Group #18010553 — subgroups

Subgroups — #18010553

Order 4736.

252 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 66 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	#14558611	C₂	1	1	yes	maximal, characteristic
2	#14558548	C₂	1	2	yes	maximal, complemented
3	#14558507	C₂	1	1	yes	maximal, characteristic
4	#14558408	C₂	1	2	yes	maximal, complemented
5	#14557514	C₂	1	1	yes	maximal, characteristic
6	#11924623	—	2	2	no	—
7	#11924623	—	2	2	no	—
8	C₂×#10596953	C₂²	1	1	yes	characteristic
9	D₃₇⋊M₁₆	C₂²	1	2	yes	—
10	C₂×#10596957	—	2	2	no	—
11	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
12	#11924607	—	2	2	no	—
13	#11924606	—	2	2	no	—
14	C₂²⋊#64438	—	2	2	no	—
15	C₂×#10596964	C₂²	1	1	yes	characteristic
16	M₁₆×D₃₇	C₂²	1	2	yes	—
17	#11924454	C₄	1	2	yes	—
18	#11924453	C₄	1	1	yes	characteristic
19	#100×C₃₇	C₄	1	1	yes	characteristic
20	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₃₇	Q₈	1	2	yes	—
22	C₂²×Dic₃₇	D₄	1	2	yes	—
23	#10596964	—	2	2	no	—
24	#10596964	—	2	2	no	—
25	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
26	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
27	#10596957	—	2	4	no	—
28	#10596957	—	2	2	no	—
29	C₂×#64438	—	2	2	no	—
30	C₂×#64438	—	2	2	no	—
31	C₂×#64438	—	2	1	no	—
32	C₂×#64438	—	2	1	no	—
33	#10596953	—	2	2	no	—
34	#10596953	D₄	1	2	yes	—
35	#10596953	D₄	1	2	yes	—
36	D₃₇⋊C₈	—	2	2	no	—
37	D₃₇⋊C₈	—	2	2	no	—
38	M₁₆×C₃₇	C₂×C₄	1	2	yes	—
39	#10596934	C₂×C₄	1	2	yes	—
40	#10596929	—	2	2	no	—
41	C₈×D₃₇	—	2	2	no	—
42	C₂³×C₃₇	C₄⋊C₄	1	2	yes	—
43	C₂²×D₃₇	C₂×Q₈	1	1	yes	characteristic
44	D₄×C₃₇	—	2	2	no	—
45	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²⋊D₃₇	—	2	4	no	—
47	C₂×Dic₃₇	—	2	2	no	—
48	C₂×Dic₃₇	—	2	2	no	—
49	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
50	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
51	C₄×D₃₇	C₂×D₄	1	2	yes	—
52	C₄×D₃₇	#41	1	2	yes	—
53	Dic₇₄	—	2	2	no	—
54	#64438	—	2	2	no	—
55	#64438	—	2	2	no	—
56	#64438	—	4	1	no	—
57	#64438	—	4	1	no	—
58	C₈×C₃₇	—	2	2	no	—
59	#64436	—	2	2	no	—
60	C₂²×C₃₇	—	2	2	no	—
61	C₂²×C₃₇	—	4	2	no	—
62	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×D₃₇	—	4	1	no	—
64	C₂×D₃₇	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₃₇	C₄×D₄	1	2	yes	—
66	Dic₃₇	—	2	2	no	—
67	Dic₃₇	#120	1	1	yes	characteristic
68	Dic₃₇	#123	1	1	yes	characteristic
69	#1858	—	37	1	no	maximal, Sylow
70	C₂×C₃₇	—	2	1	no	—
71	C₂×C₃₇	—	4	2	no	—
72	C₂×C₃₇	#385	1	1	yes	characteristic
73	D₃₇	—	2	2	no	—
74	#568	—	37	1	no	—
75	#430	—	37	2	no	—
76	#413	—	37	1	no	—
77	#408	—	37	1	no	—
78	#341	—	37	2	no	—
79	C₃₇	#1858	1	1	yes	characteristic, complemented, Sylow
80	C₂×#41	—	37	1	no	—
81	#131	—	74	2	no	—
82	#131	—	74	2	no	—
83	C₂×M₁₆	—	37	2	no	—
84	C₂×M₁₆	—	37	2	no	—
85	C₂×M₁₆	—	37	1	no	—
86	C₂×M₁₆	—	74	2	no	—
87	C₂²×C₈	—	37	1	no	—
88	#108	—	74	2	no	—
89	#108	—	74	2	no	—
90	#101	—	37	2	no	—
91	#100	—	37	1	no	—
92	#100	#3866	1	1	yes	characteristic
93	C₂²⋊C₈	—	74	2	no	—
94	#41	—	74	2	no	—
95	#41	—	74	2	no	—
96	C₂×Q₈	—	37	1	no	—
97	C₂×D₄	—	37	2	no	—
98	C₂×D₄	C₂×#3866	1	1	yes	characteristic
99	C₂²×C₄	—	37	1	no	—
100	C₂²×C₄	—	37	2	no	—
101	M₁₆	—	37	2	no	—
102	M₁₆	—	37	2	no	—
103	M₁₆	—	37	2	no	—
104	M₁₆	—	74	2	no	—
105	M₁₆	—	74	2	no	—
106	M₁₆	—	74	4	no	—
107	M₁₆	—	74	2	no	—
108	M₁₆	C₂×#3866	1	2	yes	—
109	C₂×C₈	—	74	2	no	—
110	C₂×C₈	—	74	2	no	—
111	C₂×C₈	—	74	2	no	—
112	C₂×C₈	—	74	2	no	—
113	C₂×C₈	—	74	2	no	—
114	C₂×C₈	—	74	1	no	—
115	C₂×C₈	—	74	1	no	—
116	C₂³	—	37	1	no	—
117	C₂³	C₄⋊#3866	1	2	yes	—
118	Q₈	—	74	2	no	—
119	D₄	—	2	2	no	—
120	D₄	—	74	4	no	—
121	C₂×C₄	—	37	1	no	—
122	C₂×C₄	—	37	2	no	—
123	C₂×C₄	—	37	2	no	—
124	C₂×C₄	—	37	2	no	—
125	C₂×C₄	—	74	2	no	—
126	C₂×C₄	—	74	2	no	—
127	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	74	2	no	—
130	C₈	—	74	2	no	—
131	C₈	—	74	2	no	—
132	C₈	—	148	1	no	—
133	C₈	—	148	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	2	no	—
136	C₂²	—	37	2	no	—
137	C₂²	—	148	1	no	—
138	C₂²	C₂×C₄⋊#3866	1	1	yes	characteristic
139	C₄	—	37	1	no	—
140	C₄	—	37	1	no	—
141	C₄	—	74	2	no	—
142	C₄	D₄×#3866	1	2	yes	—
143	C₂	—	2	1	no	—
144	C₂	—	4	2	no	—
145	C₂	—	74	2	no	—
146	C₂	#14558480	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558611

C₂

yes

maximal, characteristic

#14558548

C₂

yes

maximal, complemented

#14558507

C₂

yes

maximal, characteristic

#14558408

C₂

yes

maximal, complemented

#14557514

C₂

yes

maximal, characteristic

#11924623

—

#11924623

—

C₂×#10596953

C₂²

yes

characteristic

D₃₇⋊M₁₆

C₂²

yes

—

C₂×#10596957

—

C₂×D₃₇⋊C₈

C₂²

yes

—

—

—

C₂²

yes

characteristic

M₁₆×D₃₇

C₂²

yes

—

#11924454

C₄

yes

—

#11924453

C₄

yes

characteristic

#100×C₃₇

C₄

yes

characteristic

C₂×D₄×C₃₇

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₃₇

Q₈

yes

—

C₂²×Dic₃₇

D₄

yes

—

#10596964

—

#10596964

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

C₂×C₄

yes

—

—

—

—

—

—

—

D₄

yes

—

#10596953

D₄

yes

—

D₃₇⋊C₈

—

D₃₇⋊C₈

—

M₁₆×C₃₇

C₂×C₄

yes

—

#10596934

C₂×C₄

yes

—

#10596929

—

C₈×D₃₇

—

C₂³×C₃₇

C₄⋊C₄

yes

—

C₂²×D₃₇

C₂×Q₈

yes

characteristic

D₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

C₂×D₄

yes

—

C₄×D₃₇

#41

yes

—

Dic₇₄

—

—

—

—

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

C₄×D₄

yes

—

Dic₃₇

—

Dic₃₇

yes

Dic₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#385

yes

characteristic

D₃₇

—

—

—

—

—

—

C₃₇

#1858

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

yes

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

C₂×#3866

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

C₂×#3866

yes

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂³

—

117

C₂³

C₄⋊#3866

yes

—

118

Q₈

—

119

D₄

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂²×#3866

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

148

—

133

C₈

—

148

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

148

—

138

C₂²

C₂×C₄⋊#3866

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

D₄×#3866

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#14558480

yes

characteristic

	subgroup	G/H
center	C₂	#14558480
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	#100×C₃₇	C₄
socle	C₂×C₃₇	#385
prime radical	C₂×#10596964	C₂²
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#100	#3866
hypocenter	C₃₇	#1858
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

subgroup

G/H

center

C₂

#14558480

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

#100×C₃₇

C₄

C₂×C₃₇

C₂²

C₂×C₄×D₃₇

C₂³

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #100

N	H ≅ G/N	type
2. #14558548	C₂	semidirect
4. #14558408	C₂	semidirect
79. C₃₇	#1858	semidirect

H ≅ G/N

type

2. #14558548

C₂

semidirect

4. #14558408

C₂

semidirect

79. C₃₇

#1858

semidirect

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

subgroup	index
#14558611	2
#14558548	2
#14558507	2
#14558408	2
#14557514	2
#1858	37

subgroup

index

Subgroups — #18010553

Conjugacy classes of subgroups

Complements

Maximal subgroups