Group #37533102 — subgroups

Subgroups — #37533102

Order 23168.

150 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 53 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	#26041675	C₂	1	1	yes	maximal, characteristic
2	#26041575	C₂	1	1	yes	maximal, characteristic
3	#26041486	C₂	1	1	yes	maximal, characteristic
4	#26041403	C₂	1	1	yes	maximal, characteristic
5	#26041029	C₂	1	1	yes	maximal, characteristic
6	#26041027	C₂	1	1	yes	maximal, characteristic
7	#332×C₁₈₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#15668439	—	2	1	no	—
9	#19153816	C₂²	1	1	yes	characteristic
10	C₄×Dic₃₆₂	C₂²	1	1	yes	characteristic
11	#19153750	—	2	1	no	—
12	#19153747	—	2	1	no	—
13	#19153747	—	2	1	no	—
14	C₄⋊#13089678	C₂²	1	1	yes	characteristic
15	#19153745	—	2	1	no	—
16	#19153741	—	2	1	no	—
17	#19153740	—	2	1	no	—
18	C₄×#13089678	C₂²	1	1	yes	characteristic
19	C₈×Dic₁₈₁	—	2	1	no	—
20	C₄⋊Q₈×C₁₈₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₈₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₆₂	—	2	1	no	—
24	C₂×Dic₃₆₂	D₄	1	1	yes	characteristic
25	#15668439	—	4	2	no	—
26	C₂×#13089678	—	2	1	no	—
27	C₂×#13089678	D₄	1	1	yes	characteristic
28	C₂×#13089678	D₄	1	1	yes	characteristic
29	C₄⋊Dic₁₈₁	—	2	1	no	—
30	C₄⋊Dic₁₈₁	—	2	1	no	—
31	C₄⋊Dic₁₈₁	D₄	1	1	yes	characteristic
32	#15668428	—	2	1	no	—
33	C₄×Dic₁₈₁	—	2	1	no	—
34	C₂×Q₈×C₁₈₁	—	2	1	no	—
35	C₂×C₈×C₁₈₁	—	2	1	no	—
36	C₂×C₈×C₁₈₁	—	2	1	no	—
37	C₄⋊C₄×C₁₈₁	—	2	1	no	—
38	C₄⋊C₄×C₁₈₁	—	2	1	no	—
39	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
40	Dic₃₆₂	—	2	1	no	—
41	Dic₃₆₂	—	8	1	no	—
42	Dic₃₆₂	SD₁₆	1	2	yes	—
43	C₂×Dic₁₈₁	—	2	1	no	—
44	C₂×Dic₁₈₁	—	2	1	no	—
45	C₂×Dic₁₈₁	—	4	1	no	—
46	#13089678	—	2	1	no	—
47	#13089678	—	4	1	no	—
48	#13089678	Q₁₆	1	2	yes	complemented
49	Q₈×C₁₈₁	—	4	1	no	—
50	Q₈×C₁₈₁	—	4	1	no	—
51	C₂×C₄×C₁₈₁	—	4	1	no	—
52	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₈₁	#41	1	1	yes	characteristic
55	C₈×C₁₈₁	—	4	1	no	—
56	C₈×C₁₈₁	—	4	1	no	—
57	C₂²×C₁₈₁	#121	1	1	yes	characteristic
58	C₄×C₁₈₁	—	2	1	no	—
59	C₄×C₁₈₁	—	8	1	no	—
60	C₄×C₁₈₁	C₂×SD₁₆	1	1	yes	characteristic
61	C₄×C₁₈₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₄×C₁₈₁	#135	1	1	yes	characteristic
63	C₄×C₁₈₁	#137	1	1	yes	characteristic
64	Dic₁₈₁	—	2	2	no	—
65	Dic₁₈₁	—	4	1	no	—
66	Dic₁₈₁	—	8	1	no	—
67	C₂×C₁₈₁	#462	1	1	yes	characteristic
68	C₂×C₁₈₁	#465	1	1	yes	characteristic
69	C₂×C₁₈₁	#471	1	1	yes	characteristic
70	C₁₈₁	#1672	1	1	yes	characteristic, complemented, Sylow
71	#1672	—	181	1	no	maximal, Sylow
72	#498	—	181	1	no	—
73	#477	—	181	1	no	—
74	#462	—	181	1	no	—
75	C₈×Q₈	—	181	1	no	—
76	#334	—	181	1	no	—
77	#333	—	181	1	no	—
78	#332	D₁₈₁	1	1	yes	characteristic
79	C₂×Q₁₆	—	362	1	no	—
80	C₄⋊Q₈	—	181	1	no	—
81	C₄⋊Q₈	C₂×D₁₈₁	1	1	yes	characteristic
82	C₄×Q₈	—	181	1	no	—
83	#107	—	362	1	no	—
84	#106	—	362	1	no	—
85	#106	—	362	1	no	—
86	C₄⋊C₈	—	362	1	no	—
87	C₄⋊C₈	—	181	1	no	—
88	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
89	C₄⋊C₈	C₂×D₁₈₁	1	1	yes	characteristic
90	#103	—	362	1	no	—
91	#103	—	362	1	no	—
92	C₄×C₈	—	362	1	no	—
93	C₄×C₈	—	181	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	362	1	no	—
96	C₂×Q₈	—	181	1	no	—
97	Q₁₆	—	724	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	362	1	no	—
101	C₂×C₈	—	181	1	no	—
102	C₂×C₈	—	181	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	362	1	no	—
106	C₄⋊C₄	—	362	1	no	—
107	C₄⋊C₄	—	362	1	no	—
108	C₄⋊C₄	—	181	1	no	—
109	C₄²	—	362	1	no	—
110	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	362	1	no	—
114	Q₈	—	1448	1	no	—
115	Q₈	—	181	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	362	1	no	—
118	C₂×C₄	—	362	1	no	—
119	C₂×C₄	—	724	1	no	—
120	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
121	C₂×C₄	D₄×D₁₈₁	1	1	yes	characteristic
122	C₂×C₄	#15668449	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	4	1	no	—
125	C₈	—	362	1	no	—
126	C₈	—	181	2	no	—
127	C₈	—	724	1	no	—
128	C₂²	#19153789	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	362	2	no	—
132	C₄	—	1448	1	no	—
133	C₄	—	724	1	no	—
134	C₄	SD₁₆×D₁₈₁	1	1	yes	characteristic
135	C₄	Q₁₆×D₁₈₁	1	1	yes	characteristic
136	C₄	#19153835	1	1	yes	characteristic
137	C₄	#19153845	1	1	yes	characteristic
138	C₂	#26041488	1	1	yes	characteristic
139	C₂	#26041507	1	1	yes	characteristic
140	C₂	#26041545	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26041675

C₂

yes

maximal, characteristic

#26041575

C₂

yes

maximal, characteristic

#26041486

C₂

yes

maximal, characteristic

#26041403

C₂

yes

maximal, characteristic

#26041029

C₂

yes

maximal, characteristic

#26041027

C₂

yes

maximal, characteristic

#332×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668439

—

#19153816

C₂²

yes

characteristic

C₄×Dic₃₆₂

C₂²

yes

—

—

—

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₈×Dic₁₈₁

—

C₄⋊Q₈×C₁₈₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₈₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₈₁

C₂²

yes

characteristic

C₂×Dic₃₆₂

—

C₂×Dic₃₆₂

D₄

yes

—

—

D₄

yes

characteristic

C₂×#13089678

D₄

yes

characteristic

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

D₄

yes

characteristic

#15668428

—

C₄×Dic₁₈₁

—

C₂×Q₈×C₁₈₁

—

C₂×C₈×C₁₈₁

—

C₂×C₈×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄⋊C₄×C₁₈₁

—

C₄²×C₁₈₁

C₂³

yes

characteristic

Dic₃₆₂

—

Dic₃₆₂

—

Dic₃₆₂

SD₁₆

yes

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

#13089678

—

#13089678

—

#13089678

Q₁₆

yes

complemented

Q₈×C₁₈₁

—

Q₈×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

#41

yes

characteristic

C₈×C₁₈₁

—

C₈×C₁₈₁

—

C₂²×C₁₈₁

#121

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₂×SD₁₆

yes

characteristic

C₄×C₁₈₁

C₂×Q₁₆

yes

C₄×C₁₈₁

yes

C₄×C₁₈₁

yes

Dic₁₈₁

—

Dic₁₈₁

—

Dic₁₈₁

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₁₈₁

yes

characteristic, complemented, Sylow

#1672

—

181

maximal, Sylow

#498

—

181

—

#477

—

181

—

#462

—

181

—

C₈×Q₈

—

181

—

#334

—

181

—

#333

—

181

—

#332

D₁₈₁

yes

characteristic

C₂×Q₁₆

—

362

—

C₄⋊Q₈

—

181

—

C₄⋊Q₈

C₂×D₁₈₁

yes

characteristic

C₄×Q₈

—

181

—

#107

—

362

—

#106

—

362

—

#106

—

362

—

C₄⋊C₈

—

362

—

C₄⋊C₈

—

181

—

C₄⋊C₈

C₂×D₁₈₁

yes

characteristic

C₄⋊C₈

C₂×D₁₈₁

yes

characteristic

#103

—

362

—

#103

—

362

—

C₄×C₈

—

362

—

C₄×C₈

—

181

—

C₂×Q₈

—

C₂×Q₈

—

362

—

C₂×Q₈

—

181

—

Q₁₆

—

724

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

362

—

101

C₂×C₈

—

181

—

102

C₂×C₈

—

181

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

362

—

106

C₄⋊C₄

—

362

—

107

C₄⋊C₄

—

362

—

108

C₄⋊C₄

—

181

—

109

C₄²

—

362

—

110

C₄²

C₂²×D₁₈₁

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

362

—

114

Q₈

—

1448

—

115

Q₈

—

181

—

116

C₂×C₄

—

117

C₂×C₄

—

362

—

118

C₂×C₄

—

362

—

119

C₂×C₄

—

724

—

120

C₂×C₄

D₄×D₁₈₁

yes

characteristic

121

C₂×C₄

D₄×D₁₈₁

yes

characteristic

122

C₂×C₄

#15668449

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

362

—

126

C₈

—

181

—

127

C₈

—

724

—

128

C₂²

#19153789

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

362

—

132

C₄

—

1448

—

133

C₄

—

724

—

134

C₄

SD₁₆×D₁₈₁

yes

characteristic

135

C₄

Q₁₆×D₁₈₁

yes

characteristic

136

C₄

#19153835

yes

characteristic

137

C₄

#19153845

yes

characteristic

138

C₂

#26041488

yes

characteristic

139

C₂

#26041507

yes

characteristic

140

C₂

#26041545

yes

characteristic

	subgroup	G/H
center	C₂²	#19153789
commutator, cosocle	C₄²×C₁₈₁	C₂³
Frattini	C₄²	C₂²×D₁₈₁
Fitting	#332×C₁₈₁	C₂
socle, prime radical	C₂²×C₁₈₁	#121
hypercenter	#332	D₁₈₁
hypocenter	C₁₈₁	#1672
LCS	G ▷ C₄²×C₁₈₁ ▷ C₂²×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#19153789

commutator, cosocle

C₄²×C₁₈₁

C₂³

Frattini

C₄²

C₂²×D₁₈₁

Fitting

#332×C₁₈₁

C₂

C₂²×C₁₈₁

D₁₈₁

C₁₈₁

G ▷ C₄²×C₁₈₁ ▷ C₂²×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
48. #13089678	Q₁₆	semidirect
70. C₁₈₁	#1672	semidirect

H ≅ G/N

type

48. #13089678

Q₁₆

semidirect

70. C₁₈₁

#1672

semidirect

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

subgroup	index
#26041675	2
#26041575	2
#26041486	2
#26041403	2
#26041029	2
#26041027	2
#332×C₁₈₁	2
#1672	181

subgroup

index

#332×C₁₈₁

181

Subgroups — #37533102

Conjugacy classes of subgroups

Complements

Maximal subgroups