Group #18003530 — subgroups

Subgroups — #18003530

Order 4736.

364 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 197 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	#14558090	C₂	1	1	yes	maximal, characteristic
2	#14557663	C₂	1	4	yes	maximal, complemented
3	C₈×Dic₇₄	C₂	1	4	yes	maximal, complemented
4	#14557559	C₂	1	2	yes	maximal
5	#14557557	C₂	1	1	yes	maximal, characteristic
6	#14557488	C₂	1	2	yes	maximal
7	C₂×C₄×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#10596934	C₂²	1	2	yes	—
9	#11924536	C₄	1	2	yes	—
10	#11924528	C₄	1	2	yes	—
11	#11924515	C₄	1	4	yes	—
12	C₄×Dic₇₄	C₂²	1	4	yes	—
13	#11924495	C₂²	1	2	yes	—
14	C₄⋊#64436	C₂²	1	4	yes	—
15	#11924465	C₂²	1	8	yes	—
16	#11924446	C₂²	1	4	yes	—
17	C₈×Dic₃₇	C₂²	1	4	yes	—
18	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₃₇	C₂²	1	2	yes	—
22	C₄×C₈×C₃₇	C₂²	1	2	yes	—
23	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₇₄	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₃₇	—	2	2	no	—
27	C₂×C₈×C₃₇	Q₈	1	4	yes	complemented
28	C₂×C₈×C₃₇	C₂³	1	2	yes	—
29	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
31	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₃₇	C₂³	1	2	yes	—
34	C₂²⋊Dic₃₇	C₂×C₄	1	4	yes	—
35	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
36	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
37	#10596936	C₂×C₄	1	8	yes	—
38	C₄×Dic₃₇	C₂³	1	4	yes	—
39	#10596934	—	2	4	no	—
40	C₂×#64436	C₂³	1	4	yes	—
41	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₃₇	—	2	4	no	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
46	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₃₇	#41	1	2	yes	—
48	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
49	C₂×Dic₃₇	C₂²×C₄	1	8	yes	—
50	Dic₇₄	—	2	8	no	—
51	C₈×C₃₇	—	2	2	no	—
52	C₈×C₃₇	C₂×Q₈	1	2	yes	—
53	C₈×C₃₇	C₂×Q₈	1	2	yes	—
54	#64436	—	2	4	no	—
55	C₂²×C₃₇	—	2	2	no	—
56	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
57	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₃₇	—	2	2	no	—
59	C₄×C₃₇	C₄×Q₈	1	2	yes	—
60	C₄×C₃₇	#131	1	4	yes	—
61	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
63	Dic₃₇	—	2	8	no	—
64	#2960	—	37	1	no	maximal, Sylow
65	C₂×C₃₇	—	2	2	no	—
66	C₂×C₃₇	C₂×C₄×Q₈	1	1	yes	characteristic
67	C₂×C₃₇	C₂×#131	1	2	yes	—
68	#533	—	37	1	no	—
69	#446	—	37	4	no	—
70	C₈×Q₈	—	37	4	no	—
71	#424	—	37	2	no	—
72	#423	—	37	1	no	—
73	#405	—	37	2	no	—
74	C₂×C₄×C₈	D₃₇	1	1	yes	characteristic
75	C₃₇	#2960	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	37	2	no	—
77	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
78	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
79	C₄⋊Q₈	—	37	2	no	—
80	#125	—	37	2	no	—
81	C₂²⋊Q₈	—	37	4	no	—
82	C₄×Q₈	—	37	4	no	—
83	#117	—	37	2	no	—
84	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
85	C₄⋊C₈	—	37	8	no	—
86	C₄⋊C₈	—	37	4	no	—
87	#97	—	37	4	no	—
88	C₄×C₈	—	37	4	no	—
89	C₄×C₈	C₂×D₃₇	1	2	yes	—
90	C₄×C₈	C₂×D₃₇	1	2	yes	—
91	C₂×Q₈	—	37	4	no	—
92	C₂²×C₄	C₄×D₃₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
94	M₁₆	—	74	4	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	37	4	no	—
97	C₂×C₈	Dic₇₄	1	4	yes	complemented
98	C₂×C₈	C₂²×D₃₇	1	2	yes	—
99	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
100	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
101	C₄⋊C₄	—	37	8	no	—
102	C₄⋊C₄	—	37	4	no	—
103	C₄⋊C₄	—	37	4	no	—
104	C₂²⋊C₄	—	37	4	no	—
105	C₄²	—	37	4	no	—
106	C₄²	C₄×D₃₇	1	1	yes	characteristic
107	C₄²	C₄×D₃₇	1	1	yes	characteristic
108	C₄²	C₂²×D₃₇	1	2	yes	—
109	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
110	Q₈	—	74	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	37	8	no	—
113	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
115	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
117	C₂×C₄	#10596962	1	2	yes	—
118	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
119	C₈	—	2	2	no	—
120	C₈	—	74	4	no	—
121	C₈	C₂×Dic₇₄	1	2	yes	—
122	C₈	C₂×Dic₇₄	1	2	yes	—
123	C₂²	—	2	2	no	—
124	C₂²	C₄×Dic₇₄	1	2	yes	—
125	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	74	8	no	—
128	C₄	C₄×Dic₇₄	1	2	yes	—
129	C₄	#11924542	1	4	yes	—
130	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
131	C₄	C₂×#10596962	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×C₄×Dic₇₄	1	1	yes	characteristic
134	C₂	C₂×#11924542	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558090

C₂

yes

maximal, characteristic

#14557663

C₂

yes

maximal, complemented

C₈×Dic₇₄

C₂

yes

maximal, complemented

#14557559

C₂

yes

maximal

#14557557

C₂

yes

maximal, characteristic

#14557488

C₂

yes

maximal

C₂×C₄×C₈×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

—

#11924536

C₄

yes

—

#11924528

C₄

yes

—

#11924515

C₄

yes

—

C₄×Dic₇₄

C₂²

yes

—

#11924495

C₂²

yes

—

C₄⋊#64436

C₂²

yes

—

#11924465

C₂²

yes

—

#11924446

C₂²

yes

—

C₈×Dic₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₄×C₈×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

C₂×C₄

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

Q₈

yes

complemented

C₂×C₈×C₃₇

C₂³

yes

—

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

C₂³

yes

—

C₂²⋊Dic₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

C₂×C₄

yes

—

C₄×Dic₃₇

C₂³

yes

—

#10596934

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

—

Dic₇₄

—

C₈×C₃₇

—

C₈×C₃₇

C₂×Q₈

yes

—

C₈×C₃₇

C₂×Q₈

yes

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₄×Q₈

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₄×Q₈

yes

—

C₄×C₃₇

#131

yes

—

C₄×C₃₇

C₂²×Q₈

yes

characteristic

C₄×C₃₇

C₂×#41

yes

characteristic

Dic₃₇

—

#2960

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₃₇

C₂×#131

yes

—

#533

—

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

D₃₇

yes

characteristic

C₃₇

#2960

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

—

C₄×C₈

C₂×D₃₇

yes

—

C₂×Q₈

—

C₂²×C₄

C₄×D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₇₄

yes

complemented

C₂×C₈

C₂²×D₃₇

yes

—

C₂×C₈

C₂²×D₃₇

yes

characteristic

100

C₂×C₈

C₂²×D₃₇

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₄×D₃₇

yes

characteristic

107

C₄²

C₄×D₃₇

yes

characteristic

108

C₄²

C₂²×D₃₇

yes

—

109

C₂³

C₂×C₄×D₃₇

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×Dic₇₄

yes

—

114

C₂×C₄

C₂×C₄×D₃₇

yes

—

115

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

116

C₂×C₄

C₂×C₄×D₃₇

yes

—

117

C₂×C₄

#10596962

yes

—

118

C₂×C₄

C₂³×D₃₇

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

C₂×Dic₇₄

yes

—

122

C₈

C₂×Dic₇₄

yes

—

123

C₂²

—

124

C₂²

C₄×Dic₇₄

yes

—

125

C₂²

C₂²×C₄×D₃₇

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

C₄×Dic₇₄

yes

—

129

C₄

#11924542

yes

—

130

C₄

C₂²×Dic₇₄

yes

characteristic

131

C₄

C₂×#10596962

yes

characteristic

132

C₂

—

133

C₂

C₂×C₄×Dic₇₄

yes

characteristic

134

C₂

C₂×#11924542

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×C₄×C₈×C₃₇	C₂
prime radical	C₂³×C₃₇	C₂²×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂×C₄×C₈	D₃₇
hypocenter	C₃₇	#2960
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂²×D₃₇

commutator, socle

C₂²×C₃₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂×C₄×C₈×C₃₇

C₂

prime radical

C₂³×C₃₇

C₂²×C₄

cosocle

C₂×C₄×C₃₇

C₂⁴

hypercenter

C₂×C₄×C₈

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #14557663	C₂	semidirect
3. C₈×Dic₇₄	C₂	semidirect
27. C₂×C₈×C₃₇	Q₈	semidirect
75. C₃₇	#2960	semidirect
97. C₂×C₈	Dic₇₄	semidirect

H ≅ G/N

type

2. #14557663

C₂

semidirect

3. C₈×Dic₇₄

C₂

semidirect

27. C₂×C₈×C₃₇

Q₈

semidirect

75. C₃₇

#2960

semidirect

97. C₂×C₈

Dic₇₄

semidirect

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

subgroup	index
#14558090	2
#14557663	2
C₈×Dic₇₄	2
#14557559	2
#14557557	2
#14557488	2
C₂×C₄×C₈×C₃₇	2
#2960	37

subgroup

index

C₈×Dic₇₄

C₂×C₄×C₈×C₃₇

Subgroups — #18003530

Conjugacy classes of subgroups

Complements

Maximal subgroups