Group #37508328 — subgroups

Subgroups — #37508328

Order 22912.

172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 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	#26038849	C₂	1	2	yes	maximal
2	#26038809	C₂	1	1	yes	maximal, characteristic
3	#26038181	C₂	1	1	yes	maximal, characteristic
4	#26038181	C₂	1	2	yes	maximal
5	#564×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	#19153113	C₂²	1	1	yes	characteristic
7	#19153046	—	2	2	no	—
8	#19153044	—	2	2	no	—
9	#19153036	—	2	2	no	—
10	#19153024	C₂²	1	1	yes	characteristic
11	#19153024	C₂²	1	2	yes	—
12	#125×C₁₇₉	—	2	2	no	—
13	#125×C₁₇₉	—	2	2	no	—
14	#125×C₁₇₉	—	2	2	no	—
15	#125×C₁₇₉	C₂²	1	1	yes	characteristic
16	#125×C₁₇₉	C₂²	1	2	yes	—
17	C₂×Dic₃₅₈	—	2	1	no	—
18	C₂×C₁₇₉⋊C₈	—	2	2	no	—
19	C₂×C₁₇₉⋊C₈	—	2	1	no	—
20	C₄⋊Dic₁₇₉	—	2	2	no	—
21	C₄⋊C₄×C₁₇₉	—	2	2	no	—
22	C₄⋊C₄×C₁₇₉	—	2	2	no	—
23	C₄⋊C₄×C₁₇₉	—	2	2	no	—
24	C₄⋊C₄×C₁₇₉	—	2	2	no	—
25	C₄⋊C₄×C₁₇₉	—	2	2	no	—
26	C₄⋊C₄×C₁₇₉	—	2	2	no	—
27	C₄⋊C₄×C₁₇₉	D₄	1	2	yes	—
28	C₄⋊C₄×C₁₇₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₇₉	D₄	1	2	yes	—
30	C₄²×C₁₇₉	—	2	2	no	—
31	C₄²×C₁₇₉	C₂³	1	1	yes	characteristic
32	Dic₃₅₈	—	4	2	no	—
33	C₂×Dic₁₇₉	—	4	1	no	—
34	C₁₇₉⋊C₈	—	2	4	no	—
35	C₁₇₉⋊C₈	—	4	1	no	—
36	C₂×C₄×C₁₇₉	—	2	2	no	—
37	C₂×C₄×C₁₇₉	—	2	2	no	—
38	C₂×C₄×C₁₇₉	—	2	2	no	—
39	C₂×C₄×C₁₇₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₁₇₉	#120	1	1	yes	characteristic
42	C₄×C₁₇₉	—	2	1	no	—
43	C₄×C₁₇₉	—	2	2	no	—
44	C₄×C₁₇₉	—	4	2	no	—
45	C₄×C₁₇₉	—	4	2	no	—
46	C₄×C₁₇₉	—	4	2	no	—
47	Dic₁₇₉	—	8	1	no	—
48	C₂×C₁₇₉	#454	1	1	yes	characteristic
49	C₂×C₁₇₉	#456	1	2	yes	—
50	C₁₇₉	#1663	1	1	yes	characteristic, complemented, Sylow
51	#1663	—	179	1	no	maximal, Sylow
52	#564	D₁₇₉	1	1	yes	characteristic
53	#501	—	179	2	no	—
54	#491	—	179	1	no	—
55	#330	—	179	2	no	—
56	#330	—	179	1	no	—
57	C₄⋊Q₈	—	179	1	no	—
58	#125	—	2	2	no	—
59	#125	—	2	2	no	—
60	#125	—	2	2	no	—
61	#125	C₂×D₁₇₉	1	1	yes	characteristic
62	#125	C₂×D₁₇₉	1	2	yes	—
63	#107	—	358	2	no	—
64	#106	—	358	2	no	—
65	#103	—	358	2	no	—
66	#97	—	179	1	no	—
67	#97	—	179	2	no	—
68	C₂×Q₈	—	358	1	no	—
69	C₂×C₈	—	358	2	no	—
70	C₂×C₈	—	358	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	358	2	no	—
78	C₄⋊C₄	C₂²⋊D₁₇₉	1	2	yes	—
79	C₄⋊C₄	C₂²⋊D₁₇₉	1	2	yes	—
80	C₄⋊C₄	C₂²⋊D₁₇₉	1	2	yes	—
81	C₄²	—	2	2	no	—
82	C₄²	C₂²×D₁₇₉	1	1	yes	characteristic
83	Q₈	—	716	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	716	1	no	—
88	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	2	yes	—
89	C₂×C₄	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
90	C₈	—	358	4	no	—
91	C₈	—	716	1	no	—
92	C₂²	#19153082	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	4	2	no	—
97	C₄	—	4	2	no	—
98	C₄	—	1432	1	no	—
99	C₂	#26038612	1	1	yes	characteristic
100	C₂	#26038622	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26038849

C₂

yes

maximal

#26038809

C₂

yes

maximal, characteristic

#26038181

C₂

yes

maximal, characteristic

#26038181

C₂

yes

maximal

#564×C₁₇₉

C₂

yes

maximal, characteristic

#19153113

C₂²

yes

—

—

—

C₂²

yes

characteristic

#19153024

C₂²

yes

—

#125×C₁₇₉

—

#125×C₁₇₉

—

#125×C₁₇₉

—

#125×C₁₇₉

C₂²

yes

characteristic

#125×C₁₇₉

C₂²

yes

—

C₂×Dic₃₅₈

—

C₂×C₁₇₉⋊C₈

—

C₂×C₁₇₉⋊C₈

—

C₄⋊Dic₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

—

C₄⋊C₄×C₁₇₉

D₄

yes

—

C₄⋊C₄×C₁₇₉

D₄

yes

—

C₄⋊C₄×C₁₇₉

D₄

yes

—

C₄²×C₁₇₉

—

C₄²×C₁₇₉

C₂³

yes

characteristic

Dic₃₅₈

—

C₂×Dic₁₇₉

—

C₁₇₉⋊C₈

—

C₁₇₉⋊C₈

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

—

C₂×C₄×C₁₇₉

C₂×D₄

yes

characteristic

C₂²×C₁₇₉

#120

yes

characteristic

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

C₄×C₁₇₉

—

Dic₁₇₉

—

C₂×C₁₇₉

#454

yes

characteristic

C₂×C₁₇₉

#456

yes

—

C₁₇₉

#1663

yes

characteristic, complemented, Sylow

#1663

—

179

maximal, Sylow

#564

D₁₇₉

yes

characteristic

#501

—

179

—

#491

—

179

—

#330

—

179

—

#330

—

179

—

C₄⋊Q₈

—

179

—

—

—

—

C₂×D₁₇₉

yes

characteristic

#125

C₂×D₁₇₉

yes

—

#107

—

358

—

#106

—

358

—

#103

—

358

—

#97

—

179

—

#97

—

179

—

C₂×Q₈

—

358

—

C₂×C₈

—

358

—

C₂×C₈

—

358

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

358

—

C₄⋊C₄

C₂²⋊D₁₇₉

yes

—

C₄⋊C₄

C₂²⋊D₁₇₉

yes

—

C₄⋊C₄

C₂²⋊D₁₇₉

yes

—

C₄²

—

C₄²

C₂²×D₁₇₉

yes

characteristic

Q₈

—

716

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

716

—

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₇₉

yes

characteristic

C₈

—

358

—

C₈

—

716

—

C₂²

#19153082

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

1432

—

C₂

#26038612

yes

characteristic

100

C₂

#26038622

yes

—

	subgroup	G/H
center	C₂²	#19153082
commutator, cosocle	C₄²×C₁₇₉	C₂³
Frattini	C₄²	C₂²×D₁₇₉
Fitting	#564×C₁₇₉	C₂
socle, prime radical	C₂²×C₁₇₉	#120
hypercenter	#564	D₁₇₉
hypocenter	C₁₇₉	#1663
LCS	G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #564

subgroup

G/H

center

C₂²

#19153082

commutator, cosocle

C₄²×C₁₇₉

C₂³

Frattini

C₄²

C₂²×D₁₇₉

Fitting

#564×C₁₇₉

C₂

C₂²×C₁₇₉

D₁₇₉

C₁₇₉

G ▷ C₄²×C₁₇₉ ▷ C₂²×C₁₇₉ ▷ C₁₇₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #564

N	H ≅ G/N	type
50. C₁₇₉	#1663	semidirect

H ≅ G/N

type

50. C₁₇₉

#1663

semidirect

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

subgroup	index
#26038849	2
#26038809	2
#26038181	2
#564×C₁₇₉	2
#1663	179

subgroup

index

#564×C₁₇₉

179

Subgroups — #37508328

Conjugacy classes of subgroups

Complements

Maximal subgroups