Group #18010705 — subgroups

Subgroups — #18010705

Order 4736.

200 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 70 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×#11924606	C₂	1	1	yes	maximal, characteristic
2	#14558540	C₂	1	2	yes	maximal
3	C₂×#11924590	C₂	1	1	yes	maximal, characteristic
4	#14558403	C₂	1	2	yes	maximal
5	#14557570	C₂	1	1	yes	maximal, characteristic
6	C₂×#10596953	C₂²	1	2	yes	—
7	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×#3866	C₂²	1	1	yes	characteristic
9	#11924606	—	2	2	no	—
10	#11924603	—	2	2	no	—
11	C₄⋊#64438	—	2	2	no	—
12	#11924590	—	2	2	no	—
13	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
14	#11924494	C₂²	1	2	yes	—
15	#11924470	C₄	1	1	yes	characteristic, complemented
16	#11924467	C₄	1	2	yes	complemented
17	#107×C₃₇	C₄	1	1	yes	characteristic, complemented
18	C₂²×#3866	—	2	1	no	—
19	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
20	C₂×#64438	—	2	2	no	—
21	C₂×#64438	D₄	1	2	yes	—
22	C₄×#3866	—	2	1	no	—
23	C₄×#3866	Q₈	1	2	yes	—
24	#10596953	—	2	2	no	—
25	#10596953	—	2	2	no	—
26	D₃₇⋊C₈	—	2	1	no	—
27	D₃₇⋊C₈	—	2	1	no	—
28	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
30	#10596938	—	2	2	no	—
31	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
32	C₄×Dic₃₇	—	2	2	no	—
33	C₂×#64436	C₂×C₄	1	1	yes	characteristic
34	C₈×D₃₇	D₄	1	2	yes	—
35	C₈×D₃₇	Q₈	1	2	yes	—
36	C₂²×D₃₇	#41	1	1	yes	characteristic
37	C₂×#3866	—	2	2	no	—
38	C₂×#3866	—	4	1	no	—
39	C₂×#3866	—	4	1	no	—
40	C₂×C₄×C₃₇	—	2	2	no	—
41	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₃₇	—	2	2	no	—
43	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
44	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
45	C₄×D₃₇	C₂×Q₈	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	C₄⋊C₄	1	2	yes	—
51	#64436	C₄⋊C₄	1	2	yes	—
52	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
53	C₂×D₃₇	—	2	1	no	—
54	C₂×D₃₇	—	2	1	no	—
55	C₂×D₃₇	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂×D₃₇	#125	1	1	yes	characteristic
57	#3866	—	4	2	no	—
58	C₄×C₃₇	—	4	2	no	—
59	C₄×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₃₇	C₄×Q₈	1	1	yes	characteristic
61	Dic₃₇	—	4	2	no	—
62	Dic₃₇	#121	1	1	yes	characteristic
63	Dic₃₇	C₄⋊Q₈	1	1	yes	characteristic
64	#1946	—	37	1	no	maximal, Sylow
65	C₂×C₃₇	#389	1	1	yes	characteristic
66	C₂×C₃₇	#444	1	1	yes	characteristic
67	C₂×C₃₇	#444	1	1	yes	characteristic
68	D₃₇	—	2	1	no	—
69	D₃₇	—	2	1	no	—
70	C₂×#108	—	37	1	no	—
71	#427	—	37	1	no	—
72	#424	—	37	2	no	—
73	C₂×#97	—	37	1	no	—
74	#339	—	37	2	no	—
75	C₃₇	#1946	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	37	2	no	—
77	C₂²×C₈	—	37	1	no	—
78	C₂²×C₈	—	37	1	no	—
79	#117	—	37	2	no	—
80	C₂×C₄²	—	37	1	no	—
81	#108	—	74	2	no	—
82	#107	—	37	1	no	—
83	#107	#3866	1	1	yes	characteristic, complemented
84	#106	—	37	2	no	—
85	C₄⋊C₈	—	74	2	no	—
86	C₄⋊C₈	—	74	2	no	—
87	#97	—	74	2	no	—
88	C₂²×C₄	—	37	1	no	—
89	C₂²×C₄	—	74	1	no	—
90	M₁₆	—	74	2	no	—
91	M₁₆	—	74	2	no	—
92	C₂×C₈	—	37	1	no	—
93	C₂×C₈	—	37	2	no	—
94	C₂×C₈	—	37	2	no	—
95	C₂×C₈	—	37	2	no	—
96	C₂×C₈	—	74	1	no	—
97	C₂×C₈	—	74	1	no	—
98	C₂×C₈	—	74	2	no	—
99	C₂×C₈	C₂×#3866	1	1	yes	characteristic
100	C₄⋊C₄	—	37	2	no	—
101	C₄⋊C₄	C₂×#3866	1	2	yes	—
102	C₂²⋊C₄	—	74	2	no	—
103	C₄²	—	37	2	no	—
104	C₄²	—	74	2	no	—
105	C₄²	—	74	1	no	—
106	C₂³	—	37	1	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	37	1	no	—
109	C₂×C₄	—	37	1	no	—
110	C₂×C₄	—	37	1	no	—
111	C₂×C₄	—	37	1	no	—
112	C₂×C₄	—	37	1	no	—
113	C₂×C₄	—	74	2	no	—
114	C₂×C₄	—	74	2	no	—
115	C₂×C₄	—	148	1	no	—
116	C₂×C₄	—	148	1	no	—
117	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
118	C₈	—	37	2	no	—
119	C₈	—	74	2	no	—
120	C₈	—	148	2	no	—
121	C₈	C₄⋊#3866	1	2	yes	—
122	C₂²	—	37	1	no	—
123	C₂²	—	37	1	no	—
124	C₂²	—	74	1	no	—
125	C₂²	—	74	1	no	—
126	C₂²	D₄×#3866	1	1	yes	characteristic
127	C₄	—	4	2	no	—
128	C₄	—	37	1	no	—
129	C₄	—	37	1	no	—
130	C₄	—	148	2	no	—
131	C₄	—	148	2	no	—
132	C₄	C₂×C₄⋊#3866	1	1	yes	characteristic
133	C₄	Q₈×#3866	1	1	yes	characteristic
134	C₂	—	74	1	no	—
135	C₂	—	74	1	no	—
136	C₂	#14558484	1	1	yes	characteristic
137	C₂	#14558579	1	1	yes	characteristic
138	C₂	#14558580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924606

C₂

yes

maximal, characteristic

#14558540

C₂

yes

maximal

C₂×#11924590

C₂

yes

maximal, characteristic

#14558403

C₂

yes

maximal

#14557570

C₂

yes

maximal, characteristic

C₂×#10596953

C₂²

yes

—

C₂×D₃₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×#3866

C₂²

yes

—

—

—

—

C₂×C₈×D₃₇

C₂²

yes

characteristic

#11924494

C₂²

yes

—

#11924470

C₄

yes

characteristic, complemented

#11924467

C₄

yes

complemented

#107×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×#64438

—

C₂×#64438

D₄

yes

—

C₄×#3866

—

C₄×#3866

Q₈

yes

—

#10596953

—

#10596953

—

D₃₇⋊C₈

—

D₃₇⋊C₈

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

#10596938

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

—

C₂×#64436

C₂×C₄

yes

characteristic

C₈×D₃₇

D₄

yes

—

C₈×D₃₇

Q₈

yes

—

C₂²×D₃₇

yes

—

—

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

C₂×Q₈

yes

characteristic

C₄×D₃₇

C₂×Q₈

yes

C₄×D₃₇

yes

—

—

C₈×C₃₇

C₄⋊C₄

yes

—

#64436

C₄⋊C₄

yes

—

C₂²×C₃₇

C₄×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₂²⋊Q₈

yes

C₂×D₃₇

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₇

C₄×Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

#121

yes

characteristic

Dic₃₇

C₄⋊Q₈

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

D₃₇

—

C₂×#108

—

#427

—

#424

—

C₂×#97

—

#339

—

C₃₇

#1946

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

C₂×C₄²

—

—

—

yes

characteristic, complemented

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

characteristic

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂×#3866

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₂³

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

148

—

116

C₂×C₄

—

148

—

117

C₂×C₄

C₂²×#3866

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

148

—

121

C₈

C₄⋊#3866

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

D₄×#3866

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

148

—

131

C₄

—

148

—

132

C₄

C₂×C₄⋊#3866

yes

characteristic

133

C₄

Q₈×#3866

yes

characteristic

134

C₂

—

135

C₂

—

136

C₂

#14558484

yes

characteristic

137

C₂

#14558579

yes

characteristic

138

C₂

#14558580

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	#107×C₃₇	C₄
socle	C₂²×C₃₇	C₄×D₄
prime radical	C₂²×D₃₇	#41
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#107	#3866
hypocenter	C₃₇	#1946
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

#107×C₃₇

C₄

socle

C₂²×C₃₇

C₄×D₄

prime radical

C₂²×D₃₇

#41

cosocle

C₂×C₄×D₃₇

C₂³

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
15. #11924470	C₄	semidirect
16. #11924467	C₄	semidirect
17. #107×C₃₇	C₄	semidirect
75. C₃₇	#1946	semidirect
83. #107	#3866	semidirect

H ≅ G/N

type

15. #11924470

C₄

semidirect

16. #11924467

C₄

semidirect

17. #107×C₃₇

C₄

semidirect

75. C₃₇

#1946

semidirect

83. #107

#3866

semidirect

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

subgroup	index
C₂×#11924606	2
#14558540	2
C₂×#11924590	2
#14558403	2
#14557570	2
#1946	37

subgroup

index

Subgroups — #18010705

Conjugacy classes of subgroups

Complements

Maximal subgroups