Group #18005127 — subgroups

Subgroups — #18005127

Order 4736.

340 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 165 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	#14558161	C₂	1	2	yes	maximal
2	#14557942	C₂	1	4	yes	maximal, complemented
3	#14557935	C₂	1	2	yes	maximal, complemented
4	#14557931	C₂	1	2	yes	maximal, complemented
5	#14557575	C₂	1	2	yes	maximal
6	#14557571	C₂	1	1	yes	maximal, characteristic
7	C₂×#11924468	C₂	1	1	yes	maximal, characteristic
8	C₃₇×#405	C₂	1	1	yes	maximal, characteristic
9	#11924536	C₂²	1	4	yes	—
10	#11924535	—	2	2	no	—
11	#11924528	C₂²	1	4	yes	—
12	#11924527	—	2	2	no	—
13	#11924513	—	2	4	no	—
14	#11924495	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
16	#11924470	C₂²	1	2	yes	—
17	#11924470	C₂²	1	4	yes	—
18	#11924468	C₂²	1	2	yes	—
19	#11924468	C₂²	1	2	yes	—
20	#11924468	C₂²	1	1	yes	characteristic
21	#11924468	C₂²	1	1	yes	characteristic
22	#11924468	C₂²	1	4	yes	—
23	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
25	#117×C₃₇	C₂²	1	1	yes	characteristic
26	#97×C₃₇	C₂²	1	2	yes	—
27	C₄×C₈×C₃₇	C₂²	1	2	yes	—
28	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
29	C₂²×Dic₃₇	—	2	2	no	—
30	C₂×Dic₇₄	—	2	4	no	—
31	M₁₆×C₃₇	Q₈	1	4	yes	complemented
32	C₂×C₈×C₃₇	Q₈	1	4	yes	complemented
33	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₃₇	C₂³	1	2	yes	—
35	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₃₇	D₄	1	2	yes	—
37	C₂²⋊C₄×C₃₇	D₄	1	2	yes	—
38	C₄²×C₃₇	C₂³	1	2	yes	—
39	C₂²⋊Dic₃₇	—	2	2	no	—
40	C₄⋊Dic₃₇	—	2	2	no	—
41	C₄⋊Dic₃₇	—	2	4	no	—
42	C₄⋊Dic₃₇	C₂³	1	4	yes	—
43	C₄⋊Dic₃₇	C₂³	1	2	yes	—
44	C₄⋊Dic₃₇	C₂³	1	2	yes	—
45	#10596936	—	2	4	no	—
46	#10596936	—	2	4	no	—
47	C₄×Dic₃₇	—	2	2	no	—
48	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₃₇	—	2	2	no	—
55	C₂×Dic₃₇	—	2	2	no	—
56	C₂×Dic₃₇	—	2	4	no	—
57	C₂×Dic₃₇	—	8	2	no	—
58	Dic₇₄	—	8	4	no	—
59	C₈×C₃₇	C₂×Q₈	1	2	yes	—
60	C₈×C₃₇	C₂×Q₈	1	2	yes	—
61	C₈×C₃₇	C₂×Q₈	1	4	yes	—
62	C₂²×C₃₇	—	2	2	no	—
63	C₂²×C₃₇	C₄⋊Q₈	1	2	yes	—
64	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₃₇	—	2	4	no	—
66	C₄×C₃₇	C₄⋊Q₈	1	2	yes	—
67	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
69	Dic₃₇	—	4	2	no	—
70	Dic₃₇	—	4	4	no	—
71	Dic₃₇	—	4	2	no	—
72	#3163	—	37	1	no	maximal, Sylow
73	C₂×C₃₇	—	2	2	no	—
74	C₂×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₃₇	#577	1	2	yes	—
76	#544	—	37	2	no	—
77	#501	—	37	4	no	—
78	#499	—	37	2	no	—
79	#498	—	37	2	no	—
80	#428	—	37	2	no	—
81	#427	—	37	1	no	—
82	C₂×#106	—	37	1	no	—
83	#405	D₃₇	1	1	yes	characteristic
84	C₃₇	#3163	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
87	C₄⋊Q₈	—	37	4	no	—
88	C₄⋊Q₈	—	74	2	no	—
89	#125	—	37	4	no	—
90	#125	—	74	2	no	—
91	C₂²⋊Q₈	—	74	4	no	—
92	#117	—	37	2	no	—
93	#117	C₂×D₃₇	1	1	yes	characteristic
94	C₂×C₄⋊C₄	—	37	2	no	—
95	#107	—	37	2	no	—
96	#107	—	37	4	no	—
97	#106	—	37	2	no	—
98	#106	—	37	2	no	—
99	#106	—	37	1	no	—
100	#106	—	37	1	no	—
101	#106	—	37	4	no	—
102	#97	C₂×D₃₇	1	2	yes	—
103	C₄×C₈	C₂×D₃₇	1	2	yes	—
104	C₂×Q₈	—	74	4	no	—
105	C₂²×C₄	—	74	2	no	—
106	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
107	M₁₆	Dic₇₄	1	4	yes	complemented
108	C₂×C₈	Dic₇₄	1	4	yes	complemented
109	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
110	C₂×C₈	C₂²×D₃₇	1	2	yes	—
111	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
112	C₄⋊C₄	—	37	4	no	—
113	C₄⋊C₄	—	37	2	no	—
114	C₄⋊C₄	—	37	2	no	—
115	C₄⋊C₄	—	74	4	no	—
116	C₄⋊C₄	—	74	4	no	—
117	C₄⋊C₄	—	74	2	no	—
118	C₄⋊C₄	—	74	4	no	—
119	C₄⋊C₄	D₁₄₈	1	2	yes	—
120	C₂²⋊C₄	—	74	2	no	—
121	C₂²⋊C₄	D₁₄₈	1	2	yes	—
122	C₄²	—	74	2	no	—
123	C₄²	C₂²×D₃₇	1	2	yes	—
124	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
125	Q₈	—	296	4	no	—
126	C₂×C₄	—	296	2	no	—
127	C₂×C₄	—	74	2	no	—
128	C₂×C₄	—	74	2	no	—
129	C₂×C₄	—	74	4	no	—
130	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
131	C₂×C₄	C₂×Dic₇₄	1	2	yes	—
132	C₂×C₄	C₂×D₁₄₈	1	4	yes	—
133	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
134	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
135	C₈	C₂×Dic₇₄	1	2	yes	—
136	C₈	C₂×Dic₇₄	1	2	yes	—
137	C₈	C₂×Dic₇₄	1	4	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	#11924536	1	2	yes	—
140	C₂²	C₂²×D₁₄₈	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	148	4	no	—
143	C₄	—	148	2	no	—
144	C₄	—	148	2	no	—
145	C₄	#11924536	1	2	yes	—
146	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
147	C₄	C₂²×Dic₇₄	1	1	yes	characteristic
148	C₂	—	2	2	no	—
149	C₂	C₂×#11924536	1	1	yes	characteristic
150	C₂	#14558345	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558161

C₂

yes

maximal

#14557942

C₂

yes

maximal, complemented

#14557935

C₂

yes

maximal, complemented

#14557931

C₂

yes

maximal, complemented

#14557575

C₂

yes

maximal

#14557571

C₂

yes

maximal, characteristic

C₂×#11924468

C₂

yes

maximal, characteristic

C₃₇×#405

C₂

yes

maximal, characteristic

#11924536

C₂²

yes

—

#11924535

—

#11924528

C₂²

yes

—

#11924527

—

#11924513

—

#11924495

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

#11924470

C₂²

yes

—

#11924470

C₂²

yes

—

#11924468

C₂²

yes

—

#11924468

C₂²

yes

—

#11924468

C₂²

yes

characteristic

#11924468

C₂²

yes

characteristic

#11924468

C₂²

yes

—

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

M₁₆×C₃₇

Q₈

yes

complemented

C₂×C₈×C₃₇

Q₈

yes

complemented

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₂×C₈×C₃₇

C₂³

yes

—

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₃₇

D₄

yes

—

C₂²⋊C₄×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

C₄⋊Dic₃₇

C₂³

yes

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

#10596936

—

C₄×Dic₃₇

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

C₈×C₃₇

C₂×Q₈

yes

—

C₈×C₃₇

C₂×Q₈

yes

—

C₈×C₃₇

C₂×Q₈

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₄⋊Q₈

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₄⋊Q₈

yes

—

C₄×C₃₇

C₂²×Q₈

yes

characteristic

C₄×C₃₇

C₂²×Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

#3163

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₇

#577

yes

—

—

—

—

—

—

—

—

D₃₇

yes

characteristic

C₃₇

#3163

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×D₃₇

yes

characteristic

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

—

#117

C₂×D₃₇

yes

characteristic

C₂×C₄⋊C₄

—

—

—

—

—

—

100

#106

—

101

#106

—

102

#97

C₂×D₃₇

yes

—

103

C₄×C₈

C₂×D₃₇

yes

—

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²×D₃₇

yes

characteristic

107

M₁₆

Dic₇₄

yes

complemented

108

C₂×C₈

Dic₇₄

yes

complemented

109

C₂×C₈

C₂²×D₃₇

yes

characteristic

110

C₂×C₈

C₂²×D₃₇

yes

—

111

C₂×C₈

C₂²×D₃₇

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

D₁₄₈

yes

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

D₁₄₈

yes

—

122

C₄²

—

123

C₄²

C₂²×D₃₇

yes

—

124

C₂³

C₂×D₁₄₈

yes

characteristic

125

Q₈

—

296

—

126

C₂×C₄

—

296

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×Dic₇₄

yes

—

131

C₂×C₄

C₂×Dic₇₄

yes

—

132

C₂×C₄

C₂×D₁₄₈

yes

—

133

C₂×C₄

C₂×D₁₄₈

yes

characteristic

134

C₂×C₄

C₂³×D₃₇

yes

characteristic

135

C₈

C₂×Dic₇₄

yes

—

136

C₈

C₂×Dic₇₄

yes

—

137

C₈

C₂×Dic₇₄

yes

—

138

C₂²

—

139

C₂²

#11924536

yes

—

140

C₂²

C₂²×D₁₄₈

yes

characteristic

141

C₄

—

142

C₄

—

148

—

143

C₄

—

148

—

144

C₄

—

148

—

145

C₄

#11924536

yes

—

146

C₄

C₂²×Dic₇₄

yes

characteristic

147

C₄

C₂²×Dic₇₄

yes

characteristic

148

C₂

—

149

C₂

C₂×#11924536

yes

characteristic

150

C₂

#14558345

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₄₈
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#405	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	#405	D₃₇
hypocenter	C₃₇	#3163
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

subgroup

G/H

center

C₂²

C₂²×D₁₄₈

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#405

C₂

socle

C₂²×C₃₇

C₂²×D₄

prime radical

C₂³×C₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405

N	H ≅ G/N	type
2. #14557942	C₂	semidirect
3. #14557935	C₂	semidirect
4. #14557931	C₂	semidirect
31. M₁₆×C₃₇	Q₈	semidirect
32. C₂×C₈×C₃₇	Q₈	semidirect
84. C₃₇	#3163	semidirect
107. M₁₆	Dic₇₄	semidirect
108. C₂×C₈	Dic₇₄	semidirect

H ≅ G/N

type

2. #14557942

C₂

semidirect

3. #14557935

C₂

semidirect

4. #14557931

C₂

semidirect

31. M₁₆×C₃₇

Q₈

semidirect

32. C₂×C₈×C₃₇

Q₈

semidirect

84. C₃₇

#3163

semidirect

107. M₁₆

Dic₇₄

semidirect

108. C₂×C₈

Dic₇₄

semidirect

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

subgroup	index
#14558161	2
#14557942	2
#14557935	2
#14557931	2
#14557575	2
#14557571	2
C₂×#11924468	2
C₃₇×#405	2
#3163	37

subgroup

index

C₃₇×#405

Subgroups — #18005127

Conjugacy classes of subgroups

Complements

Maximal subgroups