Group #17997178 — subgroups

Subgroups — #17997178

Order 4736.

1104 conjugacy classes of subgroups; only the 309 normal subgroups are shown.

normal subgroups only

This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 309 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×#11924495	C₂	1	2	yes	maximal, complemented
2	#14557436	C₂	1	8	yes	maximal, complemented
3	#14557389	C₂	1	4	yes	maximal, complemented
4	C₂³×D₄×C₃₇	C₂	1	1	yes	maximal, characteristic
5	#11924495	—	2	8	no	—
6	C₂×C₄⋊Dic₃₇	C₂²	1	4	yes	complemented
7	C₂×C₂²⋊Dic₃₇	C₂²	1	4	yes	—
8	C₂×C₂²⋊Dic₃₇	C₂²	1	16	yes	complemented
9	C₂×C₄×Dic₃₇	C₂²	1	4	yes	complemented
10	C₂⁵×C₃₇	C₂²	1	2	yes	—
11	C₂²×D₄×C₃₇	—	2	8	no	—
12	C₂²×D₄×C₃₇	C₄	1	8	yes	complemented
13	C₂²×D₄×C₃₇	C₂²	1	4	yes	—
14	C₂³×C₄×C₃₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₃₇	—	2	8	no	—
16	C₂⁴×C₃₇	—	4	8	no	—
17	C₂⁴×C₃₇	C₂×C₄	1	8	yes	—
18	C₂⁴×C₃₇	C₂³	1	4	yes	—
19	C₂⁴×C₃₇	C₂³	1	1	yes	characteristic
20	C₂×D₄×C₃₇	—	2	32	no	—
21	C₂×D₄×C₃₇	—	2	8	no	—
22	C₂×D₄×C₃₇	—	2	8	no	—
23	C₂×D₄×C₃₇	C₂×C₄	1	16	yes	—
24	C₂²×C₄×C₃₇	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₃₇	D₄	1	8	yes	—
26	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
27	C₂²×Dic₃₇	C₂³	1	8	yes	—
28	C₂²⋊Dic₃₇	—	2	8	no	—
29	C₂²⋊Dic₃₇	—	4	16	no	—
30	C₄⋊Dic₃₇	—	2	8	no	—
31	C₄×Dic₃₇	—	2	8	no	—
32	C₂³×C₃₇	—	2	4	no	—
33	C₂³×C₃₇	—	2	16	no	—
34	C₂³×C₃₇	—	2	8	no	—
35	C₂³×C₃₇	—	2	16	no	—
36	C₂³×C₃₇	—	4	16	no	—
37	C₂³×C₃₇	—	4	32	no	—
38	C₂³×C₃₇	C₂²×C₄	1	8	yes	—
39	C₂³×C₃₇	C₂²×C₄	1	2	yes	—
40	C₂³×C₃₇	C₂×D₄	1	4	yes	—
41	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
42	D₄×C₃₇	—	2	32	no	—
43	C₂×C₄×C₃₇	C₂²⋊C₄	1	16	yes	complemented
44	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₃₇	C₂×D₄	1	4	yes	—
47	C₂×Dic₃₇	—	2	8	no	—
48	C₂×Dic₃₇	—	2	16	no	—
49	C₂²×C₃₇	—	2	4	no	—
50	C₂²×C₃₇	—	2	8	no	—
51	C₂²×C₃₇	—	2	8	no	—
52	C₂²×C₃₇	—	2	16	no	—
53	C₂²×C₃₇	—	4	32	no	—
54	C₂²×C₃₇	—	4	16	no	—
55	C₂²×C₃₇	C₂×C₂²⋊C₄	1	4	yes	—
56	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
57	C₂²×C₃₇	C₂²×D₄	1	2	yes	—
58	C₂²×C₃₇	H(5,𝔽₂)	1	4	yes	—
59	C₄×C₃₇	C₂×C₂²⋊C₄	1	8	yes	—
60	Dic₃₇	—	4	8	no	—
61	#2309	—	37	1	no	maximal, Sylow
62	C₂×C₃₇	—	2	4	no	—
63	C₂×C₃₇	—	4	8	no	—
64	C₂×C₃₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃₇	#518	1	2	yes	—
66	C₂×C₃₇	#535	1	4	yes	—
67	C₂³×D₄	D₃₇	1	1	yes	characteristic
68	C₂×#117	—	37	2	no	—
69	#390	—	37	8	no	—
70	#379	—	37	4	no	—
71	C₃₇	#2309	1	1	yes	characteristic, complemented, Sylow
72	C₂⁵	C₂×D₃₇	1	2	yes	—
73	C₂²×D₄	—	2	8	no	—
74	C₂²×D₄	Dic₃₇	1	8	yes	complemented
75	C₂²×D₄	C₂×D₃₇	1	4	yes	—
76	C₂³×C₄	C₂×D₃₇	1	1	yes	characteristic
77	#117	—	74	8	no	—
78	C₂×C₄⋊C₄	—	37	4	no	—
79	C₂×C₂²⋊C₄	—	37	4	no	—
80	C₂×C₂²⋊C₄	—	37	16	no	—
81	C₂×C₄²	—	37	4	no	—
82	C₂⁴	—	2	8	no	—
83	C₂⁴	—	4	8	no	—
84	C₂⁴	C₂×Dic₃₇	1	8	yes	—
85	C₂⁴	C₂²×D₃₇	1	4	yes	—
86	C₂⁴	C₂²×D₃₇	1	1	yes	characteristic
87	C₂×D₄	—	2	32	no	—
88	C₂×D₄	—	2	8	no	—
89	C₂×D₄	—	2	8	no	—
90	C₂×D₄	C₂×Dic₃₇	1	16	yes	—
91	C₂²×C₄	—	37	8	no	—
92	C₂²×C₄	C₂×Dic₃₇	1	4	yes	—
93	C₂²×C₄	C₂²⋊D₃₇	1	8	yes	—
94	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
95	C₄⋊C₄	—	74	8	no	—
96	C₂²⋊C₄	—	74	8	no	—
97	C₂²⋊C₄	—	148	16	no	—
98	C₄²	—	74	8	no	—
99	C₂³	—	2	4	no	—
100	C₂³	—	2	16	no	—
101	C₂³	—	2	8	no	—
102	C₂³	—	2	16	no	—
103	C₂³	—	4	16	no	—
104	C₂³	—	4	32	no	—
105	C₂³	C₂²×Dic₃₇	1	8	yes	—
106	C₂³	C₂²×Dic₃₇	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₃₇	1	4	yes	—
108	C₂³	C₂³×D₃₇	1	1	yes	characteristic
109	D₄	—	2	32	no	—
110	C₂×C₄	—	74	8	no	—
111	C₂×C₄	—	74	16	no	—
112	C₂×C₄	C₂²⋊Dic₃₇	1	16	yes	complemented
113	C₂×C₄	C₂²×Dic₃₇	1	4	yes	—
114	C₂×C₄	C₂×C₂²⋊D₃₇	1	4	yes	—
115	C₂×C₄	C₂×C₂²⋊D₃₇	1	4	yes	—
116	C₂²	—	2	4	no	—
117	C₂²	—	2	8	no	—
118	C₂²	—	2	8	no	—
119	C₂²	—	2	16	no	—
120	C₂²	—	4	32	no	—
121	C₂²	—	4	16	no	—
122	C₂²	C₂×C₂²⋊Dic₃₇	1	4	yes	—
123	C₂²	C₂³×Dic₃₇	1	1	yes	characteristic
124	C₂²	C₂²×C₂²⋊D₃₇	1	2	yes	—
125	C₂²	#11924581	1	4	yes	—
126	C₄	—	148	8	no	—
127	C₄	C₂×C₂²⋊Dic₃₇	1	8	yes	—
128	C₂	—	2	4	no	—
129	C₂	—	4	8	no	—
130	C₂	C₂²×C₂²⋊Dic₃₇	1	1	yes	characteristic
131	C₂	#14558008	1	2	yes	—
132	C₂	#14558100	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924495

C₂

yes

maximal, complemented

#14557436

C₂

yes

maximal, complemented

#14557389

C₂

yes

maximal, complemented

C₂³×D₄×C₃₇

C₂

yes

maximal, characteristic

#11924495

—

C₂×C₄⋊Dic₃₇

C₂²

yes

complemented

C₂×C₂²⋊Dic₃₇

C₂²

yes

—

C₂×C₂²⋊Dic₃₇

C₂²

yes

complemented

C₂×C₄×Dic₃₇

C₂²

yes

complemented

C₂⁵×C₃₇

C₂²

yes

—

C₂²×D₄×C₃₇

—

C₂²×D₄×C₃₇

C₄

yes

complemented

C₂²×D₄×C₃₇

C₂²

yes

—

C₂³×C₄×C₃₇

C₂²

yes

characteristic

C₂⁴×C₃₇

—

C₂⁴×C₃₇

—

C₂⁴×C₃₇

C₂×C₄

yes

—

C₂⁴×C₃₇

C₂³

yes

—

C₂⁴×C₃₇

C₂³

yes

characteristic

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×Dic₃₇

C₂³

yes

—

C₂²⋊Dic₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂²×C₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

—

C₂³×C₃₇

C₂⁴

yes

characteristic

D₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

complemented

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂²×C₃₇

C₂²×D₄

yes

—

C₂²×C₃₇

H(5,𝔽₂)

yes

—

C₄×C₃₇

C₂×C₂²⋊C₄

yes

—

Dic₃₇

—

#2309

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₂²×C₂²⋊C₄

yes

characteristic

C₂×C₃₇

#518

yes

—

C₂×C₃₇

#535

yes

—

C₂³×D₄

D₃₇

yes

—

—

—

C₃₇

#2309

yes

characteristic, complemented, Sylow

C₂⁵

C₂×D₃₇

yes

—

C₂²×D₄

—

C₂²×D₄

Dic₃₇

yes

complemented

C₂²×D₄

C₂×D₃₇

yes

—

C₂³×C₄

C₂×D₃₇

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂×Dic₃₇

yes

—

C₂⁴

C₂²×D₃₇

yes

—

C₂⁴

C₂²×D₃₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×Dic₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₃₇

yes

—

C₂²×C₄

C₂²⋊D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

148

—

C₄²

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

C₂²×Dic₃₇

yes

—

106

C₂³

C₂²×Dic₃₇

yes

—

107

C₂³

C₂×C₂²⋊D₃₇

yes

—

108

C₂³

C₂³×D₃₇

yes

characteristic

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²⋊Dic₃₇

yes

complemented

113

C₂×C₄

C₂²×Dic₃₇

yes

—

114

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

115

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₂²⋊Dic₃₇

yes

—

123

C₂²

C₂³×Dic₃₇

yes

characteristic

124

C₂²

C₂²×C₂²⋊D₃₇

yes

—

125

C₂²

#11924581

yes

—

126

C₄

—

148

—

127

C₄

C₂×C₂²⋊Dic₃₇

yes

—

128

C₂

—

129

C₂

—

130

C₂

C₂²×C₂²⋊Dic₃₇

yes

characteristic

131

C₂

#14558008

yes

—

132

C₂

#14558100

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₇
commutator	C₂²×C₃₇	C₂³×C₄
Fitting, prime radical	C₂³×D₄×C₃₇	C₂
socle, cosocle	C₂³×C₃₇	C₂⁴
hypercenter	C₂³×D₄	D₃₇
hypocenter	C₃₇	#2309
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂³×D₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₇

commutator

C₂²×C₃₇

C₂³×C₄

Fitting, prime radical

C₂³×D₄×C₃₇

C₂

socle, cosocle

C₂³×C₃₇

C₂⁴

hypercenter

C₂³×D₄

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂³ ◁ C₂³×D₄

N	H ≅ G/N	type
1. C₂×#11924495	C₂	semidirect
2. #14557436	C₂	semidirect
3. #14557389	C₂	semidirect
6. C₂×C₄⋊Dic₃₇	C₂²	semidirect
8. C₂×C₂²⋊Dic₃₇	C₂²	semidirect
9. C₂×C₄×Dic₃₇	C₂²	semidirect
12. C₂²×D₄×C₃₇	C₄	semidirect
43. C₂×C₄×C₃₇	C₂²⋊C₄	semidirect
71. C₃₇	#2309	semidirect
74. C₂²×D₄	Dic₃₇	semidirect
112. C₂×C₄	C₂²⋊Dic₃₇	semidirect

H ≅ G/N

type

1. C₂×#11924495

C₂

semidirect

2. #14557436

C₂

semidirect

3. #14557389

C₂

semidirect

6. C₂×C₄⋊Dic₃₇

C₂²

semidirect

8. C₂×C₂²⋊Dic₃₇

C₂²

semidirect

9. C₂×C₄×Dic₃₇

C₂²

semidirect

12. C₂²×D₄×C₃₇

C₄

semidirect

43. C₂×C₄×C₃₇

C₂²⋊C₄

semidirect

71. C₃₇

#2309

semidirect

74. C₂²×D₄

Dic₃₇

semidirect

112. C₂×C₄

C₂²⋊Dic₃₇

semidirect

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

subgroup	index
C₂×#11924495	2
#14557436	2
#14557389	2
C₂³×D₄×C₃₇	2
#2309	37

subgroup

index

C₂³×D₄×C₃₇

Subgroups — #17997178

Conjugacy classes of subgroups

Complements

Maximal subgroups