Group #38742338 — subgroups

Subgroups — #38742338

Order 25216.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#27161477	C₂	1	1	yes	maximal, characteristic, complemented
2	#27161451	C₂	1	2	yes	maximal, complemented
3	#27161425	C₂	1	2	yes	maximal, complemented
4	#27160800	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691346	—	2	4	no	—
7	#20263326	C₂²	1	2	yes	—
8	#20263258	—	2	2	no	—
9	#20263247	—	2	4	no	—
10	C₄×#13096802	C₂²	1	2	yes	—
11	C₂×D₈×C₁₉₇	—	2	2	no	—
12	#127×C₁₉₇	C₂²	1	1	yes	characteristic
13	#127×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₉₄	—	2	2	no	—
16	#15691346	—	4	8	no	—
17	C₂×#13096802	D₄	1	4	yes	—
18	C₄⋊Dic₁₉₇	—	2	2	no	—
19	C₄⋊Dic₁₉₇	—	2	2	no	—
20	C₂×D₄×C₁₉₇	—	2	1	no	—
21	C₂×D₄×C₁₉₇	—	2	1	no	—
22	C₂×D₄×C₁₉₇	—	2	2	no	—
23	C₂×D₄×C₁₉₇	—	2	2	no	—
24	D₈×C₁₉₇	—	4	4	no	—
25	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
26	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
27	Dic₃₉₄	—	4	4	no	—
28	C₂×Dic₁₉₇	—	4	2	no	—
29	#13096802	—	2	4	no	—
30	C₂³×C₁₉₇	—	4	1	no	—
31	C₂³×C₁₉₇	—	4	1	no	—
32	D₄×C₁₉₇	—	4	2	no	—
33	D₄×C₁₉₇	—	4	2	no	—
34	D₄×C₁₉₇	—	4	4	no	—
35	D₄×C₁₉₇	—	4	4	no	—
36	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₉₇	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₉₇	—	8	1	no	—
40	C₂²×C₁₉₇	—	8	1	no	—
41	C₂²×C₁₉₇	—	8	2	no	—
42	C₂²×C₁₉₇	—	8	2	no	—
43	C₂²×C₁₉₇	#127	1	1	yes	characteristic
44	C₄×C₁₉₇	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₉₇	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₁₉₇	—	8	2	no	—
47	C₂×C₁₉₇	—	16	1	no	—
48	C₂×C₁₉₇	—	16	1	no	—
49	C₂×C₁₉₇	#492	1	1	yes	characteristic
50	C₂×C₁₉₇	#496	1	2	yes	—
51	C₁₉₇	#1715	1	1	yes	characteristic, complemented, Sylow
52	#1715	—	197	1	no	maximal, Sylow
53	#498	—	197	1	no	—
54	#493	D₁₉₇	1	1	yes	characteristic
55	#492	—	197	2	no	—
56	#486	—	197	2	no	—
57	#322	—	197	1	no	—
58	C₂×SD₁₆	—	394	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	197	2	no	—
61	#127	C₂×D₁₉₇	1	1	yes	characteristic
62	#127	C₂×D₁₉₇	1	1	yes	characteristic
63	#106	—	394	2	no	—
64	#102	—	394	4	no	—
65	C₄×C₈	—	197	2	no	—
66	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
67	C₂×Q₈	—	394	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	1	no	—
72	SD₁₆	—	788	8	no	—
73	D₈	—	4	4	no	—
74	C₂×C₈	—	197	4	no	—
75	C₂×C₈	C₂²⋊D₁₉₇	1	2	yes	—
76	C₄⋊C₄	—	394	2	no	—
77	C₄⋊C₄	—	394	2	no	—
78	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
79	C₂³	—	4	1	no	—
80	C₂³	—	4	1	no	—
81	Q₈	—	788	4	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	4	no	—
85	D₄	—	4	4	no	—
86	C₂×C₄	—	788	2	no	—
87	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
89	C₈	—	394	4	no	—
90	C₈	#15691346	1	4	yes	complemented
91	C₂²	—	8	1	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	8	2	no	—
94	C₂²	—	8	2	no	—
95	C₂²	#20263322	1	1	yes	characteristic
96	C₄	—	1576	2	no	—
97	C₄	C₂×#15691346	1	2	yes	—
98	C₄	#20263350	1	4	yes	—
99	C₂	—	16	1	no	—
100	C₂	—	16	1	no	—
101	C₂	#27161451	1	1	yes	characteristic
102	C₂	#27161467	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27161477

C₂

yes

maximal, characteristic, complemented

#27161451

C₂

yes

maximal, complemented

#27161425

C₂

yes

maximal, complemented

#27160800

C₂

yes

maximal, characteristic, complemented

#493×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691346

—

#20263326

C₂²

yes

—

#20263258

—

#20263247

—

C₄×#13096802

C₂²

yes

—

C₂×D₈×C₁₉₇

—

#127×C₁₉₇

C₂²

yes

characteristic

#127×C₁₉₇

C₂²

yes

characteristic

C₄×C₈×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

—

#15691346

—

C₂×#13096802

D₄

yes

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

—

D₈×C₁₉₇

—

C₂×C₈×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

C₂³

yes

characteristic

Dic₃₉₄

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

—

C₂³×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₈×C₁₉₇

SD₁₆

yes

complemented

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#127

yes

characteristic

C₄×C₁₉₇

C₂×SD₁₆

yes

—

C₄×C₁₉₇

AGL(1,ℤ/8)

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#492

yes

characteristic

C₂×C₁₉₇

#496

yes

—

C₁₉₇

#1715

yes

characteristic, complemented, Sylow

#1715

—

197

maximal, Sylow

#498

—

197

—

#493

D₁₉₇

yes

characteristic

#492

—

197

—

#486

—

197

—

#322

—

197

—

C₂×SD₁₆

—

394

—

C₂×D₈

—

C₄⋊Q₈

—

197

—

#127

C₂×D₁₉₇

yes

characteristic

#127

C₂×D₁₉₇

yes

characteristic

#106

—

394

—

#102

—

394

—

C₄×C₈

—

197

—

C₄×C₈

C₂×D₁₉₇

yes

characteristic

C₂×Q₈

—

394

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

788

—

D₈

—

C₂×C₈

—

197

—

C₂×C₈

C₂²⋊D₁₉₇

yes

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄²

C₂²×D₁₉₇

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

788

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

788

—

C₂×C₄

D₄×D₁₉₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₈

—

394

—

C₈

#15691346

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#20263322

yes

characteristic

C₄

—

1576

—

C₄

C₂×#15691346

yes

—

C₄

#20263350

yes

—

C₂

—

100

C₂

—

101

C₂

#27161451

yes

characteristic

102

C₂

#27161467

yes

—

	subgroup	G/H
center	C₂²	#20263322
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting, prime radical	#493×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#127
hypercenter	#493	D₁₉₇
hypocenter	C₁₉₇	#1715
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#20263322

commutator, cosocle

C₄²×C₁₉₇

C₂³

Frattini

C₄²

C₂²×D₁₉₇

Fitting, prime radical

#493×C₁₉₇

C₂

C₂²×C₁₉₇

D₁₉₇

C₁₉₇

G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #27161477	C₂	semidirect
2. #27161451	C₂	semidirect
3. #27161425	C₂	semidirect
4. #27160800	C₂	semidirect
38. C₈×C₁₉₇	SD₁₆	semidirect
51. C₁₉₇	#1715	semidirect
90. C₈	#15691346	semidirect

H ≅ G/N

type

1. #27161477

C₂

semidirect

2. #27161451

C₂

semidirect

3. #27161425

C₂

semidirect

4. #27160800

C₂

semidirect

38. C₈×C₁₉₇

SD₁₆

semidirect

51. C₁₉₇

#1715

semidirect

90. C₈

#15691346

semidirect

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

subgroup	index
#27161477	2
#27161451	2
#27161425	2
#27160800	2
#493×C₁₉₇	2
#1715	197

subgroup

index

#493×C₁₉₇

197

Subgroups — #38742338

Conjugacy classes of subgroups

Complements

Maximal subgroups