Group #17996128 — subgroups

Subgroups — #17996128

Order 4736.

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

normal subgroups only

This group has 290 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	C₂×#11924468	C₂	1	2	yes	maximal, complemented
2	C₂×#11924458	C₂	1	2	yes	maximal
3	#14557435	C₂	1	1	yes	maximal, characteristic
4	#14557403	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₄₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₃₇	—	2	2	no	—
8	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
10	C₂×#10596938	—	2	2	no	—
11	#11924468	—	2	4	no	—
12	#11924458	—	2	4	no	—
13	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	C₂³×D₃₇	—	2	1	no	—
16	C₂²×C₄×C₃₇	D₄	1	2	yes	—
17	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₇	—	2	1	no	—
19	C₂²×Dic₃₇	—	2	2	no	—
20	C₂×D₁₄₈	—	2	2	no	—
21	C₂×D₁₄₈	—	2	2	no	—
22	C₂×D₁₄₈	—	2	1	no	—
23	C₂×D₁₄₈	Q₈	1	2	yes	—
24	C₂×C₈×C₃₇	—	2	2	no	—
25	C₂×C₈×C₃₇	—	2	2	no	—
26	C₂×C₈×C₃₇	D₄	1	4	yes	—
27	C₄²×C₃₇	—	2	2	no	—
28	#10596938	—	4	4	no	—
29	C₄⋊Dic₃₇	—	2	2	no	—
30	C₄⋊Dic₃₇	—	2	2	no	—
31	C₄⋊Dic₃₇	—	2	2	no	—
32	C₄⋊Dic₃₇	—	4	4	no	—
33	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₃₇	—	2	2	no	—
35	C₂²×D₃₇	—	4	1	no	—
36	C₂²×D₃₇	—	4	1	no	—
37	C₂²×D₃₇	—	4	2	no	—
38	C₂²×D₃₇	—	4	2	no	—
39	C₂×C₄×C₃₇	—	2	2	no	—
40	C₂×C₄×C₃₇	—	2	2	no	—
41	C₂×C₄×C₃₇	SD₁₆	1	4	yes	—
42	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₃₇	#41	1	2	yes	—
45	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
46	C₂×Dic₃₇	—	4	1	no	—
47	C₂×Dic₃₇	—	4	2	no	—
48	C₂×Dic₃₇	—	4	2	no	—
49	C₂×Dic₃₇	—	4	4	no	—
50	D₁₄₈	—	2	4	no	—
51	D₁₄₈	—	4	2	no	—
52	C₈×C₃₇	—	2	4	no	—
53	C₂²×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₃₇	#123	1	1	yes	characteristic
55	C₂²×C₃₇	#127	1	1	yes	characteristic
56	C₂²×C₃₇	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
58	C₂×D₃₇	—	4	2	no	—
59	C₂×D₃₇	—	4	4	no	—
60	C₂×D₃₇	—	8	2	no	—
61	C₂×D₃₇	—	8	1	no	—
62	C₂×D₃₇	—	8	1	no	—
63	C₄×C₃₇	—	2	4	no	—
64	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₃₇	#123	1	2	yes	—
66	Dic₃₇	—	8	1	no	—
67	Dic₃₇	—	8	2	no	—
68	#2094	—	37	1	no	maximal, Sylow
69	C₂×C₃₇	#399	1	1	yes	characteristic
70	C₂×C₃₇	#476	1	2	yes	—
71	C₂×C₃₇	#481	1	2	yes	—
72	C₂×C₃₇	#492	1	1	yes	characteristic
73	C₂×C₃₇	#496	1	1	yes	characteristic
74	D₃₇	—	8	2	no	—
75	C₂×#106	—	37	2	no	—
76	C₂×#102	—	37	2	no	—
77	#390	—	37	1	no	—
78	#384	—	37	1	no	—
79	#336	D₃₇	1	1	yes	characteristic, complemented
80	C₃₇	#2094	1	1	yes	characteristic, complemented, Sylow
81	C₂²×D₄	—	37	1	no	—
82	C₂²×C₈	C₂×D₃₇	1	2	yes	—
83	C₂×C₄⋊C₄	—	37	2	no	—
84	C₂×C₄⋊C₄	—	37	1	no	—
85	C₂×C₄⋊C₄	—	74	2	no	—
86	C₂×C₂²⋊C₄	—	74	2	no	—
87	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
88	#106	—	74	4	no	—
89	#102	—	74	4	no	—
90	C₂⁴	—	74	1	no	—
91	C₂×D₄	—	37	2	no	—
92	C₂×D₄	—	74	2	no	—
93	C₂×D₄	—	74	2	no	—
94	C₂×D₄	—	74	1	no	—
95	C₂²×C₄	—	74	1	no	—
96	C₂²×C₄	—	74	2	no	—
97	C₂²×C₄	D₁₄₈	1	2	yes	—
98	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	—	2	2	no	—
101	C₂×C₈	D₁₄₈	1	4	yes	—
102	C₄⋊C₄	—	74	2	no	—
103	C₄⋊C₄	—	74	2	no	—
104	C₄⋊C₄	—	74	2	no	—
105	C₄⋊C₄	—	148	4	no	—
106	C₂²⋊C₄	—	148	4	no	—
107	C₄²	—	2	2	no	—
108	C₂³	—	74	2	no	—
109	C₂³	—	148	1	no	—
110	C₂³	—	148	1	no	—
111	C₂³	—	148	2	no	—
112	C₂³	—	148	2	no	—
113	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
114	D₄	—	74	4	no	—
115	D₄	—	148	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	148	1	no	—
119	C₂×C₄	—	148	2	no	—
120	C₂×C₄	—	148	2	no	—
121	C₂×C₄	—	148	4	no	—
122	C₂×C₄	#10596930	1	4	yes	—
123	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
124	C₂×C₄	#10596964	1	2	yes	—
125	C₂×C₄	#10596964	1	1	yes	characteristic
126	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₂²	—	296	2	no	—
129	C₂²	—	296	1	no	—
130	C₂²	—	296	1	no	—
131	C₂²	—	148	2	no	—
132	C₂²	—	148	4	no	—
133	C₂²	#11924516	1	1	yes	characteristic
134	C₂²	#11924520	1	1	yes	characteristic
135	C₂²	#11924533	1	1	yes	characteristic
136	C₂²	C₂×#10596930	1	2	yes	—
137	C₂²	#11924564	1	2	yes	—
138	C₄	—	2	4	no	—
139	C₄	—	296	1	no	—
140	C₄	—	296	2	no	—
141	C₄	#11924516	1	2	yes	—
142	C₄	#11924520	1	2	yes	—
143	C₂	—	296	2	no	—
144	C₂	#14557470	1	1	yes	characteristic
145	C₂	#14557824	1	2	yes	—
146	C₂	#14557852	1	2	yes	—
147	C₂	#14557906	1	1	yes	characteristic
148	C₂	#14557922	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924468

C₂

yes

maximal, complemented

C₂×#11924458

C₂

yes

maximal

#14557435

C₂

yes

maximal, characteristic

#14557403

C₂

yes

maximal, characteristic, complemented

C₃₇×#336

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₄₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

—

—

C₂²×C₈×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂³×D₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

Q₈

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

C₄²×C₃₇

—

#10596938

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

SD₁₆

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

#41

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

C₈×C₃₇

—

C₂²×C₃₇

C₂²⋊Q₈

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

C₂×SD₁₆

yes

—

C₂²×C₃₇

AGL(1,ℤ/8)

yes

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

#123

yes

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#481

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

—

—

—

—

D₃₇

yes

characteristic, complemented

C₃₇

#2094

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×C₈

C₂×D₃₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#106

—

#102

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₄₈

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

D₁₄₈

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

148

—

106

C₂²⋊C₄

—

148

—

107

C₄²

—

108

C₂³

—

109

C₂³

—

148

—

110

C₂³

—

148

—

111

C₂³

—

148

—

112

C₂³

—

148

—

113

C₂³

C₂×D₁₄₈

yes

characteristic

114

D₄

—

115

D₄

—

148

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

148

—

119

C₂×C₄

—

148

—

120

C₂×C₄

—

148

—

121

C₂×C₄

—

148

—

122

C₂×C₄

#10596930

yes

—

123

C₂×C₄

C₂×D₁₄₈

yes

—

124

C₂×C₄

#10596964

yes

—

125

C₂×C₄

#10596964

yes

characteristic

126

C₂×C₄

Q₈×D₃₇

yes

characteristic

127

C₈

—

128

C₂²

—

296

—

129

C₂²

—

296

—

130

C₂²

—

296

—

131

C₂²

—

148

—

132

C₂²

—

148

—

133

C₂²

#11924516

yes

characteristic

134

C₂²

#11924520

yes

characteristic

135

C₂²

#11924533

yes

characteristic

136

C₂²

C₂×#10596930

yes

—

137

C₂²

#11924564

yes

—

138

C₄

—

139

C₄

—

296

—

140

C₄

—

296

—

141

C₄

#11924516

yes

—

142

C₄

#11924520

yes

—

143

C₂

—

296

—

144

C₂

#14557470

yes

characteristic

145

C₂

#14557824

yes

—

146

C₂

#14557852

yes

—

147

C₂

#14557906

yes

characteristic

148

C₂

#14557922

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₄₈
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#336	C₂
socle	C₂³×C₃₇	C₂×D₄
prime radical	C₂²×D₁₄₈	C₂²
hypercenter	#336	D₃₇
hypocenter	C₃₇	#2094
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₁₄₈

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#336

C₂

socle

C₂³×C₃₇

C₂×D₄

prime radical

C₂²×D₁₄₈

C₂²

D₃₇

C₃₇

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
1. C₂×#11924468	C₂	semidirect
4. #14557403	C₂	semidirect
5. C₃₇×#336	C₂	semidirect
79. #336	D₃₇	semidirect
80. C₃₇	#2094	semidirect

H ≅ G/N

type

1. C₂×#11924468

C₂

semidirect

4. #14557403

C₂

semidirect

5. C₃₇×#336

C₂

semidirect

79. #336

D₃₇

semidirect

80. C₃₇

#2094

semidirect

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

subgroup	index
C₂×#11924468	2
C₂×#11924458	2
#14557435	2
#14557403	2
C₃₇×#336	2
#2094	37

subgroup

index

C₃₇×#336

Subgroups — #17996128

Conjugacy classes of subgroups

Complements

Maximal subgroups