Group #37534693 — subgroups

Subgroups — #37534693

Order 23168.

276 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#19153727	C₂	1	1	yes	maximal, characteristic
3	#26041041	C₂	1	4	yes	maximal
4	#382×C₁₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₆₂	—	2	1	no	—
6	C₂²×#13089678	C₂²	1	2	yes	—
7	#19153816	—	2	2	no	—
8	#19153816	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₈₁	—	2	1	no	—
10	C₂×C₄⋊Dic₁₈₁	C₂²	1	2	yes	—
11	#19153727	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₈₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
14	#95×C₁₈₁	—	2	1	no	—
15	#95×C₁₈₁	—	2	1	no	—
16	C₂×Dic₃₆₂	—	2	4	no	—
17	C₂×Dic₃₆₂	—	4	4	no	—
18	C₂²×Dic₁₈₁	—	2	2	no	—
19	C₂×#13089678	—	2	2	no	—
20	C₂×#13089678	—	2	2	no	—
21	C₂×#13089678	Q₈	1	4	yes	complemented
22	C₄⋊Dic₁₈₁	—	2	4	no	—
23	C₄⋊Dic₁₈₁	—	2	2	no	—
24	C₄⋊Dic₁₈₁	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₈₁	—	2	2	no	—
26	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₈₁	—	2	4	no	—
30	C₄²×C₁₈₁	C₂×C₄	1	2	yes	—
31	C₄²×C₁₈₁	D₄	1	2	yes	—
32	Dic₃₆₂	—	4	8	no	—
33	C₂×Dic₁₈₁	—	2	4	no	—
34	C₂×Dic₁₈₁	—	4	4	no	—
35	#13089678	—	2	4	no	—
36	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₈₁	—	2	2	no	—
38	C₂×C₄×C₁₈₁	—	2	2	no	—
39	C₂×C₄×C₁₈₁	—	4	2	no	—
40	C₂×C₄×C₁₈₁	—	4	2	no	—
41	C₂×C₄×C₁₈₁	—	4	2	no	—
42	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₈₁	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₈₁	#41	1	2	yes	—
48	C₂²×C₁₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₈₁	#124	1	1	yes	characteristic
50	C₂²×C₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₈₁	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₁₈₁	#137	1	2	yes	—
53	C₄×C₁₈₁	—	2	2	no	—
54	C₄×C₁₈₁	—	2	2	no	—
55	C₄×C₁₈₁	—	8	2	no	—
56	C₄×C₁₈₁	C₄×Q₈	1	2	yes	—
57	C₄×C₁₈₁	C₂²⋊Q₈	1	2	yes	—
58	Dic₁₈₁	—	4	4	no	—
59	C₂×C₁₈₁	#391	1	1	yes	characteristic
60	C₂×C₁₈₁	#418	1	1	yes	characteristic
61	C₂×C₁₈₁	#419	1	1	yes	characteristic
62	C₂×C₁₈₁	#489	1	2	yes	—
63	C₂×C₁₈₁	#501	1	2	yes	—
64	C₁₈₁	#1993	1	1	yes	characteristic, complemented, Sylow
65	#1993	—	181	1	no	maximal, Sylow
66	C₂×C₄⋊Q₈	—	181	1	no	—
67	C₂×#97	—	181	1	no	—
68	#382	D₁₈₁	1	1	yes	characteristic
69	#340	—	181	4	no	—
70	C₂²×Q₈	—	362	1	no	—
71	C₂²×C₈	—	181	2	no	—
72	C₄⋊Q₈	—	362	2	no	—
73	C₄⋊Q₈	—	181	4	no	—
74	C₂×C₄⋊C₄	—	362	1	no	—
75	C₂×C₄⋊C₄	—	181	2	no	—
76	C₂×C₄⋊C₄	C₂×D₁₈₁	1	2	yes	—
77	C₂×C₄²	C₂×D₁₈₁	1	1	yes	characteristic
78	#97	—	362	2	no	—
79	#95	—	2	1	no	—
80	#95	—	2	1	no	—
81	C₂×Q₈	—	362	4	no	—
82	C₂×Q₈	—	724	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	362	2	no	—
85	C₂²×C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₈₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
88	C₂×C₈	—	362	2	no	—
89	C₂×C₈	—	362	2	no	—
90	C₂×C₈	—	181	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	362	4	no	—
93	C₄⋊C₄	—	362	2	no	—
94	C₄⋊C₄	—	181	4	no	—
95	C₄²	C₄×D₁₈₁	1	2	yes	—
96	C₄²	D₇₂₄	1	2	yes	—
97	C₂³	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
98	Q₈	—	724	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	362	4	no	—
105	C₂×C₄	—	724	4	no	—
106	C₂×C₄	#15668426	1	2	yes	—
107	C₂×C₄	#15668426	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₁₈₁	1	1	yes	characteristic
109	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₁₈₁	1	2	yes	—
111	C₂×C₄	#15668448	1	2	yes	—
112	C₈	—	362	4	no	—
113	C₂²	C₂×#15668426	1	1	yes	characteristic
114	C₂²	#19153803	1	1	yes	characteristic
115	C₂²	#19153814	1	1	yes	characteristic
116	C₂²	#19153841	1	2	yes	—
117	C₂²	#19153846	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	8	2	no	—
121	C₄	—	724	4	no	—
122	C₄	#19153781	1	2	yes	—
123	C₄	#19153796	1	2	yes	—
124	C₂	#26041184	1	1	yes	characteristic
125	C₂	#26041283	1	1	yes	characteristic
126	C₂	#26041287	1	1	yes	characteristic
127	C₂	#26041638	1	2	yes	—
128	C₂	#26041689	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19153816

C₂

yes

maximal, characteristic

C₂×#19153727

C₂

yes

maximal, characteristic

#26041041

C₂

yes

maximal

#382×C₁₈₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₆₂

—

C₂²×#13089678

C₂²

yes

—

#19153816

—

#19153816

C₄

yes

complemented

C₂×C₄⋊Dic₁₈₁

—

C₂×C₄⋊Dic₁₈₁

C₂²

yes

—

#19153727

—

C₂×C₄⋊C₄×C₁₈₁

C₂²

yes

—

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

#95×C₁₈₁

—

#95×C₁₈₁

—

C₂×Dic₃₆₂

—

C₂×Dic₃₆₂

—

C₂²×Dic₁₈₁

—

C₂×#13089678

—

C₂×#13089678

—

C₂×#13089678

Q₈

yes

complemented

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

C₂×C₄

yes

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

D₄

yes

characteristic

C₂²×C₄×C₁₈₁

D₄

yes

characteristic

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₈₁

—

C₄²×C₁₈₁

C₂×C₄

yes

—

C₄²×C₁₈₁

D₄

yes

—

Dic₃₆₂

—

C₂×Dic₁₈₁

—

C₂×Dic₁₈₁

—

#13089678

—

C₂³×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×Q₈

yes

—

C₂×C₄×C₁₈₁

#41

yes

—

C₂²×C₁₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₈₁

#124

yes

characteristic

C₂²×C₁₈₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₈₁

AGL(1,ℤ/8)

yes

—

C₂²×C₁₈₁

#137

yes

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₄×Q₈

yes

—

C₄×C₁₈₁

C₂²⋊Q₈

yes

—

Dic₁₈₁

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

—

C₂×C₁₈₁

#501

yes

—

C₁₈₁

#1993

yes

characteristic, complemented, Sylow

#1993

—

181

maximal, Sylow

C₂×C₄⋊Q₈

—

181

—

C₂×#97

—

181

—

#382

D₁₈₁

yes

characteristic

#340

—

181

—

C₂²×Q₈

—

362

—

C₂²×C₈

—

181

—

C₄⋊Q₈

—

362

—

C₄⋊Q₈

—

181

—

C₂×C₄⋊C₄

—

362

—

C₂×C₄⋊C₄

—

181

—

C₂×C₄⋊C₄

C₂×D₁₈₁

yes

—

C₂×C₄²

C₂×D₁₈₁

yes

characteristic

#97

—

362

—

#95

—

#95

—

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

—

362

—

C₂×C₈

—

362

—

C₂×C₈

—

181

—

C₄⋊C₄

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

181

—

C₄²

C₄×D₁₈₁

yes

—

C₄²

D₇₂₄

yes

—

C₂³

C₂×C₂²⋊D₁₈₁

yes

characteristic

Q₈

—

724

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

362

—

105

C₂×C₄

—

724

—

106

C₂×C₄

#15668426

yes

—

107

C₂×C₄

#15668426

yes

—

108

C₂×C₄

C₂×C₄×D₁₈₁

yes

characteristic

109

C₂×C₄

C₂×D₇₂₄

yes

characteristic

110

C₂×C₄

Q₈×D₁₈₁

yes

—

111

C₂×C₄

#15668448

yes

—

112

C₈

—

362

—

113

C₂²

C₂×#15668426

yes

characteristic

114

C₂²

#19153803

yes

characteristic

115

C₂²

#19153814

yes

characteristic

116

C₂²

#19153841

yes

—

117

C₂²

#19153846

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

724

—

122

C₄

#19153781

yes

—

123

C₄

#19153796

yes

—

124

C₂

#26041184

yes

characteristic

125

C₂

#26041283

yes

characteristic

126

C₂

#26041287

yes

characteristic

127

C₂

#26041638

yes

—

128

C₂

#26041689

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₈₁
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	#382×C₁₈₁	C₂
socle, prime radical	C₂³×C₁₈₁	C₂×D₄
cosocle	C₂²×C₄×C₁₈₁	C₂³
hypercenter	#382	D₁₈₁
hypocenter	C₁₈₁	#1993
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₈₁

commutator

C₂×C₄×C₁₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

#382×C₁₈₁

C₂

socle, prime radical

C₂³×C₁₈₁

C₂×D₄

cosocle

C₂²×C₄×C₁₈₁

C₂³

D₁₈₁

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #19153816	C₄	semidirect
21. C₂×#13089678	Q₈	semidirect
64. C₁₈₁	#1993	semidirect

H ≅ G/N

type

8. #19153816

C₄

semidirect

21. C₂×#13089678

Q₈

semidirect

64. C₁₈₁

#1993

semidirect

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

subgroup	index
C₂×#19153816	2
C₂×#19153727	2
#26041041	2
#382×C₁₈₁	2
#1993	181

subgroup

index

#382×C₁₈₁

181

Subgroups — #37534693

Conjugacy classes of subgroups

Complements

Maximal subgroups