Group #37534702 — subgroups

Subgroups — #37534702

Order 23168.

244 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 93 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	#26041836	C₂	1	1	yes	maximal, characteristic
2	#26041303	C₂	1	1	yes	maximal, characteristic
3	#26041039	C₂	1	4	yes	maximal
4	#376×C₁₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668433	C₂²	1	2	yes	—
6	#19153816	C₄	1	2	yes	complemented
7	#19153808	C₄	1	2	yes	complemented
8	#19153795	—	2	2	no	—
9	C₄×Dic₃₆₂	—	2	2	no	—
10	#19153775	C₂²	1	2	yes	—
11	C₄⋊#13089678	—	2	2	no	—
12	C₂×C₄²×C₁₈₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
14	#95×C₁₈₁	—	2	2	no	—
15	C₂×Dic₃₆₂	—	2	2	no	—
16	#15668433	—	2	4	no	—
17	C₂×#13089678	—	2	2	no	—
18	C₄⋊Dic₁₈₁	—	2	2	no	—
19	C₄⋊Dic₁₈₁	C₂×C₄	1	4	yes	—
20	#15668428	—	2	4	no	—
21	C₂²⋊Dic₁₈₁	—	2	2	no	—
22	C₄×Dic₁₈₁	—	2	2	no	—
23	C₂²×C₄×C₁₈₁	—	2	2	no	—
24	C₂²×C₄×C₁₈₁	D₄	1	2	yes	—
25	C₂²×C₄×C₁₈₁	C₂³	1	1	yes	characteristic
26	C₄²×C₁₈₁	—	2	2	no	—
27	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₈₁	D₄	1	2	yes	—
30	C₄²×C₁₈₁	Q₈	1	4	yes	complemented
31	Dic₃₆₂	—	4	4	no	—
32	C₂×Dic₁₈₁	—	2	4	no	—
33	#13089678	—	4	2	no	—
34	C₂³×C₁₈₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₈₁	—	2	4	no	—
36	C₂×C₄×C₁₈₁	—	4	2	no	—
37	C₂×C₄×C₁₈₁	—	4	4	no	—
38	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₈₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₈₁	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₁₈₁	#41	1	2	yes	—
44	C₂²×C₁₈₁	—	2	2	no	—
45	C₂²×C₁₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₈₁	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₁₈₁	—	2	2	no	—
48	C₄×C₁₈₁	—	2	2	no	—
49	C₄×C₁₈₁	—	4	4	no	—
50	C₄×C₁₈₁	C₄×Q₈	1	2	yes	—
51	C₄×C₁₈₁	#124	1	1	yes	characteristic
52	C₄×C₁₈₁	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₈₁	—	4	4	no	—
54	C₂×C₁₈₁	—	2	2	no	—
55	C₂×C₁₈₁	#391	1	1	yes	characteristic
56	C₂×C₁₈₁	#421	1	2	yes	—
57	C₁₈₁	#1995	1	1	yes	characteristic, complemented, Sylow
58	#1995	—	181	1	no	maximal, Sylow
59	#533	—	181	1	no	—
60	#423	—	181	1	no	—
61	#376	D₁₈₁	1	1	yes	characteristic
62	#339	—	181	4	no	—
63	C₂×M₁₆	—	181	2	no	—
64	C₄⋊Q₈	—	181	2	no	—
65	#125	—	181	2	no	—
66	C₂²⋊Q₈	—	362	2	no	—
67	C₄×Q₈	—	362	2	no	—
68	#117	—	181	2	no	—
69	C₂×C₄²	C₂×D₁₈₁	1	2	yes	—
70	C₂×C₄²	C₂×D₁₈₁	1	1	yes	characteristic
71	C₄⋊C₈	—	362	2	no	—
72	#95	—	2	2	no	—
73	C₂×Q₈	—	362	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	C₂²⋊D₁₈₁	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₈₁	1	1	yes	characteristic
77	M₁₆	—	362	4	no	—
78	C₂×C₈	—	362	2	no	—
79	C₄⋊C₄	—	362	4	no	—
80	C₄⋊C₄	—	362	2	no	—
81	C₄⋊C₄	—	181	4	no	—
82	C₂²⋊C₄	—	362	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	362	2	no	—
85	C₄²	C₄×D₁₈₁	1	1	yes	characteristic
86	C₄²	C₄×D₁₈₁	1	1	yes	characteristic
87	C₄²	D₇₂₄	1	2	yes	—
88	C₄²	Dic₃₆₂	1	4	yes	complemented
89	C₂³	C₂×C₂²⋊D₁₈₁	1	1	yes	characteristic
90	Q₈	—	724	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	—	362	4	no	—
95	C₂×C₄	#15668426	1	2	yes	—
96	C₂×C₄	#15668426	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₁₈₁	1	1	yes	characteristic
98	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₃₆₂	1	2	yes	—
100	C₂×C₄	#15668449	1	2	yes	—
101	C₈	—	724	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×#15668426	1	1	yes	characteristic
104	C₂²	#19153795	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	724	4	no	—
109	C₄	C₄×Dic₃₆₂	1	2	yes	—
110	C₄	#19153804	1	1	yes	characteristic
111	C₄	#19153816	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#26041185	1	1	yes	characteristic
114	C₂	#26041297	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26041836

C₂

yes

maximal, characteristic

#26041303

C₂

yes

maximal, characteristic

#26041039

C₂

yes

maximal

#376×C₁₈₁

C₂

yes

maximal, characteristic

C₂×#15668433

C₂²

yes

—

#19153816

C₄

yes

complemented

#19153808

C₄

yes

complemented

#19153795

—

C₄×Dic₃₆₂

—

#19153775

C₂²

yes

—

C₄⋊#13089678

—

C₂×C₄²×C₁₈₁

C₂²

yes

—

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

#95×C₁₈₁

—

C₂×Dic₃₆₂

—

#15668433

—

C₂×#13089678

—

C₄⋊Dic₁₈₁

—

C₄⋊Dic₁₈₁

C₂×C₄

yes

—

#15668428

—

C₂²⋊Dic₁₈₁

—

C₄×Dic₁₈₁

—

C₂²×C₄×C₁₈₁

—

C₂²×C₄×C₁₈₁

D₄

yes

—

C₂²×C₄×C₁₈₁

C₂³

yes

characteristic

C₄²×C₁₈₁

—

C₄²×C₁₈₁

C₂×C₄

yes

characteristic

C₄²×C₁₈₁

C₂×C₄

yes

characteristic

C₄²×C₁₈₁

D₄

yes

—

C₄²×C₁₈₁

Q₈

yes

complemented

Dic₃₆₂

—

C₂×Dic₁₈₁

—

#13089678

—

C₂³×C₁₈₁

C₂×D₄

yes

characteristic

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

—

C₂²×C₁₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₈₁

C₂²⋊Q₈

yes

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

C₄×Q₈

yes

—

C₄×C₁₈₁

#124

yes

characteristic

C₄×C₁₈₁

C₄⋊Q₈

yes

characteristic

Dic₁₈₁

—

C₂×C₁₈₁

—

C₂×C₁₈₁

#391

yes

characteristic

C₂×C₁₈₁

#421

yes

—

C₁₈₁

#1995

yes

characteristic, complemented, Sylow

#1995

—

181

maximal, Sylow

#533

—

181

—

#423

—

181

—

#376

D₁₈₁

yes

characteristic

#339

—

181

—

C₂×M₁₆

—

181

—

C₄⋊Q₈

—

181

—

#125

—

181

—

C₂²⋊Q₈

—

362

—

C₄×Q₈

—

362

—

#117

—

181

—

C₂×C₄²

C₂×D₁₈₁

yes

—

C₂×C₄²

C₂×D₁₈₁

yes

characteristic

C₄⋊C₈

—

362

—

#95

—

C₂×Q₈

—

362

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₈₁

yes

—

C₂²×C₄

C₂²×D₁₈₁

yes

characteristic

M₁₆

—

362

—

C₂×C₈

—

362

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

362

—

C₄⋊C₄

—

181

—

C₂²⋊C₄

—

362

—

C₄²

—

C₄²

—

362

—

C₄²

C₄×D₁₈₁

yes

characteristic

C₄²

C₄×D₁₈₁

yes

characteristic

C₄²

D₇₂₄

yes

—

C₄²

Dic₃₆₂

yes

complemented

C₂³

C₂×C₂²⋊D₁₈₁

yes

characteristic

Q₈

—

724

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

362

—

C₂×C₄

#15668426

yes

—

C₂×C₄

#15668426

yes

—

C₂×C₄

C₂×C₄×D₁₈₁

yes

characteristic

C₂×C₄

C₂×D₇₂₄

yes

characteristic

C₂×C₄

C₂×Dic₃₆₂

yes

—

100

C₂×C₄

#15668449

yes

—

101

C₈

—

724

—

102

C₂²

—

103

C₂²

C₂×#15668426

yes

characteristic

104

C₂²

#19153795

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

724

—

109

C₄

C₄×Dic₃₆₂

yes

—

110

C₄

#19153804

yes

characteristic

111

C₄

#19153816

yes

characteristic

112

C₂

—

113

C₂

#26041185

yes

characteristic

114

C₂

#26041297

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₂₄
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₈₁
Fitting	#376×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₈₁	C₂×D₄
cosocle	C₂²×C₄×C₁₈₁	C₂³
hypercenter	#376	D₁₈₁
hypocenter	C₁₈₁	#1995
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₇₂₄

commutator

C₂×C₄×C₁₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₈₁

Fitting

#376×C₁₈₁

C₂

socle

C₂²×C₁₈₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₈₁

C₂×D₄

cosocle

C₂²×C₄×C₁₈₁

C₂³

D₁₈₁

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
6. #19153816	C₄	semidirect
7. #19153808	C₄	semidirect
30. C₄²×C₁₈₁	Q₈	semidirect
57. C₁₈₁	#1995	semidirect
88. C₄²	Dic₃₆₂	semidirect

H ≅ G/N

type

6. #19153816

C₄

semidirect

7. #19153808

C₄

semidirect

30. C₄²×C₁₈₁

Q₈

semidirect

57. C₁₈₁

#1995

semidirect

88. C₄²

Dic₃₆₂

semidirect

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

subgroup	index
#26041836	2
#26041303	2
#26041039	2
#376×C₁₈₁	2
#1995	181

subgroup

index

#376×C₁₈₁

181

Subgroups — #37534702

Conjugacy classes of subgroups

Complements

Maximal subgroups