Group #17997322 — subgroups

Subgroups — #17997322

Order 4736.

740 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 237 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×D₄×Dic₃₇	C₂	1	2	yes	maximal, complemented
2	#14557436	C₂	1	2	yes	maximal, complemented
3	#14557413	C₂	1	8	yes	maximal, complemented
4	#14557399	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557379	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#127×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₃₇	C₂²	1	4	yes	—
8	D₄×Dic₃₇	—	2	8	no	—
9	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
10	C₂×C₂²⋊Dic₃₇	C₂²	1	4	yes	—
11	C₂×C₂²⋊Dic₃₇	C₂²	1	8	yes	—
12	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
14	#11924443	C₂²	1	4	yes	—
15	#11924443	C₂²	1	4	yes	—
16	C₂²×D₄×C₃₇	C₂²	1	2	yes	—
17	C₂²×D₄×C₃₇	C₂²	1	4	yes	—
18	#127×C₃₇	C₄	1	8	yes	complemented
19	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
20	C₂⁴×C₃₇	C₂³	1	4	yes	—
21	C₂×D₄×C₃₇	—	2	4	no	—
22	C₂×D₄×C₃₇	—	2	4	no	—
23	C₂×D₄×C₃₇	—	4	8	no	—
24	C₂×D₄×C₃₇	C₂×C₄	1	8	yes	complemented
25	C₂×D₄×C₃₇	C₂×C₄	1	16	yes	complemented
26	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
28	C₂²×Dic₃₇	—	2	8	no	—
29	C₂²×Dic₃₇	—	4	8	no	—
30	C₂²×Dic₃₇	C₂³	1	2	yes	—
31	C₂²×Dic₃₇	C₂³	1	4	yes	—
32	C₂²×Dic₃₇	C₂³	1	2	yes	—
33	C₄²×C₃₇	C₂×C₄	1	4	yes	complemented
34	C₂²⋊Dic₃₇	—	2	8	no	—
35	C₂²⋊Dic₃₇	—	4	8	no	—
36	C₄⋊Dic₃₇	—	2	4	no	—
37	C₄×Dic₃₇	—	2	2	no	—
38	C₄×Dic₃₇	D₄	1	4	yes	complemented
39	C₂³×C₃₇	—	2	4	no	—
40	C₂³×C₃₇	—	2	4	no	—
41	C₂³×C₃₇	—	2	8	no	—
42	C₂³×C₃₇	—	2	8	no	—
43	C₂³×C₃₇	C₂²×C₄	1	8	yes	—
44	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
45	D₄×C₃₇	—	2	16	no	—
46	D₄×C₃₇	—	4	16	no	—
47	C₂×C₄×C₃₇	—	2	4	no	—
48	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₃₇	#41	1	2	yes	—
52	C₂×Dic₃₇	—	2	4	no	—
53	C₂×Dic₃₇	—	2	4	no	—
54	C₂×Dic₃₇	—	2	2	no	—
55	C₂×Dic₃₇	—	2	4	no	—
56	C₂×Dic₃₇	—	2	8	no	—
57	C₂×Dic₃₇	—	4	16	no	—
58	C₂×Dic₃₇	C₂×D₄	1	4	yes	—
59	C₂²×C₃₇	—	2	8	no	—
60	C₂²×C₃₇	—	2	16	no	—
61	C₂²×C₃₇	—	4	4	no	—
62	C₂²×C₃₇	—	4	4	no	—
63	C₂²×C₃₇	—	4	8	no	—
64	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
65	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
66	C₂²×C₃₇	C₂×#41	1	1	yes	characteristic
67	C₂²×C₃₇	H(5,𝔽₂)	1	4	yes	—
68	C₄×C₃₇	—	2	4	no	—
69	C₄×C₃₇	C₄×D₄	1	4	yes	—
70	Dic₃₇	—	2	4	no	—
71	Dic₃₇	—	4	4	no	—
72	Dic₃₇	—	4	2	no	—
73	#2328	—	37	1	no	maximal, Sylow
74	C₂×C₃₇	—	4	8	no	—
75	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
76	C₂×C₃₇	#518	1	2	yes	—
77	C₂×C₃₇	#535	1	2	yes	—
78	C₂×C₃₇	#540	1	2	yes	—
79	C₂×#127	D₃₇	1	1	yes	characteristic
80	C₂×C₄×D₄	—	37	2	no	—
81	#390	—	37	2	no	—
82	#386	—	37	8	no	—
83	#382	—	37	1	no	—
84	#376	—	37	1	no	—
85	C₃₇	#2328	1	1	yes	characteristic, complemented, Sylow
86	C₂²×D₄	C₂×D₃₇	1	2	yes	—
87	C₂²×D₄	C₂×D₃₇	1	4	yes	—
88	C₂³×C₄	—	37	4	no	—
89	#127	Dic₃₇	1	8	yes	complemented
90	C₄×D₄	—	74	8	no	—
91	C₂×C₄⋊C₄	—	37	2	no	—
92	C₂×C₂²⋊C₄	—	37	4	no	—
93	C₂×C₂²⋊C₄	—	37	8	no	—
94	C₂×C₄²	—	37	1	no	—
95	C₂×C₄²	—	37	1	no	—
96	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
97	#95	—	37	4	no	—
98	#95	—	37	4	no	—
99	C₂⁴	C₂²×D₃₇	1	4	yes	—
100	C₂×D₄	—	2	4	no	—
101	C₂×D₄	—	2	4	no	—
102	C₂×D₄	—	4	8	no	—
103	C₂×D₄	C₂×Dic₃₇	1	8	yes	complemented
104	C₂×D₄	C₂×Dic₃₇	1	16	yes	complemented
105	C₂²×C₄	—	37	2	no	—
106	C₂²×C₄	—	37	2	no	—
107	C₂²×C₄	—	37	4	no	—
108	C₂²×C₄	—	74	8	no	—
109	C₂²×C₄	—	148	8	no	—
110	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
112	C₄⋊C₄	—	74	4	no	—
113	C₂²⋊C₄	—	74	8	no	—
114	C₂²⋊C₄	—	148	8	no	—
115	C₄²	—	37	4	no	—
116	C₄²	—	74	2	no	—
117	C₄²	C₂×Dic₃₇	1	4	yes	complemented
118	C₂³	—	2	4	no	—
119	C₂³	—	2	4	no	—
120	C₂³	—	2	8	no	—
121	C₂³	—	2	8	no	—
122	C₂³	C₂²×Dic₃₇	1	8	yes	—
123	C₂³	C₂³×D₃₇	1	1	yes	characteristic
124	D₄	—	2	16	no	—
125	D₄	—	4	16	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	37	4	no	—
128	C₂×C₄	—	74	4	no	—
129	C₂×C₄	—	74	4	no	—
130	C₂×C₄	—	74	2	no	—
131	C₂×C₄	—	74	4	no	—
132	C₂×C₄	—	74	8	no	—
133	C₂×C₄	—	148	16	no	—
134	C₂×C₄	D₄×D₃₇	1	2	yes	—
135	C₂×C₄	#10596964	1	2	yes	—
136	C₂×C₄	C₂²×Dic₃₇	1	4	yes	—
137	C₂×C₄	C₂²×Dic₃₇	1	2	yes	—
138	C₂²	—	2	8	no	—
139	C₂²	—	2	16	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	4	4	no	—
142	C₂²	—	4	8	no	—
143	C₂²	C₂³×Dic₃₇	1	1	yes	characteristic
144	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
145	C₂²	C₂×#10596964	1	1	yes	characteristic
146	C₂²	#11924581	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	74	4	no	—
149	C₄	—	148	4	no	—
150	C₄	—	148	2	no	—
151	C₄	D₄×Dic₃₇	1	4	yes	—
152	C₂	—	4	8	no	—
153	C₂	C₂×D₄×Dic₃₇	1	1	yes	characteristic
154	C₂	#14558008	1	2	yes	—
155	C₂	#14558096	1	2	yes	—
156	C₂	#14558137	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×Dic₃₇

C₂

yes

maximal, complemented

#14557436

C₂

yes

maximal, complemented

#14557413

C₂

yes

maximal, complemented

#14557399

C₂

yes

maximal, characteristic, complemented

#14557379

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₃₇

C₂

yes

maximal, characteristic

C₂³×Dic₃₇

C₂²

yes

—

D₄×Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₂²⋊Dic₃₇

C₂²

yes

—

C₂×C₂²⋊Dic₃₇

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924443

C₂²

yes

—

#11924443

C₂²

yes

—

C₂²×D₄×C₃₇

C₂²

yes

—

C₂²×D₄×C₃₇

C₂²

yes

—

#127×C₃₇

C₄

yes

complemented

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂⁴×C₃₇

C₂³

yes

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂×C₄

yes

complemented

C₂×D₄×C₃₇

C₂×C₄

yes

complemented

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

—

C₂²×Dic₃₇

C₂³

yes

—

C₂²×Dic₃₇

C₂³

yes

—

C₂²×Dic₃₇

C₂³

yes

—

C₄²×C₃₇

C₂×C₄

yes

complemented

C₂²⋊Dic₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

complemented

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂²×C₄

yes

—

C₂³×C₃₇

C₂⁴

yes

characteristic

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂²×C₃₇

C₂×#41

yes

characteristic

C₂²×C₃₇

H(5,𝔽₂)

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

#2328

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₄×D₄

yes

characteristic

C₂×C₃₇

#518

yes

—

C₂×C₃₇

#535

yes

—

C₂×C₃₇

#540

yes

—

C₂×#127

D₃₇

yes

characteristic

C₂×C₄×D₄

—

—

—

—

—

C₃₇

#2328

yes

characteristic, complemented, Sylow

C₂²×D₄

C₂×D₃₇

yes

—

C₂²×D₄

C₂×D₃₇

yes

—

C₂³×C₄

—

#127

Dic₃₇

yes

complemented

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

#95

—

#95

—

C₂⁴

C₂²×D₃₇

yes

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

C₂×Dic₃₇

yes

complemented

104

C₂×D₄

C₂×Dic₃₇

yes

complemented

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

148

—

110

C₂²×C₄

C₂²×D₃₇

yes

—

111

C₂²×C₄

C₂²×D₃₇

yes

characteristic

112

C₄⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

148

—

115

C₄²

—

116

C₄²

—

117

C₄²

C₂×Dic₃₇

yes

complemented

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

C₂²×Dic₃₇

yes

—

123

C₂³

C₂³×D₃₇

yes

characteristic

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

148

—

134

C₂×C₄

D₄×D₃₇

yes

—

135

C₂×C₄

#10596964

yes

—

136

C₂×C₄

C₂²×Dic₃₇

yes

—

137

C₂×C₄

C₂²×Dic₃₇

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂³×Dic₃₇

yes

characteristic

144

C₂²

C₂×D₄×D₃₇

yes

characteristic

145

C₂²

C₂×#10596964

yes

characteristic

146

C₂²

#11924581

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

148

—

150

C₄

—

148

—

151

C₄

D₄×Dic₃₇

yes

—

152

C₂

—

153

C₂

C₂×D₄×Dic₃₇

yes

characteristic

154

C₂

#14558008

yes

—

155

C₂

#14558096

yes

—

156

C₂

#14558137

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting, prime radical	C₂×#127×C₃₇	C₂
socle, cosocle	C₂³×C₃₇	C₂⁴
hypercenter	C₂×#127	D₃₇
hypocenter	C₃₇	#2328
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₇

commutator

C₂²×C₃₇

C₂³×C₄

Fitting, prime radical

C₂×#127×C₃₇

C₂

socle, cosocle

C₂³×C₃₇

C₂⁴

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×D₄×Dic₃₇	C₂	semidirect
2. #14557436	C₂	semidirect
3. #14557413	C₂	semidirect
4. #14557399	C₂	semidirect
5. #14557379	C₂	semidirect
18. #127×C₃₇	C₄	semidirect
24. C₂×D₄×C₃₇	C₂×C₄	semidirect
25. C₂×D₄×C₃₇	C₂×C₄	semidirect
33. C₄²×C₃₇	C₂×C₄	semidirect
38. C₄×Dic₃₇	D₄	semidirect
85. C₃₇	#2328	semidirect
89. #127	Dic₃₇	semidirect
103. C₂×D₄	C₂×Dic₃₇	semidirect
104. C₂×D₄	C₂×Dic₃₇	semidirect
117. C₄²	C₂×Dic₃₇	semidirect

H ≅ G/N

type

1. C₂×D₄×Dic₃₇

C₂

semidirect

2. #14557436

C₂

semidirect

3. #14557413

C₂

semidirect

4. #14557399

C₂

semidirect

5. #14557379

C₂

semidirect

18. #127×C₃₇

C₄

semidirect

24. C₂×D₄×C₃₇

C₂×C₄

semidirect

25. C₂×D₄×C₃₇

C₂×C₄

semidirect

33. C₄²×C₃₇

C₂×C₄

semidirect

38. C₄×Dic₃₇

D₄

semidirect

85. C₃₇

#2328

semidirect

89. #127

Dic₃₇

semidirect

103. C₂×D₄

C₂×Dic₃₇

semidirect

104. C₂×D₄

C₂×Dic₃₇

semidirect

117. C₄²

C₂×Dic₃₇

semidirect

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

subgroup	index
C₂×D₄×Dic₃₇	2
#14557436	2
#14557413	2
#14557399	2
#14557379	2
C₂×#127×C₃₇	2
#2328	37

subgroup

index

C₂×D₄×Dic₃₇

C₂×#127×C₃₇

Subgroups — #17997322

Conjugacy classes of subgroups

Complements

Maximal subgroups