Group #38742617 — subgroups

Subgroups — #38742617

Order 25216.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#20263261	C₂	1	2	yes	maximal
2	M₁₆×Dic₁₉₇	C₂	1	2	yes	maximal
3	C₂×#20263236	C₂	1	1	yes	maximal, characteristic
4	#27160844	C₂	1	1	yes	maximal, characteristic
5	#341×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691342	C₂²	1	2	yes	—
7	C₂²×#13096802	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
9	#20263261	C₄	1	8	yes	complemented
10	#20263236	—	2	2	no	—
11	#20263236	C₄	1	4	yes	—
12	C₈×Dic₁₉₇	—	2	2	no	—
13	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₉₇	Q₈	1	2	yes	—
16	#15691342	—	2	2	no	—
17	#15691342	C₂×C₄	1	4	yes	—
18	C₂×#13096802	—	2	2	no	—
19	C₂×#13096802	C₂×C₄	1	2	yes	—
20	C₂×#13096802	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₉₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₉₇	D₄	1	2	yes	—
23	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₉₇	—	2	2	no	—
25	M₁₆×C₁₉₇	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₁₉₇	—	2	2	no	—
27	C₂×C₈×C₁₉₇	—	2	2	no	—
28	C₂×C₈×C₁₉₇	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₉₇	—	4	2	no	—
30	C₂×Dic₁₉₇	C₄⋊C₄	1	4	yes	—
31	#13096802	—	2	4	no	—
32	#13096802	C₄²	1	4	yes	—
33	C₂³×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₇	#41	1	2	yes	—
38	C₈×C₁₉₇	—	2	2	no	—
39	C₈×C₁₉₇	—	2	4	no	—
40	C₂²×C₁₉₇	—	2	2	no	—
41	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₉₇	C₄×Q₈	1	2	yes	—
43	C₄×C₁₉₇	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₁₉₇	#117	1	1	yes	characteristic
45	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
46	Dic₁₉₇	—	2	4	no	—
47	C₂×C₁₉₇	—	2	2	no	—
48	C₂×C₁₉₇	C₄×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₁₉₇	#430	1	2	yes	—
50	C₁₉₇	#1778	1	1	yes	characteristic, complemented, Sylow
51	#1778	—	197	1	no	maximal, Sylow
52	C₂×#108	—	197	2	no	—
53	C₄×M₁₆	—	197	2	no	—
54	C₂×#97	—	197	1	no	—
55	#341	—	197	1	no	—
56	#341	D₁₉₇	1	1	yes	characteristic
57	C₂×M₁₆	—	197	2	no	—
58	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
59	C₂²×C₈	—	197	1	no	—
60	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
61	C₂×C₄²	—	197	1	no	—
62	#108	—	197	8	no	—
63	#97	—	394	2	no	—
64	#97	—	197	4	no	—
65	C₄×C₈	—	394	2	no	—
66	C₂²×C₄	—	197	2	no	—
67	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	394	2	no	—
70	M₁₆	—	197	4	no	—
71	M₁₆	C₄×D₁₉₇	1	4	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	394	2	no	—
75	C₂×C₈	—	197	2	no	—
76	C₂×C₈	—	197	4	no	—
77	C₂×C₈	C₄×D₁₉₇	1	2	yes	—
78	C₄²	—	197	2	no	—
79	C₄²	—	197	2	no	—
80	C₂³	Q₈×D₁₉₇	1	1	yes	characteristic
81	C₂×C₄	—	788	2	no	—
82	C₂×C₄	—	197	4	no	—
83	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
85	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
86	C₂×C₄	#15691357	1	2	yes	—
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	—	394	4	no	—
90	C₈	—	197	4	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₄⋊C₄×D₁₉₇	1	1	yes	characteristic
93	C₂²	#20263291	1	2	yes	—
94	C₄	—	394	4	no	—
95	C₄	C₄²×D₁₉₇	1	1	yes	characteristic
96	C₄	#20263282	1	1	yes	characteristic
97	C₄	#20263286	1	2	yes	—
98	C₂	—	2	2	no	—
99	C₂	#27160930	1	1	yes	characteristic
100	C₂	#27161126	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263261

C₂

yes

maximal

M₁₆×Dic₁₉₇

C₂

yes

maximal

C₂×#20263236

C₂

yes

maximal, characteristic

#27160844

C₂

yes

maximal, characteristic

#341×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

C₂²×#13096802

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

#20263261

C₄

yes

complemented

#20263236

—

#20263236

C₄

yes

—

C₈×Dic₁₉₇

—

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂²×C₈×C₁₉₇

C₂²

yes

characteristic

C₂²×Dic₁₉₇

Q₈

yes

—

#15691342

—

#15691342

C₂×C₄

yes

—

C₂×#13096802

—

C₂×#13096802

C₂×C₄

yes

—

C₂×#13096802

C₂×C₄

yes

—

C₄×Dic₁₉₇

C₂×C₄

yes

—

C₄×Dic₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

C₂×C₄

yes

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

C₂×C₄

yes

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

C₄⋊C₄

yes

—

#13096802

—

#13096802

C₄²

yes

—

C₂³×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂²×C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₇

C₄×Q₈

yes

—

C₄×C₁₉₇

C₂×C₄²

yes

characteristic

C₄×C₁₉₇

#117

yes

characteristic

C₄×C₁₉₇

C₄×D₄

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₉₇

#430

yes

—

C₁₉₇

#1778

yes

characteristic, complemented, Sylow

#1778

—

197

maximal, Sylow

C₂×#108

—

197

—

C₄×M₁₆

—

197

—

C₂×#97

—

197

—

#341

—

197

—

#341

D₁₉₇

yes

characteristic

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

C₂²×C₈

—

197

—

C₂²×C₈

C₂×D₁₉₇

yes

characteristic

C₂×C₄²

—

197

—

#108

—

197

—

#97

—

394

—

#97

—

197

—

C₄×C₈

—

394

—

C₂²×C₄

—

197

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

394

—

M₁₆

—

197

—

M₁₆

C₄×D₁₉₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

394

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

C₂×C₈

C₄×D₁₉₇

yes

—

C₄²

—

197

—

C₄²

—

197

—

C₂³

Q₈×D₁₉₇

yes

characteristic

C₂×C₄

—

788

—

C₂×C₄

—

197

—

C₂×C₄

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₉₇

yes

—

C₂×C₄

D₄×D₁₉₇

yes

characteristic

C₂×C₄

#15691357

yes

—

C₈

—

C₈

—

C₈

—

394

—

C₈

—

197

—

C₂²

—

C₂²

C₄⋊C₄×D₁₉₇

yes

characteristic

C₂²

#20263291

yes

—

C₄

—

394

—

C₄

C₄²×D₁₉₇

yes

C₄

yes

C₄

yes

—

C₂

—

C₂

#27160930

yes

characteristic

100

C₂

#27161126

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₉₇
commutator	C₄×C₁₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#341×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₉₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#341	D₁₉₇
hypocenter	C₁₉₇	#1778
LCS	G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₉₇

commutator

C₄×C₁₉₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#341×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₉₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #20263261	C₄	semidirect
50. C₁₉₇	#1778	semidirect

H ≅ G/N

type

9. #20263261

C₄

semidirect

50. C₁₉₇

#1778

semidirect

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

subgroup	index
C₂×#20263261	2
M₁₆×Dic₁₉₇	2
C₂×#20263236	2
#27160844	2
#341×C₁₉₇	2
#1778	197

subgroup

index

M₁₆×Dic₁₉₇

#341×C₁₉₇

197

Subgroups — #38742617

Conjugacy classes of subgroups

Complements

Maximal subgroups