Group #37548917 — subgroups

Subgroups — #37548917

Order 23168.

248 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₄×D₁₈₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₈₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#13089685⋊C₈	C₂	1	4	yes	maximal, complemented
4	C₄×C₈×D₁₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₈₁⋊C₈	C₄	1	2	yes	—
6	C₂×D₁₈₁⋊C₈	C₄	1	2	yes	—
7	C₄×#13089685	C₄	1	2	yes	—
8	C₄×#13089685	C₄	1	2	yes	—
9	C₄×#13089685	C₂²	1	2	yes	—
10	C₄×#13089685	C₂²	1	2	yes	—
11	C₂×C₈×D₁₈₁	C₄	1	2	yes	—
12	C₄²×D₁₈₁	C₂²	1	1	yes	characteristic
13	C₄×#13089678	C₄	1	1	yes	characteristic
14	C₈×Dic₁₈₁	C₂²	1	2	yes	—
15	C₄×C₈×C₁₈₁	C₄	1	1	yes	characteristic
16	D₁₈₁⋊C₈	—	2	4	no	—
17	D₁₈₁⋊C₈	C₈	1	8	yes	complemented
18	C₂×#13089685	C₂×C₄	1	4	yes	—
19	C₂×#13089685	C₂×C₄	1	4	yes	—
20	C₂×C₄×D₁₈₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×D₁₈₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₈₁	C₂×C₄	1	1	yes	characteristic
23	C₂×#13089678	C₂×C₄	1	2	yes	—
24	C₈×D₁₈₁	—	2	4	no	—
25	C₄×Dic₁₈₁	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₁₈₁	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₈₁	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₈₁	C₂×C₄	1	2	yes	—
29	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
30	#13089685	—	2	4	no	—
31	#13089685	C₂×C₈	1	8	yes	complemented
32	C₂²×D₁₈₁	C₄²	1	1	yes	characteristic
33	C₂×Dic₁₈₁	C₄²	1	1	yes	characteristic
34	C₂×Dic₁₈₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×Dic₁₈₁	C₂²×C₄	1	1	yes	characteristic
36	C₄×D₁₈₁	C₂×C₈	1	4	yes	—
37	C₄×D₁₈₁	M₁₆	1	4	yes	—
38	C₄×D₁₈₁	M₁₆	1	4	yes	—
39	#13089678	—	2	2	no	—
40	C₂×C₄×C₁₈₁	C₄²	1	1	yes	characteristic
41	C₂×C₄×C₁₈₁	C₄²	1	1	yes	characteristic
42	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₁₈₁	—	2	2	no	—
44	C₂²×C₁₈₁	C₂×C₄²	1	1	yes	characteristic
45	C₂×D₁₈₁	C₄×C₈	1	2	yes	—
46	C₂×D₁₈₁	#97	1	2	yes	—
47	C₂×D₁₈₁	#97	1	2	yes	—
48	C₄×C₁₈₁	C₄×C₈	1	2	yes	—
49	C₄×C₁₈₁	#97	1	2	yes	—
50	C₄×C₁₈₁	C₂×M₁₆	1	2	yes	—
51	Dic₁₈₁	#97	1	2	yes	—
52	Dic₁₈₁	C₂²×C₈	1	2	yes	—
53	Dic₁₈₁	C₂×M₁₆	1	2	yes	—
54	C₂×C₁₈₁	C₂×C₄×C₈	1	1	yes	characteristic
55	C₂×C₁₈₁	C₂×#97	1	1	yes	characteristic
56	C₂×C₁₈₁	C₂×#97	1	1	yes	characteristic
57	D₁₈₁	#322	1	4	yes	complemented
58	C₁₈₁	C₂×#322	1	1	yes	characteristic, complemented, Sylow
59	C₂×#322	—	181	1	no	maximal, Sylow
60	C₂×C₄×C₈	—	181	1	no	—
61	C₂×C₄×C₈	—	181	1	no	—
62	C₂×C₄×C₈	—	181	1	no	—
63	#322	—	181	4	no	—
64	C₂²×C₈	—	181	2	no	—
65	C₂²×C₈	—	181	2	no	—
66	C₂²×C₈	—	181	2	no	—
67	C₂×C₄²	—	181	1	no	—
68	C₄×C₈	—	181	2	no	—
69	C₄×C₈	—	181	2	no	—
70	C₄×C₈	—	181	2	no	—
71	C₄×C₈	—	181	2	no	—
72	C₄×C₈	—	181	2	no	—
73	C₄×C₈	—	181	1	no	—
74	C₄×C₈	#10624382	1	1	yes	characteristic
75	C₂²×C₄	—	181	1	no	—
76	C₂²×C₄	—	181	1	no	—
77	C₂²×C₄	—	181	1	no	—
78	C₂×C₈	—	362	4	no	—
79	C₂×C₈	—	362	4	no	—
80	C₂×C₈	—	181	4	no	—
81	C₂×C₈	—	181	2	no	—
82	C₂×C₈	—	181	4	no	—
83	C₂×C₈	—	181	8	no	—
84	C₂×C₈	C₂×#10624382	1	2	yes	—
85	C₄²	—	181	1	no	—
86	C₄²	—	181	1	no	—
87	C₄²	—	181	1	no	—
88	C₄²	C₂×#10624382	1	1	yes	characteristic
89	C₂³	—	181	1	no	—
90	C₂×C₄	—	181	1	no	—
91	C₂×C₄	—	181	1	no	—
92	C₂×C₄	—	181	1	no	—
93	C₂×C₄	—	181	4	no	—
94	C₂×C₄	—	181	4	no	—
95	C₂×C₄	—	181	4	no	—
96	C₂×C₄	C₄×#10624382	1	1	yes	characteristic
97	C₂×C₄	C₄×#10624382	1	1	yes	characteristic
98	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	362	4	no	—
101	C₈	—	362	2	no	—
102	C₈	—	181	8	no	—
103	C₂²	—	181	2	no	—
104	C₂²	—	181	2	no	—
105	C₂²	—	181	2	no	—
106	C₂²	C₂×C₄×#10624382	1	1	yes	characteristic
107	C₄	—	181	2	no	—
108	C₄	—	181	2	no	—
109	C₄	—	181	2	no	—
110	C₄	C₈×#10624382	1	2	yes	—
111	C₄	#19153870	1	2	yes	—
112	C₄	D₁₈₁⋊M₁₆	1	2	yes	—
113	C₂	—	181	4	no	—
114	C₂	C₂×C₈×#10624382	1	1	yes	characteristic
115	C₂	C₂×#19153870	1	1	yes	characteristic
116	C₂	#26042238	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₁₈₁⋊C₈

C₂

yes

maximal, characteristic

C₄×D₁₈₁⋊C₈

C₂

yes

maximal, characteristic

#13089685⋊C₈

C₂

yes

maximal, complemented

C₄×C₈×D₁₈₁

C₂

yes

maximal, characteristic

C₂×D₁₈₁⋊C₈

C₄

yes

—

C₂×D₁₈₁⋊C₈

C₄

yes

—

C₄×#13089685

C₄

yes

—

C₄×#13089685

C₄

yes

—

C₄×#13089685

C₂²

yes

—

C₄×#13089685

C₂²

yes

—

C₂×C₈×D₁₈₁

C₄

yes

—

C₄²×D₁₈₁

C₂²

yes

characteristic

C₄×#13089678

C₄

yes

characteristic

C₈×Dic₁₈₁

C₂²

yes

—

C₄×C₈×C₁₈₁

C₄

yes

characteristic

D₁₈₁⋊C₈

—

D₁₈₁⋊C₈

C₈

yes

complemented

C₂×#13089685

C₂×C₄

yes

—

C₂×#13089685

C₂×C₄

yes

—

C₂×C₄×D₁₈₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₈₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₈₁

C₂×C₄

yes

characteristic

C₂×#13089678

C₂×C₄

yes

—

C₈×D₁₈₁

—

C₄×Dic₁₈₁

C₂×C₄

yes

characteristic

C₄×Dic₁₈₁

C₂×C₄

yes

characteristic

C₄×Dic₁₈₁

C₂³

yes

characteristic

C₂×C₈×C₁₈₁

C₂×C₄

yes

—

C₄²×C₁₈₁

C₂×C₄

yes

characteristic

#13089685

—

#13089685

C₂×C₈

yes

complemented

C₂²×D₁₈₁

C₄²

yes

characteristic

C₂×Dic₁₈₁

C₄²

yes

characteristic

C₂×Dic₁₈₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₈₁

C₂²×C₄

yes

characteristic

C₄×D₁₈₁

C₂×C₈

yes

—

C₄×D₁₈₁

M₁₆

yes

—

C₄×D₁₈₁

M₁₆

yes

—

#13089678

—

C₂×C₄×C₁₈₁

C₄²

yes

characteristic

C₂×C₄×C₁₈₁

C₄²

yes

characteristic

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₈×C₁₈₁

—

C₂²×C₁₈₁

C₂×C₄²

yes

characteristic

C₂×D₁₈₁

C₄×C₈

yes

—

C₂×D₁₈₁

#97

yes

—

C₂×D₁₈₁

#97

yes

—

C₄×C₁₈₁

C₄×C₈

yes

—

C₄×C₁₈₁

#97

yes

—

C₄×C₁₈₁

C₂×M₁₆

yes

—

Dic₁₈₁

#97

yes

—

Dic₁₈₁

C₂²×C₈

yes

—

Dic₁₈₁

C₂×M₁₆

yes

—

C₂×C₁₈₁

C₂×C₄×C₈

yes

characteristic

C₂×C₁₈₁

C₂×#97

yes

characteristic

C₂×C₁₈₁

C₂×#97

yes

D₁₈₁

yes

C₁₈₁

yes

characteristic, complemented, Sylow

C₂×#322

—

181

maximal, Sylow

C₂×C₄×C₈

—

181

—

C₂×C₄×C₈

—

181

—

C₂×C₄×C₈

—

181

—

#322

—

181

—

C₂²×C₈

—

181

—

C₂²×C₈

—

181

—

C₂²×C₈

—

181

—

C₂×C₄²

—

181

—

C₄×C₈

—

181

—

C₄×C₈

—

181

—

C₄×C₈

—

181

—

C₄×C₈

—

181

—

C₄×C₈

—

181

—

C₄×C₈

—

181

—

C₄×C₈

#10624382

yes

characteristic

C₂²×C₄

—

181

—

C₂²×C₄

—

181

—

C₂²×C₄

—

181

—

C₂×C₈

—

362

—

C₂×C₈

—

362

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

C₂×#10624382

yes

—

C₄²

—

181

—

C₄²

—

181

—

C₄²

—

181

—

C₄²

C₂×#10624382

yes

characteristic

C₂³

—

181

—

C₂×C₄

—

181

—

C₂×C₄

—

181

—

C₂×C₄

—

181

—

C₂×C₄

—

181

—

C₂×C₄

—

181

—

C₂×C₄

—

181

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

100

C₈

—

362

—

101

C₈

—

362

—

102

C₈

—

181

—

103

C₂²

—

181

—

104

C₂²

—

181

—

105

C₂²

—

181

—

106

C₂²

C₂×C₄×#10624382

yes

characteristic

107

C₄

—

181

—

108

C₄

—

181

—

109

C₄

—

181

—

110

C₄

C₈×#10624382

yes

—

111

C₄

#19153870

yes

—

112

C₄

D₁₈₁⋊M₁₆

yes

—

113

C₂

—

181

—

114

C₂

C₂×C₈×#10624382

yes

characteristic

115

C₂

C₂×#19153870

yes

characteristic

116

C₂

#26042238

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×#10624382
commutator	C₂×C₁₈₁	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#10624382
Fitting	C₄×C₈×C₁₈₁	C₄
socle	C₂²×C₁₈₁	C₂×C₄²
prime radical	C₂²×D₁₈₁	C₄²
cosocle	C₄×Dic₁₈₁	C₂³
hypercenter	C₄×C₈	#10624382
hypocenter	C₁₈₁	C₂×#322
LCS	G ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₄²

C₂×#10624382

commutator

C₂×C₁₈₁

C₂×C₄×C₈

Frattini

C₂×C₄

C₂²×#10624382

Fitting

C₄×C₈×C₁₈₁

C₄

socle

C₂²×C₁₈₁

C₂×C₄²

prime radical

C₂²×D₁₈₁

C₄²

cosocle

C₄×Dic₁₈₁

C₂³

C₄×C₈

C₁₈₁

G ▷ C₂×C₁₈₁ ▷ C₁₈₁

C₁ ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
3. #13089685⋊C₈	C₂	semidirect
17. D₁₈₁⋊C₈	C₈	semidirect
31. #13089685	C₂×C₈	semidirect
57. D₁₈₁	#322	semidirect
58. C₁₈₁	C₂×#322	semidirect

H ≅ G/N

type

3. #13089685⋊C₈

C₂

semidirect

17. D₁₈₁⋊C₈

C₈

semidirect

31. #13089685

C₂×C₈

semidirect

57. D₁₈₁

#322

semidirect

58. C₁₈₁

C₂×#322

semidirect

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

subgroup	index
C₄×D₁₈₁⋊C₈	2
#13089685⋊C₈	2
C₄×C₈×D₁₈₁	2
C₂×#322	181

subgroup

index

C₄×D₁₈₁⋊C₈

#13089685⋊C₈

C₄×C₈×D₁₈₁

C₂×#322

181

Subgroups — #37548917

Conjugacy classes of subgroups

Complements

Maximal subgroups