Group #37550086 — subgroups

Subgroups — #37550086

Order 23168.

138 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 70 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⋊#15668452	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#15668452	C₂	1	2	yes	maximal
3	#26042167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#15668452	C₂	1	1	yes	maximal, characteristic, complemented
5	#26041235	C₂	1	1	yes	maximal, characteristic
6	C₂×#15668452	C₂²	1	2	yes	—
7	C₂×#15668452	C₂²	1	2	yes	—
8	C₂×#15668433	C₄	1	1	yes	characteristic
9	C₂²×#13089678	C₂²	1	1	yes	characteristic
10	#19153727	C₄	1	1	yes	characteristic
11	#19153727	C₄	1	1	yes	characteristic
12	C₄×#13089678	C₂²	1	1	yes	characteristic
13	C₄×#13089678	C₂²	1	1	yes	characteristic
14	#117×C₁₈₁	C₄	1	1	yes	characteristic
15	#15668452	—	2	2	no	—
16	#15668452	—	2	2	no	—
17	#15668433	—	2	2	no	—
18	C₂×#13089678	—	2	2	no	—
19	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
20	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
21	C₂×#13089678	C₂×C₄	1	1	yes	characteristic
22	C₂×#13089678	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₁₈₁	C₂×C₄	1	1	yes	characteristic
24	C₄⋊C₄×C₁₈₁	C₈	1	2	yes	—
25	C₂²⋊C₄×C₁₈₁	C₈	1	2	yes	—
26	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₈₁	C₂×C₄	1	1	yes	characteristic
28	#13089678	—	2	1	no	—
29	#13089678	—	2	1	no	—
30	#13089678	M₁₆	1	2	yes	—
31	#13089678	#41	1	2	yes	—
32	C₂³×C₁₈₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₈₁	—	2	1	no	—
34	C₂×C₄×C₁₈₁	—	2	1	no	—
35	C₂×C₄×C₁₈₁	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₁₈₁	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₁₈₁	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₁₈₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₁₈₁	—	2	1	no	—
40	C₂²×C₁₈₁	—	2	2	no	—
41	C₂²×C₁₈₁	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₈₁	—	2	1	no	—
43	C₄×C₁₈₁	—	2	2	no	—
44	C₄×C₁₈₁	—	2	2	no	—
45	C₄×C₁₈₁	#117	1	1	yes	characteristic
46	C₄×C₁₈₁	C₂×M₁₆	1	1	yes	characteristic
47	C₂×C₁₈₁	—	4	1	no	—
48	C₂×C₁₈₁	#431	1	1	yes	characteristic
49	C₂×C₁₈₁	#504	1	2	yes	—
50	C₁₈₁	#2164	1	1	yes	characteristic, complemented, Sylow
51	#2164	—	181	1	no	maximal, Sylow
52	#405	—	181	1	no	—
53	C₄⋊C₁₆	—	181	2	no	—
54	C₂²⋊C₁₆	—	181	2	no	—
55	#346	—	181	1	no	—
56	C₄×C₁₆	—	181	1	no	—
57	C₂×M₁₆	—	181	1	no	—
58	C₂²×C₈	—	181	1	no	—
59	#117	#10624382	1	1	yes	characteristic
60	C₂×C₁₆	—	181	2	no	—
61	C₂×C₁₆	—	181	2	no	—
62	#97	—	181	1	no	—
63	#97	—	181	1	no	—
64	C₄×C₈	—	181	1	no	—
65	C₄×C₈	—	181	1	no	—
66	C₂²×C₄	C₂×#10624382	1	1	yes	characteristic
67	M₁₆	—	362	2	no	—
68	C₂×C₈	—	362	2	no	—
69	C₂×C₈	—	181	1	no	—
70	C₂×C₈	—	181	1	no	—
71	C₂×C₈	—	181	1	no	—
72	C₂×C₈	—	181	1	no	—
73	C₄⋊C₄	#13089685	1	2	yes	—
74	C₂²⋊C₄	#13089685	1	2	yes	—
75	C₄²	C₂×#10624382	1	1	yes	characteristic
76	C₄²	C₂×#10624382	1	1	yes	characteristic
77	C₁₆	—	362	2	no	—
78	C₁₆	—	362	2	no	—
79	C₂³	C₂×#13089685	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	C₂×#13089685	1	2	yes	—
83	C₂×C₄	C₂×#13089685	1	2	yes	—
84	C₂×C₄	C₂×#13089685	1	1	yes	characteristic
85	C₂×C₄	C₂²×#10624382	1	1	yes	characteristic
86	C₈	—	362	1	no	—
87	C₈	—	362	1	no	—
88	C₈	—	181	2	no	—
89	C₈	—	181	2	no	—
90	C₂²	—	2	1	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₂²×#13089685	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	#19153895	1	1	yes	characteristic
97	C₄	D₁₈₁⋊M₁₆	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	#26042295	1	1	yes	characteristic
100	C₂	#26042337	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#15668452

C₂

yes

maximal, complemented

C₂²⋊#15668452

C₂

yes

maximal

#26042167

C₂

yes

maximal, characteristic, complemented

C₄×#15668452

C₂

yes

maximal, characteristic, complemented

#26041235

C₂

yes

maximal, characteristic

C₂×#15668452

C₂²

yes

—

C₂×#15668452

C₂²

yes

—

C₂×#15668433

C₄

yes

characteristic

C₂²×#13089678

C₂²

yes

characteristic

#19153727

C₄

yes

characteristic

#19153727

C₄

yes

characteristic

C₄×#13089678

C₂²

yes

characteristic

C₄×#13089678

C₂²

yes

characteristic

#117×C₁₈₁

C₄

yes

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#13089678

C₂×C₄

yes

characteristic

C₂×#13089678

C₂×C₄

yes

characteristic

C₂×#13089678

C₂³

yes

characteristic

C₂²×C₄×C₁₈₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₈₁

C₈

yes

—

C₂²⋊C₄×C₁₈₁

C₈

yes

—

C₄²×C₁₈₁

C₂×C₄

yes

characteristic

C₄²×C₁₈₁

C₂×C₄

yes

—

—

M₁₆

yes

—

#13089678

#41

yes

—

C₂³×C₁₈₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

—

C₂×C₄×C₁₈₁

C₂×C₈

yes

—

C₂×C₄×C₁₈₁

C₂×C₈

yes

—

C₂×C₄×C₁₈₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₈₁

C₂²×C₄

yes

characteristic

C₂²×C₁₈₁

—

C₂²×C₁₈₁

—

C₂²×C₁₈₁

C₂²×C₈

yes

characteristic

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

—

C₄×C₁₈₁

#117

yes

characteristic

C₄×C₁₈₁

C₂×M₁₆

yes

characteristic

C₂×C₁₈₁

—

C₂×C₁₈₁

#431

yes

characteristic

C₂×C₁₈₁

#504

yes

—

C₁₈₁

#2164

yes

characteristic, complemented, Sylow

#2164

—

181

maximal, Sylow

#405

—

181

—

C₄⋊C₁₆

—

181

—

C₂²⋊C₁₆

—

181

—

#346

—

181

—

C₄×C₁₆

—

181

—

C₂×M₁₆

—

181

—

C₂²×C₈

—

181

—

#117

#10624382

yes

characteristic

C₂×C₁₆

—

181

—

C₂×C₁₆

—

181

—

#97

—

181

—

#97

—

181

—

C₄×C₈

—

181

—

C₄×C₈

—

181

—

C₂²×C₄

C₂×#10624382

yes

characteristic

M₁₆

—

362

—

C₂×C₈

—

362

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₂×C₈

—

181

—

C₄⋊C₄

#13089685

yes

—

C₂²⋊C₄

#13089685

yes

—

C₄²

yes

C₄²

yes

C₁₆

—

362

—

C₁₆

—

362

—

C₂³

C₂×#13089685

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#13089685

yes

—

C₂×C₄

C₂×#13089685

yes

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

362

—

C₈

—

362

—

C₈

—

181

—

C₈

—

181

—

C₂²

—

C₂²

—

C₂²

C₂²×#13089685

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#19153895

yes

characteristic

C₄

D₁₈₁⋊M₁₆

yes

characteristic

C₂

—

C₂

#26042295

yes

characteristic

100

C₂

#26042337

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#10624382
commutator, socle	C₂²×C₁₈₁	C₂²×C₈
Fitting	#117×C₁₈₁	C₄
prime radical	C₂³×C₁₈₁	C₂×C₈
cosocle	C₂×#13089678	C₂³
hypercenter	#117	#10624382
hypocenter	C₁₈₁	#2164
LCS	G ▷ C₂²×C₁₈₁ ▷ C₁₈₁
UCS	C₁ ◁ C₂×C₄ ◁ #117

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#10624382

commutator, socle

C₂²×C₁₈₁

C₂²×C₈

Fitting

#117×C₁₈₁

C₄

prime radical

C₂³×C₁₈₁

C₂×C₈

C₂³

C₁₈₁

G ▷ C₂²×C₁₈₁ ▷ C₁₈₁

C₁ ◁ C₂×C₄ ◁ #117

N	H ≅ G/N	type
1. C₄⋊#15668452	C₂	semidirect
3. #26042167	C₂	semidirect
4. C₄×#15668452	C₂	semidirect
50. C₁₈₁	#2164	semidirect

H ≅ G/N

type

1. C₄⋊#15668452

C₂

semidirect

3. #26042167

C₂

semidirect

4. C₄×#15668452

C₂

semidirect

50. C₁₈₁

#2164

semidirect

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

subgroup	index
C₄⋊#15668452	2
C₂²⋊#15668452	2
#26042167	2
C₄×#15668452	2
#26041235	2
#2164	181

subgroup

index

181

Subgroups — #37550086

Conjugacy classes of subgroups

Complements

Maximal subgroups