Group #17996153 — subgroups

Subgroups — #17996153

Order 4736.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#14557431	C₂	1	1	yes	maximal, characteristic
2	#14557295	C₂	1	2	yes	maximal
3	#14557295	C₂	1	2	yes	maximal
4	#14557288	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64436	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₃₇	C₂²	1	1	yes	characteristic
9	C₂×#10596936	—	2	1	no	—
10	#11924443	—	2	1	no	—
11	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
12	C₂×C₄⋊C₄×C₃₇	—	2	1	no	—
13	C₂×C₄⋊C₄×C₃₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄×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	1	no	—
20	C₂²×Dic₃₇	—	2	1	no	—
21	C₄⋊C₄×C₃₇	—	2	4	no	—
22	C₄⋊C₄×C₃₇	—	4	2	no	—
23	C₄⋊C₄×C₃₇	—	4	2	no	—
24	C₄²×C₃₇	—	2	2	no	—
25	C₄⋊Dic₃₇	—	2	1	no	—
26	C₄⋊Dic₃₇	—	2	2	no	—
27	C₄⋊Dic₃₇	Q₈	1	2	yes	—
28	#10596936	—	4	2	no	—
29	C₂×#64436	—	2	2	no	—
30	C₂×#64436	—	2	2	no	—
31	C₂×#64436	—	2	2	no	—
32	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₃₇	—	2	1	no	—
34	C₂×C₄×C₃₇	—	2	1	no	—
35	C₂×C₄×C₃₇	—	2	1	no	—
36	C₂×C₄×C₃₇	—	2	1	no	—
37	C₂×C₄×C₃₇	—	4	2	no	—
38	C₂×C₄×C₃₇	—	4	2	no	—
39	C₂×C₄×C₃₇	—	4	2	no	—
40	C₂×C₄×C₃₇	D₈	1	2	yes	—
41	C₂×C₄×C₃₇	SD₁₆	1	2	yes	—
42	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₃₇	#41	1	2	yes	—
44	C₂×C₄×C₃₇	#41	1	2	yes	—
45	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
46	C₂×Dic₃₇	—	2	2	no	—
47	C₂×Dic₃₇	—	4	1	no	—
48	C₂×Dic₃₇	—	4	1	no	—
49	C₂×Dic₃₇	—	4	1	no	—
50	C₂×Dic₃₇	—	4	1	no	—
51	C₂×Dic₃₇	—	4	1	no	—
52	#64436	—	4	2	no	—
53	C₂²×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₃₇	#123	1	1	yes	characteristic
55	C₂²×C₃₇	#124	1	1	yes	characteristic
56	C₂²×C₃₇	C₂×D₈	1	1	yes	characteristic
57	C₂²×C₃₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂²×C₃₇	#137	1	1	yes	characteristic
59	C₂²×C₃₇	#137	1	1	yes	characteristic
60	C₄×C₃₇	—	2	2	no	—
61	C₄×C₃₇	—	2	2	no	—
62	C₄×C₃₇	—	8	2	no	—
63	C₄×C₃₇	#125	1	2	yes	—
64	C₄×C₃₇	#126	1	2	yes	—
65	Dic₃₇	—	4	2	no	—
66	Dic₃₇	—	8	1	no	—
67	#2099	—	37	1	no	maximal, Sylow
68	C₂×C₃₇	#400	1	1	yes	characteristic
69	C₂×C₃₇	#474	1	1	yes	characteristic
70	C₂×C₃₇	#475	1	1	yes	characteristic
71	C₂×C₃₇	#480	1	1	yes	characteristic
72	C₂×C₃₇	#483	1	1	yes	characteristic
73	C₂×C₃₇	#486	1	1	yes	characteristic
74	C₂×C₃₇	#491	1	1	yes	characteristic
75	#389	—	37	1	no	—
76	#384	D₃₇	1	1	yes	characteristic
77	#340	—	37	2	no	—
78	#340	—	37	2	no	—
79	#336	—	37	1	no	—
80	C₃₇	#2099	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	37	2	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	37	1	no	—
85	C₂×C₄⋊C₄	—	37	1	no	—
86	C₂×C₄⋊C₄	—	74	1	no	—
87	C₂×C₄⋊C₄	C₂×D₃₇	1	2	yes	—
88	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
89	#95	—	74	1	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	74	1	no	—
92	C₂²×C₄	—	74	1	no	—
93	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
94	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
96	C₂×C₈	—	74	2	no	—
97	C₂×C₈	—	74	2	no	—
98	C₂×C₈	—	74	2	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	37	2	no	—
103	C₄⋊C₄	—	74	1	no	—
104	C₄⋊C₄	—	74	2	no	—
105	C₄⋊C₄	—	148	2	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	74	2	no	—
116	C₂×C₄	—	148	1	no	—
117	C₂×C₄	—	148	1	no	—
118	C₂×C₄	—	148	1	no	—
119	C₂×C₄	—	148	1	no	—
120	C₂×C₄	—	148	1	no	—
121	C₂×C₄	#10596939	1	2	yes	—
122	C₂×C₄	#10596940	1	2	yes	—
123	C₂×C₄	#10596962	1	2	yes	—
124	C₂×C₄	#10596964	1	2	yes	—
125	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
126	C₂×C₄	#10596966	1	1	yes	characteristic
127	C₈	—	148	2	no	—
128	C₂²	#11924512	1	1	yes	characteristic
129	C₂²	#11924519	1	1	yes	characteristic
130	C₂²	#11924523	1	1	yes	characteristic
131	C₂²	C₂×#10596939	1	1	yes	characteristic
132	C₂²	C₂×#10596940	1	1	yes	characteristic
133	C₂²	#11924567	1	1	yes	characteristic
134	C₂²	#11924567	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	8	2	no	—
138	C₄	—	296	1	no	—
139	C₄	—	148	2	no	—
140	C₄	#11924526	1	2	yes	—
141	C₄	#11924531	1	2	yes	—
142	C₂	#14557471	1	1	yes	characteristic
143	C₂	#14557810	1	1	yes	characteristic
144	C₂	#14557815	1	1	yes	characteristic
145	C₂	#14557846	1	1	yes	characteristic
146	C₂	#14557863	1	1	yes	characteristic
147	C₂	#14557879	1	1	yes	characteristic
148	C₂	#14557901	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557431

C₂

yes

maximal, characteristic

#14557295

C₂

yes

maximal

#14557295

C₂

yes

maximal

#14557288

C₂

yes

maximal, characteristic

C₃₇×#384

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₇

C₂²

yes

characteristic

C₂×#10596936

—

#11924443

—

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

Q₈

yes

—

—

—

—

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

D₈

yes

—

C₂×C₄×C₃₇

SD₁₆

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

#41

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

#64436

—

C₂²×C₃₇

C₂²⋊Q₈

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

C₂×D₈

yes

characteristic

C₂²×C₃₇

C₂×SD₁₆

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#125

yes

—

C₄×C₃₇

#126

yes

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

yes

—

—

—

C₃₇

#2099

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₇

yes

characteristic

C₂²×C₄

C₂²⋊D₃₇

yes

characteristic

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

148

—

106

C₄²

—

107

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

148

—

117

C₂×C₄

—

148

—

118

C₂×C₄

—

148

—

119

C₂×C₄

—

148

—

120

C₂×C₄

—

148

—

121

C₂×C₄

#10596939

yes

—

122

C₂×C₄

#10596940

yes

—

123

C₂×C₄

#10596962

yes

—

124

C₂×C₄

#10596964

yes

—

125

C₂×C₄

Q₈×D₃₇

yes

characteristic

126

C₂×C₄

#10596966

yes

characteristic

127

C₈

—

148

—

128

C₂²

#11924512

yes

characteristic

129

C₂²

#11924519

yes

characteristic

130

C₂²

#11924523

yes

characteristic

131

C₂²

C₂×#10596939

yes

characteristic

132

C₂²

C₂×#10596940

yes

characteristic

133

C₂²

#11924567

yes

characteristic

134

C₂²

#11924567

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

296

—

139

C₄

—

148

—

140

C₄

#11924526

yes

—

141

C₄

#11924531

yes

—

142

C₂

#14557471

yes

characteristic

143

C₂

#14557810

yes

characteristic

144

C₂

#14557815

yes

characteristic

145

C₂

#14557846

yes

characteristic

146

C₂

#14557863

yes

characteristic

147

C₂

#14557879

yes

characteristic

148

C₂

#14557901

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#384	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	#384	D₃₇
hypocenter	C₃₇	#2099
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#384

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
80. C₃₇	#2099	semidirect

H ≅ G/N

type

80. C₃₇

#2099

semidirect

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

subgroup	index
#14557431	2
#14557295	2
#14557288	2
C₃₇×#384	2
#2099	37

subgroup

index

C₃₇×#384

Subgroups — #17996153

Conjugacy classes of subgroups

Complements

Maximal subgroups