Group #18011257 — subgroups

Subgroups — #18011257

Order 4736.

440 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 198 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	#14558612	C₂	1	2	yes	maximal
2	#14558548	C₂	1	6	yes	maximal, complemented
3	D₃₇⋊#108	C₂	1	3	yes	maximal, complemented
4	C₂×#11924606	C₂	1	3	yes	maximal, complemented
5	#131×D₃₇	C₂	1	1	yes	maximal, characteristic
6	#11924624	—	2	2	no	—
7	#11924623	—	2	6	no	—
8	C₂×#10596953	C₂²	1	6	yes	—
9	D₃₇⋊M₁₆	C₂²	1	6	yes	—
10	#11924607	C₂²	1	6	yes	—
11	#11924606	C₂²	1	3	yes	—
12	#11924606	C₂²	1	6	yes	—
13	#11924606	C₂²	1	1	yes	characteristic
14	#41×D₃₇	C₂²	1	1	yes	characteristic
15	#11924544	C₄	1	3	yes	—
16	#11924543	C₄	1	1	yes	characteristic
17	#11924542	C₄	1	3	yes	—
18	M₁₆×D₃₇	C₂²	1	3	yes	—
19	C₂×C₈×D₃₇	C₂²	1	3	yes	—
20	#131×C₃₇	C₄	1	1	yes	characteristic
21	#41×C₃₇	C₂×C₄	1	1	yes	characteristic
22	#10596966	D₄	1	1	yes	characteristic
23	Q₈×D₃₇	Q₈	1	1	yes	characteristic
24	#10596964	D₄	1	3	yes	—
25	D₄×D₃₇	Q₈	1	3	yes	—
26	#10596962	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₃₇	C₂³	1	3	yes	—
28	#10596957	—	2	6	no	—
29	C₂×#64438	—	2	6	no	—
30	#10596953	—	2	6	no	—
31	#10596953	C₂³	1	6	yes	—
32	#10596953	C₂³	1	2	yes	—
33	D₃₇⋊C₈	—	2	6	no	—
34	M₁₆×C₃₇	C₂×C₄	1	3	yes	—
35	C₂×C₈×C₃₇	C₂×C₄	1	3	yes	—
36	#10596934	C₂×C₄	1	3	yes	—
37	C₂×#64436	C₂×C₄	1	3	yes	—
38	#10596929	C₂×C₄	1	6	yes	—
39	C₈×D₃₇	C₂×C₄	1	3	yes	—
40	C₈×D₃₇	C₂×C₄	1	3	yes	—
41	C₈×D₃₇	C₂³	1	1	yes	characteristic
42	C₈×D₃₇	C₂³	1	3	yes	—
43	C₂²×D₃₇	C₂×Q₈	1	3	yes	—
44	Q₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
45	D₄×C₃₇	C₄⋊C₄	1	3	yes	—
46	C₂×C₄×C₃₇	C₂²×C₄	1	3	yes	—
47	C₂²⋊D₃₇	C₄⋊C₄	1	6	yes	—
48	C₂×Dic₃₇	C₂×D₄	1	3	yes	—
49	D₁₄₈	C₄⋊C₄	1	3	yes	—
50	C₄×D₃₇	C₂²×C₄	1	3	yes	—
51	C₄×D₃₇	C₂×D₄	1	3	yes	—
52	C₄×D₃₇	C₂×Q₈	1	3	yes	—
53	C₄×D₃₇	C₂⁴	1	1	yes	characteristic
54	Dic₇₄	C₄⋊C₄	1	3	yes	—
55	#64438	—	2	2	no	—
56	#64438	—	2	6	no	—
57	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
58	C₈×C₃₇	C₂²×C₄	1	3	yes	—
59	#64436	C₂²×C₄	1	1	yes	characteristic
60	#64436	C₂²×C₄	1	3	yes	—
61	C₂²×C₃₇	C₂×C₄⋊C₄	1	3	yes	—
62	C₂×D₃₇	—	4	3	no	—
63	C₂×D₃₇	C₂×C₄⋊C₄	1	3	yes	—
64	C₂×D₃₇	C₂²×Q₈	1	1	yes	characteristic
65	C₄×C₃₇	C₂×C₄⋊C₄	1	3	yes	—
66	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
67	Dic₃₇	C₂×C₄⋊C₄	1	3	yes	—
68	Dic₃₇	C₂²×D₄	1	1	yes	characteristic
69	#2916	—	37	1	no	maximal, Sylow
70	C₂×C₃₇	—	2	3	no	—
71	C₂×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
72	D₃₇	—	2	1	no	—
73	D₃₇	—	2	3	no	—
74	#568	—	37	2	no	—
75	C₂×#131	—	37	1	no	—
76	#430	—	37	6	no	—
77	C₂×#108	—	37	3	no	—
78	C₂×#108	—	37	3	no	—
79	C₃₇	#2916	1	1	yes	characteristic, complemented, Sylow
80	C₂×#41	—	37	1	no	—
81	#131	—	37	1	no	—
82	#131	—	37	3	no	—
83	#131	—	37	3	no	—
84	#131	—	74	6	no	—
85	#131	—	74	2	no	—
86	#131	#3866	1	1	yes	characteristic
87	C₂×M₁₆	—	37	3	no	—
88	C₂×M₁₆	—	37	6	no	—
89	C₂×M₁₆	—	37	6	no	—
90	C₂²×C₈	—	37	3	no	—
91	#108	—	37	3	no	—
92	#108	—	37	1	no	—
93	#108	—	37	6	no	—
94	#108	—	37	6	no	—
95	#41	—	37	1	no	—
96	#41	—	37	3	no	—
97	#41	—	37	3	no	—
98	#41	C₂×#3866	1	1	yes	characteristic
99	C₂×Q₈	—	37	1	no	—
100	C₂×D₄	—	37	3	no	—
101	C₂²×C₄	—	37	3	no	—
102	M₁₆	—	37	3	no	—
103	M₁₆	—	37	6	no	—
104	M₁₆	—	37	6	no	—
105	M₁₆	—	37	2	no	—
106	M₁₆	—	74	6	no	—
107	M₁₆	—	74	6	no	—
108	M₁₆	C₂×#3866	1	3	yes	—
109	C₂×C₈	—	37	3	no	—
110	C₂×C₈	—	37	3	no	—
111	C₂×C₈	—	37	3	no	—
112	C₂×C₈	—	37	1	no	—
113	C₂×C₈	—	37	3	no	—
114	C₂×C₈	—	74	6	no	—
115	C₂×C₈	—	74	6	no	—
116	C₂×C₈	C₂×#3866	1	3	yes	—
117	C₂³	—	37	3	no	—
118	Q₈	—	37	3	no	—
119	Q₈	C₄⋊#3866	1	1	yes	characteristic
120	D₄	—	37	6	no	—
121	D₄	—	37	3	no	—
122	D₄	C₄⋊#3866	1	3	yes	—
123	C₂×C₄	—	37	3	no	—
124	C₂×C₄	—	37	3	no	—
125	C₂×C₄	—	37	3	no	—
126	C₂×C₄	—	37	1	no	—
127	C₂×C₄	—	37	3	no	—
128	C₂×C₄	C₂²×#3866	1	3	yes	—
129	C₈	—	37	1	no	—
130	C₈	—	37	3	no	—
131	C₈	—	74	2	no	—
132	C₈	—	74	6	no	—
133	C₈	C₂²×#3866	1	1	yes	characteristic
134	C₈	C₂²×#3866	1	3	yes	—
135	C₂²	—	37	1	no	—
136	C₂²	—	37	3	no	—
137	C₂²	—	148	3	no	—
138	C₂²	C₂×C₄⋊#3866	1	3	yes	—
139	C₄	—	37	1	no	—
140	C₄	—	37	3	no	—
141	C₄	C₂×C₄⋊#3866	1	3	yes	—
142	C₄	C₂³×#3866	1	1	yes	characteristic
143	C₂	—	2	3	no	—
144	C₂	—	74	1	no	—
145	C₂	—	74	3	no	—
146	C₂	C₂²×C₄⋊#3866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558612

C₂

yes

maximal

#14558548

C₂

yes

maximal, complemented

D₃₇⋊#108

C₂

yes

maximal, complemented

C₂×#11924606

C₂

yes

maximal, complemented

#131×D₃₇

C₂

yes

maximal, characteristic

#11924624

—

#11924623

—

C₂×#10596953

C₂²

yes

—

D₃₇⋊M₁₆

C₂²

yes

—

#11924607

C₂²

yes

—

#11924606

C₂²

yes

—

#11924606

C₂²

yes

—

#11924606

C₂²

yes

characteristic

#41×D₃₇

C₂²

yes

characteristic

#11924544

C₄

yes

—

#11924543

C₄

yes

characteristic

#11924542

C₄

yes

—

M₁₆×D₃₇

C₂²

yes

—

C₂×C₈×D₃₇

C₂²

yes

—

#131×C₃₇

C₄

yes

characteristic

#41×C₃₇

C₂×C₄

yes

characteristic

#10596966

D₄

yes

characteristic

Q₈×D₃₇

Q₈

yes

characteristic

#10596964

D₄

yes

—

D₄×D₃₇

Q₈

yes

—

#10596962

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

—

—

—

C₂³

yes

—

#10596953

C₂³

yes

—

D₃₇⋊C₈

—

M₁₆×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

#10596934

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

—

#10596929

C₂×C₄

yes

—

C₈×D₃₇

C₂×C₄

yes

—

C₈×D₃₇

C₂×C₄

yes

—

C₈×D₃₇

C₂³

yes

characteristic

C₈×D₃₇

C₂³

yes

—

C₂²×D₃₇

C₂×Q₈

yes

—

Q₈×C₃₇

C₄⋊C₄

yes

characteristic

D₄×C₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂²⋊D₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

—

D₁₄₈

C₄⋊C₄

yes

—

C₄×D₃₇

C₂²×C₄

yes

—

C₄×D₃₇

C₂×D₄

yes

—

C₄×D₃₇

C₂×Q₈

yes

—

C₄×D₃₇

C₂⁴

yes

characteristic

Dic₇₄

C₄⋊C₄

yes

—

#64438

—

#64438

—

C₈×C₃₇

C₂²×C₄

yes

characteristic

C₈×C₃₇

C₂²×C₄

yes

—

#64436

C₂²×C₄

yes

characteristic

#64436

C₂²×C₄

yes

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

—

C₂×D₃₇

—

C₂×D₃₇

C₂×C₄⋊C₄

yes

—

C₂×D₃₇

C₂²×Q₈

yes

characteristic

C₄×C₃₇

C₂×C₄⋊C₄

yes

—

C₄×C₃₇

C₂³×C₄

yes

characteristic

Dic₃₇

C₂×C₄⋊C₄

yes

—

Dic₃₇

C₂²×D₄

yes

characteristic

#2916

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂²×C₄⋊C₄

yes

characteristic

D₃₇

—

D₃₇

—

—

—

—

—

—

C₃₇

#2916

yes

characteristic, complemented, Sylow

C₂×#41

—

—

—

—

—

—

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

—

—

—

yes

C₂×Q₈

—

100

C₂×D₄

—

101

C₂²×C₄

—

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

C₂×C₈

C₂×#3866

yes

—

117

C₂³

—

118

Q₈

—

119

Q₈

C₄⋊#3866

yes

characteristic

120

D₄

—

121

D₄

—

122

D₄

C₄⋊#3866

yes

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂²×#3866

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

C₂²×#3866

yes

characteristic

134

C₈

C₂²×#3866

yes

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

148

—

138

C₂²

C₂×C₄⋊#3866

yes

—

139

C₄

—

140

C₄

—

141

C₄

C₂×C₄⋊#3866

yes

—

142

C₄

C₂³×#3866

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

C₂²×C₄⋊#3866

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₄⋊#3866
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₄	C₂³×#3866
Fitting	#131×C₃₇	C₄
socle	C₂×C₃₇	C₂²×C₄⋊C₄
prime radical	#41×D₃₇	C₂²
cosocle	C₄×D₃₇	C₂⁴
hypercenter	#131	#3866
hypocenter	C₃₇	#2916
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×C₄⋊#3866

commutator

C₄×C₃₇

C₂³×C₄

Frattini

C₄

C₂³×#3866

Fitting

#131×C₃₇

C₄

socle

C₂×C₃₇

C₂²×C₄⋊C₄

prime radical

#41×D₃₇

C₂²

cosocle

C₄×D₃₇

C₂⁴

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
2. #14558548	C₂	semidirect
3. D₃₇⋊#108	C₂	semidirect
4. C₂×#11924606	C₂	semidirect
79. C₃₇	#2916	semidirect

H ≅ G/N

type

2. #14558548

C₂

semidirect

3. D₃₇⋊#108

C₂

semidirect

4. C₂×#11924606

C₂

semidirect

79. C₃₇

#2916

semidirect

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

subgroup	index
#14558612	2
#14558548	2
D₃₇⋊#108	2
C₂×#11924606	2
#131×D₃₇	2
#2916	37

subgroup

index

D₃₇⋊#108

#131×D₃₇

Subgroups — #18011257

Conjugacy classes of subgroups

Complements

Maximal subgroups