Group #37544227 — subgroups

Subgroups — C₂×#26041682 #37544227

Order 23168.

424 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 197 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×#19153816	C₂	1	1	yes	maximal, characteristic
2	#26041682	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19153749	C₂	1	4	yes	maximal
4	C₂×C₄×#13089678	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₆₂	—	2	1	no	—
7	C₂²×#13089678	C₂²	1	2	yes	—
8	#19153816	—	2	2	no	—
9	#19153816	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₈₁	—	2	1	no	—
11	C₂×C₄⋊Dic₁₈₁	C₂²	1	2	yes	—
12	#19153749	C₂²	1	16	yes	—
13	C₄×#13089678	C₂²	1	4	yes	—
14	C₂²×Q₈×C₁₈₁	—	2	1	no	—
15	C₄⋊Q₈×C₁₈₁	—	2	2	no	—
16	C₄⋊Q₈×C₁₈₁	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₈₁	—	2	1	no	—
18	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
20	C₂×Dic₃₆₂	—	2	4	no	—
21	C₂×Dic₃₆₂	—	4	4	no	—
22	C₂²×Dic₁₈₁	—	2	2	no	—
23	C₂×#13089678	Q₈	1	8	yes	complemented
24	C₂×#13089678	C₂³	1	4	yes	—
25	C₄⋊Dic₁₈₁	—	2	4	no	—
26	C₄⋊Dic₁₈₁	—	2	2	no	—
27	C₄⋊Dic₁₈₁	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₈₁	—	2	4	no	—
29	C₂×Q₈×C₁₈₁	—	4	4	no	—
30	C₂²×C₄×C₁₈₁	—	2	2	no	—
31	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₈₁	—	2	4	no	—
35	C₄⋊C₄×C₁₈₁	—	2	2	no	—
36	C₄⋊C₄×C₁₈₁	C₂³	1	4	yes	—
37	C₄²×C₁₈₁	D₄	1	2	yes	—
38	C₄²×C₁₈₁	C₂³	1	2	yes	—
39	Dic₃₆₂	—	4	8	no	—
40	C₂×Dic₁₈₁	—	2	4	no	—
41	C₂×Dic₁₈₁	—	4	4	no	—
42	#13089678	C₂×Q₈	1	8	yes	complemented
43	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₁₈₁	—	4	8	no	—
45	C₂×C₄×C₁₈₁	—	2	4	no	—
46	C₂×C₄×C₁₈₁	—	4	4	no	—
47	C₂×C₄×C₁₈₁	D₈	1	4	yes	—
48	C₂×C₄×C₁₈₁	Q₁₆	1	4	yes	—
49	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₈₁	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₈₁	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₈₁	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₁₈₁	C₂⁴	1	1	yes	characteristic
54	C₂²×C₁₈₁	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₁₈₁	C₂×D₈	1	2	yes	—
56	C₂²×C₁₈₁	C₂×Q₁₆	1	2	yes	—
57	C₂²×C₁₈₁	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₈₁	—	4	4	no	—
59	C₄×C₁₈₁	C₄⋊Q₈	1	2	yes	—
60	C₄×C₁₈₁	C₂×D₈	1	4	yes	—
61	C₄×C₁₈₁	C₂×Q₁₆	1	4	yes	—
62	C₄×C₁₈₁	C₂²×Q₈	1	2	yes	—
63	Dic₁₈₁	—	4	4	no	—
64	C₂×C₁₈₁	#500	1	4	yes	complemented
65	C₂×C₁₈₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₁₈₁	C₂²×D₈	1	1	yes	characteristic
67	C₂×C₁₈₁	C₂²×Q₁₆	1	1	yes	characteristic
68	C₁₈₁	C₂×#500	1	1	yes	characteristic, complemented, Sylow
69	C₂×#500	—	181	1	no	maximal, Sylow
70	C₂×C₄⋊Q₈	—	181	1	no	—
71	C₂×C₄⋊Q₈	D₁₈₁	1	1	yes	characteristic
72	#500	—	181	8	no	—
73	C₂×#107	—	181	4	no	—
74	C₂×C₄×C₈	—	181	1	no	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×Q₈	—	362	1	no	—
77	C₂²×C₈	—	181	2	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	362	2	no	—
80	C₄⋊Q₈	—	181	4	no	—
81	C₄⋊Q₈	C₂×D₁₈₁	1	4	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	362	1	no	—
84	C₂×C₄⋊C₄	—	181	2	no	—
85	C₂×C₄⋊C₄	C₂×D₁₈₁	1	2	yes	—
86	C₂×C₄²	C₂×D₁₈₁	1	1	yes	characteristic
87	#107	—	181	16	no	—
88	C₄×C₈	—	181	4	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	4	4	no	—
91	C₂×Q₈	—	362	4	no	—
92	C₂×Q₈	—	724	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	362	2	no	—
95	C₂²×C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
98	C₂×C₈	—	181	8	no	—
99	C₂×C₈	—	181	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	362	4	no	—
103	C₄⋊C₄	—	362	2	no	—
104	C₄⋊C₄	—	181	4	no	—
105	C₄⋊C₄	C₂²×D₁₈₁	1	4	yes	—
106	C₄²	C₂²⋊D₁₈₁	1	2	yes	—
107	C₄²	C₂²×D₁₈₁	1	2	yes	—
108	C₂³	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
109	Q₈	—	4	8	no	—
110	Q₈	—	724	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	362	4	no	—
114	C₂×C₄	—	724	4	no	—
115	C₂×C₄	#15668434	1	4	yes	—
116	C₂×C₄	#15668439	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₈₁	1	2	yes	—
120	C₂×C₄	Q₈×D₁₈₁	1	4	yes	—
121	C₂×C₄	C₂³×D₁₈₁	1	1	yes	characteristic
122	C₈	—	181	8	no	—
123	C₂²	#19153814	1	2	yes	—
124	C₂²	C₂×#15668434	1	2	yes	—
125	C₂²	C₂×#15668439	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₁₈₁	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	724	4	no	—
129	C₄	#19153814	1	2	yes	—
130	C₄	C₂×#15668434	1	4	yes	—
131	C₄	C₂×#15668439	1	4	yes	—
132	C₄	C₂×Q₈×D₁₈₁	1	2	yes	—
133	C₂	#26041682	1	4	yes	direct product factor
134	C₂	C₂×#19153814	1	1	yes	characteristic
135	C₂	C₂²×#15668434	1	1	yes	characteristic
136	C₂	C₂²×#15668439	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153816

C₂

yes

maximal, characteristic

#26041682

C₂

yes

maximal, direct product factor

C₂×#19153749

C₂

yes

maximal

C₂×C₄×#13089678

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₈₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₆₂

—

C₂²×#13089678

C₂²

yes

—

#19153816

—

#19153816

C₂²

yes

—

C₂×C₄⋊Dic₁₈₁

—

C₂×C₄⋊Dic₁₈₁

C₂²

yes

—

#19153749

C₂²

yes

—

C₄×#13089678

C₂²

yes

—

C₂²×Q₈×C₁₈₁

—

C₄⋊Q₈×C₁₈₁

—

C₄⋊Q₈×C₁₈₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₈₁

—

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

—

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

C₂×Dic₃₆₂

—

C₂×Dic₃₆₂

—

C₂²×Dic₁₈₁

—

C₂×#13089678

Q₈

yes

complemented

C₂×#13089678

C₂³

yes

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

C₂³

yes

—

C₂×Q₈×C₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

D₄

yes

characteristic

C₂²×C₄×C₁₈₁

D₄

yes

characteristic

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

C₂³

yes

—

C₄²×C₁₈₁

D₄

yes

—

C₄²×C₁₈₁

C₂³

yes

—

Dic₃₆₂

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

#13089678

C₂×Q₈

yes

complemented

C₂³×C₁₈₁

C₂×D₄

yes

characteristic

Q₈×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

D₈

yes

—

C₂×C₄×C₁₈₁

Q₁₆

yes

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

—

C₂×C₄×C₁₈₁

C₂×Q₈

yes

—

C₂×C₄×C₁₈₁

C₂⁴

yes

characteristic

C₂²×C₁₈₁

C₄⋊Q₈

yes

—

C₂²×C₁₈₁

C₂×D₈

yes

—

C₂²×C₁₈₁

C₂×Q₁₆

yes

—

C₂²×C₁₈₁

C₂²×D₄

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

C₄⋊Q₈

yes

—

C₄×C₁₈₁

C₂×D₈

yes

—

C₄×C₁₈₁

C₂×Q₁₆

yes

—

C₄×C₁₈₁

C₂²×Q₈

yes

—

Dic₁₈₁

—

C₂×C₁₈₁

#500

yes

complemented

C₂×C₁₈₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₈₁

C₂²×D₈

yes

characteristic

C₂×C₁₈₁

C₂²×Q₁₆

yes

characteristic

C₁₈₁

C₂×#500

yes

characteristic, complemented, Sylow

C₂×#500

—

181

maximal, Sylow

C₂×C₄⋊Q₈

—

181

—

C₂×C₄⋊Q₈

D₁₈₁

yes

characteristic

#500

—

181

—

C₂×#107

—

181

—

C₂×C₄×C₈

—

181

—

C₂²×Q₈

—

C₂²×Q₈

—

362

—

C₂²×C₈

—

181

—

C₄⋊Q₈

—

C₄⋊Q₈

—

362

—

C₄⋊Q₈

—

181

—

C₄⋊Q₈

C₂×D₁₈₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

362

—

C₂×C₄⋊C₄

—

181

—

C₂×C₄⋊C₄

C₂×D₁₈₁

yes

—

C₂×C₄²

C₂×D₁₈₁

yes

characteristic

#107

—

181

—

C₄×C₈

—

181

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

362

—

C₂×Q₈

—

724

—

C₂²×C₄

—

C₂²×C₄

—

362

—

C₂²×C₄

C₂²⋊D₁₈₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₈₁

yes

characteristic

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

C₂×C₈

—

181

—

C₂×C₈

—

181

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

362

—

103

C₄⋊C₄

—

362

—

104

C₄⋊C₄

—

181

—

105

C₄⋊C₄

C₂²×D₁₈₁

yes

—

106

C₄²

C₂²⋊D₁₈₁

yes

—

107

C₄²

C₂²×D₁₈₁

yes

—

108

C₂³

C₂×C₂²⋊D₁₈₁

yes

characteristic

109

Q₈

—

110

Q₈

—

724

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

362

—

114

C₂×C₄

—

724

—

115

C₂×C₄

#15668434

yes

—

116

C₂×C₄

#15668439

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₈₁

yes

—

120

C₂×C₄

Q₈×D₁₈₁

yes

—

121

C₂×C₄

C₂³×D₁₈₁

yes

characteristic

122

C₈

—

181

—

123

C₂²

#19153814

yes

—

124

C₂²

C₂×#15668434

yes

—

125

C₂²

C₂×#15668439

yes

—

126

C₂²

C₂²×C₂²⋊D₁₈₁

yes

characteristic

127

C₄

—

128

C₄

—

724

—

129

C₄

#19153814

yes

—

130

C₄

C₂×#15668434

yes

—

131

C₄

C₂×#15668439

yes

—

132

C₄

C₂×Q₈×D₁₈₁

yes

—

133

C₂

#26041682

yes

direct product factor

134

C₂

C₂×#19153814

yes

characteristic

135

C₂

C₂²×#15668434

yes

characteristic

136

C₂

C₂²×#15668439

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₈₁
commutator, cosocle	C₂×C₄×C₁₈₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₈₁
Fitting	C₂×C₄⋊Q₈×C₁₈₁	C₂
socle, prime radical	C₂³×C₁₈₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₈₁
hypocenter	C₁₈₁	C₂×#500
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₈₁

commutator, cosocle

C₂×C₄×C₁₈₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₈₁

Fitting

C₂×C₄⋊Q₈×C₁₈₁

C₂

socle, prime radical

C₂³×C₁₈₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₈₁

hypocenter

C₁₈₁

C₂×#500

LCS

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #26041682	C₂	direct (N × H)
23. C₂×#13089678	Q₈	semidirect
42. #13089678	C₂×Q₈	semidirect
64. C₂×C₁₈₁	#500	semidirect
68. C₁₈₁	C₂×#500	semidirect
133. C₂	#26041682	direct (N × H)

H ≅ G/N

type

2. #26041682

C₂

direct (N × H)

23. C₂×#13089678

Q₈

semidirect

42. #13089678

C₂×Q₈

semidirect

64. C₂×C₁₈₁

#500

semidirect

68. C₁₈₁

C₂×#500

semidirect

133. C₂

#26041682

direct (N × H)

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

subgroup	index
C₂×#19153816	2
#26041682	2
C₂×#19153749	2
C₂×C₄×#13089678	2
C₂×C₄⋊Q₈×C₁₈₁	2
C₂×#500	181

subgroup

index

C₂×C₄⋊Q₈×C₁₈₁

181

Subgroups — C₂×#26041682 #37544227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#26041682 #37544227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26041682 #37544227