Group #18010715 — subgroups

Subgroups — #18010715

Order 4736.

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

normal subgroups only

This group has 184 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	#14558548	C₂	1	2	yes	maximal
2	C₂×#11924606	C₂	1	1	yes	maximal, characteristic
3	#14558496	C₂	1	1	yes	maximal, characteristic
4	#14558416	C₂	1	2	yes	maximal
5	#108×D₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596953	C₂²	1	2	yes	—
7	D₃₇⋊M₁₆	C₂²	1	1	yes	characteristic
8	#11924615	C₂²	1	1	yes	characteristic
9	#11924607	—	2	2	no	—
10	#11924606	—	2	2	no	—
11	#11924606	—	2	1	no	—
12	#11924606	—	2	1	no	—
13	#11924590	—	2	1	no	—
14	C₈×#3866	—	2	1	no	—
15	M₁₆×D₃₇	C₂²	1	2	yes	—
16	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
17	#11924472	C₄	1	1	yes	characteristic, complemented
18	#11924471	C₄	1	2	yes	complemented
19	#108×C₃₇	C₄	1	1	yes	characteristic, complemented
20	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
21	C₂²⋊#3866	Q₈	1	2	yes	—
22	#10596957	—	2	1	no	—
23	C₂×#64438	—	2	2	no	—
24	C₄⋊#3866	—	2	1	no	—
25	C₄×#3866	—	2	1	no	—
26	#10596953	—	2	2	no	—
27	#10596953	—	2	2	no	—
28	#10596953	D₄	1	2	yes	—
29	D₃₇⋊C₈	—	2	1	no	—
30	M₁₆×C₃₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
32	#10596934	C₂×C₄	1	2	yes	—
33	C₂×#64436	C₂×C₄	1	1	yes	characteristic
34	#10596929	—	2	2	no	—
35	C₈×D₃₇	—	2	2	no	—
36	C₈×D₃₇	D₄	1	1	yes	characteristic
37	C₈×D₃₇	D₄	1	1	yes	characteristic
38	C₈×D₃₇	Q₈	1	1	yes	characteristic
39	C₈×D₃₇	Q₈	1	1	yes	characteristic
40	C₂²×D₃₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×#3866	—	2	2	no	—
42	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₃₇	#41	1	1	yes	characteristic
44	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
45	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
46	C₄×D₃₇	C₂×Q₈	1	1	yes	characteristic
47	C₄×D₃₇	#41	1	1	yes	characteristic
48	#64438	—	2	2	no	—
49	#64438	—	4	2	no	—
50	C₈×C₃₇	—	2	2	no	—
51	C₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
52	C₈×C₃₇	C₄⋊C₄	1	1	yes	characteristic
53	#64436	—	2	2	no	—
54	#64436	C₄⋊C₄	1	1	yes	characteristic
55	#64436	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₃₇	C₄×Q₈	1	1	yes	characteristic
57	C₂×D₃₇	—	4	1	no	—
58	C₂×D₃₇	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×D₃₇	C₄⋊Q₈	1	1	yes	characteristic
60	#3866	—	4	2	no	—
61	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
63	Dic₃₇	#121	1	1	yes	characteristic
64	Dic₃₇	#125	1	1	yes	characteristic
65	#1953	—	37	1	no	maximal, Sylow
66	C₂×C₃₇	—	2	1	no	—
67	C₂×C₃₇	#389	1	1	yes	characteristic
68	D₃₇	—	2	1	no	—
69	D₃₇	—	2	1	no	—
70	#430	—	37	2	no	—
71	C₂×#108	—	37	1	no	—
72	C₂×#108	—	37	1	no	—
73	#405	—	37	1	no	—
74	#344	—	37	2	no	—
75	C₃₇	#1953	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	37	1	no	—
77	C₂×M₁₆	—	37	2	no	—
78	C₂×M₁₆	—	37	2	no	—
79	C₂²×C₈	—	37	1	no	—
80	#117	—	37	1	no	—
81	#108	—	37	2	no	—
82	#108	—	37	1	no	—
83	#108	—	74	1	no	—
84	#108	—	74	1	no	—
85	#108	—	74	2	no	—
86	#108	—	74	2	no	—
87	#108	#3866	1	1	yes	characteristic, complemented
88	#97	—	74	1	no	—
89	C₄×C₈	—	74	1	no	—
90	C₂²×C₄	—	37	1	no	—
91	M₁₆	—	37	2	no	—
92	M₁₆	—	37	2	no	—
93	M₁₆	—	74	2	no	—
94	M₁₆	—	74	1	no	—
95	M₁₆	—	74	2	no	—
96	M₁₆	—	74	2	no	—
97	M₁₆	C₂×#3866	1	2	yes	—
98	C₂×C₈	—	37	1	no	—
99	C₂×C₈	—	37	1	no	—
100	C₂×C₈	—	37	1	no	—
101	C₂×C₈	—	37	1	no	—
102	C₂×C₈	—	37	1	no	—
103	C₂×C₈	—	74	2	no	—
104	C₂×C₈	—	74	1	no	—
105	C₂×C₈	—	74	2	no	—
106	C₂×C₈	C₂×#3866	1	1	yes	characteristic
107	C₄⋊C₄	—	74	1	no	—
108	C₂²⋊C₄	—	37	2	no	—
109	C₄²	—	74	1	no	—
110	C₂³	—	37	1	no	—
111	C₂×C₄	—	37	1	no	—
112	C₂×C₄	—	37	1	no	—
113	C₂×C₄	—	37	1	no	—
114	C₂×C₄	—	37	1	no	—
115	C₂×C₄	—	37	1	no	—
116	C₂×C₄	—	74	2	no	—
117	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	37	1	no	—
120	C₈	—	37	1	no	—
121	C₈	—	74	2	no	—
122	C₈	—	74	2	no	—
123	C₈	—	148	2	no	—
124	C₈	C₄⋊#3866	1	1	yes	characteristic
125	C₈	C₄⋊#3866	1	1	yes	characteristic
126	C₂²	—	37	1	no	—
127	C₂²	—	37	1	no	—
128	C₂²	—	148	1	no	—
129	C₂²	Q₈×#3866	1	1	yes	characteristic
130	C₄	—	37	1	no	—
131	C₄	—	37	1	no	—
132	C₄	—	148	2	no	—
133	C₄	C₂×C₄⋊#3866	1	1	yes	characteristic
134	C₄	D₄×#3866	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	74	1	no	—
137	C₂	—	74	1	no	—
138	C₂	#14558484	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558548

C₂

yes

maximal

C₂×#11924606

C₂

yes

maximal, characteristic

#14558496

C₂

yes

maximal, characteristic

#14558416

C₂

yes

maximal

#108×D₃₇

C₂

yes

maximal, characteristic

C₂×#10596953

C₂²

yes

—

D₃₇⋊M₁₆

C₂²

yes

characteristic

#11924615

C₂²

yes

—

—

—

—

—

—

M₁₆×D₃₇

C₂²

yes

—

C₂×C₈×D₃₇

C₂²

yes

characteristic

#11924472

C₄

yes

characteristic, complemented

#11924471

C₄

yes

complemented

#108×C₃₇

C₄

yes

characteristic, complemented

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂²⋊#3866

Q₈

yes

—

—

—

—

—

—

—

D₄

yes

—

D₃₇⋊C₈

—

M₁₆×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

#10596934

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

characteristic

#10596929

—

C₈×D₃₇

—

C₈×D₃₇

D₄

yes

characteristic

C₈×D₃₇

D₄

yes

characteristic

C₈×D₃₇

Q₈

yes

characteristic

C₈×D₃₇

Q₈

yes

characteristic

C₂²×D₃₇

C₂×Q₈

yes

characteristic

C₂×#3866

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

#41

yes

characteristic

C₄×D₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

C₂×Q₈

yes

C₄×D₃₇

yes

—

—

C₈×C₃₇

—

C₈×C₃₇

C₄⋊C₄

yes

characteristic

C₈×C₃₇

C₄⋊C₄

yes

characteristic

#64436

—

#64436

C₄⋊C₄

yes

characteristic

#64436

C₄⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×Q₈

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₂²⋊Q₈

yes

characteristic

C₂×D₃₇

C₄⋊Q₈

yes

characteristic

#3866

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₄×D₄

yes

Dic₃₇

yes

Dic₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

#389

yes

characteristic

D₃₇

—

D₃₇

—

—

—

—

—

—

C₃₇

#1953

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

—

—

—

yes

characteristic, complemented

#97

—

C₄×C₈

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂×#3866

yes

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×#3866

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₂³

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²×#3866

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

148

—

124

C₈

C₄⋊#3866

yes

characteristic

125

C₈

C₄⋊#3866

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

148

—

129

C₂²

Q₈×#3866

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

148

—

133

C₄

C₂×C₄⋊#3866

yes

characteristic

134

C₄

D₄×#3866

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

—

138

C₂

#14558484

yes

characteristic

	subgroup	G/H
center	C₂	#14558484
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	#108×C₃₇	C₄
socle	C₂×C₃₇	#389
prime radical	C₂²×D₃₇	C₂×Q₈
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#108	#3866
hypocenter	C₃₇	#1953
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₂

#14558484

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

#108×C₃₇

C₄

socle

C₂×C₃₇

#389

prime radical

C₂²×D₃₇

C₂×Q₈

cosocle

C₂×C₄×D₃₇

C₂³

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
17. #11924472	C₄	semidirect
18. #11924471	C₄	semidirect
19. #108×C₃₇	C₄	semidirect
75. C₃₇	#1953	semidirect
87. #108	#3866	semidirect

H ≅ G/N

type

17. #11924472

C₄

semidirect

18. #11924471

C₄

semidirect

19. #108×C₃₇

C₄

semidirect

75. C₃₇

#1953

semidirect

87. #108

#3866

semidirect

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

subgroup	index
#14558548	2
C₂×#11924606	2
#14558496	2
#14558416	2
#108×D₃₇	2
#1953	37

subgroup

index

#108×D₃₇

Subgroups — #18010715

Conjugacy classes of subgroups

Complements

Maximal subgroups