Group #37527164 — subgroups

Subgroups — #37527164

Order 22976.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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	#26040014	C₂	1	1	yes	maximal, characteristic, complemented
2	#26040010	C₂	1	2	yes	maximal, complemented
3	#26040008	C₂	1	2	yes	maximal, complemented
4	#26040001	C₂	1	2	yes	maximal
5	#26039991	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×Dic₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#26039980	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊D₃₅₉	C₂	1	1	yes	maximal, characteristic
9	#26039974	C₂	1	1	yes	maximal, characteristic, complemented
10	#26039973	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊Q₈×C₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₇₁₈	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₃₅₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₃₅₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₃₅₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₃₅₉	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₃₅₉	C₂²	1	2	yes	—
18	Dic₃₅₉⋊C₄	C₂²	1	1	yes	characteristic
19	Dic₃₅₉⋊C₄	C₂²	1	2	yes	—
20	Dic₃₅₉⋊C₄	C₂²	1	2	yes	—
21	C₂²⋊Dic₃₅₉	C₂²	1	2	yes	—
22	C₂²⋊Dic₃₅₉	C₂²	1	1	yes	characteristic
23	(C₂×D₃₅₉)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₃₅₉)⋊C₄	C₂²	1	1	yes	characteristic
25	(C₂×D₃₅₉)⋊C₄	C₂²	1	2	yes	—
26	C₄×Dic₃₅₉	C₂²	1	2	yes	—
27	C₄×Dic₃₅₉	C₂²	1	2	yes	—
28	C₄×Dic₃₅₉	C₂²	1	1	yes	characteristic
29	C₄×Dic₃₅₉	C₂²	1	2	yes	—
30	C₂×Q₈×C₃₅₉	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₃₅₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₃₅₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×C₃₅₉	C₂²	1	2	yes	—
34	C₂²⋊C₄×C₃₅₉	C₂²	1	2	yes	—
35	C₂²×D₃₅₉	C₂³	1	1	yes	characteristic
36	Dic₇₁₈	—	2	4	no	—
37	C₂²⋊D₃₅₉	—	2	2	no	—
38	C₂×Dic₃₅₉	C₂³	1	2	yes	—
39	C₂×Dic₃₅₉	C₂³	1	1	yes	characteristic
40	C₂×Dic₃₅₉	C₂³	1	1	yes	characteristic
41	C₂×Dic₃₅₉	C₂³	1	2	yes	—
42	C₂×Dic₃₅₉	C₂³	1	1	yes	characteristic
43	C₄×D₃₅₉	—	2	2	no	—
44	C₂³×C₃₅₉	C₂³	1	1	yes	characteristic
45	Q₈×C₃₅₉	—	2	2	no	—
46	C₂×C₄×C₃₅₉	—	2	2	no	—
47	C₂×C₄×C₃₅₉	C₂³	1	2	yes	—
48	C₂×C₄×C₃₅₉	C₂³	1	2	yes	—
49	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
51	C₂²×C₃₅₉	—	2	1	no	—
52	C₂²×C₃₅₉	—	2	1	no	—
53	C₂²×C₃₅₉	—	2	1	no	—
54	C₂²×C₃₅₉	C₂⁴	1	1	yes	characteristic
55	C₂×D₃₅₉	—	2	1	no	—
56	C₂×D₃₅₉	—	2	1	no	—
57	C₂×D₃₅₉	—	2	1	no	—
58	C₄×C₃₅₉	—	2	1	no	—
59	C₄×C₃₅₉	—	2	2	no	—
60	C₄×C₃₅₉	—	2	2	no	—
61	C₄×C₃₅₉	#41	1	2	yes	—
62	Dic₃₅₉	—	2	1	no	—
63	Dic₃₅₉	—	2	2	no	—
64	Dic₃₅₉	—	2	2	no	—
65	Dic₃₅₉	—	2	1	no	—
66	Dic₃₅₉	#41	1	2	yes	—
67	C₂×C₃₅₉	—	4	1	no	—
68	C₂×C₃₅₉	C₂×#41	1	1	yes	characteristic
69	C₂×C₃₅₉	C₂×#41	1	1	yes	characteristic
70	C₂×C₃₅₉	2₋¹⁺⁴	1	1	yes	characteristic
71	D₃₅₉	—	4	1	no	—
72	C₃₅₉	#556	1	1	yes	characteristic, complemented, Sylow
73	#556	—	359	1	no	maximal, Sylow
74	C₄⋊Q₈	—	359	1	no	—
75	#126	—	359	2	no	—
76	#126	—	359	2	no	—
77	#124	—	359	2	no	—
78	C₂²⋊Q₈	—	359	1	no	—
79	C₂²⋊Q₈	D₃₅₉	1	1	yes	characteristic, complemented
80	C₄×Q₈	—	359	1	no	—
81	C₄×Q₈	—	359	2	no	—
82	C₄×D₄	—	359	1	no	—
83	#117	—	359	1	no	—
84	#117	—	359	1	no	—
85	C₂×Q₈	—	359	2	no	—
86	C₂×Q₈	C₂×D₃₅₉	1	1	yes	characteristic
87	C₂×D₄	—	359	1	no	—
88	C₂²×C₄	—	359	1	no	—
89	C₂²×C₄	C₂×D₃₅₉	1	1	yes	characteristic
90	C₄⋊C₄	—	359	1	no	—
91	C₄⋊C₄	—	359	1	no	—
92	C₄⋊C₄	—	359	1	no	—
93	C₄⋊C₄	—	359	2	no	—
94	C₄⋊C₄	—	359	2	no	—
95	C₄⋊C₄	—	359	2	no	—
96	C₄⋊C₄	C₂×D₃₅₉	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₃₅₉	1	2	yes	—
98	C₂²⋊C₄	—	359	2	no	—
99	C₂²⋊C₄	—	359	1	no	—
100	C₂²⋊C₄	—	359	2	no	—
101	C₂²⋊C₄	—	359	1	no	—
102	C₂²⋊C₄	—	359	2	no	—
103	C₂²⋊C₄	C₂×D₃₅₉	1	2	yes	—
104	C₄²	—	359	2	no	—
105	C₄²	—	359	2	no	—
106	C₄²	—	359	1	no	—
107	C₄²	—	359	2	no	—
108	C₂³	—	359	1	no	—
109	C₂³	C₂²×D₃₅₉	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	Q₈	—	718	4	no	—
112	D₄	—	718	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	359	1	no	—
115	C₂×C₄	—	359	2	no	—
116	C₂×C₄	—	359	1	no	—
117	C₂×C₄	—	359	1	no	—
118	C₂×C₄	—	359	2	no	—
119	C₂×C₄	—	718	2	no	—
120	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₃₅₉	1	2	yes	—
122	C₂×C₄	C₂²×D₃₅₉	1	2	yes	—
123	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	718	1	no	—
128	C₂²	—	718	1	no	—
129	C₂²	—	718	1	no	—
130	C₂²	C₂³×D₃₅₉	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	359	2	no	—
135	C₄	—	718	2	no	—
136	C₄	—	718	1	no	—
137	C₄	—	718	2	no	—
138	C₄	—	718	1	no	—
139	C₄	#19153340	1	2	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	1436	1	no	—
142	C₂	#41×D₃₅₉	1	1	yes	characteristic
143	C₂	C₂×#19153340	1	1	yes	characteristic
144	C₂	#26040063	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26040014

C₂

yes

maximal, characteristic, complemented

#26040010

C₂

yes

maximal, complemented

#26040008

C₂

yes

maximal, complemented

#26040001

C₂

yes

maximal

#26039991

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₃₅₉

C₂

yes

maximal, characteristic, complemented

#26039980

C₂

yes

maximal, complemented

C₄×C₂²⋊D₃₅₉

C₂

yes

maximal, characteristic

#26039974

C₂

yes

maximal, characteristic, complemented

#26039973

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₃₅₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₇₁₈

C₂²

yes

—

C₂×C₂²⋊D₃₅₉

C₂²

yes

characteristic

C₂×C₄×D₃₅₉

C₂²

yes

characteristic

C₄⋊Dic₃₅₉

C₂²

yes

characteristic

C₄⋊Dic₃₅₉

C₂²

yes

characteristic

C₄⋊Dic₃₅₉

C₂²

yes

—

Dic₃₅₉⋊C₄

C₂²

yes

characteristic

Dic₃₅₉⋊C₄

C₂²

yes

—

Dic₃₅₉⋊C₄

C₂²

yes

—

C₂²⋊Dic₃₅₉

C₂²

yes

—

C₂²⋊Dic₃₅₉

C₂²

yes

characteristic

(C₂×D₃₅₉)⋊C₄

C₂²

yes

—

(C₂×D₃₅₉)⋊C₄

C₂²

yes

characteristic

(C₂×D₃₅₉)⋊C₄

C₂²

yes

—

C₄×Dic₃₅₉

C₂²

yes

—

C₄×Dic₃₅₉

C₂²

yes

—

C₄×Dic₃₅₉

C₂²

yes

characteristic

C₄×Dic₃₅₉

C₂²

yes

—

C₂×Q₈×C₃₅₉

C₂²

yes

characteristic

C₂²×C₄×C₃₅₉

C₂²

yes

characteristic

C₄⋊C₄×C₃₅₉

C₂²

yes

characteristic

C₄⋊C₄×C₃₅₉

C₂²

yes

—

C₂²⋊C₄×C₃₅₉

C₂²

yes

—

C₂²×D₃₅₉

C₂³

yes

characteristic

Dic₇₁₈

—

C₂²⋊D₃₅₉

—

C₂×Dic₃₅₉

C₂³

yes

—

C₂×Dic₃₅₉

C₂³

yes

characteristic

C₂×Dic₃₅₉

C₂³

yes

characteristic

C₂×Dic₃₅₉

C₂³

yes

—

C₂×Dic₃₅₉

C₂³

yes

characteristic

C₄×D₃₅₉

—

C₂³×C₃₅₉

C₂³

yes

characteristic

Q₈×C₃₅₉

—

C₂×C₄×C₃₅₉

—

C₂×C₄×C₃₅₉

C₂³

yes

—

C₂×C₄×C₃₅₉

C₂³

yes

—

C₂×C₄×C₃₅₉

C₂³

yes

characteristic

C₂×C₄×C₃₅₉

C₂³

yes

characteristic

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

#41

yes

—

Dic₃₅₉

—

Dic₃₅₉

—

Dic₃₅₉

—

Dic₃₅₉

—

Dic₃₅₉

#41

yes

—

C₂×C₃₅₉

—

C₂×C₃₅₉

C₂×#41

yes

characteristic

C₂×C₃₅₉

C₂×#41

yes

characteristic

C₂×C₃₅₉

2₋¹⁺⁴

yes

characteristic

D₃₅₉

—

C₃₅₉

#556

yes

characteristic, complemented, Sylow

#556

—

359

maximal, Sylow

C₄⋊Q₈

—

359

—

#126

—

359

—

#126

—

359

—

#124

—

359

—

C₂²⋊Q₈

—

359

—

C₂²⋊Q₈

D₃₅₉

yes

characteristic, complemented

C₄×Q₈

—

359

—

C₄×Q₈

—

359

—

C₄×D₄

—

359

—

#117

—

359

—

#117

—

359

—

C₂×Q₈

—

359

—

C₂×Q₈

C₂×D₃₅₉

yes

characteristic

C₂×D₄

—

359

—

C₂²×C₄

—

359

—

C₂²×C₄

C₂×D₃₅₉

yes

characteristic

C₄⋊C₄

—

359

—

C₄⋊C₄

—

359

—

C₄⋊C₄

—

359

—

C₄⋊C₄

—

359

—

C₄⋊C₄

—

359

—

C₄⋊C₄

—

359

—

C₄⋊C₄

C₂×D₃₅₉

yes

characteristic

C₄⋊C₄

C₂×D₃₅₉

yes

—

C₂²⋊C₄

—

359

—

C₂²⋊C₄

—

359

—

100

C₂²⋊C₄

—

359

—

101

C₂²⋊C₄

—

359

—

102

C₂²⋊C₄

—

359

—

103

C₂²⋊C₄

C₂×D₃₅₉

yes

—

104

C₄²

—

359

—

105

C₄²

—

359

—

106

C₄²

—

359

—

107

C₄²

—

359

—

108

C₂³

—

359

—

109

C₂³

C₂²×D₃₅₉

yes

characteristic

110

Q₈

—

111

Q₈

—

718

—

112

D₄

—

718

—

113

C₂×C₄

—

114

C₂×C₄

—

359

—

115

C₂×C₄

—

359

—

116

C₂×C₄

—

359

—

117

C₂×C₄

—

359

—

118

C₂×C₄

—

359

—

119

C₂×C₄

—

718

—

120

C₂×C₄

C₂²×D₃₅₉

yes

characteristic

121

C₂×C₄

C₂²×D₃₅₉

yes

—

122

C₂×C₄

C₂²×D₃₅₉

yes

—

123

C₂×C₄

C₂²×D₃₅₉

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

718

—

128

C₂²

—

718

—

129

C₂²

—

718

—

130

C₂²

C₂³×D₃₅₉

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

359

—

135

C₄

—

718

—

136

C₄

—

718

—

137

C₄

—

718

—

138

C₄

—

718

—

139

C₄

#19153340

yes

—

140

C₂

—

141

C₂

—

1436

—

142

C₂

#41×D₃₅₉

yes

characteristic

143

C₂

C₂×#19153340

yes

characteristic

144

C₂

#26040063

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₅₉
commutator, socle, cosocle	C₂²×C₃₅₉	C₂⁴
Fitting	C₂²⋊Q₈×C₃₅₉	C₂
prime radical	C₂×C₂²⋊D₃₅₉	C₂²
hypercenter	C₂²⋊Q₈	D₃₅₉
hypocenter	C₃₅₉	#556
LCS	G ▷ C₂²×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₅₉

commutator, socle, cosocle

C₂²×C₃₅₉

C₂⁴

Fitting

C₂²⋊Q₈×C₃₅₉

C₂

prime radical

C₂×C₂²⋊D₃₅₉

C₂²

hypercenter

C₂²⋊Q₈

D₃₅₉

C₃₅₉

G ▷ C₂²×C₃₅₉ ▷ C₃₅₉

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #26040014	C₂	semidirect
2. #26040010	C₂	semidirect
3. #26040008	C₂	semidirect
5. #26039991	C₂	semidirect
6. Q₈×Dic₃₅₉	C₂	semidirect
7. #26039980	C₂	semidirect
9. #26039974	C₂	semidirect
10. #26039973	C₂	semidirect
11. C₂²⋊Q₈×C₃₅₉	C₂	semidirect
72. C₃₅₉	#556	semidirect
79. C₂²⋊Q₈	D₃₅₉	semidirect

H ≅ G/N

type

1. #26040014

C₂

semidirect

2. #26040010

C₂

semidirect

3. #26040008

C₂

semidirect

5. #26039991

C₂

semidirect

6. Q₈×Dic₃₅₉

C₂

semidirect

7. #26039980

C₂

semidirect

9. #26039974

C₂

semidirect

10. #26039973

C₂

semidirect

11. C₂²⋊Q₈×C₃₅₉

C₂

semidirect

72. C₃₅₉

#556

semidirect

79. C₂²⋊Q₈

D₃₅₉

semidirect

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

subgroup	index
#26040014	2
#26040010	2
#26040008	2
#26040001	2
#26039991	2
Q₈×Dic₃₅₉	2
#26039980	2
C₄×C₂²⋊D₃₅₉	2
#26039974	2
#26039973	2
C₂²⋊Q₈×C₃₅₉	2
#556	359

subgroup

index

Q₈×Dic₃₅₉

C₄×C₂²⋊D₃₅₉

C₂²⋊Q₈×C₃₅₉

359

Subgroups — #37527164

Conjugacy classes of subgroups

Complements

Maximal subgroups