Group #37530003 — subgroups

Subgroups — #2060×C₁₈₁ #37530003

Order 23168.

246 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 92 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	#544×C₁₈₁	C₂	1	1	yes	maximal, characteristic
2	#428×C₁₈₁	C₂	1	2	yes	maximal
3	#384×C₁₈₁	C₂	1	1	yes	maximal, characteristic
4	#343×C₁₈₁	C₂	1	1	yes	maximal, characteristic
5	#339×C₁₈₁	C₂	1	2	yes	maximal
6	C₂×M₁₆×C₁₈₁	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₁₈₁	—	2	2	no	—
8	#125×C₁₈₁	—	2	2	no	—
9	C₂²⋊Q₈×C₁₈₁	—	2	2	no	—
10	#117×C₁₈₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₁₈₁	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
15	#107×C₁₈₁	—	2	2	no	—
16	#106×C₁₈₁	—	2	2	no	—
17	C₂×Q₈×C₁₈₁	—	2	2	no	—
18	C₂²×C₄×C₁₈₁	—	2	1	no	—
19	C₂²×C₄×C₁₈₁	—	2	1	no	—
20	C₂²×C₄×C₁₈₁	D₄	1	2	yes	—
21	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₈₁	—	2	2	no	—
23	M₁₆×C₁₈₁	Q₈	1	4	yes	complemented
24	C₂×C₈×C₁₈₁	—	2	2	no	—
25	C₄⋊C₄×C₁₈₁	—	2	2	no	—
26	C₄⋊C₄×C₁₈₁	—	2	4	no	—
27	C₄⋊C₄×C₁₈₁	—	2	2	no	—
28	C₄⋊C₄×C₁₈₁	—	2	2	no	—
29	C₄⋊C₄×C₁₈₁	—	4	4	no	—
30	C₄⋊C₄×C₁₈₁	D₄	1	2	yes	—
31	C₄⋊C₄×C₁₈₁	D₄	1	2	yes	—
32	C₂²⋊C₄×C₁₈₁	—	2	2	no	—
33	C₄²×C₁₈₁	—	2	2	no	—
34	C₄²×C₁₈₁	—	2	1	no	—
35	C₄²×C₁₈₁	Q₈	1	2	yes	complemented
36	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₁₈₁	—	8	2	no	—
38	C₂×C₄×C₁₈₁	—	2	2	no	—
39	C₂×C₄×C₁₈₁	—	2	2	no	—
40	C₂×C₄×C₁₈₁	—	2	2	no	—
41	C₂×C₄×C₁₈₁	—	2	2	no	—
42	C₂×C₄×C₁₈₁	—	2	2	no	—
43	C₂×C₄×C₁₈₁	—	2	2	no	—
44	C₂×C₄×C₁₈₁	—	4	2	no	—
45	C₂×C₄×C₁₈₁	—	4	2	no	—
46	C₂×C₄×C₁₈₁	—	4	1	no	—
47	C₂×C₄×C₁₈₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₈₁	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₈₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₈₁	#41	1	1	yes	characteristic
51	C₈×C₁₈₁	—	2	4	no	—
52	C₂²×C₁₈₁	—	2	1	no	—
53	C₂²×C₁₈₁	—	2	1	no	—
54	C₂²×C₁₈₁	#120	1	1	yes	characteristic
55	C₂²×C₁₈₁	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
57	C₄×C₁₈₁	—	2	4	no	—
58	C₄×C₁₈₁	—	4	2	no	—
59	C₄×C₁₈₁	—	4	2	no	—
60	C₄×C₁₈₁	—	4	2	no	—
61	C₄×C₁₈₁	—	8	1	no	—
62	C₄×C₁₈₁	C₂²⋊Q₈	1	2	yes	—
63	C₄×C₁₈₁	C₄⋊Q₈	1	2	yes	—
64	C₂×C₁₈₁	—	2	1	no	—
65	C₂×C₁₈₁	—	2	1	no	—
66	C₂×C₁₈₁	#395	1	1	yes	characteristic
67	C₂×C₁₈₁	#453	1	1	yes	characteristic
68	C₂×C₁₈₁	#456	1	1	yes	characteristic
69	C₁₈₁	#2060	1	1	yes	characteristic, direct product factor, Sylow
70	#2060	C₁₈₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	#544	C₂×C₁₈₁	1	1	yes	characteristic
72	#428	C₂×C₁₈₁	1	2	yes	—
73	#384	C₂×C₁₈₁	1	1	yes	characteristic
74	#343	C₂×C₁₈₁	1	1	yes	characteristic
75	#339	C₂×C₁₈₁	1	2	yes	—
76	C₂×M₁₆	C₂²×C₁₈₁	1	2	yes	—
77	C₄⋊Q₈	—	2	2	no	—
78	#125	—	2	2	no	—
79	C₂²⋊Q₈	—	2	2	no	—
80	#117	C₂²×C₁₈₁	1	2	yes	—
81	C₂×C₄⋊C₄	—	2	2	no	—
82	C₂×C₄⋊C₄	C₂²×C₁₈₁	1	1	yes	characteristic
83	C₂×C₄⋊C₄	C₂²×C₁₈₁	1	1	yes	characteristic
84	C₂×C₄²	C₂²×C₁₈₁	1	1	yes	characteristic
85	#107	—	2	2	no	—
86	#106	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	D₄×C₁₈₁	1	2	yes	—
91	C₂²×C₄	C₂³×C₁₈₁	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	Q₈×C₁₈₁	1	4	yes	complemented
94	C₂×C₈	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	4	4	no	—
100	C₄⋊C₄	D₄×C₁₈₁	1	2	yes	—
101	C₄⋊C₄	D₄×C₁₈₁	1	2	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	1	no	—
105	C₄²	Q₈×C₁₈₁	1	2	yes	complemented
106	C₂³	C₂×D₄×C₁₈₁	1	1	yes	characteristic
107	Q₈	—	8	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	C₂×D₄×C₁₈₁	1	2	yes	—
118	C₂×C₄	C₂×Q₈×C₁₈₁	1	2	yes	—
119	C₂×C₄	C₂×Q₈×C₁₈₁	1	1	yes	characteristic
120	C₂×C₄	#41×C₁₈₁	1	1	yes	characteristic
121	C₈	—	2	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#120×C₁₈₁	1	1	yes	characteristic
125	C₂²	C₂²⋊Q₈×C₁₈₁	1	1	yes	characteristic
126	C₂²	C₄⋊Q₈×C₁₈₁	1	1	yes	characteristic
127	C₄	—	2	4	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	4	2	no	—
131	C₄	—	8	1	no	—
132	C₄	C₂²⋊Q₈×C₁₈₁	1	2	yes	—
133	C₄	C₄⋊Q₈×C₁₈₁	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#395×C₁₈₁	1	1	yes	characteristic
137	C₂	#453×C₁₈₁	1	1	yes	characteristic
138	C₂	#456×C₁₈₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#544×C₁₈₁

C₂

yes

maximal, characteristic

#428×C₁₈₁

C₂

yes

maximal

#384×C₁₈₁

C₂

yes

maximal, characteristic

#343×C₁₈₁

C₂

yes

maximal, characteristic

#339×C₁₈₁

C₂

yes

maximal

C₂×M₁₆×C₁₈₁

C₂²

yes

—

C₄⋊Q₈×C₁₈₁

—

#125×C₁₈₁

—

C₂²⋊Q₈×C₁₈₁

—

#117×C₁₈₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₈₁

—

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

characteristic

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

#107×C₁₈₁

—

#106×C₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

D₄

yes

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

M₁₆×C₁₈₁

—

M₁₆×C₁₈₁

Q₈

yes

complemented

C₂×C₈×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

D₄

yes

—

C₄⋊C₄×C₁₈₁

D₄

yes

—

C₂²⋊C₄×C₁₈₁

—

C₄²×C₁₈₁

—

C₄²×C₁₈₁

—

C₄²×C₁₈₁

Q₈

yes

complemented

C₂³×C₁₈₁

C₂×D₄

yes

characteristic

Q₈×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

—

C₂×C₄×C₁₈₁

C₂×Q₈

yes

—

C₂×C₄×C₁₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₈₁

#41

yes

characteristic

C₈×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

#120

yes

characteristic

C₂²×C₁₈₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₈₁

C₄⋊Q₈

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂²⋊Q₈

yes

—

C₄×C₁₈₁

C₄⋊Q₈

yes

—

C₂×C₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₁₈₁

yes

characteristic, direct product factor, Sylow

#2060

C₁₈₁

yes

maximal, characteristic, direct product factor, Sylow

#544

C₂×C₁₈₁

yes

characteristic

#428

C₂×C₁₈₁

yes

—

#384

C₂×C₁₈₁

yes

characteristic

#343

C₂×C₁₈₁

yes

characteristic

#339

C₂×C₁₈₁

yes

—

C₂×M₁₆

C₂²×C₁₈₁

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂²×C₁₈₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₈₁

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₈₁

yes

characteristic

C₂×C₄²

C₂²×C₁₈₁

yes

characteristic

#107

—

#106

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₈₁

yes

—

C₂²×C₄

C₂³×C₁₈₁

yes

characteristic

M₁₆

—

M₁₆

Q₈×C₁₈₁

yes

complemented

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

D₄×C₁₈₁

yes

—

101

C₄⋊C₄

D₄×C₁₈₁

yes

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

Q₈×C₁₈₁

yes

complemented

106

C₂³

C₂×D₄×C₁₈₁

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂×D₄×C₁₈₁

yes

—

118

C₂×C₄

C₂×Q₈×C₁₈₁

yes

—

119

C₂×C₄

C₂×Q₈×C₁₈₁

yes

characteristic

120

C₂×C₄

#41×C₁₈₁

yes

characteristic

121

C₈

—

122

C₂²

—

123

C₂²

—

124

C₂²

#120×C₁₈₁

yes

characteristic

125

C₂²

C₂²⋊Q₈×C₁₈₁

yes

characteristic

126

C₂²

C₄⋊Q₈×C₁₈₁

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂²⋊Q₈×C₁₈₁

yes

—

133

C₄

C₄⋊Q₈×C₁₈₁

yes

—

134

C₂

—

135

C₂

—

136

C₂

#395×C₁₈₁

yes

characteristic

137

C₂

#453×C₁₈₁

yes

characteristic

138

C₂

#456×C₁₈₁

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₈₁	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₈₁
prime radical	C₂³×C₁₈₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₈₁ ◁ C₂²×C₄×C₁₈₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₈₁

#120

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₈₁

prime radical

C₂³×C₁₈₁

C₂×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₈₁ ◁ C₂²×C₄×C₁₈₁ ◁ G

N	H ≅ G/N	type
23. M₁₆×C₁₈₁	Q₈	semidirect
35. C₄²×C₁₈₁	Q₈	semidirect
69. C₁₈₁	#2060	direct (N × H)
70. #2060	C₁₈₁	direct (N × H)
93. M₁₆	Q₈×C₁₈₁	semidirect
105. C₄²	Q₈×C₁₈₁	semidirect

H ≅ G/N

type

23. M₁₆×C₁₈₁

Q₈

semidirect

35. C₄²×C₁₈₁

Q₈

semidirect

69. C₁₈₁

#2060

direct (N × H)

70. #2060

C₁₈₁

direct (N × H)

93. M₁₆

Q₈×C₁₈₁

semidirect

105. C₄²

Q₈×C₁₈₁

semidirect

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

subgroup	index
#544×C₁₈₁	2
#428×C₁₈₁	2
#384×C₁₈₁	2
#343×C₁₈₁	2
#339×C₁₈₁	2
#2060	181

subgroup

index

#544×C₁₈₁

#428×C₁₈₁

#384×C₁₈₁

#343×C₁₈₁

#339×C₁₈₁

#2060

181

Subgroups — #2060×C₁₈₁ #37530003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2060×C181 #37530003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2060×C₁₈₁ #37530003