Group #37527053 — subgroups

Subgroups — #37527053

Order 22976.

192 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 95 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	#26040013	C₂	1	1	yes	maximal, characteristic, complemented
2	#26040008	C₂	1	2	yes	maximal
3	#26040008	C₂	1	2	yes	maximal
4	#26040006	C₂	1	1	yes	maximal, characteristic, complemented
5	#26040006	C₂	1	2	yes	maximal, complemented
6	#26040005	C₂	1	2	yes	maximal, complemented
7	#26039994	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#26039980	C₂	1	1	yes	maximal, characteristic, complemented
10	#26039974	C₂	1	1	yes	maximal, characteristic
11	C₂²⋊Q₈×C₃₅₉	C₂	1	1	yes	maximal, characteristic
12	C₂×Dic₇₁₈	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₃₅₉	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₃₅₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₃₅₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₃₅₉	C₂²	1	2	yes	—
17	C₄⋊Dic₃₅₉	C₂²	1	2	yes	—
18	Dic₃₅₉⋊C₄	C₂²	1	2	yes	—
19	Dic₃₅₉⋊C₄	C₂²	1	2	yes	—
20	Dic₃₅₉⋊C₄	C₂²	1	2	yes	—
21	Dic₃₅₉⋊C₄	C₂²	1	2	yes	—
22	Dic₃₅₉⋊C₄	C₂²	1	2	yes	—
23	C₂²⋊Dic₃₅₉	C₂²	1	1	yes	characteristic
24	C₂²⋊Dic₃₅₉	C₂²	1	1	yes	characteristic
25	C₂²⋊Dic₃₅₉	C₂²	1	2	yes	—
26	C₄×Dic₃₅₉	C₂²	1	1	yes	characteristic
27	C₄×Dic₃₅₉	C₂²	1	1	yes	characteristic
28	C₄×Dic₃₅₉	C₂²	1	2	yes	—
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	Dic₇₁₈	—	2	2	no	—
36	C₂×Dic₃₅₉	C₂³	1	1	yes	characteristic
37	C₂×Dic₃₅₉	C₂³	1	1	yes	characteristic
38	C₂×Dic₃₅₉	C₂³	1	1	yes	characteristic
39	C₂×Dic₃₅₉	C₂³	1	2	yes	—
40	C₂×Dic₃₅₉	C₂³	1	1	yes	characteristic
41	C₂×Dic₃₅₉	C₂³	1	2	yes	—
42	C₂³×C₃₅₉	C₂³	1	1	yes	characteristic
43	Q₈×C₃₅₉	—	2	2	no	—
44	C₂×C₄×C₃₅₉	—	2	2	no	—
45	C₂×C₄×C₃₅₉	C₂³	1	2	yes	—
46	C₂×C₄×C₃₅₉	C₂³	1	2	yes	—
47	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
49	C₂²×C₃₅₉	—	2	1	no	—
50	C₂²×C₃₅₉	—	2	1	no	—
51	C₂²×C₃₅₉	—	2	1	no	—
52	C₂²×C₃₅₉	C₂⁴	1	1	yes	characteristic
53	C₄×C₃₅₉	—	2	1	no	—
54	C₄×C₃₅₉	—	2	2	no	—
55	C₄×C₃₅₉	—	2	2	no	—
56	C₄×C₃₅₉	#41	1	2	yes	—
57	Dic₃₅₉	—	2	2	no	—
58	Dic₃₅₉	—	2	1	no	—
59	Dic₃₅₉	—	2	1	no	—
60	Dic₃₅₉	—	2	1	no	—
61	Dic₃₅₉	—	2	1	no	—
62	Dic₃₅₉	—	2	2	no	—
63	C₂×C₃₅₉	—	4	1	no	—
64	C₂×C₃₅₉	C₂×#41	1	1	yes	characteristic
65	C₂×C₃₅₉	2₋¹⁺⁴	1	1	yes	characteristic
66	C₂×C₃₅₉	2₋¹⁺⁴	1	1	yes	characteristic
67	C₃₅₉	#541	1	1	yes	characteristic, complemented, Sylow
68	#541	—	359	1	no	maximal, Sylow
69	C₄⋊Q₈	—	359	1	no	—
70	#126	—	359	2	no	—
71	#126	—	359	2	no	—
72	#125	—	359	2	no	—
73	#125	—	359	1	no	—
74	#125	—	359	2	no	—
75	C₂²⋊Q₈	—	359	1	no	—
76	C₂²⋊Q₈	D₃₅₉	1	1	yes	characteristic
77	C₄×Q₈	—	359	1	no	—
78	C₄×Q₈	—	359	1	no	—
79	#117	—	359	1	no	—
80	C₂×Q₈	—	359	1	no	—
81	C₂×Q₈	C₂×D₃₅₉	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₃₅₉	1	1	yes	characteristic
83	C₄⋊C₄	—	359	1	no	—
84	C₄⋊C₄	—	359	1	no	—
85	C₄⋊C₄	—	359	1	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	2	no	—
91	C₄⋊C₄	—	359	2	no	—
92	C₄⋊C₄	—	359	2	no	—
93	C₄⋊C₄	C₂×D₃₅₉	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₃₅₉	1	2	yes	—
95	C₂²⋊C₄	—	359	1	no	—
96	C₂²⋊C₄	—	359	1	no	—
97	C₂²⋊C₄	—	359	2	no	—
98	C₂²⋊C₄	C₂×D₃₅₉	1	2	yes	—
99	C₄²	—	359	1	no	—
100	C₄²	—	359	1	no	—
101	C₄²	—	359	2	no	—
102	C₄²	—	359	2	no	—
103	C₂³	C₂²×D₃₅₉	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	Q₈	—	718	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	359	1	no	—
108	C₂×C₄	—	359	1	no	—
109	C₂×C₄	—	359	1	no	—
110	C₂×C₄	—	359	1	no	—
111	C₂×C₄	—	359	2	no	—
112	C₂×C₄	—	359	2	no	—
113	C₂×C₄	C₂²×D₃₅₉	1	2	yes	—
114	C₂×C₄	C₂²×D₃₅₉	1	2	yes	—
115	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂³×D₃₅₉	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	718	2	no	—
125	C₄	—	718	1	no	—
126	C₄	—	718	1	no	—
127	C₄	—	718	1	no	—
128	C₄	—	718	1	no	—
129	C₄	—	718	2	no	—
130	C₄	#19153340	1	2	yes	—
131	C₂	—	4	1	no	—
132	C₂	C₂×#19153340	1	1	yes	characteristic
133	C₂	#26040062	1	1	yes	characteristic
134	C₂	#26040063	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26040013

C₂

yes

maximal, characteristic, complemented

#26040008

C₂

yes

maximal

#26040008

C₂

yes

maximal

#26040006

C₂

yes

maximal, characteristic, complemented

#26040006

C₂

yes

maximal, complemented

#26040005

C₂

yes

maximal, complemented

#26039994

C₂

yes

maximal, characteristic

Q₈×Dic₃₅₉

C₂

yes

maximal, characteristic, complemented

#26039980

C₂

yes

maximal, characteristic, complemented

#26039974

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₃₅₉

C₂

yes

maximal, characteristic

C₂×Dic₇₁₈

C₂²

yes

characteristic

C₄⋊Dic₃₅₉

C₂²

yes

characteristic

C₄⋊Dic₃₅₉

C₂²

yes

characteristic

C₄⋊Dic₃₅₉

C₂²

yes

characteristic

C₄⋊Dic₃₅₉

C₂²

yes

—

C₄⋊Dic₃₅₉

C₂²

yes

—

Dic₃₅₉⋊C₄

C₂²

yes

—

Dic₃₅₉⋊C₄

C₂²

yes

—

Dic₃₅₉⋊C₄

C₂²

yes

—

Dic₃₅₉⋊C₄

C₂²

yes

—

Dic₃₅₉⋊C₄

C₂²

yes

—

C₂²⋊Dic₃₅₉

C₂²

yes

characteristic

C₂²⋊Dic₃₅₉

C₂²

yes

characteristic

C₂²⋊Dic₃₅₉

C₂²

yes

—

C₄×Dic₃₅₉

C₂²

yes

characteristic

C₄×Dic₃₅₉

C₂²

yes

characteristic

C₄×Dic₃₅₉

C₂²

yes

—

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

—

Dic₇₁₈

—

C₂×Dic₃₅₉

C₂³

yes

characteristic

C₂×Dic₃₅₉

C₂³

yes

characteristic

C₂×Dic₃₅₉

C₂³

yes

characteristic

C₂×Dic₃₅₉

C₂³

yes

—

C₂×Dic₃₅₉

C₂³

yes

characteristic

C₂×Dic₃₅₉

C₂³

yes

—

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

C₄×C₃₅₉

—

C₄×C₃₅₉

—

C₄×C₃₅₉

—

C₄×C₃₅₉

#41

yes

—

Dic₃₅₉

—

Dic₃₅₉

—

Dic₃₅₉

—

Dic₃₅₉

—

Dic₃₅₉

—

Dic₃₅₉

—

C₂×C₃₅₉

—

C₂×C₃₅₉

C₂×#41

yes

characteristic

C₂×C₃₅₉

2₋¹⁺⁴

yes

characteristic

C₂×C₃₅₉

2₋¹⁺⁴

yes

characteristic

C₃₅₉

#541

yes

characteristic, complemented, Sylow

#541

—

359

maximal, Sylow

C₄⋊Q₈

—

359

—

#126

—

359

—

#126

—

359

—

#125

—

359

—

#125

—

359

—

#125

—

359

—

C₂²⋊Q₈

—

359

—

C₂²⋊Q₈

D₃₅₉

yes

characteristic

C₄×Q₈

—

359

—

C₄×Q₈

—

359

—

#117

—

359

—

C₂×Q₈

—

359

—

C₂×Q₈

C₂×D₃₅₉

yes

characteristic

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

—

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

—

C₂²⋊C₄

—

359

—

C₂²⋊C₄

C₂×D₃₅₉

yes

—

C₄²

—

359

—

100

C₄²

—

359

—

101

C₄²

—

359

—

102

C₄²

—

359

—

103

C₂³

C₂²×D₃₅₉

yes

characteristic

104

Q₈

—

105

Q₈

—

718

—

106

C₂×C₄

—

107

C₂×C₄

—

359

—

108

C₂×C₄

—

359

—

109

C₂×C₄

—

359

—

110

C₂×C₄

—

359

—

111

C₂×C₄

—

359

—

112

C₂×C₄

—

359

—

113

C₂×C₄

C₂²×D₃₅₉

yes

—

114

C₂×C₄

C₂²×D₃₅₉

yes

—

115

C₂×C₄

C₂²×D₃₅₉

yes

characteristic

116

C₂×C₄

C₂²×D₃₅₉

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂³×D₃₅₉

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

718

—

125

C₄

—

718

—

126

C₄

—

718

—

127

C₄

—

718

—

128

C₄

—

718

—

129

C₄

—

718

—

130

C₄

#19153340

yes

—

131

C₂

—

132

C₂

C₂×#19153340

yes

characteristic

133

C₂

#26040062

yes

characteristic

134

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	C₂³×C₃₅₉	C₂³
hypercenter	C₂²⋊Q₈	D₃₅₉
hypocenter	C₃₅₉	#541
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

C₂³×C₃₅₉

C₂³

hypercenter

C₂²⋊Q₈

D₃₅₉

C₃₅₉

G ▷ C₂²×C₃₅₉ ▷ C₃₅₉

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #26040013	C₂	semidirect
4. #26040006	C₂	semidirect
5. #26040006	C₂	semidirect
6. #26040005	C₂	semidirect
8. Q₈×Dic₃₅₉	C₂	semidirect
9. #26039980	C₂	semidirect
67. C₃₅₉	#541	semidirect

H ≅ G/N

type

1. #26040013

C₂

semidirect

4. #26040006

C₂

semidirect

5. #26040006

C₂

semidirect

6. #26040005

C₂

semidirect

8. Q₈×Dic₃₅₉

C₂

semidirect

9. #26039980

C₂

semidirect

67. C₃₅₉

#541

semidirect

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

subgroup	index
#26040013	2
#26040008	2
#26040006	2
#26040005	2
#26039994	2
Q₈×Dic₃₅₉	2
#26039980	2
#26039974	2
C₂²⋊Q₈×C₃₅₉	2
#541	359

subgroup

index

Q₈×Dic₃₅₉

C₂²⋊Q₈×C₃₅₉

359

Subgroups — #37527053

Conjugacy classes of subgroups

Complements

Maximal subgroups