Group #38743705 — subgroups

Subgroups — #38743705

Order 25216.

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

normal subgroups only

This group has 228 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	C₂×#20263318	C₂	1	1	yes	maximal, characteristic
2	#27161103	C₂	1	1	yes	maximal, characteristic
3	#27160847	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#15691342	C₂²	1	2	yes	—
6	#20263318	—	2	1	no	—
7	#20263318	—	2	1	no	—
8	#20263318	—	2	2	no	—
9	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
10	C₂×#15691337	C₄	1	4	yes	—
11	C₄⋊#13096802	—	2	2	no	—
12	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
14	#97×C₁₉₇	—	2	1	no	—
15	C₄×C₈×C₁₉₇	—	2	1	no	—
16	C₂²×Dic₁₉₇	C₂×C₄	1	4	yes	—
17	#15691342	—	2	4	no	—
18	C₂×#13096802	—	2	2	no	—
19	C₄⋊Dic₁₉₇	—	2	2	no	—
20	C₄⋊Dic₁₉₇	—	2	2	no	—
21	#15691337	—	2	4	no	—
22	#15691337	—	2	4	no	—
23	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₉₇	—	2	2	no	—
26	M₁₆×C₁₉₇	Q₈	1	4	yes	—
27	C₂×C₈×C₁₉₇	—	2	2	no	—
28	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
29	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
30	C₄²×C₁₉₇	D₄	1	2	yes	—
31	C₂×Dic₁₉₇	—	2	4	no	—
32	C₂×Dic₁₉₇	—	4	4	no	—
33	#13096802	—	4	2	no	—
34	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₉₇	—	4	2	no	—
36	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₉₇	#41	1	2	yes	—
42	C₈×C₁₉₇	—	2	4	no	—
43	C₂²×C₁₉₇	—	2	2	no	—
44	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₉₇	C₄×Q₈	1	2	yes	—
46	C₄×C₁₉₇	—	2	2	no	—
47	C₄×C₁₉₇	—	2	2	no	—
48	C₄×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₁₉₇	#124	1	1	yes	characteristic
50	C₄×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₁₉₇	—	4	4	no	—
52	C₂×C₁₉₇	—	2	2	no	—
53	C₂×C₁₉₇	#391	1	1	yes	characteristic
54	C₂×C₁₉₇	#413	1	2	yes	—
55	C₁₉₇	#1998	1	1	yes	characteristic, complemented, Sylow
56	#1998	—	197	1	no	maximal, Sylow
57	C₂×#125	—	197	1	no	—
58	#423	—	197	1	no	—
59	C₄×M₁₆	D₁₉₇	1	1	yes	characteristic
60	#343	—	197	4	no	—
61	C₂×M₁₆	—	197	2	no	—
62	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
63	#125	—	394	1	no	—
64	#125	—	394	1	no	—
65	#125	—	394	2	no	—
66	C₂×C₄⋊C₄	—	197	4	no	—
67	C₂×C₄⋊C₄	—	197	2	no	—
68	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
69	C₄⋊C₈	—	394	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	—	197	4	no	—
73	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	394	4	no	—
77	M₁₆	Dic₃₉₄	1	4	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	394	2	no	—
80	C₄⋊C₄	—	394	4	no	—
81	C₄⋊C₄	—	394	4	no	—
82	C₄⋊C₄	—	394	2	no	—
83	C₄⋊C₄	—	394	2	no	—
84	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
85	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
86	C₄²	D₇₈₈	1	2	yes	—
87	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	788	4	no	—
90	C₂×C₄	—	394	4	no	—
91	C₂×C₄	#15691335	1	2	yes	—
92	C₂×C₄	#15691335	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
94	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
95	C₂×C₄	C₂×Dic₃₉₄	1	2	yes	—
96	C₂×C₄	#15691358	1	2	yes	—
97	C₈	—	2	4	no	—
98	C₈	—	788	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×#15691335	1	1	yes	characteristic
101	C₂²	C₄×Dic₃₉₄	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	788	4	no	—
105	C₄	#20263305	1	2	yes	—
106	C₄	#20263314	1	1	yes	characteristic
107	C₄	#20263326	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#27160985	1	1	yes	characteristic
110	C₂	#27161061	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263318

C₂

yes

maximal, characteristic

#27161103

C₂

yes

maximal, characteristic

#27160847

C₂

yes

maximal

C₄×M₁₆×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

#20263318

—

#20263318

—

#20263318

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

—

C₂×#15691337

C₄

yes

—

C₄⋊#13096802

—

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

#97×C₁₉₇

—

C₄×C₈×C₁₉₇

—

C₂²×Dic₁₉₇

C₂×C₄

yes

—

#15691342

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

#15691337

—

#15691337

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

Q₈

yes

—

C₂×C₈×C₁₉₇

—

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

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

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

C₄×Q₈

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂²⋊Q₈

yes

—

C₄×C₁₉₇

#124

yes

characteristic

C₄×C₁₉₇

C₄⋊Q₈

yes

characteristic

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#391

yes

characteristic

C₂×C₁₉₇

#413

yes

—

C₁₉₇

#1998

yes

characteristic, complemented, Sylow

#1998

—

197

maximal, Sylow

C₂×#125

—

197

—

#423

—

197

—

C₄×M₁₆

D₁₉₇

yes

characteristic

#343

—

197

—

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

#125

—

394

—

#125

—

394

—

#125

—

394

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

394

—

#97

—

C₄×C₈

—

C₂²×C₄

—

197

—

C₂²×C₄

C₄×D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

394

—

M₁₆

Dic₃₉₄

yes

—

C₂×C₈

—

C₂×C₈

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄²

C₂²⋊D₁₉₇

yes

characteristic

C₄²

C₂²⋊D₁₉₇

yes

characteristic

C₄²

D₇₈₈

yes

—

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

788

—

C₂×C₄

—

394

—

C₂×C₄

#15691335

yes

—

C₂×C₄

#15691335

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₂×C₄

C₂×D₇₈₈

yes

characteristic

C₂×C₄

C₂×Dic₃₉₄

yes

—

C₂×C₄

#15691358

yes

—

C₈

—

C₈

—

788

—

C₂²

—

100

C₂²

C₂×#15691335

yes

characteristic

101

C₂²

C₄×Dic₃₉₄

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

788

—

105

C₄

#20263305

yes

—

106

C₄

#20263314

yes

characteristic

107

C₄

#20263326

yes

characteristic

108

C₂

—

109

C₂

#27160985

yes

characteristic

110

C₂

#27161061

yes

—

	subgroup	G/H
center	C₂²	C₂×#15691335
commutator, prime radical	C₂³×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₄×M₁₆×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₄×M₁₆	D₁₉₇
hypocenter	C₁₉₇	#1998
LCS	G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#15691335

commutator, prime radical

C₂³×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₄×M₁₆×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

hypercenter

C₄×M₁₆

D₁₉₇

hypocenter

C₁₉₇

#1998

LCS

G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
55. C₁₉₇	#1998	semidirect

H ≅ G/N

type

55. C₁₉₇

#1998

semidirect

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

subgroup	index
C₂×#20263318	2
#27161103	2
#27160847	2
C₄×M₁₆×C₁₉₇	2
#1998	197

subgroup

index

C₄×M₁₆×C₁₉₇

197

Subgroups — #38743705

Conjugacy classes of subgroups

Complements

Maximal subgroups