Group #18010779 — subgroups

Subgroups — #18010779

Order 4736.

260 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 78 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	#14558492	C₂	1	1	yes	maximal, characteristic
2	#14558479	C₂	1	1	yes	maximal, characteristic
3	#14558406	C₂	1	4	yes	maximal
4	#125×D₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×D₃₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#3866	—	2	2	no	—
7	C₂×C₄⋊#3866	C₂²	1	2	yes	—
8	#11924591	—	2	2	no	—
9	#11924528	C₄	1	1	yes	characteristic, complemented
10	#11924527	C₄	1	2	yes	complemented
11	#11924526	—	2	2	no	—
12	C₄⋊C₄×D₃₇	—	2	2	no	—
13	C₄⋊C₄×D₃₇	C₂²	1	2	yes	—
14	C₄²×D₃₇	C₂²	1	1	yes	characteristic
15	#125×C₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂²×#3866	—	2	2	no	—
17	C₂×C₄×D₃₇	—	2	2	no	—
18	C₂×C₄×D₃₇	D₄	1	2	yes	—
19	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
20	C₂×#64438	—	2	2	no	—
21	C₄⋊#3866	—	2	4	no	—
22	C₄⋊#3866	—	4	4	no	—
23	D₃₇⋊C₈	—	2	2	no	—
24	D₃₇⋊C₈	Q₈	1	4	yes	—
25	C₄⋊C₄×C₃₇	—	2	2	no	—
26	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
27	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
28	C₄⋊Dic₃₇	—	2	2	no	—
29	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
30	#10596936	—	2	2	no	—
31	#10596936	—	2	2	no	—
32	#10596936	—	2	2	no	—
33	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
34	C₄×Dic₃₇	D₄	1	2	yes	—
35	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
36	C₂×#3866	—	4	2	no	—
37	C₂×#3866	—	4	4	no	—
38	C₂×C₄×C₃₇	—	2	2	no	—
39	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₃₇	—	2	2	no	—
42	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
43	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
44	C₄×D₃₇	—	2	2	no	—
45	C₄×D₃₇	—	2	2	no	—
46	C₄×D₃₇	—	4	2	no	—
47	C₄×D₃₇	—	4	2	no	—
48	C₄×D₃₇	C₂×Q₈	1	2	yes	—
49	C₄×D₃₇	#41	1	2	yes	—
50	#64438	—	2	4	no	—
51	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×D₃₇	#124	1	1	yes	characteristic
53	C₂×D₃₇	C₄⋊Q₈	1	1	yes	characteristic
54	C₂×D₃₇	AGL(1,ℤ/8)	1	2	yes	—
55	C₂×D₃₇	#137	1	2	yes	—
56	#3866	—	8	2	no	—
57	C₄×C₃₇	—	2	2	no	—
58	C₄×C₃₇	—	4	2	no	—
59	C₄×C₃₇	C₄×Q₈	1	2	yes	—
60	Dic₃₇	—	2	2	no	—
61	Dic₃₇	—	4	2	no	—
62	Dic₃₇	C₂²⋊Q₈	1	2	yes	—
63	#1992	—	37	1	no	maximal, Sylow
64	C₂×C₃₇	#391	1	1	yes	characteristic
65	C₂×C₃₇	#417	1	2	yes	—
66	D₃₇	#490	1	1	yes	characteristic
67	D₃₇	#491	1	1	yes	characteristic
68	D₃₇	#501	1	2	yes	—
69	C₂×#125	—	37	1	no	—
70	C₂×#97	—	37	1	no	—
71	#384	—	37	1	no	—
72	#340	—	37	4	no	—
73	C₃₇	#1992	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₈	—	37	2	no	—
75	#125	—	37	2	no	—
76	#125	—	37	1	no	—
77	#125	—	74	2	no	—
78	#125	#3866	1	1	yes	characteristic, complemented
79	C₂×C₄⋊C₄	—	37	2	no	—
80	C₂×C₄⋊C₄	—	37	2	no	—
81	C₂×C₄⋊C₄	—	74	2	no	—
82	C₂×C₄⋊C₄	—	74	2	no	—
83	C₂×C₄²	—	37	1	no	—
84	#97	—	74	2	no	—
85	C₂²×C₄	—	37	1	no	—
86	C₂²×C₄	—	37	2	no	—
87	C₂²×C₄	—	74	2	no	—
88	C₂²×C₄	—	74	2	no	—
89	C₂×C₈	—	37	4	no	—
90	C₂×C₈	—	74	2	no	—
91	C₂×C₈	—	74	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	37	2	no	—
94	C₄⋊C₄	—	74	2	no	—
95	C₄⋊C₄	—	74	2	no	—
96	C₄⋊C₄	—	74	2	no	—
97	C₄⋊C₄	—	74	4	no	—
98	C₄⋊C₄	—	74	2	no	—
99	C₄⋊C₄	—	148	4	no	—
100	C₄⋊C₄	C₂×#3866	1	2	yes	—
101	C₄²	—	37	2	no	—
102	C₄²	—	37	1	no	—
103	C₄²	C₂×#3866	1	1	yes	characteristic
104	C₂³	—	37	1	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	37	1	no	—
107	C₂×C₄	—	37	2	no	—
108	C₂×C₄	—	37	2	no	—
109	C₂×C₄	—	37	2	no	—
110	C₂×C₄	—	74	2	no	—
111	C₂×C₄	—	74	2	no	—
112	C₂×C₄	—	74	2	no	—
113	C₂×C₄	—	148	2	no	—
114	C₂×C₄	—	148	2	no	—
115	C₂×C₄	—	148	2	no	—
116	C₂×C₄	—	148	4	no	—
117	C₂×C₄	C₂²⋊#3866	1	2	yes	—
118	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
119	C₈	—	74	4	no	—
120	C₂²	—	37	1	no	—
121	C₂²	—	37	1	no	—
122	C₂²	—	37	2	no	—
123	C₂²	—	37	2	no	—
124	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	4	2	no	—
127	C₄	—	37	2	no	—
128	C₄	—	296	2	no	—
129	C₄	—	74	2	no	—
130	C₄	—	148	2	no	—
131	C₄	Q₈×#3866	1	2	yes	—
132	C₂	—	37	1	no	—
133	C₂	—	37	1	no	—
134	C₂	—	37	2	no	—
135	C₂	(C₂×#3866)⋊Q₈	1	1	yes	characteristic
136	C₂	#14558524	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558492

C₂

yes

maximal, characteristic

#14558479

C₂

yes

maximal, characteristic

#14558406

C₂

yes

maximal

#125×D₃₇

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₈

C₂²

yes

—

C₂×C₄⋊#3866

—

C₂×C₄⋊#3866

C₂²

yes

—

#11924591

—

#11924528

C₄

yes

characteristic, complemented

#11924527

C₄

yes

complemented

#11924526

—

C₄⋊C₄×D₃₇

—

C₄⋊C₄×D₃₇

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

#125×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

—

—

D₃₇⋊C₈

—

D₃₇⋊C₈

Q₈

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

—

#10596936

—

#10596936

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

—

C₂²×D₃₇

C₂×D₄

yes

characteristic

C₂×#3866

—

C₂×#3866

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

C₂×Q₈

yes

—

C₄×D₃₇

#41

yes

—

#64438

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₃₇

#124

yes

characteristic

C₂×D₃₇

C₄⋊Q₈

yes

characteristic

C₂×D₃₇

AGL(1,ℤ/8)

yes

—

C₂×D₃₇

#137

yes

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×Q₈

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

C₂²⋊Q₈

yes

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

D₃₇

yes

D₃₇

yes

D₃₇

yes

—

C₂×#125

—

C₂×#97

—

#384

—

#340

—

C₃₇

#1992

yes

characteristic, complemented, Sylow

C₂²×C₈

—

—

—

—

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#97

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

148

—

100

C₄⋊C₄

C₂×#3866

yes

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂×#3866

yes

characteristic

104

C₂³

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

148

—

114

C₂×C₄

—

148

—

115

C₂×C₄

—

148

—

116

C₂×C₄

—

148

—

117

C₂×C₄

C₂²⋊#3866

yes

—

118

C₂×C₄

C₂²×#3866

yes

characteristic

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊#3866

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

296

—

129

C₄

—

130

C₄

—

148

—

131

C₄

Q₈×#3866

yes

—

132

C₂

—

133

C₂

—

134

C₂

—

135

C₂

(C₂×#3866)⋊Q₈

yes

characteristic

136

C₂

#14558524

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	#125×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₃₇	C₂×D₄
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	#125	#3866
hypocenter	C₃₇	#1992
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

#125×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₃₇

C₂×D₄

cosocle

C₂×C₄×D₃₇

C₂³

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
9. #11924528	C₄	semidirect
10. #11924527	C₄	semidirect
15. #125×C₃₇	C₄	semidirect
73. C₃₇	#1992	semidirect
78. #125	#3866	semidirect

H ≅ G/N

type

9. #11924528

C₄

semidirect

10. #11924527

C₄

semidirect

15. #125×C₃₇

C₄

semidirect

73. C₃₇

#1992

semidirect

78. #125

#3866

semidirect

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

subgroup	index
#14558492	2
#14558479	2
#14558406	2
#125×D₃₇	2
#1992	37

subgroup

index

#125×D₃₇

Subgroups — #18010779

Conjugacy classes of subgroups

Complements

Maximal subgroups