Group #17994254 — subgroups

Subgroups — #17994254

Order 4736.

186 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14557912	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557883	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557881	C₂	1	1	yes	maximal, characteristic
4	#14557881	C₂	1	1	yes	maximal, characteristic
5	#14557880	C₂	1	1	yes	maximal, characteristic
6	#14557255	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10596942	—	2	2	no	—
9	#11924536	C₂²	1	1	yes	characteristic
10	#11924533	C₂²	1	1	yes	characteristic
11	#11924460	—	2	2	no	—
12	#11924458	—	2	2	no	—
13	#11924456	—	2	2	no	—
14	#11924456	—	2	2	no	—
15	C₄×#64436	C₂²	1	1	yes	characteristic
16	C₄×#64436	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₃₇	C₂²	1	1	yes	characteristic
19	#106×C₃₇	—	2	2	no	—
20	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₃₇	—	2	1	no	—
22	C₂×Q₈×C₃₇	—	2	1	no	—
23	C₂×D₁₄₈	—	2	1	no	—
24	C₂×D₁₄₈	—	2	1	no	—
25	C₂×D₁₄₈	—	2	1	no	—
26	C₂×Dic₇₄	—	2	1	no	—
27	C₂×C₈×C₃₇	D₄	1	2	yes	—
28	C₄⋊C₄×C₃₇	—	2	1	no	—
29	C₄⋊C₄×C₃₇	—	2	1	no	—
30	C₄⋊C₄×C₃₇	—	2	1	no	—
31	C₄⋊C₄×C₃₇	—	2	1	no	—
32	C₄²×C₃₇	C₂³	1	1	yes	characteristic
33	#10596942	—	4	4	no	—
34	C₄⋊Dic₃₇	—	2	1	no	—
35	C₄⋊Dic₃₇	—	2	1	no	—
36	C₂×#64436	D₄	1	2	yes	—
37	C₂×#64436	D₄	1	2	yes	—
38	C₂²×D₃₇	—	4	1	no	—
39	Q₈×C₃₇	—	4	2	no	—
40	Q₈×C₃₇	—	4	2	no	—
41	C₂×C₄×C₃₇	—	4	1	no	—
42	C₂×C₄×C₃₇	—	4	1	no	—
43	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₃₇	—	4	1	no	—
47	D₁₄₈	—	4	2	no	—
48	D₁₄₈	—	4	2	no	—
49	D₁₄₈	—	4	2	no	—
50	Dic₇₄	—	4	2	no	—
51	C₈×C₃₇	—	2	2	no	—
52	#64436	—	2	2	no	—
53	#64436	—	2	2	no	—
54	C₂²×C₃₇	#127	1	1	yes	characteristic
55	C₂×D₃₇	—	8	1	no	—
56	C₂×D₃₇	—	8	1	no	—
57	C₂×D₃₇	—	8	1	no	—
58	C₄×C₃₇	—	8	1	no	—
59	C₄×C₃₇	—	8	1	no	—
60	C₄×C₃₇	#135	1	2	yes	—
61	C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₃₇	#137	1	2	yes	—
63	Dic₃₇	—	8	1	no	—
64	#1720	—	37	1	no	maximal, Sylow
65	C₂×C₃₇	#495	1	1	yes	characteristic
66	C₂×C₃₇	#496	1	1	yes	characteristic
67	C₂×C₃₇	#497	1	1	yes	characteristic
68	D₃₇	—	16	1	no	—
69	#498	D₃₇	1	1	yes	characteristic, complemented
70	#494	—	37	1	no	—
71	#487	—	37	1	no	—
72	#486	—	37	1	no	—
73	#486	—	37	1	no	—
74	#486	—	37	1	no	—
75	#322	—	37	1	no	—
76	C₃₇	#1720	1	1	yes	characteristic, complemented, Sylow
77	C₂×Q₁₆	—	74	2	no	—
78	C₄⋊Q₈	—	37	1	no	—
79	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
80	C₄⋊Q₈	C₂×D₃₇	1	1	yes	characteristic
81	#127	—	37	1	no	—
82	#106	—	2	2	no	—
83	#103	—	74	2	no	—
84	#102	—	74	2	no	—
85	#102	—	74	2	no	—
86	#102	—	74	2	no	—
87	C₄×C₈	—	37	1	no	—
88	C₄×C₈	—	37	1	no	—
89	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	74	1	no	—
93	C₂×D₄	—	74	1	no	—
94	C₂×D₄	—	74	1	no	—
95	C₂×D₄	—	74	1	no	—
96	Q₁₆	—	148	4	no	—
97	C₂×C₈	—	37	2	no	—
98	C₂×C₈	—	37	2	no	—
99	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	74	1	no	—
105	C₄⋊C₄	—	74	1	no	—
106	C₄²	C₂²×D₃₇	1	1	yes	characteristic
107	C₂³	—	148	1	no	—
108	Q₈	—	4	2	no	—
109	Q₈	—	4	2	no	—
110	Q₈	—	148	2	no	—
111	D₄	—	148	2	no	—
112	D₄	—	148	2	no	—
113	D₄	—	148	2	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	148	1	no	—
117	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
118	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	74	2	no	—
122	C₈	—	74	2	no	—
123	C₂²	—	296	1	no	—
124	C₂²	—	296	1	no	—
125	C₂²	—	296	1	no	—
126	C₂²	#11924532	1	1	yes	characteristic
127	C₄	—	8	1	no	—
128	C₄	—	8	1	no	—
129	C₄	—	296	1	no	—
130	C₄	#11924557	1	2	yes	—
131	C₄	#11924563	1	2	yes	—
132	C₄	#11924567	1	2	yes	—
133	C₂	—	592	1	no	—
134	C₂	#14557914	1	1	yes	characteristic
135	C₂	#14557919	1	1	yes	characteristic
136	C₂	#14557925	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14557912

C₂

yes

maximal, characteristic, complemented

#14557883

C₂

yes

maximal, characteristic, complemented

#14557881

C₂

yes

maximal, characteristic

#14557881

C₂

yes

maximal, characteristic

#14557880

C₂

yes

maximal, characteristic

#14557255

C₂

yes

maximal, characteristic, complemented

C₃₇×#498

C₂

yes

maximal, characteristic, complemented

C₂×#10596942

—

#11924536

C₂²

yes

characteristic

#11924533

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#64436

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₂²

yes

characteristic

#106×C₃₇

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×Dic₇₄

—

C₂×C₈×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596942

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₂×#64436

D₄

yes

—

C₂×#64436

D₄

yes

—

C₂²×D₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

#64436

—

C₂²×C₃₇

#127

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#135

yes

—

C₄×C₃₇

AGL(1,ℤ/8)

yes

—

C₄×C₃₇

#137

yes

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

—

#498

D₃₇

yes

characteristic, complemented

—

—

—

—

—

—

C₃₇

#1720

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

characteristic

C₄⋊Q₈

C₂×D₃₇

yes

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

148

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₇

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

C₂²×D₃₇

yes

characteristic

107

C₂³

—

148

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

148

—

111

D₄

—

148

—

112

D₄

—

148

—

113

D₄

—

148

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

148

—

117

C₂×C₄

D₄×D₃₇

yes

characteristic

118

C₂×C₄

D₄×D₃₇

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

296

—

124

C₂²

—

296

—

125

C₂²

—

296

—

126

C₂²

#11924532

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

296

—

130

C₄

#11924557

yes

—

131

C₄

#11924563

yes

—

132

C₄

#11924567

yes

—

133

C₂

—

592

—

134

C₂

#14557914

yes

characteristic

135

C₂

#14557919

yes

characteristic

136

C₂

#14557925

yes

characteristic

	subgroup	G/H
center	C₂²	#11924532
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#498	C₂
socle	C₂²×C₃₇	#127
prime radical	#11924533	C₂²
hypercenter	#498	D₃₇
hypocenter	C₃₇	#1720
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#11924532

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#498

C₂

C₂²×C₃₇

C₂²

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
1. #14557912	C₂	semidirect
2. #14557883	C₂	semidirect
6. #14557255	C₂	semidirect
7. C₃₇×#498	C₂	semidirect
69. #498	D₃₇	semidirect
76. C₃₇	#1720	semidirect

H ≅ G/N

type

1. #14557912

C₂

semidirect

2. #14557883

C₂

semidirect

6. #14557255

C₂

semidirect

7. C₃₇×#498

C₂

semidirect

69. #498

D₃₇

semidirect

76. C₃₇

#1720

semidirect

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

subgroup	index
#14557912	2
#14557883	2
#14557881	2
#14557880	2
#14557255	2
C₃₇×#498	2
#1720	37

subgroup

index

C₃₇×#498

Subgroups — #17994254

Conjugacy classes of subgroups

Complements

Maximal subgroups