	#20238587	C₂²	1	2	yes	—
22	#20238587	C₂²	1	2	yes	—
23	#20238587	C₂²	1	2	yes	—
24	#20238587	C₂²	1	2	yes	—
25	C₄×Dic₃₇₃	C₂²	1	2	yes	—
26	C₄⋊C₄×C₃₇₃	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₃₇₃	C₂²	1	2	yes	—
28	C₄⋊C₄×C₃₇₃	C₂²	1	2	yes	—
29	C₄⋊C₄×C₃₇₃	C₂²	1	1	yes	characteristic
30	C₄²×C₃₇₃	C₂²	1	1	yes	characteristic
31	C₂²×D₃₇₃	C₂³	1	2	yes	—
32	Dic₇₄₆	—	4	2	no	—
33	Dic₇₄₆	—	4	1	no	—
34	D₁₄₉₂	—	4	1	no	—
35	C₂×Dic₃₇₃	C₂³	1	2	yes	—
36	C₂×Dic₃₇₃	C₂³	1	2	yes	—
37	C₂×Dic₃₇₃	C₂³	1	2	yes	—
38	C₄×D₃₇₃	—	4	2	no	—
39	C₂×C₄×C₃₇₃	C₂³	1	2	yes	—
40	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₃₇₃	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₃₇₃	C₂³	1	2	yes	—
44	C₂²×C₃₇₃	C₂⁴	1	1	yes	characteristic
45	C₂×D₃₇₃	—	2	2	no	—
46	C₂×D₃₇₃	—	2	2	no	—
47	C₂×D₃₇₃	—	2	2	no	—
48	C₄×C₃₇₃	—	2	2	no	—
49	C₄×C₃₇₃	—	2	1	no	—
50	C₄×C₃₇₃	—	2	1	no	—
51	C₄×C₃₇₃	—	2	2	no	—
52	C₄×C₃₇₃	—	2	1	no	—
53	Dic₃₇₃	—	2	2	no	—
54	Dic₃₇₃	—	2	2	no	—
55	Dic₃₇₃	—	2	2	no	—
56	C₂×C₃₇₃	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₃₇₃	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂×C₃₇₃	2₋¹⁺⁴	1	1	yes	characteristic
59	D₃₇₃	—	4	2	no	—
60	C₃₇₃	#563	1	1	yes	characteristic, complemented, Sylow
61	#563	—	373	1	no	maximal, Sylow
62	C₄⋊Q₈	—	373	2	no	—
63	#126	—	373	2	no	—
64	#126	—	373	2	no	—
65	#125	—	373	1	no	—
66	#125	D₃₇₃	1	1	yes	characteristic, complemented
67	#124	—	373	1	no	—
68	#123	—	373	2	no	—
69	C₂²⋊Q₈	—	373	2	no	—
70	C₂²⋊Q₈	—	373	2	no	—
71	C₂×Q₈	—	373	2	no	—
72	C₂×Q₈	—	373	1	no	—
73	C₂×D₄	—	373	1	no	—
74	C₂²×C₄	—	373	2	no	—
75	C₄⋊C₄	—	373	1	no	—
76	C₄⋊C₄	—	373	2	no	—
77	C₄⋊C₄	—	373	2	no	—
78	C₄⋊C₄	—	373	2	no	—
79	C₄⋊C₄	—	373	2	no	—
80	C₄⋊C₄	—	373	1	no	—
81	C₄⋊C₄	C₂×D₃₇₃	1	1	yes	characteristic
82	C₄⋊C₄	C₂×D₃₇₃	1	2	yes	—
83	C₄⋊C₄	C₂×D₃₇₃	1	2	yes	—
84	C₄⋊C₄	C₂×D₃₇₃	1	1	yes	characteristic
85	C₂²⋊C₄	—	373	2	no	—
86	C₂²⋊C₄	—	373	2	no	—
87	C₂²⋊C₄	—	373	2	no	—
88	C₂²⋊C₄	—	373	2	no	—
89	C₂²⋊C₄	—	373	2	no	—
90	C₄²	—	373	2	no	—
91	C₄²	C₂×D₃₇₃	1	1	yes	characteristic
92	C₂³	—	373	2	no	—
93	Q₈	—	1492	2	no	—
94	Q₈	—	1492	1	no	—
95	D₄	—	1492	1	no	—
96	C₂×C₄	—	373	2	no	—
97	C₂×C₄	—	373	2	no	—
98	C₂×C₄	—	373	2	no	—
99	C₂×C₄	—	1492	2	no	—
100	C₂×C₄	C₂²×D₃₇₃	1	2	yes	—
101	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₃₇₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₃₇₃	1	2	yes	—
105	C₂²	—	746	2	no	—
106	C₂²	—	746	2	no	—
107	C₂²	—	746	2	no	—
108	C₂²	C₂³×D₃₇₃	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	746	2	no	—
115	C₄	—	746	2	no	—
116	C₄	—	746	2	no	—
117	C₂	—	1492	2	no	—
118	C₂	#26045838	1	1	yes	characteristic
119	C₂	#26045839	1	1	yes	characteristic
120	C₂	#26045840	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26045791

C₂

yes

maximal, complemented

#26045787

C₂

yes

maximal, complemented

#26045787

C₂

yes

maximal, complemented

#26045784

C₂

yes

maximal, characteristic, complemented

#26045780

C₂

yes

maximal, characteristic

#26045777

C₂

yes

maximal

#26045772

C₂

yes

maximal, complemented

#26045770

C₂

yes

maximal, complemented

#125×C₃₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₄₆

C₂²

yes

—

C₂×Dic₇₄₆

C₂²

yes

characteristic

C₂×D₁₄₉₂

C₂²

yes

characteristic

C₂×C₄×D₃₇₃

C₂²

yes

—

C₄⋊Dic₃₇₃

C₂²

yes

characteristic

C₄⋊Dic₃₇₃

C₂²

yes

—

C₄⋊Dic₃₇₃

C₂²

yes

characteristic

#20238589

C₂²

yes

—

#20238589

C₂²

yes

—

#20238589

C₂²

yes

—

#20238587

C₂²

yes

—

#20238587

C₂²

yes

—

#20238587

C₂²

yes

—

#20238587

C₂²

yes

—

#20238587

C₂²

yes

—

C₄×Dic₃₇₃

C₂²

yes

—

C₄⋊C₄×C₃₇₃

C₂²

yes

characteristic

C₄⋊C₄×C₃₇₃

C₂²

yes

—

C₄⋊C₄×C₃₇₃

C₂²

yes

—

C₄⋊C₄×C₃₇₃

C₂²

yes

characteristic

C₄²×C₃₇₃

C₂²

yes

characteristic

C₂²×D₃₇₃

C₂³

yes

—

Dic₇₄₆

—

Dic₇₄₆

—

D₁₄₉₂

—

C₂×Dic₃₇₃

C₂³

yes

—

C₂×Dic₃₇₃

C₂³

yes

—

C₂×Dic₃₇₃

C₂³

yes

—

C₄×D₃₇₃

—

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂×C₄×C₃₇₃

C₂³

yes

characteristic

C₂×C₄×C₃₇₃

C₂³

yes

characteristic

C₂×C₄×C₃₇₃

C₂³

yes

characteristic

C₂×C₄×C₃₇₃

C₂³

yes

—

C₂²×C₃₇₃

C₂⁴

yes

characteristic

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₂×D₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

C₄×C₃₇₃

—

Dic₃₇₃

—

Dic₃₇₃

—

Dic₃₇₃

—

C₂×C₃₇₃

H(5,𝔽₂)

yes

characteristic

C₂×C₃₇₃

2₋¹⁺⁴

yes

characteristic

C₂×C₃₇₃

2₋¹⁺⁴

yes

characteristic

D₃₇₃

—

C₃₇₃

#563

yes

characteristic, complemented, Sylow

#563

—

373

maximal, Sylow

C₄⋊Q₈

—

373

—

#126

—

373

—

#126

—

373

—

#125

—

373

—

#125

D₃₇₃

yes

characteristic, complemented

#124

—

373

—

#123

—

373

—

C₂²⋊Q₈

—

373

—

C₂²⋊Q₈

—

373

—

C₂×Q₈

—

373

—

C₂×Q₈

—

373

—

C₂×D₄

—

373

—

C₂²×C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

—

373

—

C₄⋊C₄

C₂×D₃₇₃

yes

characteristic

C₄⋊C₄

C₂×D₃₇₃

yes

—

C₄⋊C₄

C₂×D₃₇₃

yes

—

C₄⋊C₄

C₂×D₃₇₃

yes

characteristic

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₂²⋊C₄

—

373

—

C₄²

—

373

—

C₄²

C₂×D₃₇₃

yes

characteristic

C₂³

—

373

—

Q₈

—

1492

—

Q₈

—

1492

—

D₄

—

1492

—

C₂×C₄

—

373

—

C₂×C₄

—

373

—

C₂×C₄

—

373

—

C₂×C₄

—

1492

—

100

C₂×C₄

C₂²×D₃₇₃

yes

—

101

C₂×C₄

C₂²×D₃₇₃

yes

characteristic

102

C₂×C₄

C₂²×D₃₇₃

yes

characteristic

103

C₂×C₄

C₂²×D₃₇₃

yes

characteristic

104

C₂×C₄

C₂²×D₃₇₃

yes

—

105

C₂²

—

746

—

106

C₂²

—

746

—

107

C₂²

—

746

—

108

C₂²

C₂³×D₃₇₃

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

746

—

115

C₄

—

746

—

116

C₄

—

746

—

117

C₂

—

1492

—

118

C₂

#26045838

yes

characteristic

119

C₂

#26045839

yes

characteristic

120

C₂

#26045840

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇₃
commutator, socle, cosocle	C₂²×C₃₇₃	C₂⁴
Fitting	#125×C₃₇₃	C₂
prime radical	C₂×D₁₄₉₂	C₂²
hypercenter	#125	D₃₇₃
hypocenter	C₃₇₃	#563
LCS	G ▷ C₂²×C₃₇₃ ▷ C₃₇₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₇₃

commutator, socle, cosocle

C₂²×C₃₇₃

C₂⁴

Fitting

#125×C₃₇₃

C₂

prime radical

C₂×D₁₄₉₂

C₂²

D₃₇₃

C₃₇₃

G ▷ C₂²×C₃₇₃ ▷ C₃₇₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #26045791	C₂	semidirect
2. #26045787	C₂	semidirect
3. #26045787	C₂	semidirect
4. #26045784	C₂	semidirect
7. #26045772	C₂	semidirect
8. #26045770	C₂	semidirect
9. #125×C₃₇₃	C₂	semidirect
60. C₃₇₃	#563	semidirect
66. #125	D₃₇₃	semidirect

H ≅ G/N

type

1. #26045791

C₂

semidirect

2. #26045787

C₂

semidirect

3. #26045787

C₂

semidirect

4. #26045784

C₂

semidirect

7. #26045772

C₂

semidirect

8. #26045770

C₂

semidirect

9. #125×C₃₇₃

C₂

semidirect

60. C₃₇₃

#563

semidirect

66. #125

D₃₇₃

semidirect

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

subgroup	index
#26045791	2
#26045787	2
#26045784	2
#26045780	2
#26045777	2
#26045772	2
#26045770	2
#125×C₃₇₃	2
#563	373

subgroup

index

#125×C₃₇₃

373

Subgroups — #37560176

Conjugacy classes of subgroups

Complements

Maximal subgroups