Group #38742466 — subgroups

Subgroups — #38742466

Order 25216.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#27161037	C₂	1	1	yes	maximal, characteristic
2	#27161035	C₂	1	2	yes	maximal
3	#27160837	C₂	1	1	yes	maximal, characteristic
4	#27160835	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691342	C₂²	1	2	yes	—
7	C₂²×#13096802	C₂²	1	1	yes	characteristic
8	#20263285	C₂²	1	1	yes	characteristic
9	#20263237	C₄	1	4	yes	complemented
10	#20263236	C₄	1	2	yes	complemented
11	C₄×#13096802	C₄	1	4	yes	complemented
12	C₈×Dic₁₉₇	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
14	#117×C₁₉₇	C₂²	1	2	yes	—
15	#15691342	—	2	2	no	—
16	#15691342	—	2	2	no	—
17	C₂×#13096802	C₂×C₄	1	4	yes	—
18	C₂×#13096802	C₂×C₄	1	1	yes	characteristic
19	C₂×#13096802	C₂×C₄	1	2	yes	—
20	C₂×#13096802	C₂×C₄	1	1	yes	characteristic
21	C₂×#13096802	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₁₉₇	—	2	1	no	—
23	C₂²⋊Dic₁₉₇	—	2	1	no	—
24	C₄×Dic₁₉₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₉₇	—	2	2	no	—
27	C₂×C₈×C₁₉₇	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₁₉₇	—	2	2	no	—
29	C₂²⋊C₄×C₁₉₇	—	2	2	no	—
30	C₄²×C₁₉₇	C₂×C₄	1	4	yes	—
31	C₂×Dic₁₉₇	—	2	2	no	—
32	#13096802	—	2	4	no	—
33	#13096802	C₄²	1	1	yes	characteristic
34	#13096802	C₄²	1	1	yes	characteristic
35	#13096802	C₄²	1	2	yes	—
36	C₂³×C₁₉₇	#41	1	1	yes	characteristic
37	C₂×C₄×C₁₉₇	—	2	4	no	—
38	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₇	#41	1	2	yes	—
41	C₂×C₄×C₁₉₇	#41	1	1	yes	characteristic
42	C₈×C₁₉₇	—	2	2	no	—
43	C₂²×C₁₉₇	—	4	1	no	—
44	C₂²×C₁₉₇	#117	1	2	yes	—
45	C₂²×C₁₉₇	#117	1	1	yes	characteristic
46	C₄×C₁₉₇	—	4	4	no	—
47	C₄×C₁₉₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₉₇	#117	1	1	yes	characteristic
49	C₄×C₁₉₇	#117	1	2	yes	—
50	Dic₁₉₇	—	4	2	no	—
51	C₂×C₁₉₇	—	2	1	no	—
52	C₂×C₁₉₇	—	2	2	no	—
53	C₂×C₁₉₇	#376	1	1	yes	characteristic
54	C₁₉₇	#1747	1	1	yes	characteristic, complemented, Sylow
55	#1747	—	197	1	no	maximal, Sylow
56	#405	—	197	1	no	—
57	#405	—	197	2	no	—
58	#338	—	197	1	no	—
59	H(3,ℤ/4)	—	197	2	no	—
60	H(3,ℤ/4)	D₁₉₇	1	1	yes	characteristic
61	C₂×M₁₆	—	197	2	no	—
62	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
63	C₂²×C₈	—	197	1	no	—
64	#117	—	197	1	no	—
65	#117	C₂×D₁₉₇	1	2	yes	—
66	#97	—	197	2	no	—
67	#97	—	197	4	no	—
68	C₄×C₈	—	197	2	no	—
69	C₄×C₈	—	197	4	no	—
70	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	394	2	no	—
73	M₁₆	—	394	2	no	—
74	C₂×C₈	—	197	4	no	—
75	C₂×C₈	—	197	1	no	—
76	C₂×C₈	—	197	2	no	—
77	C₂×C₈	—	197	1	no	—
78	C₂×C₈	—	197	2	no	—
79	C₂×C₈	C₄×D₁₉₇	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	394	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	394	1	no	—
84	C₄²	—	197	2	no	—
85	C₄²	C₄×D₁₉₇	1	4	yes	—
86	C₂³	#15691357	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	394	2	no	—
89	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
91	C₂×C₄	#15691357	1	2	yes	—
92	C₂×C₄	#15691359	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	394	4	no	—
95	C₈	—	197	1	no	—
96	C₈	—	197	1	no	—
97	C₈	—	197	2	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#20263282	1	2	yes	—
100	C₂²	#20263284	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	788	2	no	—
103	C₄	C₄²×D₁₉₇	1	1	yes	characteristic
104	C₄	#20263282	1	1	yes	characteristic
105	C₄	#20263284	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#27160924	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161037

C₂

yes

maximal, characteristic

#27161035

C₂

yes

maximal

#27160837

C₂

yes

maximal, characteristic

#27160835

C₂

yes

maximal

H(3,ℤ/4)×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

C₂²×#13096802

C₂²

yes

characteristic

#20263285

C₂²

yes

characteristic

#20263237

C₄

yes

complemented

#20263236

C₄

yes

complemented

C₄×#13096802

C₄

yes

complemented

C₈×Dic₁₉₇

C₄

yes

complemented

C₂×M₁₆×C₁₉₇

C₂²

yes

characteristic

#117×C₁₉₇

C₂²

yes

—

#15691342

—

#15691342

—

C₂×#13096802

C₂×C₄

yes

—

C₂×#13096802

C₂×C₄

yes

characteristic

C₂×#13096802

C₂×C₄

yes

—

C₂×#13096802

C₂×C₄

yes

characteristic

C₂×#13096802

C₂×C₄

yes

—

C₄⋊Dic₁₉₇

—

C₂²⋊Dic₁₉₇

—

C₄×Dic₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₉₇

—

C₂²⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

C₂×C₄

yes

—

C₂×Dic₁₉₇

—

#13096802

—

#13096802

C₄²

yes

characteristic

#13096802

C₄²

yes

characteristic

#13096802

C₄²

yes

—

C₂³×C₁₉₇

#41

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

#41

yes

characteristic

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#117

yes

—

C₂²×C₁₉₇

#117

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂×C₄²

yes

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#376

yes

characteristic

C₁₉₇

#1747

yes

characteristic, complemented, Sylow

#1747

—

197

maximal, Sylow

#405

—

197

—

#405

—

197

—

#338

—

197

—

H(3,ℤ/4)

—

197

—

H(3,ℤ/4)

D₁₉₇

yes

characteristic

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

characteristic

C₂²×C₈

—

197

—

#117

—

197

—

#117

C₂×D₁₉₇

yes

—

#97

—

197

—

#97

—

197

—

C₄×C₈

—

197

—

C₄×C₈

—

197

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

394

—

M₁₆

—

394

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

C₂×C₈

C₄×D₁₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

394

—

C₂²⋊C₄

—

C₂²⋊C₄

—

394

—

C₄²

—

197

—

C₄²

C₄×D₁₉₇

yes

—

C₂³

#15691357

yes

characteristic

C₂×C₄

—

C₂×C₄

—

394

—

C₂×C₄

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₉₇

yes

—

C₂×C₄

#15691357

yes

—

C₂×C₄

#15691359

yes

characteristic

C₈

—

C₈

—

394

—

C₈

—

197

—

C₈

—

197

—

C₈

—

197

—

C₂²

—

C₂²

#20263282

yes

—

100

C₂²

#20263284

yes

characteristic

101

C₄

—

102

C₄

—

788

—

103

C₄

C₄²×D₁₉₇

yes

characteristic

104

C₄

#20263282

yes

characteristic

105

C₄

#20263284

yes

—

106

C₂

—

107

C₂

—

108

C₂

#27160924

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₁₉₇
commutator	C₄×C₁₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	H(3,ℤ/4)×C₁₉₇	C₂
socle	C₂×C₁₉₇	#376
prime radical	C₂³×C₁₉₇	#41
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	H(3,ℤ/4)	D₁₉₇
hypocenter	C₁₉₇	#1747
LCS	G ▷ C₄×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₁₉₇

commutator

C₄×C₁₉₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

H(3,ℤ/4)×C₁₉₇

C₂

C₂×C₁₉₇

C₂³×C₁₉₇

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₄×C₁₉₇ ▷ C₁₉₇

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #20263237	C₄	semidirect
10. #20263236	C₄	semidirect
11. C₄×#13096802	C₄	semidirect
12. C₈×Dic₁₉₇	C₄	semidirect
54. C₁₉₇	#1747	semidirect

H ≅ G/N

type

9. #20263237

C₄

semidirect

10. #20263236

C₄

semidirect

11. C₄×#13096802

C₄

semidirect

12. C₈×Dic₁₉₇

C₄

semidirect

54. C₁₉₇

#1747

semidirect

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

subgroup	index
#27161037	2
#27161035	2
#27160837	2
#27160835	2
H(3,ℤ/4)×C₁₉₇	2
#1747	197

subgroup

index

H(3,ℤ/4)×C₁₉₇

197

Subgroups — #38742466

Conjugacy classes of subgroups

Complements

Maximal subgroups