Group #18010401 — subgroups

Subgroups — #18010401

Order 4736.

232 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 88 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	#14558543	C₂	1	1	yes	maximal, characteristic
2	#14558499	C₂	1	2	yes	maximal
3	C₂×C₈×#3866	C₂	1	1	yes	maximal, characteristic
4	#14558405	C₂	1	2	yes	maximal
5	#108×D₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×D₃₇⋊C₈	C₂²	1	1	yes	characteristic
7	#11924615	C₂²	1	2	yes	—
8	C₂×C₄×#3866	C₂²	1	1	yes	characteristic
9	#11924605	C₄	1	2	yes	complemented
10	#11924604	C₄	1	2	yes	complemented
11	#11924590	—	2	2	no	—
12	C₈×#3866	—	2	2	no	—
13	C₈×#3866	C₄	1	2	yes	complemented
14	C₈×#3866	C₄	1	2	yes	complemented
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	C₂²×#3866	—	2	1	no	—
21	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
22	C₂²⋊#3866	—	2	2	no	—
23	C₂×#64438	Q₈	1	2	yes	—
24	C₄⋊#3866	C₂×C₄	1	4	yes	—
25	C₄×#3866	—	2	2	no	—
26	C₄×#3866	—	2	1	no	—
27	C₄×#3866	C₂×C₄	1	2	yes	—
28	D₃₇⋊C₈	C₂×C₄	1	2	yes	—
29	D₃₇⋊C₈	D₄	1	1	yes	characteristic
30	D₃₇⋊C₈	D₄	1	1	yes	characteristic
31	M₁₆×C₃₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
33	#10596934	C₂×C₄	1	2	yes	—
34	C₂×#64436	C₂×C₄	1	1	yes	characteristic
35	#10596929	—	2	2	no	—
36	C₈×D₃₇	—	2	2	no	—
37	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
38	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
39	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
40	C₈×D₃₇	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₃₇	#41	1	1	yes	characteristic
42	C₂×#3866	—	2	4	no	—
43	C₂×#3866	—	2	2	no	—
44	C₂×#3866	—	2	4	no	—
45	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₃₇	C₂×Q₈	1	1	yes	characteristic
47	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
48	C₄×D₃₇	C₂²×C₄	1	1	yes	characteristic
49	C₄×D₃₇	C₂×D₄	1	1	yes	characteristic
50	C₄×D₃₇	#41	1	1	yes	characteristic
51	#64438	C₄⋊C₄	1	2	yes	—
52	#64438	C₄⋊C₄	1	2	yes	—
53	C₈×C₃₇	—	2	2	no	—
54	C₈×C₃₇	C₄²	1	1	yes	characteristic
55	C₈×C₃₇	C₄²	1	1	yes	characteristic
56	#64436	—	2	2	no	—
57	#64436	C₄²	1	1	yes	characteristic
58	#64436	C₄²	1	1	yes	characteristic
59	C₂²×C₃₇	C₄×Q₈	1	1	yes	characteristic
60	C₂×D₃₇	—	2	2	no	—
61	C₂×D₃₇	#117	1	1	yes	characteristic
62	C₂×D₃₇	C₄×D₄	1	1	yes	characteristic
63	#3866	—	2	4	no	—
64	#3866	—	4	4	no	—
65	C₄×C₃₇	C₂×C₄²	1	1	yes	characteristic
66	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
67	Dic₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
68	Dic₃₇	C₄×Q₈	1	1	yes	characteristic
69	#1772	—	37	1	no	maximal, Sylow
70	C₂×C₃₇	—	2	1	no	—
71	C₂×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
72	D₃₇	—	2	1	no	—
73	D₃₇	#443	1	2	yes	—
74	C₂×#108	—	37	1	no	—
75	#427	—	37	1	no	—
76	#405	—	37	2	no	—
77	C₂×C₄×C₈	—	37	1	no	—
78	#339	—	37	2	no	—
79	C₃₇	#1772	1	1	yes	characteristic, complemented, Sylow
80	C₂×M₁₆	—	37	2	no	—
81	C₂²×C₈	—	37	1	no	—
82	C₂²×C₈	—	37	1	no	—
83	#117	—	37	2	no	—
84	C₂×C₄²	—	37	1	no	—
85	#108	—	37	2	no	—
86	#108	—	37	1	no	—
87	#108	#3866	1	1	yes	characteristic, complemented
88	#107	—	37	2	no	—
89	#106	—	37	2	no	—
90	#97	—	74	2	no	—
91	C₄×C₈	—	37	2	no	—
92	C₄×C₈	—	37	2	no	—
93	C₄×C₈	—	74	2	no	—
94	C₂²×C₄	—	37	1	no	—
95	C₂²×C₄	—	74	1	no	—
96	M₁₆	—	37	2	no	—
97	M₁₆	—	74	2	no	—
98	M₁₆	C₂×#3866	1	2	yes	—
99	C₂×C₈	—	37	2	no	—
100	C₂×C₈	—	37	1	no	—
101	C₂×C₈	—	37	1	no	—
102	C₂×C₈	—	37	2	no	—
103	C₂×C₈	—	37	1	no	—
104	C₂×C₈	—	37	1	no	—
105	C₂×C₈	—	37	1	no	—
106	C₂×C₈	—	37	1	no	—
107	C₂×C₈	—	37	1	no	—
108	C₂×C₈	—	74	2	no	—
109	C₂×C₈	C₂×#3866	1	1	yes	characteristic
110	C₄⋊C₄	—	37	4	no	—
111	C₂²⋊C₄	—	74	2	no	—
112	C₄²	—	37	2	no	—
113	C₄²	—	74	2	no	—
114	C₄²	—	74	1	no	—
115	C₂³	—	37	1	no	—
116	C₂×C₄	—	37	1	no	—
117	C₂×C₄	—	37	1	no	—
118	C₂×C₄	—	37	1	no	—
119	C₂×C₄	—	37	1	no	—
120	C₂×C₄	—	37	1	no	—
121	C₂×C₄	—	74	2	no	—
122	C₂×C₄	—	74	4	no	—
123	C₂×C₄	—	74	4	no	—
124	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	37	2	no	—
127	C₈	—	37	2	no	—
128	C₈	—	37	1	no	—
129	C₈	—	37	1	no	—
130	C₈	—	74	2	no	—
131	C₈	C₄×#3866	1	1	yes	characteristic
132	C₈	C₄×#3866	1	1	yes	characteristic
133	C₂²	—	37	1	no	—
134	C₂²	—	37	1	no	—
135	C₂²	—	74	2	no	—
136	C₂²	Q₈×#3866	1	1	yes	characteristic
137	C₄	—	37	1	no	—
138	C₄	—	37	1	no	—
139	C₄	—	74	4	no	—
140	C₄	—	148	4	no	—
141	C₄	C₂×C₄×#3866	1	1	yes	characteristic
142	C₄	D₄×#3866	1	1	yes	characteristic
143	C₂	—	2	1	no	—
144	C₂	—	37	2	no	—
145	C₂	—	74	1	no	—
146	C₂	C₄⋊C₄×#3866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558543

C₂

yes

maximal, characteristic

#14558499

C₂

yes

maximal

C₂×C₈×#3866

C₂

yes

maximal, characteristic

#14558405

C₂

yes

maximal

#108×D₃₇

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₈

C₂²

yes

characteristic

#11924615

C₂²

yes

—

C₂×C₄×#3866

C₂²

yes

characteristic

#11924605

C₄

yes

complemented

#11924604

C₄

yes

—

—

C₄

yes

complemented

C₈×#3866

C₄

yes

complemented

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

C₂²×#3866

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂²⋊#3866

—

C₂×#64438

Q₈

yes

—

C₄⋊#3866

C₂×C₄

yes

—

C₄×#3866

—

C₄×#3866

—

C₄×#3866

C₂×C₄

yes

—

D₃₇⋊C₈

C₂×C₄

yes

—

D₃₇⋊C₈

D₄

yes

characteristic

D₃₇⋊C₈

D₄

yes

characteristic

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

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

characteristic

C₈×D₃₇

C₂×C₄

yes

C₂²×D₃₇

yes

—

—

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

C₂×Q₈

yes

characteristic

C₄×D₃₇

C₂²×C₄

yes

characteristic

C₄×D₃₇

C₂²×C₄

yes

characteristic

C₄×D₃₇

C₂×D₄

yes

C₄×D₃₇

yes

C₄⋊C₄

yes

—

#64438

C₄⋊C₄

yes

—

C₈×C₃₇

—

C₈×C₃₇

C₄²

yes

characteristic

C₈×C₃₇

C₄²

yes

characteristic

#64436

—

#64436

C₄²

yes

characteristic

#64436

C₄²

yes

characteristic

C₂²×C₃₇

C₄×Q₈

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

#117

yes

characteristic

C₂×D₃₇

C₄×D₄

yes

characteristic

#3866

—

#3866

—

C₄×C₃₇

C₂×C₄²

yes

characteristic

C₄×C₃₇

C₄×D₄

yes

characteristic

Dic₃₇

C₂×C₄⋊C₄

yes

characteristic

Dic₃₇

C₄×Q₈

yes

characteristic

#1772

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₄×C₄⋊C₄

yes

characteristic

D₃₇

—

D₃₇

#443

yes

—

C₂×#108

—

#427

—

#405

—

C₂×C₄×C₈

—

#339

—

C₃₇

#1772

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

C₂×C₄²

—

—

—

yes

characteristic, complemented

#107

—

#106

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

C₂×#3866

yes

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂×#3866

yes

characteristic

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₂³

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂²×#3866

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

C₄×#3866

yes

characteristic

132

C₈

C₄×#3866

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

Q₈×#3866

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

148

—

141

C₄

C₂×C₄×#3866

yes

characteristic

142

C₄

D₄×#3866

yes

characteristic

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

C₄⋊C₄×#3866

yes

characteristic

	subgroup	G/H
center	C₄	D₄×#3866
commutator	C₄×C₃₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×#3866
Fitting	#108×C₃₇	C₄
socle	C₂×C₃₇	C₄×C₄⋊C₄
prime radical	C₂²×D₃₇	#41
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#108	#3866
hypocenter	C₃₇	#1772
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₄

D₄×#3866

commutator

C₄×C₃₇

C₂×C₄²

Frattini

C₂×C₄

C₂²×#3866

Fitting

#108×C₃₇

C₄

socle

C₂×C₃₇

C₄×C₄⋊C₄

prime radical

C₂²×D₃₇

#41

cosocle

C₂×C₄×D₃₇

C₂³

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
9. #11924605	C₄	semidirect
10. #11924604	C₄	semidirect
13. C₈×#3866	C₄	semidirect
14. C₈×#3866	C₄	semidirect
17. #11924472	C₄	semidirect
18. #11924471	C₄	semidirect
19. #108×C₃₇	C₄	semidirect
79. C₃₇	#1772	semidirect
87. #108	#3866	semidirect

H ≅ G/N

type

9. #11924605

C₄

semidirect

10. #11924604

C₄

semidirect

13. C₈×#3866

C₄

semidirect

14. C₈×#3866

C₄

semidirect

17. #11924472

C₄

semidirect

18. #11924471

C₄

semidirect

19. #108×C₃₇

C₄

semidirect

79. C₃₇

#1772

semidirect

87. #108

#3866

semidirect

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

subgroup	index
#14558543	2
#14558499	2
C₂×C₈×#3866	2
#14558405	2
#108×D₃₇	2
#1772	37

subgroup

index

C₂×C₈×#3866

#108×D₃₇

Subgroups — #18010401

Conjugacy classes of subgroups

Complements

Maximal subgroups