Group #38743689 — subgroups

Subgroups — #38743689

Order 25216.

284 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 101 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	#27161636	C₂	1	1	yes	maximal, characteristic
2	#27161103	C₂	1	1	yes	maximal, characteristic
3	#27160839	C₂	1	4	yes	maximal
4	C₄³×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#15691342	C₂²	1	2	yes	—
6	#20263326	C₄	1	2	yes	complemented
7	#20263318	C₄	1	2	yes	complemented
8	#20263305	—	2	2	no	—
9	C₄×Dic₃₉₄	—	2	2	no	—
10	#20263285	C₂²	1	2	yes	—
11	C₄⋊#13096802	—	2	2	no	—
12	C₂×C₄²×C₁₉₇	—	2	2	no	—
13	C₂×C₄²×C₁₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₉₄	—	2	2	no	—
16	#15691342	—	2	4	no	—
17	C₂×#13096802	—	2	2	no	—
18	C₄⋊Dic₁₉₇	—	2	2	no	—
19	C₄⋊Dic₁₉₇	C₂×C₄	1	4	yes	—
20	#15691337	—	2	4	no	—
21	C₂²⋊Dic₁₉₇	—	2	2	no	—
22	C₄×Dic₁₉₇	—	2	2	no	—
23	C₂²×C₄×C₁₉₇	—	2	2	no	—
24	C₂²×C₄×C₁₉₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
26	C₄²×C₁₉₇	—	2	8	no	—
27	C₄²×C₁₉₇	—	2	2	no	—
28	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₉₇	D₄	1	2	yes	—
31	C₄²×C₁₉₇	Q₈	1	4	yes	complemented
32	Dic₃₉₄	—	4	4	no	—
33	C₂×Dic₁₉₇	—	2	4	no	—
34	#13096802	—	4	2	no	—
35	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₉₇	—	2	8	no	—
37	C₂×C₄×C₁₉₇	—	2	4	no	—
38	C₂×C₄×C₁₉₇	—	2	4	no	—
39	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₁₉₇	#41	1	2	yes	—
45	C₂²×C₁₉₇	—	2	2	no	—
46	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₁₉₇	—	2	8	no	—
49	C₄×C₁₉₇	—	2	2	no	—
50	C₄×C₁₉₇	#104	1	4	yes	—
51	C₄×C₁₉₇	C₄×Q₈	1	2	yes	—
52	C₄×C₁₉₇	#124	1	1	yes	characteristic
53	C₄×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₁₉₇	—	4	4	no	—
55	C₂×C₁₉₇	—	2	2	no	—
56	C₂×C₁₉₇	#391	1	1	yes	characteristic
57	C₂×C₁₉₇	C₂×#104	1	2	yes	—
58	C₁₉₇	#1994	1	1	yes	characteristic, complemented, Sylow
59	#1994	—	197	1	no	maximal, Sylow
60	#533	—	197	1	no	—
61	#423	—	197	1	no	—
62	C₄³	D₁₉₇	1	1	yes	characteristic
63	#339	—	197	4	no	—
64	C₂×M₁₆	—	197	2	no	—
65	C₄⋊Q₈	—	197	2	no	—
66	#125	—	197	2	no	—
67	C₂²⋊Q₈	—	394	2	no	—
68	C₄×Q₈	—	394	2	no	—
69	#117	—	197	2	no	—
70	C₂×C₄²	—	2	2	no	—
71	C₂×C₄²	C₂×D₁₉₇	1	2	yes	—
72	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
73	C₄⋊C₈	—	394	2	no	—
74	C₂×Q₈	—	394	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²⋊D₁₉₇	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
78	M₁₆	—	394	4	no	—
79	C₂×C₈	—	394	2	no	—
80	C₄⋊C₄	—	394	4	no	—
81	C₄⋊C₄	—	394	2	no	—
82	C₄⋊C₄	—	197	4	no	—
83	C₂²⋊C₄	—	394	2	no	—
84	C₄²	—	2	8	no	—
85	C₄²	—	2	2	no	—
86	C₄²	—	394	2	no	—
87	C₄²	C₄×D₁₉₇	1	1	yes	characteristic
88	C₄²	C₄×D₁₉₇	1	1	yes	characteristic
89	C₄²	D₇₈₈	1	2	yes	—
90	C₄²	Dic₃₉₄	1	4	yes	complemented
91	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
92	Q₈	—	788	4	no	—
93	C₂×C₄	—	2	8	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	394	4	no	—
97	C₂×C₄	#15691335	1	2	yes	—
98	C₂×C₄	#15691335	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
100	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
101	C₂×C₄	C₂×Dic₃₉₄	1	2	yes	—
102	C₂×C₄	#15691358	1	2	yes	—
103	C₈	—	788	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×#15691335	1	1	yes	characteristic
106	C₂²	#20263305	1	2	yes	—
107	C₄	—	2	8	no	—
108	C₄	—	2	2	no	—
109	C₄	—	788	4	no	—
110	C₄	#20263254	1	4	yes	—
111	C₄	C₄×Dic₃₉₄	1	2	yes	—
112	C₄	#20263314	1	1	yes	characteristic
113	C₄	#20263326	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#27160985	1	1	yes	characteristic
116	C₂	C₂×#20263254	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27161636

C₂

yes

maximal, characteristic

#27161103

C₂

yes

maximal, characteristic

#27160839

C₂

yes

maximal

C₄³×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

#20263326

C₄

yes

complemented

#20263318

C₄

yes

complemented

#20263305

—

C₄×Dic₃₉₄

—

#20263285

C₂²

yes

—

C₄⋊#13096802

—

C₂×C₄²×C₁₉₇

—

C₂×C₄²×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

—

#15691342

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

C₂×C₄

yes

—

#15691337

—

C₂²⋊Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄²×C₁₉₇

—

C₄²×C₁₉₇

—

C₄²×C₁₉₇

C₂×C₄

yes

characteristic

C₄²×C₁₉₇

C₂×C₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

Q₈

yes

complemented

Dic₃₉₄

—

C₂×Dic₁₉₇

—

#13096802

—

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

C₂²⋊Q₈

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#104

yes

—

C₄×C₁₉₇

C₄×Q₈

yes

—

C₄×C₁₉₇

#124

yes

characteristic

C₄×C₁₉₇

C₄⋊Q₈

yes

characteristic

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#391

yes

characteristic

C₂×C₁₉₇

C₂×#104

yes

—

C₁₉₇

#1994

yes

characteristic, complemented, Sylow

#1994

—

197

maximal, Sylow

#533

—

197

—

#423

—

197

—

C₄³

D₁₉₇

yes

characteristic

#339

—

197

—

C₂×M₁₆

—

197

—

C₄⋊Q₈

—

197

—

#125

—

197

—

C₂²⋊Q₈

—

394

—

C₄×Q₈

—

394

—

#117

—

197

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₉₇

yes

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

394

—

C₂×Q₈

—

394

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

394

—

C₂×C₈

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

197

—

C₂²⋊C₄

—

394

—

C₄²

—

C₄²

—

C₄²

—

394

—

C₄²

C₄×D₁₉₇

yes

characteristic

C₄²

C₄×D₁₉₇

yes

characteristic

C₄²

D₇₈₈

yes

—

C₄²

Dic₃₉₄

yes

complemented

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

Q₈

—

788

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

394

—

C₂×C₄

#15691335

yes

—

C₂×C₄

#15691335

yes

—

C₂×C₄

C₂×C₄×D₁₉₇

yes

characteristic

100

C₂×C₄

C₂×D₇₈₈

yes

characteristic

101

C₂×C₄

C₂×Dic₃₉₄

yes

—

102

C₂×C₄

#15691358

yes

—

103

C₈

—

788

—

104

C₂²

—

105

C₂²

C₂×#15691335

yes

characteristic

106

C₂²

#20263305

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

788

—

110

C₄

#20263254

yes

—

111

C₄

C₄×Dic₃₉₄

yes

—

112

C₄

#20263314

yes

characteristic

113

C₄

#20263326

yes

characteristic

114

C₂

—

115

C₂

#27160985

yes

characteristic

116

C₂

C₂×#20263254

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₈₈
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₄³×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₉₇	C₂×D₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₄³	D₁₉₇
hypocenter	C₁₉₇	#1994
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

subgroup

G/H

center

C₂×C₄

C₂×D₇₈₈

commutator

C₂×C₄×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₄³×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₉₇

C₂×D₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

hypercenter

C₄³

D₁₉₇

hypocenter

C₁₉₇

#1994

LCS

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄³

N	H ≅ G/N	type
6. #20263326	C₄	semidirect
7. #20263318	C₄	semidirect
31. C₄²×C₁₉₇	Q₈	semidirect
58. C₁₉₇	#1994	semidirect
90. C₄²	Dic₃₉₄	semidirect

H ≅ G/N

type

6. #20263326

C₄

semidirect

7. #20263318

C₄

semidirect

31. C₄²×C₁₉₇

Q₈

semidirect

58. C₁₉₇

#1994

semidirect

90. C₄²

Dic₃₉₄

semidirect

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

subgroup	index
#27161636	2
#27161103	2
#27160839	2
C₄³×C₁₉₇	2
#1994	197

subgroup

index

C₄³×C₁₉₇

197

Subgroups — #38743689

Conjugacy classes of subgroups

Complements

Maximal subgroups