Group #18010355 — subgroups

Subgroups — #18010355

Order 4736.

284 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 86 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×#11924624	C₂	1	1	yes	maximal, characteristic
2	#14558514	C₂	1	1	yes	maximal, characteristic
3	#14558499	C₂	1	2	yes	maximal, complemented
4	#14558416	C₂	1	2	yes	maximal, complemented
5	#14557516	C₂	1	1	yes	maximal, characteristic
6	#11924624	—	2	2	no	—
7	#11924624	—	2	2	no	—
8	C₂×#10596953	C₄	1	2	yes	complemented
9	C₂×#10596953	C₂²	1	1	yes	characteristic
10	C₂×D₃₇⋊C₈	C₄	1	2	yes	complemented
11	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
12	#11924615	C₂²	1	2	yes	—
13	#11924595	C₄	1	4	yes	—
14	#11924590	—	2	2	no	—
15	C₈×#3866	—	2	2	no	—
16	C₂×#10596966	C₂²	1	1	yes	characteristic
17	M₁₆×D₃₇	C₂²	1	2	yes	—
18	#11924455	C₄	1	1	yes	characteristic, complemented
19	#11924452	C₄	1	2	yes	complemented
20	#101×C₃₇	C₄	1	1	yes	characteristic, complemented
21	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
22	#10596966	—	2	2	no	—
23	#10596966	—	2	2	no	—
24	C₂×D₁₄₈	C₂×C₄	1	2	yes	—
25	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
28	C₂²⋊#3866	C₂×C₄	1	4	yes	—
29	C₂×#64438	C₂×C₄	1	2	yes	—
30	C₂×#64438	C₂×C₄	1	1	yes	characteristic
31	C₂×#64438	C₂×C₄	1	1	yes	characteristic
32	C₄⋊#3866	—	2	2	no	—
33	C₄×#3866	—	2	2	no	—
34	#10596953	—	2	2	no	—
35	#10596953	—	2	4	no	—
36	D₃₇⋊C₈	—	2	2	no	—
37	D₃₇⋊C₈	—	2	2	no	—
38	D₃₇⋊C₈	D₄	1	2	yes	—
39	D₃₇⋊C₈	D₄	1	2	yes	—
40	M₁₆×C₃₇	C₂×C₄	1	2	yes	—
41	#10596934	C₂×C₄	1	2	yes	—
42	#10596929	—	2	2	no	—
43	C₈×D₃₇	—	2	2	no	—
44	C₂²×D₃₇	C₄²	1	2	yes	—
45	C₂²×D₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×#3866	—	2	4	no	—
47	Q₈×C₃₇	—	2	2	no	—
48	C₂×C₄×C₃₇	C₄²	1	2	yes	—
49	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₃₇	C₂²×C₄	1	1	yes	characteristic
51	D₁₄₈	—	2	4	no	—
52	D₁₄₈	—	2	2	no	—
53	C₄×D₃₇	—	2	2	no	—
54	C₄×D₃₇	—	2	2	no	—
55	C₄×D₃₇	C₂×D₄	1	2	yes	—
56	C₄×D₃₇	#41	1	2	yes	—
57	#64438	—	2	2	no	—
58	#64438	C₂²⋊C₄	1	1	yes	characteristic
59	#64438	C₂²⋊C₄	1	1	yes	characteristic
60	#64438	C₂²⋊C₄	1	2	yes	—
61	C₈×C₃₇	—	2	2	no	—
62	#64436	—	2	2	no	—
63	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
64	C₂×D₃₇	—	2	2	no	—
65	C₂×D₃₇	—	4	1	no	—
66	C₂×D₃₇	—	4	2	no	—
67	C₂×D₃₇	C₄×D₄	1	2	yes	—
68	#3866	—	4	4	no	—
69	C₄×C₃₇	—	2	2	no	—
70	C₄×C₃₇	C₄×D₄	1	2	yes	—
71	Dic₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	Dic₃₇	#117	1	1	yes	characteristic
73	#1757	—	37	1	no	maximal, Sylow
74	C₂×C₃₇	—	2	1	no	—
75	C₂×C₃₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
76	D₃₇	—	2	2	no	—
77	D₃₇	—	4	2	no	—
78	C₂×#131	—	37	1	no	—
79	#413	—	37	1	no	—
80	#410	—	37	1	no	—
81	#405	—	37	2	no	—
82	#344	—	37	2	no	—
83	C₃₇	#1757	1	1	yes	characteristic, complemented, Sylow
84	C₂×#41	—	37	1	no	—
85	#131	—	74	2	no	—
86	#131	—	74	2	no	—
87	C₂×M₁₆	—	37	1	no	—
88	C₂×M₁₆	—	37	2	no	—
89	C₂×M₁₆	—	37	2	no	—
90	C₂²×C₈	—	37	1	no	—
91	C₂²×C₈	—	37	2	no	—
92	#117	—	37	2	no	—
93	#101	—	37	1	no	—
94	#101	#3866	1	1	yes	characteristic, complemented
95	#100	—	37	2	no	—
96	#99	—	37	4	no	—
97	#97	—	74	2	no	—
98	C₄×C₈	—	74	2	no	—
99	#41	—	74	2	no	—
100	#41	—	74	2	no	—
101	C₂×Q₈	C₂×#3866	1	1	yes	characteristic
102	C₂×D₄	—	37	2	no	—
103	C₂×D₄	—	37	1	no	—
104	C₂²×C₄	—	37	1	no	—
105	C₂²×C₄	—	37	2	no	—
106	M₁₆	—	37	2	no	—
107	M₁₆	—	74	2	no	—
108	M₁₆	—	74	4	no	—
109	M₁₆	—	74	2	no	—
110	M₁₆	C₂×#3866	1	2	yes	—
111	C₂×C₈	—	37	2	no	—
112	C₂×C₈	—	37	2	no	—
113	C₂×C₈	—	37	1	no	—
114	C₂×C₈	—	37	1	no	—
115	C₂×C₈	—	37	2	no	—
116	C₂×C₈	—	74	2	no	—
117	C₂×C₈	—	74	2	no	—
118	C₂×C₈	—	74	2	no	—
119	C₄⋊C₄	—	74	2	no	—
120	C₂²⋊C₄	—	37	4	no	—
121	C₄²	—	74	2	no	—
122	C₂³	—	37	1	no	—
123	C₂³	—	37	2	no	—
124	Q₈	—	2	2	no	—
125	D₄	—	74	4	no	—
126	D₄	—	74	2	no	—
127	C₂×C₄	—	37	1	no	—
128	C₂×C₄	—	37	2	no	—
129	C₂×C₄	—	37	2	no	—
130	C₂×C₄	—	74	2	no	—
131	C₂×C₄	—	74	2	no	—
132	C₂×C₄	—	74	4	no	—
133	C₂×C₄	C₄×#3866	1	2	yes	—
134	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
135	C₈	—	2	2	no	—
136	C₈	—	37	1	no	—
137	C₈	—	37	1	no	—
138	C₈	—	37	2	no	—
139	C₈	—	74	2	no	—
140	C₈	—	74	2	no	—
141	C₂²	—	37	2	no	—
142	C₂²	—	74	2	no	—
143	C₂²	—	148	1	no	—
144	C₂²	—	148	2	no	—
145	C₂²	C₂×C₄×#3866	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	37	1	no	—
148	C₄	—	37	1	no	—
149	C₄	—	148	4	no	—
150	C₄	D₄×#3866	1	2	yes	—
151	C₂	—	2	1	no	—
152	C₂	—	74	2	no	—
153	C₂	—	148	2	no	—
154	C₂	C₂²⋊C₄×#3866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924624

C₂

yes

maximal, characteristic

#14558514

C₂

yes

maximal, characteristic

#14558499

C₂

yes

maximal, complemented

#14558416

C₂

yes

maximal, complemented

#14557516

C₂

yes

maximal, characteristic

#11924624

—

#11924624

—

C₂×#10596953

C₄

yes

complemented

C₂×#10596953

C₂²

yes

characteristic

C₂×D₃₇⋊C₈

C₄

yes

complemented

C₂×D₃₇⋊C₈

C₂²

yes

characteristic

#11924615

C₂²

yes

—

#11924595

C₄

yes

—

#11924590

—

C₈×#3866

—

C₂×#10596966

C₂²

yes

characteristic

M₁₆×D₃₇

C₂²

yes

—

#11924455

C₄

yes

characteristic, complemented

#11924452

C₄

yes

complemented

#101×C₃₇

C₄

yes

characteristic, complemented

C₂×Q₈×C₃₇

C₂×C₄

yes

characteristic

#10596966

—

#10596966

—

C₂×D₁₄₈

C₂×C₄

yes

—

C₂×D₁₄₈

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂²⋊#3866

C₂×C₄

yes

—

C₂×#64438

C₂×C₄

yes

—

C₂×#64438

C₂×C₄

yes

characteristic

C₂×#64438

C₂×C₄

yes

—

—

—

—

D₃₇⋊C₈

—

D₃₇⋊C₈

—

D₃₇⋊C₈

D₄

yes

—

D₃₇⋊C₈

D₄

yes

—

M₁₆×C₃₇

C₂×C₄

yes

—

#10596934

C₂×C₄

yes

—

#10596929

—

C₈×D₃₇

—

C₂²×D₃₇

C₄²

yes

—

C₂²×D₃₇

C₂²×C₄

yes

characteristic

C₂×#3866

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂²×C₄

yes

characteristic

D₁₄₈

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

C₂×D₄

yes

—

C₄×D₃₇

#41

yes

—

#64438

—

#64438

C₂²⋊C₄

yes

characteristic

#64438

C₂²⋊C₄

yes

characteristic

#64438

C₂²⋊C₄

yes

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₄²

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₄×D₄

yes

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

Dic₃₇

C₂×C₂²⋊C₄

yes

Dic₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

C₄×C₂²⋊C₄

yes

characteristic

D₃₇

—

D₃₇

—

—

—

—

—

—

C₃₇

#1757

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

—

—

yes

characteristic, complemented

#100

—

#99

—

#97

—

C₄×C₈

—

#41

—

100

#41

—

101

C₂×Q₈

C₂×#3866

yes

characteristic

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

C₂×#3866

yes

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₂³

—

123

C₂³

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄×#3866

yes

—

134

C₂×C₄

C₂²×#3866

yes

characteristic

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

148

—

144

C₂²

—

148

—

145

C₂²

C₂×C₄×#3866

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

148

—

150

C₄

D₄×#3866

yes

—

151

C₂

—

152

C₂

—

153

C₂

—

148

—

154

C₂

C₂²⋊C₄×#3866

yes

characteristic

	subgroup	G/H
center	C₂	C₂²⋊C₄×#3866
commutator	C₂²×C₃₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#3866
Fitting	#101×C₃₇	C₄
socle	C₂×C₃₇	C₄×C₂²⋊C₄
prime radical	C₂×#10596966	C₂²
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#101	#3866
hypocenter	C₃₇	#1757
LCS	G ▷ C₂²×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

subgroup

G/H

center

C₂

C₂²⋊C₄×#3866

commutator

C₂²×C₃₇

C₂×C₄²

Frattini

C₂×C₄

C₂²×#3866

Fitting

#101×C₃₇

C₄

socle

C₂×C₃₇

C₄×C₂²⋊C₄

prime radical

C₂×#10596966

C₂²

cosocle

C₂×C₄×D₃₇

C₂³

C₃₇

G ▷ C₂²×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #101

N	H ≅ G/N	type
3. #14558499	C₂	semidirect
4. #14558416	C₂	semidirect
8. C₂×#10596953	C₄	semidirect
10. C₂×D₃₇⋊C₈	C₄	semidirect
18. #11924455	C₄	semidirect
19. #11924452	C₄	semidirect
20. #101×C₃₇	C₄	semidirect
83. C₃₇	#1757	semidirect
94. #101	#3866	semidirect

H ≅ G/N

type

3. #14558499

C₂

semidirect

4. #14558416

C₂

semidirect

8. C₂×#10596953

C₄

semidirect

10. C₂×D₃₇⋊C₈

C₄

semidirect

18. #11924455

C₄

semidirect

19. #11924452

C₄

semidirect

20. #101×C₃₇

C₄

semidirect

83. C₃₇

#1757

semidirect

94. #101

#3866

semidirect

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

subgroup	index
C₂×#11924624	2
#14558514	2
#14558499	2
#14558416	2
#14557516	2
#1757	37

subgroup

index

Subgroups — #18010355

Conjugacy classes of subgroups

Complements

Maximal subgroups