Group #37527187 — subgroups

Subgroups — #37527187

Order 22976.

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

normal subgroups only

This group has 220 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	#26040007	C₂	1	1	yes	maximal, characteristic, complemented
2	#26040003	C₂	1	1	yes	maximal, characteristic, complemented
3	#26039999	C₂	1	2	yes	maximal, complemented
4	#26039997	C₂	1	2	yes	maximal, complemented
5	#26039992	C₂	1	2	yes	maximal, complemented
6	#26039991	C₂	1	2	yes	maximal, complemented
7	#26039988	C₂	1	2	yes	maximal, complemented
8	#26039987	C₂	1	2	yes	maximal, complemented
9	#124×C₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₇₁₈	C₂²	1	1	yes	characteristic
11	C₂×D₁₄₃₆	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₃₅₉	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₃₅₉	C₂²	1	2	yes	—
14	C₂²×Dic₃₅₉	C₂²	1	2	yes	—
15	C₂×C₄×D₃₅₉	C₂²	1	2	yes	—
16	C₄⋊Dic₃₅₉	C₂²	1	2	yes	—
17	C₄⋊Dic₃₅₉	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₃₅₉	C₂²	1	1	yes	characteristic
19	Dic₃₅₉⋊C₄	C₂²	1	2	yes	—
20	Dic₃₅₉⋊C₄	C₂²	1	2	yes	—
21	Dic₃₅₉⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₃₅₉	C₂²	1	2	yes	—
23	(C₂×D₃₅₉)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₃₅₉)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₃₅₉)⋊C₄	C₂²	1	2	yes	—
26	C₂×Q₈×C₃₅₉	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₃₅₉	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₃₅₉	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₃₅₉	C₂²	1	2	yes	—
30	C₄²×C₃₅₉	C₂²	1	1	yes	characteristic
31	C₂²×D₃₅₉	C₂³	1	2	yes	—
32	Dic₇₁₈	—	4	1	no	—
33	D₁₄₃₆	—	4	1	no	—
34	C₂²⋊D₃₅₉	—	4	2	no	—
35	C₂²⋊D₃₅₉	—	4	2	no	—
36	C₂×Dic₃₅₉	—	4	2	no	—
37	C₂×Dic₃₅₉	C₂³	1	2	yes	—
38	C₂×Dic₃₅₉	C₂³	1	2	yes	—
39	C₂×Dic₃₅₉	C₂³	1	2	yes	—
40	C₄×D₃₅₉	—	4	2	no	—
41	C₂³×C₃₅₉	C₂³	1	2	yes	—
42	Q₈×C₃₅₉	—	4	1	no	—
43	D₄×C₃₅₉	—	4	1	no	—
44	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₃₅₉	C₂³	1	2	yes	—
48	C₂²×C₃₅₉	—	2	2	no	—
49	C₂²×C₃₅₉	—	2	2	no	—
50	C₂²×C₃₅₉	—	2	2	no	—
51	C₂²×C₃₅₉	C₂⁴	1	1	yes	characteristic
52	C₂×D₃₅₉	—	2	2	no	—
53	C₂×D₃₅₉	—	2	2	no	—
54	C₂×D₃₅₉	—	2	2	no	—
55	C₄×C₃₅₉	—	2	1	no	—
56	C₄×C₃₅₉	—	2	2	no	—
57	C₄×C₃₅₉	—	2	1	no	—
58	C₄×C₃₅₉	—	2	1	no	—
59	Dic₃₅₉	—	2	2	no	—
60	Dic₃₅₉	—	2	2	no	—
61	Dic₃₅₉	—	2	2	no	—
62	C₂×C₃₅₉	—	4	2	no	—
63	C₂×C₃₅₉	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₃₅₉	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₃₅₉	2₋¹⁺⁴	1	1	yes	characteristic
66	D₃₅₉	—	4	2	no	—
67	C₃₅₉	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	359	1	no	maximal, Sylow
69	#125	—	359	1	no	—
70	#124	—	359	1	no	—
71	#124	D₃₅₉	1	1	yes	characteristic, complemented
72	#123	—	359	2	no	—
73	#123	—	359	2	no	—
74	C₂²⋊Q₈	—	359	2	no	—
75	C₂²⋊Q₈	—	359	2	no	—
76	#121	—	359	2	no	—
77	#121	—	359	2	no	—
78	C₂×Q₈	—	359	1	no	—
79	C₂×Q₈	C₂×D₃₅₉	1	1	yes	characteristic
80	C₂×D₄	—	359	2	no	—
81	C₂×D₄	—	359	2	no	—
82	C₂×D₄	—	359	1	no	—
83	C₂×D₄	C₂×D₃₅₉	1	1	yes	characteristic
84	C₂²×C₄	—	359	2	no	—
85	C₂²×C₄	—	359	2	no	—
86	C₄⋊C₄	—	359	2	no	—
87	C₄⋊C₄	—	359	2	no	—
88	C₄⋊C₄	—	359	2	no	—
89	C₄⋊C₄	—	359	2	no	—
90	C₄⋊C₄	—	359	1	no	—
91	C₄⋊C₄	—	359	1	no	—
92	C₂²⋊C₄	—	359	2	no	—
93	C₂²⋊C₄	—	359	2	no	—
94	C₂²⋊C₄	—	359	2	no	—
95	C₂²⋊C₄	—	359	2	no	—
96	C₂²⋊C₄	C₂×D₃₅₉	1	2	yes	—
97	C₂²⋊C₄	C₂×D₃₅₉	1	2	yes	—
98	C₄²	C₂×D₃₅₉	1	1	yes	characteristic
99	C₂³	—	359	2	no	—
100	C₂³	C₂²×D₃₅₉	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	1436	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	1436	1	no	—
105	D₄	—	1436	2	no	—
106	D₄	—	1436	2	no	—
107	C₂×C₄	—	359	2	no	—
108	C₂×C₄	—	359	2	no	—
109	C₂×C₄	—	359	2	no	—
110	C₂×C₄	—	1436	2	no	—
111	C₂×C₄	—	1436	2	no	—
112	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₃₅₉	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	718	2	no	—
120	C₂²	—	718	2	no	—
121	C₂²	—	718	2	no	—
122	C₂²	C₂³×D₃₅₉	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	718	2	no	—
128	C₄	—	718	2	no	—
129	C₄	—	718	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	1436	2	no	—
132	C₂	#26040060	1	1	yes	characteristic
133	C₂	#26040061	1	1	yes	characteristic
134	C₂	#26040062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26040007

C₂

yes

maximal, characteristic, complemented

#26040003

C₂

yes

maximal, characteristic, complemented

#26039999

C₂

yes

maximal, complemented

#26039997

C₂

yes

maximal, complemented

#26039992

C₂

yes

maximal, complemented

#26039991

C₂

yes

maximal, complemented

#26039988

C₂

yes

maximal, complemented

#26039987

C₂

yes

maximal, complemented

#124×C₃₅₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₁₈

C₂²

yes

characteristic

C₂×D₁₄₃₆

C₂²

yes

characteristic

C₂×C₂²⋊D₃₅₉

C₂²

yes

—

C₂×C₂²⋊D₃₅₉

C₂²

yes

—

C₂²×Dic₃₅₉

C₂²

yes

—

C₂×C₄×D₃₅₉

C₂²

yes

—

C₄⋊Dic₃₅₉

C₂²

yes

—

C₄⋊Dic₃₅₉

C₂²

yes

characteristic

C₄⋊Dic₃₅₉

C₂²

yes

characteristic

Dic₃₅₉⋊C₄

C₂²

yes

—

Dic₃₅₉⋊C₄

C₂²

yes

—

Dic₃₅₉⋊C₄

C₂²

yes

—

C₂²⋊Dic₃₅₉

C₂²

yes

—

(C₂×D₃₅₉)⋊C₄

C₂²

yes

—

(C₂×D₃₅₉)⋊C₄

C₂²

yes

—

(C₂×D₃₅₉)⋊C₄

C₂²

yes

—

C₂×Q₈×C₃₅₉

C₂²

yes

characteristic

C₂×D₄×C₃₅₉

C₂²

yes

characteristic

C₂²⋊C₄×C₃₅₉

C₂²

yes

—

C₂²⋊C₄×C₃₅₉

C₂²

yes

—

C₄²×C₃₅₉

C₂²

yes

characteristic

C₂²×D₃₅₉

C₂³

yes

—

Dic₇₁₈

—

D₁₄₃₆

—

C₂²⋊D₃₅₉

—

C₂²⋊D₃₅₉

—

C₂×Dic₃₅₉

—

C₂×Dic₃₅₉

C₂³

yes

—

C₂×Dic₃₅₉

C₂³

yes

—

C₂×Dic₃₅₉

C₂³

yes

—

C₄×D₃₅₉

—

C₂³×C₃₅₉

C₂³

yes

—

Q₈×C₃₅₉

—

D₄×C₃₅₉

—

C₂×C₄×C₃₅₉

C₂³

yes

characteristic

C₂×C₄×C₃₅₉

C₂³

yes

characteristic

C₂×C₄×C₃₅₉

C₂³

yes

characteristic

C₂×C₄×C₃₅₉

C₂³

yes

—

C₂²×C₃₅₉

—

C₂²×C₃₅₉

—

C₂²×C₃₅₉

—

C₂²×C₃₅₉

C₂⁴

yes

characteristic

C₂×D₃₅₉

—

C₂×D₃₅₉

—

C₂×D₃₅₉

—

C₄×C₃₅₉

—

C₄×C₃₅₉

—

C₄×C₃₅₉

—

C₄×C₃₅₉

—

Dic₃₅₉

—

Dic₃₅₉

—

Dic₃₅₉

—

C₂×C₃₅₉

—

C₂×C₃₅₉

yes

C₂×C₃₅₉

yes

C₂×C₃₅₉

2₋¹⁺⁴

yes

characteristic

D₃₅₉

—

C₃₅₉

#562

yes

characteristic, complemented, Sylow

#562

—

359

maximal, Sylow

#125

—

359

—

#124

—

359

—

#124

D₃₅₉

yes

characteristic, complemented

#123

—

359

—

#123

—

359

—

C₂²⋊Q₈

—

359

—

C₂²⋊Q₈

—

359

—

#121

—

359

—

#121

—

359

—

C₂×Q₈

—

359

—

C₂×Q₈

C₂×D₃₅₉

yes

characteristic

C₂×D₄

—

359

—

C₂×D₄

—

359

—

C₂×D₄

—

359

—

C₂×D₄

C₂×D₃₅₉

yes

characteristic

C₂²×C₄

—

359

—

C₂²×C₄

—

359

—

C₄⋊C₄

—

359

—

C₄⋊C₄

—

359

—

C₄⋊C₄

—

359

—

C₄⋊C₄

—

359

—

C₄⋊C₄

—

359

—

C₄⋊C₄

—

359

—

C₂²⋊C₄

—

359

—

C₂²⋊C₄

—

359

—

C₂²⋊C₄

—

359

—

C₂²⋊C₄

—

359

—

C₂²⋊C₄

C₂×D₃₅₉

yes

—

C₂²⋊C₄

C₂×D₃₅₉

yes

—

C₄²

C₂×D₃₅₉

yes

characteristic

C₂³

—

359

—

100

C₂³

C₂²×D₃₅₉

yes

—

101

Q₈

—

102

Q₈

—

1436

—

103

D₄

—

104

D₄

—

1436

—

105

D₄

—

1436

—

106

D₄

—

1436

—

107

C₂×C₄

—

359

—

108

C₂×C₄

—

359

—

109

C₂×C₄

—

359

—

110

C₂×C₄

—

1436

—

111

C₂×C₄

—

1436

—

112

C₂×C₄

C₂²×D₃₅₉

yes

characteristic

113

C₂×C₄

C₂²×D₃₅₉

yes

characteristic

114

C₂×C₄

C₂²×D₃₅₉

yes

characteristic

115

C₂×C₄

C₂²×D₃₅₉

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

718

—

120

C₂²

—

718

—

121

C₂²

—

718

—

122

C₂²

C₂³×D₃₅₉

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

718

—

128

C₄

—

718

—

129

C₄

—

718

—

130

C₂

—

131

C₂

—

1436

—

132

C₂

#26040060

yes

characteristic

133

C₂

#26040061

yes

characteristic

134

C₂

#26040062

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₅₉
commutator, socle, cosocle	C₂²×C₃₅₉	C₂⁴
Fitting	#124×C₃₅₉	C₂
hypercenter	#124	D₃₅₉
hypocenter	C₃₅₉	#562
LCS	G ▷ C₂²×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₅₉

commutator, socle, cosocle

C₂²×C₃₅₉

C₂⁴

Fitting

#124×C₃₅₉

C₂

D₃₅₉

C₃₅₉

G ▷ C₂²×C₃₅₉ ▷ C₃₅₉

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #26040007	C₂	semidirect
2. #26040003	C₂	semidirect
3. #26039999	C₂	semidirect
4. #26039997	C₂	semidirect
5. #26039992	C₂	semidirect
6. #26039991	C₂	semidirect
7. #26039988	C₂	semidirect
8. #26039987	C₂	semidirect
9. #124×C₃₅₉	C₂	semidirect
67. C₃₅₉	#562	semidirect
71. #124	D₃₅₉	semidirect

H ≅ G/N

type

1. #26040007

C₂

semidirect

2. #26040003

C₂

semidirect

3. #26039999

C₂

semidirect

4. #26039997

C₂

semidirect

5. #26039992

C₂

semidirect

6. #26039991

C₂

semidirect

7. #26039988

C₂

semidirect

8. #26039987

C₂

semidirect

9. #124×C₃₅₉

C₂

semidirect

67. C₃₅₉

#562

semidirect

71. #124

D₃₅₉

semidirect

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

subgroup	index
#26040007	2
#26040003	2
#26039999	2
#26039997	2
#26039992	2
#26039991	2
#26039988	2
#26039987	2
#124×C₃₅₉	2
#562	359

subgroup

index

#124×C₃₅₉

359

Subgroups — #37527187

Conjugacy classes of subgroups

Complements

Maximal subgroups