Group #37532492 — subgroups

Subgroups — #37532492

Order 23168.

168 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 97 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	#26041303	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#13089678	C₂	1	1	yes	maximal, characteristic
3	#26041030	C₂	1	4	yes	maximal, complemented
4	#431×C₁₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#15668433	—	2	1	no	—
6	C₂²×#13089678	—	2	1	no	—
7	C₄⋊#13089678	—	2	1	no	—
8	C₄⋊#13089678	—	2	1	no	—
9	C₄⋊#13089678	C₂²	1	2	yes	—
10	C₄⋊#13089678	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₈₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₈₁	C₄	1	2	yes	—
13	C₂²⋊C₈×C₁₈₁	C₄	1	2	yes	—
14	C₄×C₈×C₁₈₁	C₂²	1	2	yes	—
15	#15668433	—	4	2	no	—
16	C₂×#13089678	—	2	2	no	—
17	C₂×#13089678	—	2	2	no	—
18	C₂×#13089678	—	4	2	no	—
19	C₂²×C₄×C₁₈₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₈₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₈₁	Q₈	1	1	yes	characteristic
22	C₂×C₈×C₁₈₁	C₂×C₄	1	4	yes	—
23	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₈₁	D₄	1	1	yes	characteristic
25	C₄²×C₁₈₁	Q₈	1	1	yes	characteristic
26	C₄²×C₁₈₁	C₂³	1	1	yes	characteristic
27	#13089678	—	4	2	no	—
28	#13089678	—	4	2	no	—
29	C₂³×C₁₈₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₈₁	—	2	2	no	—
31	C₂×C₄×C₁₈₁	—	4	1	no	—
32	C₂×C₄×C₁₈₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₈₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₈₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₈₁	SD₁₆	1	2	yes	—
36	C₂×C₄×C₁₈₁	SD₁₆	1	2	yes	—
37	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₈₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₈₁	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₁₈₁	—	2	4	no	—
41	C₂²×C₁₈₁	—	2	1	no	—
42	C₂²×C₁₈₁	—	2	1	no	—
43	C₂²×C₁₈₁	#106	1	2	yes	—
44	C₂²×C₁₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₁₈₁	—	2	1	no	—
46	C₄×C₁₈₁	—	2	1	no	—
47	C₄×C₁₈₁	—	2	1	no	—
48	C₄×C₁₈₁	#106	1	2	yes	—
49	C₄×C₁₈₁	#131	1	2	yes	—
50	C₄×C₁₈₁	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₁₈₁	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×C₁₈₁	—	2	2	no	—
53	C₂×C₁₈₁	#424	1	1	yes	characteristic
54	C₂×C₁₈₁	C₂×#106	1	1	yes	characteristic
55	C₂×C₁₈₁	#430	1	1	yes	characteristic
56	C₁₈₁	#1570	1	1	yes	characteristic, complemented, Sylow
57	#1570	—	181	1	no	maximal, Sylow
58	#431	D₁₈₁	1	1	yes	characteristic
59	#423	—	181	1	no	—
60	C₂×C₄⋊C₈	—	181	1	no	—
61	#334	—	181	4	no	—
62	C₂×M₁₆	—	362	1	no	—
63	C₂²×C₈	—	362	1	no	—
64	C₂×C₄²	C₂×D₁₈₁	1	1	yes	characteristic
65	C₄⋊C₈	—	362	1	no	—
66	C₄⋊C₈	—	362	1	no	—
67	C₄⋊C₈	—	181	2	no	—
68	C₄⋊C₈	—	181	2	no	—
69	C₄⋊C₈	Dic₁₈₁	1	2	yes	—
70	C₂²⋊C₈	Dic₁₈₁	1	2	yes	—
71	C₄×C₈	C₂×D₁₈₁	1	2	yes	—
72	C₂²×C₄	C₂×Dic₁₈₁	1	1	yes	characteristic
73	C₂²×C₄	D₇₂₄	1	1	yes	characteristic
74	C₂²×C₄	Dic₃₆₂	1	1	yes	characteristic
75	M₁₆	—	724	2	no	—
76	C₂×C₈	—	362	2	no	—
77	C₂×C₈	—	362	2	no	—
78	C₂×C₈	—	724	2	no	—
79	C₂×C₈	C₂×Dic₁₈₁	1	4	yes	—
80	C₄²	C₂×Dic₁₈₁	1	1	yes	characteristic
81	C₄²	D₇₂₄	1	1	yes	characteristic
82	C₄²	Dic₃₆₂	1	1	yes	characteristic
83	C₄²	C₂²×D₁₈₁	1	1	yes	characteristic
84	C₂³	C₄⋊Dic₁₈₁	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	C₄⋊Dic₁₈₁	1	1	yes	characteristic
88	C₂×C₄	C₄⋊Dic₁₈₁	1	1	yes	characteristic
89	C₂×C₄	C₄⋊Dic₁₈₁	1	1	yes	characteristic
90	C₂×C₄	#15668438	1	2	yes	—
91	C₂×C₄	#15668438	1	2	yes	—
92	C₂×C₄	C₂²×Dic₁₈₁	1	1	yes	characteristic
93	C₂×C₄	C₂×D₇₂₄	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₃₆₂	1	1	yes	characteristic
95	C₈	—	2	4	no	—
96	C₈	—	724	2	no	—
97	C₈	—	724	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#19153748	1	2	yes	—
101	C₂²	C₂×C₄⋊Dic₁₈₁	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#19153748	1	2	yes	—
106	C₄	#19153823	1	2	yes	—
107	C₄	C₂×#15668438	1	1	yes	characteristic
108	C₄	C₂×#15668438	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#26041306	1	1	yes	characteristic
111	C₂	C₂×#19153748	1	1	yes	characteristic
112	C₂	#26041328	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26041303

C₂

yes

maximal, characteristic

C₂×C₄⋊#13089678

C₂

yes

maximal, characteristic

#26041030

C₂

yes

maximal, complemented

#431×C₁₈₁

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

—

C₄⋊#13089678

C₂²

yes

—

C₂×C₄²×C₁₈₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₈₁

C₄

yes

—

C₂²⋊C₈×C₁₈₁

C₄

yes

—

C₄×C₈×C₁₈₁

C₂²

yes

—

—

—

—

—

C₂²×C₄×C₁₈₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₈₁

D₄

yes

characteristic

C₂²×C₄×C₁₈₁

Q₈

yes

characteristic

C₂×C₈×C₁₈₁

C₂×C₄

yes

—

C₄²×C₁₈₁

C₂×C₄

yes

characteristic

C₄²×C₁₈₁

D₄

yes

characteristic

C₄²×C₁₈₁

Q₈

yes

characteristic

C₄²×C₁₈₁

C₂³

yes

characteristic

#13089678

—

#13089678

—

C₂³×C₁₈₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₈₁

SD₁₆

yes

—

C₂×C₄×C₁₈₁

SD₁₆

yes

—

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₈₁

C₂×Q₈

yes

characteristic

C₈×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

#106

yes

—

C₂²×C₁₈₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

#106

yes

—

C₄×C₁₈₁

#131

yes

—

C₄×C₁₈₁

C₂×SD₁₆

yes

characteristic

C₄×C₁₈₁

C₂×SD₁₆

yes

characteristic

C₂×C₁₈₁

—

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₂×C₁₈₁

yes

C₁₈₁

yes

characteristic, complemented, Sylow

#1570

—

181

maximal, Sylow

#431

D₁₈₁

yes

characteristic

#423

—

181

—

C₂×C₄⋊C₈

—

181

—

#334

—

181

—

C₂×M₁₆

—

362

—

C₂²×C₈

—

362

—

C₂×C₄²

C₂×D₁₈₁

yes

characteristic

C₄⋊C₈

—

362

—

C₄⋊C₈

—

362

—

C₄⋊C₈

—

181

—

C₄⋊C₈

—

181

—

C₄⋊C₈

Dic₁₈₁

yes

—

C₂²⋊C₈

Dic₁₈₁

yes

—

C₄×C₈

C₂×D₁₈₁

yes

—

C₂²×C₄

C₂×Dic₁₈₁

yes

characteristic

C₂²×C₄

D₇₂₄

yes

characteristic

C₂²×C₄

Dic₃₆₂

yes

characteristic

M₁₆

—

724

—

C₂×C₈

—

362

—

C₂×C₈

—

362

—

C₂×C₈

—

724

—

C₂×C₈

C₂×Dic₁₈₁

yes

—

C₄²

C₂×Dic₁₈₁

yes

characteristic

C₄²

D₇₂₄

yes

characteristic

C₄²

Dic₃₆₂

yes

characteristic

C₄²

C₂²×D₁₈₁

yes

characteristic

C₂³

C₄⋊Dic₁₈₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₈₁

yes

characteristic

C₂×C₄

C₄⋊Dic₁₈₁

yes

characteristic

C₂×C₄

C₄⋊Dic₁₈₁

yes

characteristic

C₂×C₄

#15668438

yes

—

C₂×C₄

#15668438

yes

—

C₂×C₄

C₂²×Dic₁₈₁

yes

characteristic

C₂×C₄

C₂×D₇₂₄

yes

characteristic

C₂×C₄

C₂×Dic₃₆₂

yes

characteristic

C₈

—

C₈

—

724

—

C₈

—

724

—

C₂²

—

C₂²

—

100

C₂²

#19153748

yes

—

101

C₂²

C₂×C₄⋊Dic₁₈₁

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#19153748

yes

—

106

C₄

#19153823

yes

—

107

C₄

C₂×#15668438

yes

characteristic

108

C₄

C₂×#15668438

yes

characteristic

109

C₂

—

110

C₂

#26041306

yes

characteristic

111

C₂

C₂×#19153748

yes

characteristic

112

C₂

#26041328

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₇₂₄
commutator	C₂×C₄×C₁₈₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₈₁
Fitting	#431×C₁₈₁	C₂
socle	C₂²×C₁₈₁	C₂×C₄⋊C₄
prime radical	C₂³×C₁₈₁	C₄⋊C₄
cosocle	C₄²×C₁₈₁	C₂³
hypercenter	#431	D₁₈₁
hypocenter	C₁₈₁	#1570
LCS	G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₇₂₄

commutator

C₂×C₄×C₁₈₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₈₁

Fitting

#431×C₁₈₁

C₂

socle

C₂²×C₁₈₁

C₂×C₄⋊C₄

prime radical

C₂³×C₁₈₁

C₄⋊C₄

cosocle

C₄²×C₁₈₁

C₂³

D₁₈₁

C₁₈₁

G ▷ C₂×C₄×C₁₈₁ ▷ C₂×C₁₈₁ ▷ C₁₈₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #26041030	C₂	semidirect
56. C₁₈₁	#1570	semidirect

H ≅ G/N

type

3. #26041030

C₂

semidirect

56. C₁₈₁

#1570

semidirect

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

subgroup	index
#26041303	2
C₂×C₄⋊#13089678	2
#26041030	2
#431×C₁₈₁	2
#1570	181

subgroup

index

#431×C₁₈₁

181

Subgroups — #37532492

Conjugacy classes of subgroups

Complements

Maximal subgroups