Group #17995535 — subgroups

Subgroups — #17995535

Order 4736.

280 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×C₄×Dic₇₄	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924447	C₂	1	1	yes	maximal, characteristic
3	#14557287	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₇₄	C₄	1	1	yes	characteristic
6	C₂²×#64436	C₂²	1	2	yes	—
7	C₄×Dic₇₄	—	2	2	no	—
8	C₄×Dic₇₄	—	2	2	no	—
9	C₂×C₄⋊Dic₃₇	C₄	1	1	yes	characteristic
10	C₂×#10596936	C₄	1	2	yes	—
11	C₂×C₄×Dic₃₇	C₂²	1	2	yes	—
12	#11924447	—	2	2	no	—
13	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₃₇	—	2	2	no	—
16	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
20	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
21	C₂×Dic₇₄	—	2	2	no	—
22	C₂×Dic₇₄	—	2	4	no	—
23	C₂×C₈×C₃₇	—	2	2	no	—
24	C₂×C₈×C₃₇	—	2	4	no	—
25	C₄²×C₃₇	D₄	1	2	yes	—
26	C₄²×C₃₇	D₄	1	2	yes	—
27	C₄⋊Dic₃₇	—	2	2	no	—
28	#10596936	—	2	4	no	—
29	C₄×Dic₃₇	—	2	2	no	—
30	C₄×Dic₃₇	—	2	2	no	—
31	C₂×#64436	—	2	2	no	—
32	C₂×#64436	—	2	2	no	—
33	C₂×#64436	Q₈	1	4	yes	complemented
34	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₃₇	—	2	2	no	—
36	C₂×C₄×C₃₇	—	2	2	no	—
37	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₃₇	#41	1	2	yes	—
43	C₂×Dic₃₇	—	2	2	no	—
44	C₂×Dic₃₇	—	2	2	no	—
45	C₂×Dic₃₇	—	2	2	no	—
46	C₂×Dic₃₇	—	2	4	no	—
47	C₂×Dic₃₇	M₁₆	1	4	yes	—
48	Dic₇₄	—	2	8	no	—
49	C₈×C₃₇	—	4	2	no	—
50	#64436	—	2	4	no	—
51	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₃₇	C₂×M₁₆	1	2	yes	—
54	C₂²×C₃₇	#131	1	2	yes	—
55	C₄×C₃₇	—	2	2	no	—
56	C₄×C₃₇	—	2	2	no	—
57	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₃₇	#124	1	1	yes	characteristic
59	C₄×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₃₇	—	2	4	no	—
61	Dic₃₇	—	4	2	no	—
62	#1985	—	37	1	no	maximal, Sylow
63	C₂×C₃₇	#391	1	1	yes	characteristic
64	C₂×C₃₇	#407	1	1	yes	characteristic
65	C₂×C₃₇	#408	1	1	yes	characteristic
66	C₂×C₃₇	#446	1	4	yes	—
67	C₂×C₄×Q₈	—	37	1	no	—
68	C₂×C₄⋊C₈	D₃₇	1	1	yes	characteristic
69	C₂×#97	—	37	1	no	—
70	#336	—	37	4	no	—
71	C₃₇	#1985	1	1	yes	characteristic, complemented, Sylow
72	C₂²×Q₈	—	37	1	no	—
73	C₂²×C₈	—	37	2	no	—
74	C₂²×C₈	C₂×D₃₇	1	2	yes	—
75	C₄×Q₈	—	74	2	no	—
76	C₄×Q₈	—	74	2	no	—
77	C₂×C₄⋊C₄	—	37	2	no	—
78	C₂×C₄⋊C₄	—	37	1	no	—
79	C₂×C₄²	—	37	2	no	—
80	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#97	—	74	2	no	—
83	C₂×Q₈	—	74	4	no	—
84	C₂×Q₈	—	74	2	no	—
85	C₂²×C₄	—	37	2	no	—
86	C₂²×C₄	—	37	2	no	—
87	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
88	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	37	4	no	—
93	C₂×C₈	—	74	2	no	—
94	C₂×C₈	—	74	2	no	—
95	C₄⋊C₄	—	74	4	no	—
96	C₄⋊C₄	—	74	2	no	—
97	C₄²	—	74	2	no	—
98	C₄²	—	74	2	no	—
99	C₄²	D₁₄₈	1	2	yes	—
100	C₄²	C₂²⋊D₃₇	1	2	yes	—
101	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
102	Q₈	—	74	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	37	4	no	—
106	C₂×C₄	—	74	2	no	—
107	C₂×C₄	—	74	2	no	—
108	C₂×C₄	—	74	2	no	—
109	C₂×C₄	—	74	4	no	—
110	C₂×C₄	#10596938	1	2	yes	—
111	C₂×C₄	#10596938	1	2	yes	—
112	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
113	C₂×C₄	#10596964	1	2	yes	—
114	C₂×C₄	Q₈×D₃₇	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
116	C₈	—	4	2	no	—
117	C₈	—	74	4	no	—
118	C₂²	C₂×#10596938	1	1	yes	characteristic
119	C₂²	#11924501	1	2	yes	—
120	C₂²	M₁₆×D₃₇	1	2	yes	—
121	C₂²	#11924542	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	74	4	no	—
125	C₄	—	148	2	no	—
126	C₄	#11924516	1	2	yes	—
127	C₄	#11924523	1	1	yes	characteristic
128	C₄	#11924534	1	1	yes	characteristic
129	C₂	#14557438	1	1	yes	characteristic
130	C₂	#14557496	1	1	yes	characteristic
131	C₂	#14557499	1	1	yes	characteristic
132	C₂	#14557662	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₇₄

C₂

yes

maximal, characteristic

C₂×#11924447

C₂

yes

maximal, characteristic

#14557287

C₂

yes

maximal

C₂×C₄⋊C₈×C₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₄

yes

characteristic

C₂²×#64436

C₂²

yes

—

C₄×Dic₇₄

—

C₄×Dic₇₄

—

C₂×C₄⋊Dic₃₇

C₄

yes

characteristic

C₂×#10596936

C₄

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

#11924447

—

C₂²×C₈×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

D₄

yes

—

C₄⋊Dic₃₇

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

Q₈

yes

complemented

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

M₁₆

yes

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×Q₈

yes

—

C₂²×C₃₇

C₂×M₁₆

yes

—

C₂²×C₃₇

#131

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

#124

yes

characteristic

C₄×C₃₇

C₄⋊Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

D₃₇

yes

characteristic

C₂×#97

—

#336

—

C₃₇

#1985

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃₇

yes

characteristic

C₂²×C₄

C₄×D₃₇

yes

characteristic

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

D₁₄₈

yes

—

100

C₄²

C₂²⋊D₃₇

yes

—

101

C₂³

C₂×C₄×D₃₇

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

#10596938

yes

—

111

C₂×C₄

#10596938

yes

—

112

C₂×C₄

C₂×D₁₄₈

yes

characteristic

113

C₂×C₄

#10596964

yes

—

114

C₂×C₄

Q₈×D₃₇

yes

—

115

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

116

C₈

—

117

C₈

—

118

C₂²

C₂×#10596938

yes

characteristic

119

C₂²

#11924501

yes

—

120

C₂²

M₁₆×D₃₇

yes

—

121

C₂²

#11924542

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

148

—

126

C₄

#11924516

yes

—

127

C₄

#11924523

yes

characteristic

128

C₄

#11924534

yes

characteristic

129

C₂

#14557438

yes

characteristic

130

C₂

#14557496

yes

characteristic

131

C₂

#14557499

yes

characteristic

132

C₂

#14557662

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₇
commutator, socle, prime radical	C₂³×C₃₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₃₇	C₂
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₃₇
hypocenter	C₃₇	#1985
LCS	G ▷ C₂³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₇

commutator, socle, prime radical

C₂³×C₃₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₃₇

C₂

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₃₇

C₃₇

G ▷ C₂³×C₃₇ ▷ C₃₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
33. C₂×#64436	Q₈	semidirect
71. C₃₇	#1985	semidirect

H ≅ G/N

type

33. C₂×#64436

Q₈

semidirect

71. C₃₇

#1985

semidirect

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

subgroup	index
C₂×C₄×Dic₇₄	2
C₂×#11924447	2
#14557287	2
C₂×C₄⋊C₈×C₃₇	2
#1985	37

subgroup

index

C₂×C₄×Dic₇₄

C₂×#11924447

#14557287

C₂×C₄⋊C₈×C₃₇

#1985

Subgroups — #17995535

Conjugacy classes of subgroups

Complements

Maximal subgroups