Group #18008171 — subgroups

Subgroups — C₂×#14558263 #18008171

Order 4736.

1112 conjugacy classes of subgroups; only the 581 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 581 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	#14558263	C₂	1	16	yes	maximal, direct product factor
2	C₂×#11924535	C₂	1	4	yes	maximal
3	C₂×#11924534	C₂	1	4	yes	maximal
4	C₂×C₄×Dic₇₄	C₂	1	2	yes	maximal
5	C₂×#11924501	C₂	1	4	yes	maximal
6	C₂×C₄⋊Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₇₄	C₂²	1	4	yes	—
8	C₂²×Dic₇₄	C₂²	1	2	yes	—
9	#11924535	C₂²	1	32	yes	—
10	#11924534	C₂²	1	32	yes	—
11	C₄×Dic₇₄	C₂²	1	16	yes	—
12	#11924501	C₂²	1	32	yes	—
13	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
14	C₂×#10596936	C₂²	1	8	yes	—
15	C₂×#10596936	C₂²	1	4	yes	—
16	C₂×C₄×Dic₃₇	C₂²	1	4	yes	—
17	C₂×C₄×Dic₃₇	C₂²	1	4	yes	—
18	C₂²×Q₈×C₃₇	C₂²	1	2	yes	—
19	C₄⋊Q₈×C₃₇	C₂²	1	8	yes	—
20	C₂×C₄⋊C₄×C₃₇	C₂²	1	4	yes	—
21	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
22	C₂×Q₈×C₃₇	—	4	4	no	—
23	C₂×Q₈×C₃₇	C₂³	1	8	yes	—
24	C₂²×C₄×C₃₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₃₇	C₂³	1	2	yes	—
27	C₂²×Dic₃₇	C₂³	1	4	yes	—
28	C₂²×Dic₃₇	C₂³	1	4	yes	—
29	C₂×Dic₇₄	—	4	8	no	—
30	C₂×Dic₇₄	Q₈	1	16	yes	complemented
31	C₂×Dic₇₄	C₂³	1	16	yes	—
32	C₂×Dic₇₄	C₂³	1	8	yes	—
33	C₄⋊C₄×C₃₇	C₂³	1	16	yes	—
34	C₄²×C₃₇	C₂³	1	4	yes	—
35	C₄⋊Dic₃₇	C₂³	1	8	yes	—
36	#10596936	C₂³	1	32	yes	—
37	#10596936	C₂³	1	16	yes	—
38	C₄×Dic₃₇	C₂³	1	16	yes	—
39	C₄×Dic₃₇	C₂³	1	16	yes	—
40	C₂³×C₃₇	C₂⁴	1	1	yes	characteristic
41	Q₈×C₃₇	—	4	8	no	—
42	C₂×C₄×C₃₇	—	2	8	no	—
43	C₂×C₄×C₃₇	—	2	2	no	—
44	C₂×C₄×C₃₇	C₂×Q₈	1	8	yes	—
45	C₂×C₄×C₃₇	C₂⁴	1	8	yes	—
46	C₂×C₄×C₃₇	C₂⁴	1	2	yes	—
47	C₂×C₄×C₃₇	C₂⁴	1	4	yes	—
48	C₂×Dic₃₇	—	2	8	no	—
49	C₂×Dic₃₇	C₂×Q₈	1	16	yes	—
50	C₂×Dic₃₇	C₂⁴	1	8	yes	—
51	C₂×Dic₃₇	C₂⁴	1	8	yes	—
52	Dic₇₄	—	4	16	no	—
53	Dic₇₄	C₂×Q₈	1	32	yes	complemented
54	C₂²×C₃₇	C₂²×Q₈	1	4	yes	—
55	C₂²×C₃₇	H(5,𝔽₂)	1	2	yes	—
56	C₂²×C₃₇	C₂⁵	1	1	yes	characteristic
57	C₄×C₃₇	—	2	8	no	—
58	C₄×C₃₇	—	2	2	no	—
59	C₄×C₃₇	C₂²×Q₈	1	8	yes	—
60	Dic₃₇	—	2	8	no	—
61	Dic₃₇	C₂²×Q₈	1	16	yes	—
62	C₂×Q₈²	—	37	1	no	maximal, Sylow
63	C₂×C₃₇	Q₈²	1	4	yes	complemented
64	C₂×C₃₇	C₂³×Q₈	1	2	yes	—
65	C₂×C₃₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
66	Q₈²	—	37	16	no	—
67	C₂×C₄⋊Q₈	—	37	4	no	—
68	C₂×C₄⋊Q₈	—	37	4	no	—
69	C₂×C₄⋊Q₈	D₃₇	1	1	yes	characteristic
70	C₂×C₄×Q₈	—	37	4	no	—
71	C₂×C₄×Q₈	—	37	2	no	—
72	C₃₇	C₂×Q₈²	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	37	2	no	—
74	C₂²×Q₈	—	37	4	no	—
75	C₂²×Q₈	C₂×D₃₇	1	2	yes	—
76	C₄⋊Q₈	—	37	32	no	—
77	C₄⋊Q₈	—	37	32	no	—
78	C₄⋊Q₈	C₂×D₃₇	1	8	yes	—
79	C₄×Q₈	—	37	32	no	—
80	C₄×Q₈	—	37	16	no	—
81	C₂×C₄⋊C₄	—	37	8	no	—
82	C₂×C₄⋊C₄	—	37	4	no	—
83	C₂×C₄⋊C₄	—	37	2	no	—
84	C₂×C₄⋊C₄	C₂×D₃₇	1	4	yes	—
85	C₂×C₄²	—	37	4	no	—
86	C₂×C₄²	—	37	4	no	—
87	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
88	C₂×Q₈	—	4	4	no	—
89	C₂×Q₈	—	37	16	no	—
90	C₂×Q₈	—	37	8	no	—
91	C₂×Q₈	—	37	16	no	—
92	C₂×Q₈	—	148	8	no	—
93	C₂×Q₈	C₂²×D₃₇	1	8	yes	—
94	C₂²×C₄	—	37	4	no	—
95	C₂²×C₄	—	37	4	no	—
96	C₂²×C₄	C₂²×D₃₇	1	4	yes	—
97	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₃₇	1	2	yes	—
99	C₄⋊C₄	—	37	32	no	—
100	C₄⋊C₄	—	37	16	no	—
101	C₄⋊C₄	—	37	8	no	—
102	C₄⋊C₄	C₂²×D₃₇	1	16	yes	—
103	C₄²	—	37	16	no	—
104	C₄²	—	37	16	no	—
105	C₄²	C₂²×D₃₇	1	4	yes	—
106	C₂³	C₂³×D₃₇	1	1	yes	characteristic
107	Q₈	—	4	8	no	—
108	Q₈	—	37	32	no	—
109	Q₈	—	148	16	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	37	16	no	—
113	C₂×C₄	—	37	8	no	—
114	C₂×C₄	—	37	8	no	—
115	C₂×C₄	—	74	8	no	—
116	C₂×C₄	Q₈×D₃₇	1	8	yes	—
117	C₂×C₄	C₂³×D₃₇	1	8	yes	—
118	C₂×C₄	C₂³×D₃₇	1	2	yes	—
119	C₂×C₄	C₂³×D₃₇	1	4	yes	—
120	C₂²	C₂×Q₈×D₃₇	1	4	yes	—
121	C₂²	#11924581	1	2	yes	—
122	C₂²	C₂⁴×D₃₇	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	8	no	—
125	C₄	—	37	16	no	—
126	C₄	—	74	8	no	—
127	C₄	C₂×Q₈×D₃₇	1	8	yes	—
128	C₂	#14558263	1	4	yes	direct product factor
129	C₂	C₂²×Q₈×D₃₇	1	2	yes	—
130	C₂	C₂×#11924581	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558263

C₂

yes

maximal, direct product factor

C₂×#11924535

C₂

yes

maximal

C₂×#11924534

C₂

yes

maximal

C₂×C₄×Dic₇₄

C₂

yes

maximal

C₂×#11924501

C₂

yes

maximal

C₂×C₄⋊Q₈×C₃₇

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

—

C₂²×Dic₇₄

C₂²

yes

—

#11924535

C₂²

yes

—

#11924534

C₂²

yes

—

C₄×Dic₇₄

C₂²

yes

—

#11924501

C₂²

yes

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×#10596936

C₂²

yes

—

C₂×#10596936

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

C₂²×Q₈×C₃₇

C₂²

yes

—

C₄⋊Q₈×C₃₇

C₂²

yes

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

—

C₂²×Dic₃₇

C₂³

yes

—

C₂²×Dic₃₇

C₂³

yes

—

C₂×Dic₇₄

—

C₂×Dic₇₄

Q₈

yes

complemented

C₂×Dic₇₄

C₂³

yes

—

C₂×Dic₇₄

C₂³

yes

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂³

yes

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

C₂³

yes

—

#10596936

C₂³

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂³×C₃₇

C₂⁴

yes

characteristic

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×Q₈

yes

—

C₂×Dic₃₇

C₂⁴

yes

—

C₂×Dic₃₇

C₂⁴

yes

—

Dic₇₄

—

Dic₇₄

C₂×Q₈

yes

complemented

C₂²×C₃₇

C₂²×Q₈

yes

—

C₂²×C₃₇

H(5,𝔽₂)

yes

—

C₂²×C₃₇

C₂⁵

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²×Q₈

yes

—

Dic₃₇

—

Dic₃₇

C₂²×Q₈

yes

—

C₂×Q₈²

—

maximal, Sylow

C₂×C₃₇

Q₈²

yes

complemented

C₂×C₃₇

C₂³×Q₈

yes

—

C₂×C₃₇

C₂×H(5,𝔽₂)

yes

characteristic

Q₈²

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₃₇

yes

characteristic

C₂×C₄×Q₈

—

C₂×C₄×Q₈

—

C₃₇

C₂×Q₈²

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₃₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

148

—

C₂×Q₈

C₂²×D₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

C₂²×C₄

C₂²×D₃₇

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

C₂²×D₃₇

yes

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂²×D₃₇

yes

—

106

C₂³

C₂³×D₃₇

yes

characteristic

107

Q₈

—

108

Q₈

—

109

Q₈

—

148

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

Q₈×D₃₇

yes

—

117

C₂×C₄

C₂³×D₃₇

yes

—

118

C₂×C₄

C₂³×D₃₇

yes

—

119

C₂×C₄

C₂³×D₃₇

yes

—

120

C₂²

C₂×Q₈×D₃₇

yes

—

121

C₂²

#11924581

yes

—

122

C₂²

C₂⁴×D₃₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

C₂×Q₈×D₃₇

yes

—

128

C₂

#14558263

yes

direct product factor

129

C₂

C₂²×Q₈×D₃₇

yes

—

130

C₂

C₂×#11924581

yes

characteristic

	subgroup	G/H
center	C₂³	C₂³×D₃₇
commutator, cosocle	C₂²×C₃₇	C₂⁵
Frattini	C₂²	C₂⁴×D₃₇
Fitting	C₂×C₄⋊Q₈×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₃₇
hypocenter	C₃₇	C₂×Q₈²
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂³×D₃₇

commutator, cosocle

C₂²×C₃₇

C₂⁵

Frattini

C₂²

C₂⁴×D₃₇

Fitting

C₂×C₄⋊Q₈×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₃₇

hypocenter

C₃₇

C₂×Q₈²

LCS

G ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. #14558263	C₂	direct (N × H)
30. C₂×Dic₇₄	Q₈	semidirect
53. Dic₇₄	C₂×Q₈	semidirect
63. C₂×C₃₇	Q₈²	semidirect
72. C₃₇	C₂×Q₈²	semidirect
128. C₂	#14558263	direct (N × H)

H ≅ G/N

type

1. #14558263

C₂

direct (N × H)

30. C₂×Dic₇₄

Q₈

semidirect

53. Dic₇₄

C₂×Q₈

semidirect

63. C₂×C₃₇

Q₈²

semidirect

72. C₃₇

C₂×Q₈²

semidirect

128. C₂

#14558263

direct (N × H)

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

subgroup	index
#14558263	2
C₂×#11924535	2
C₂×#11924534	2
C₂×C₄×Dic₇₄	2
C₂×#11924501	2
C₂×C₄⋊Q₈×C₃₇	2
C₂×Q₈²	37

subgroup

index

C₂×C₄×Dic₇₄

C₂×C₄⋊Q₈×C₃₇

C₂×Q₈²

Subgroups — C₂×#14558263 #18008171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14558263 #18008171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14558263 #18008171