Group #17995339 — subgroups

Subgroups — #17995339

Order 4736.

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

normal subgroups only

This group has 240 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×#11924470	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924468	C₂	1	1	yes	maximal, characteristic
3	#14557431	C₂	1	2	yes	maximal
4	#14557296	C₂	1	2	yes	maximal
5	C₂×#97×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
8	C₂×#10596936	—	2	2	no	—
9	#11924470	—	2	2	no	—
10	#11924468	C₄	1	4	yes	complemented
11	#11924443	—	2	2	no	—
12	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	#97×C₃₇	—	2	2	no	—
16	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₃₇	—	2	2	no	—
20	C₂²×Dic₃₇	—	2	2	no	—
21	C₂×C₈×C₃₇	—	2	1	no	—
22	C₂×C₈×C₃₇	—	2	1	no	—
23	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₃₇	D₄	1	2	yes	—
25	C₂×C₈×C₃₇	Q₈	1	2	yes	—
26	C₂×C₈×C₃₇	Q₈	1	2	yes	—
27	C₄²×C₃₇	—	2	1	no	—
28	C₄²×C₃₇	—	2	1	no	—
29	C₄⋊Dic₃₇	—	2	2	no	—
30	C₄⋊Dic₃₇	—	2	2	no	—
31	C₄⋊Dic₃₇	—	2	2	no	—
32	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
33	#10596936	—	4	4	no	—
34	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₃₇	—	2	1	no	—
36	C₂×C₄×C₃₇	—	2	1	no	—
37	C₂×C₄×C₃₇	—	2	2	no	—
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
44	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
45	C₂×Dic₃₇	—	2	4	no	—
46	C₂×Dic₃₇	—	4	2	no	—
47	C₂×Dic₃₇	—	4	2	no	—
48	C₂×Dic₃₇	—	4	2	no	—
49	C₂×Dic₃₇	—	4	2	no	—
50	C₂×Dic₃₇	—	4	2	no	—
51	C₈×C₃₇	—	2	2	no	—
52	C₈×C₃₇	C₄⋊C₄	1	4	yes	complemented
53	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₃₇	#121	1	1	yes	characteristic
55	C₂²×C₃₇	C₄⋊Q₈	1	1	yes	characteristic
56	C₂²×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
57	C₂²×C₃₇	#137	1	2	yes	—
58	C₄×C₃₇	—	4	1	no	—
59	C₄×C₃₇	—	4	1	no	—
60	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₄×C₃₇	C₄×Q₈	1	1	yes	characteristic
62	C₄×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₄×C₃₇	#125	1	1	yes	characteristic
64	Dic₃₇	—	4	4	no	—
65	Dic₃₇	—	8	2	no	—
66	#1944	—	37	1	no	maximal, Sylow
67	C₂×C₃₇	#389	1	1	yes	characteristic
68	C₂×C₃₇	#440	1	2	yes	—
69	C₂×C₃₇	#469	1	1	yes	characteristic
70	C₂×C₃₇	#470	1	1	yes	characteristic
71	C₂×C₃₇	#501	1	2	yes	—
72	C₂×#107	—	37	1	no	—
73	C₂×#106	—	37	1	no	—
74	C₂×#97	D₃₇	1	1	yes	characteristic
75	#389	—	37	2	no	—
76	#340	—	37	2	no	—
77	C₃₇	#1944	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
79	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
80	C₂×C₄⋊C₄	—	37	2	no	—
81	C₂×C₄⋊C₄	—	37	2	no	—
82	C₂×C₄⋊C₄	—	74	2	no	—
83	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
84	#107	—	74	2	no	—
85	#106	—	37	4	no	—
86	#97	—	2	2	no	—
87	#95	—	74	2	no	—
88	C₂²×C₄	—	74	2	no	—
89	C₂²×C₄	—	74	2	no	—
90	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
91	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	Dic₇₄	1	2	yes	—
96	C₂×C₈	Dic₇₄	1	2	yes	—
97	C₂×C₈	C₄×D₃₇	1	2	yes	—
98	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
99	C₄⋊C₄	—	37	4	no	—
100	C₄⋊C₄	—	74	2	no	—
101	C₄⋊C₄	—	74	2	no	—
102	C₄⋊C₄	—	74	2	no	—
103	C₄⋊C₄	—	148	4	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	1	no	—
106	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	74	4	no	—
112	C₂×C₄	—	148	2	no	—
113	C₂×C₄	—	148	2	no	—
114	C₂×C₄	—	148	2	no	—
115	C₂×C₄	—	148	2	no	—
116	C₂×C₄	—	148	2	no	—
117	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
118	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
119	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
120	C₂×C₄	#10596962	1	1	yes	characteristic
121	C₂×C₄	#10596962	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	#10596936	1	4	yes	complemented
125	C₂²	C₄×D₁₄₈	1	1	yes	characteristic
126	C₂²	#11924510	1	1	yes	characteristic
127	C₂²	#11924536	1	1	yes	characteristic
128	C₂²	#11924564	1	2	yes	—
129	C₂²	#11924569	1	2	yes	—
130	C₄	—	4	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	296	2	no	—
133	C₄	—	148	4	no	—
134	C₄	C₂×#10596936	1	1	yes	characteristic
135	C₄	C₄×Dic₇₄	1	1	yes	characteristic
136	C₄	#11924515	1	1	yes	characteristic
137	C₄	#11924528	1	1	yes	characteristic
138	C₂	#14557431	1	1	yes	characteristic
139	C₂	#14557635	1	2	yes	—
140	C₂	#14557786	1	1	yes	characteristic
141	C₂	#14557791	1	1	yes	characteristic
142	C₂	#14557942	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924470

C₂

yes

maximal, characteristic

C₂×#11924468

C₂

yes

maximal, characteristic

#14557431

C₂

yes

maximal

#14557296

C₂

yes

maximal

C₂×#97×C₃₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

—

#11924470

—

#11924468

C₄

yes

complemented

#11924443

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

Q₈

yes

—

C₂×C₈×C₃₇

Q₈

yes

—

C₄²×C₃₇

—

C₄²×C₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₈×C₃₇

—

C₈×C₃₇

C₄⋊C₄

yes

complemented

C₂²×C₃₇

C₄×D₄

yes

characteristic

C₂²×C₃₇

#121

yes

characteristic

C₂²×C₃₇

C₄⋊Q₈

yes

characteristic

C₂²×C₃₇

AGL(1,ℤ/8)

yes

—

C₂²×C₃₇

#137

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₄×Q₈

yes

characteristic

C₄×C₃₇

C₂²⋊Q₈

yes

characteristic

C₄×C₃₇

#125

yes

characteristic

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×#107

—

C₂×#106

—

C₂×#97

D₃₇

yes

characteristic

#389

—

#340

—

C₃₇

#1944

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₄₈

yes

characteristic

C₂²×C₄

D₁₄₈

yes

characteristic

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₇₄

yes

—

C₂×C₈

Dic₇₄

yes

—

C₂×C₈

C₄×D₃₇

yes

—

C₂×C₈

C₂²⋊D₃₇

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

148

—

104

C₄²

—

105

C₄²

—

106

C₂³

C₂×D₁₄₈

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

148

—

113

C₂×C₄

—

148

—

114

C₂×C₄

—

148

—

115

C₂×C₄

—

148

—

116

C₂×C₄

—

148

—

117

C₂×C₄

C₂×Dic₇₄

yes

characteristic

118

C₂×C₄

C₂×Dic₇₄

yes

characteristic

119

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

120

C₂×C₄

#10596962

yes

characteristic

121

C₂×C₄

#10596962

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

123

C₈

—

124

C₈

#10596936

yes

complemented

125

C₂²

C₄×D₁₄₈

yes

characteristic

126

C₂²

#11924510

yes

characteristic

127

C₂²

#11924536

yes

characteristic

128

C₂²

#11924564

yes

—

129

C₂²

#11924569

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

296

—

133

C₄

—

148

—

134

C₄

C₂×#10596936

yes

characteristic

135

C₄

C₄×Dic₇₄

yes

characteristic

136

C₄

#11924515

yes

characteristic

137

C₄

#11924528

yes

characteristic

138

C₂

#14557431

yes

characteristic

139

C₂

#14557635

yes

—

140

C₂

#14557786

yes

characteristic

141

C₂

#14557791

yes

characteristic

142

C₂

#14557942

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₄₈
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂×#97×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×#97	D₃₇
hypocenter	C₃₇	#1944
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₄₈

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂×#97×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₂×#97

D₃₇

hypocenter

C₃₇

#1944

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
10. #11924468	C₄	semidirect
52. C₈×C₃₇	C₄⋊C₄	semidirect
77. C₃₇	#1944	semidirect
124. C₈	#10596936	semidirect

H ≅ G/N

type

10. #11924468

C₄

semidirect

52. C₈×C₃₇

C₄⋊C₄

semidirect

77. C₃₇

#1944

semidirect

124. C₈

#10596936

semidirect

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

subgroup	index
C₂×#11924470	2
C₂×#11924468	2
#14557431	2
#14557296	2
C₂×#97×C₃₇	2
#1944	37

subgroup

index

C₂×#97×C₃₇

Subgroups — #17995339

Conjugacy classes of subgroups

Complements

Maximal subgroups