Group #38744224 — subgroups

Subgroups — #38744224

Order 25216.

176 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 71 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	#27161121	C₂	1	2	yes	maximal
2	C₂×#20263258	C₂	1	1	yes	maximal, characteristic
3	#27161105	C₂	1	1	yes	maximal, characteristic
4	#27160847	C₂	1	2	yes	maximal
5	#341×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691342	C₂²	1	1	yes	characteristic
7	#20263285	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
9	#20263260	—	2	2	no	—
10	#20263258	—	2	2	no	—
11	#20263258	—	2	2	no	—
12	#20263255	—	2	2	no	—
13	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₉₇	—	2	2	no	—
16	#15691342	—	2	2	no	—
17	C₂×#13096802	—	2	1	no	—
18	C₄⋊Dic₁₉₇	—	2	2	no	—
19	C₄⋊Dic₁₉₇	—	2	2	no	—
20	C₄⋊Dic₁₉₇	D₄	1	2	yes	—
21	C₂²⋊Dic₁₉₇	—	2	1	no	—
22	C₄×Dic₁₉₇	—	2	1	no	—
23	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₉₇	—	2	2	no	—
25	M₁₆×C₁₉₇	Q₈	1	4	yes	—
26	C₂×C₈×C₁₉₇	—	2	2	no	—
27	C₂×C₈×C₁₉₇	—	2	2	no	—
28	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
29	C₂×Dic₁₉₇	—	2	2	no	—
30	C₂×Dic₁₉₇	—	4	2	no	—
31	C₂×Dic₁₉₇	—	8	2	no	—
32	#13096802	—	4	1	no	—
33	C₂³×C₁₉₇	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₁₉₇	#41	1	2	yes	—
38	C₈×C₁₉₇	—	2	4	no	—
39	C₈×C₁₉₇	—	2	2	no	—
40	C₂²×C₁₉₇	—	2	2	no	—
41	C₂²×C₁₉₇	#121	1	1	yes	characteristic
42	C₂²×C₁₉₇	#125	1	2	yes	—
43	C₄×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
44	C₄×C₁₉₇	#123	1	1	yes	characteristic
45	C₄×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
46	Dic₁₉₇	—	4	2	no	—
47	Dic₁₉₇	—	8	2	no	—
48	C₂×C₁₉₇	—	2	2	no	—
49	C₂×C₁₉₇	#398	1	1	yes	characteristic
50	C₂×C₁₉₇	#471	1	2	yes	—
51	C₁₉₇	#2090	1	1	yes	characteristic, complemented, Sylow
52	#2090	—	197	1	no	maximal, Sylow
53	#428	—	197	2	no	—
54	C₂×#106	—	197	1	no	—
55	#424	—	197	1	no	—
56	#343	—	197	2	no	—
57	#341	D₁₉₇	1	1	yes	characteristic
58	C₂×M₁₆	—	197	1	no	—
59	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
60	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
61	#117	—	197	1	no	—
62	C₂×C₄⋊C₄	—	197	2	no	—
63	#107	—	394	2	no	—
64	#106	—	394	2	no	—
65	#106	—	394	2	no	—
66	C₄⋊C₈	—	394	2	no	—
67	C₂²×C₄	—	394	2	no	—
68	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	394	2	no	—
71	M₁₆	Dic₃₉₄	1	4	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	394	1	no	—
75	C₂×C₈	D₇₈₈	1	2	yes	—
76	C₄⋊C₄	—	394	2	no	—
77	C₄⋊C₄	—	394	2	no	—
78	C₄⋊C₄	—	197	2	no	—
79	C₂²⋊C₄	—	394	1	no	—
80	C₄²	—	394	1	no	—
81	C₂³	#15691359	1	1	yes	characteristic
82	C₂×C₄	—	788	2	no	—
83	C₂×C₄	—	1576	2	no	—
84	C₂×C₄	—	394	2	no	—
85	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
86	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
87	C₂×C₄	C₂×Dic₃₉₄	1	2	yes	—
88	C₂×C₄	#15691357	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	—	2	2	no	—
91	C₈	—	788	1	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#20263301	1	1	yes	characteristic
94	C₂²	#20263317	1	2	yes	—
95	C₄	—	788	2	no	—
96	C₄	—	1576	2	no	—
97	C₄	#20263303	1	2	yes	—
98	C₄	#20263310	1	1	yes	characteristic
99	C₄	#20263326	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#27161011	1	1	yes	characteristic
102	C₂	#27161343	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27161121

C₂

yes

maximal

C₂×#20263258

C₂

yes

maximal, characteristic

#27161105

C₂

yes

maximal, characteristic

#27160847

C₂

yes

maximal

#341×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

characteristic

#20263285

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₉₇

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂²×C₈×C₁₉₇

C₂²

yes

characteristic

C₂²×Dic₁₉₇

—

#15691342

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

D₄

yes

—

C₂²⋊Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

Q₈

yes

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

D₄

yes

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

#41

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#121

yes

characteristic

C₂²×C₁₉₇

#125

yes

—

C₄×C₁₉₇

C₂²⋊Q₈

yes

—

C₄×C₁₉₇

#123

yes

characteristic

C₄×C₁₉₇

C₄⋊Q₈

yes

characteristic

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#398

yes

characteristic

C₂×C₁₉₇

#471

yes

—

C₁₉₇

#2090

yes

characteristic, complemented, Sylow

#2090

—

197

maximal, Sylow

#428

—

197

—

C₂×#106

—

197

—

#424

—

197

—

#343

—

197

—

#341

D₁₉₇

yes

characteristic

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

C₂²×C₈

C₂×D₁₉₇

yes

characteristic

#117

—

197

—

C₂×C₄⋊C₄

—

197

—

#107

—

394

—

#106

—

394

—

#106

—

394

—

C₄⋊C₈

—

394

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

394

—

M₁₆

Dic₃₉₄

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

394

—

C₂×C₈

D₇₈₈

yes

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

197

—

C₂²⋊C₄

—

394

—

C₄²

—

394

—

C₂³

#15691359

yes

characteristic

C₂×C₄

—

788

—

C₂×C₄

—

1576

—

C₂×C₄

—

394

—

C₂×C₄

D₄×D₁₉₇

yes

characteristic

C₂×C₄

C₂×D₇₈₈

yes

characteristic

C₂×C₄

C₂×Dic₃₉₄

yes

—

C₂×C₄

#15691357

yes

—

C₈

—

C₈

—

C₈

—

788

—

C₂²

—

C₂²

#20263301

yes

characteristic

C₂²

#20263317

yes

—

C₄

—

788

—

C₄

—

1576

—

C₄

#20263303

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#27161011

yes

characteristic

102

C₂

#27161343

yes

—

	subgroup	G/H
center	C₂²	#20263301
commutator, cosocle	C₂²×C₄×C₁₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#341×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#121
prime radical	C₂³×C₁₉₇	#41
hypercenter	#341	D₁₉₇
hypocenter	C₁₉₇	#2090
LCS	G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂²

#20263301

commutator, cosocle

C₂²×C₄×C₁₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#341×C₁₉₇

C₂

C₂²×C₁₉₇

C₂³×C₁₉₇

D₁₉₇

C₁₉₇

G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
51. C₁₉₇	#2090	semidirect

H ≅ G/N

type

51. C₁₉₇

#2090

semidirect

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

subgroup	index
#27161121	2
C₂×#20263258	2
#27161105	2
#27160847	2
#341×C₁₉₇	2
#2090	197

subgroup

index

#341×C₁₉₇

197

Subgroups — #38744224

Conjugacy classes of subgroups

Complements

Maximal subgroups