Group #38744264 — subgroups

Subgroups — #38744264

Order 25216.

210 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 63 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	#27161107	C₂	1	2	yes	maximal, complemented
2	#27161095	C₂	1	2	yes	maximal
3	#27161047	C₂	1	1	yes	maximal, characteristic
4	#27160836	C₂	1	1	yes	maximal, characteristic, complemented
5	#430×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691358	C₂²	1	1	yes	characteristic
7	C₂²×#13096802	C₂²	1	1	yes	characteristic
8	#20263285	C₂²	1	2	yes	—
9	#20263255	—	2	4	no	—
10	#20263252	—	2	4	no	—
11	#20263238	—	2	2	no	—
12	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
13	C₂×M₁₆×C₁₉₇	C₂²	1	1	yes	characteristic
14	#108×C₁₉₇	—	2	2	no	—
15	#15691358	—	2	2	no	—
16	#15691358	—	2	2	no	—
17	C₂×Dic₃₉₄	Q₈	1	1	yes	characteristic
18	C₂×D₇₈₈	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₉₇	—	2	1	no	—
20	C₂×C₄×D₁₉₇	—	2	1	no	—
21	C₂×#13096802	—	2	1	no	—
22	C₂×#13096802	—	2	2	no	—
23	C₄⋊Dic₁₉₇	—	2	2	no	—
24	C₂²⋊Dic₁₉₇	—	2	2	no	—
25	C₄×Dic₁₉₇	D₄	1	4	yes	—
26	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₉₇	—	2	2	no	—
28	M₁₆×C₁₉₇	—	2	2	no	—
29	M₁₆×C₁₉₇	—	2	2	no	—
30	C₂×C₈×C₁₉₇	—	2	2	no	—
31	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
32	C₂²×D₁₉₇	—	2	1	no	—
33	Dic₃₉₄	—	2	2	no	—
34	D₇₈₈	—	2	2	no	—
35	C₂²⋊D₁₉₇	—	4	2	no	—
36	C₂×Dic₁₉₇	—	2	1	no	—
37	C₂×Dic₁₉₇	—	2	4	no	—
38	C₄×D₁₉₇	—	4	1	no	—
39	C₄×D₁₉₇	—	4	1	no	—
40	#13096802	—	4	1	no	—
41	C₂³×C₁₉₇	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₉₇	#41	1	2	yes	—
46	C₈×C₁₉₇	—	2	2	no	—
47	C₈×C₁₉₇	—	4	2	no	—
48	C₂²×C₁₉₇	—	4	1	no	—
49	C₂²×C₁₉₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₉₇	#123	1	2	yes	—
51	C₂×D₁₉₇	—	4	2	no	—
52	C₂×D₁₉₇	—	4	1	no	—
53	C₄×C₁₉₇	C₂²⋊Q₈	1	2	yes	—
54	C₄×C₁₉₇	#123	1	1	yes	characteristic
55	C₄×C₁₉₇	#127	1	1	yes	characteristic
56	Dic₁₉₇	—	4	4	no	—
57	Dic₁₉₇	—	4	1	no	—
58	C₂×C₁₉₇	—	2	2	no	—
59	C₂×C₁₉₇	—	2	1	no	—
60	C₂×C₁₉₇	#399	1	1	yes	characteristic
61	D₁₉₇	—	8	1	no	—
62	C₁₉₇	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	197	1	no	maximal, Sylow
64	#430	D₁₉₇	1	1	yes	characteristic, complemented
65	#424	—	197	2	no	—
66	#421	—	197	2	no	—
67	#408	—	197	1	no	—
68	H(3,ℤ/4)	—	197	1	no	—
69	C₂×#41	—	197	1	no	—
70	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
71	C₂×M₁₆	C₂×D₁₉₇	1	1	yes	characteristic
72	C₂²×C₈	—	197	1	no	—
73	#117	—	197	2	no	—
74	#108	—	2	2	no	—
75	C₄⋊C₈	—	394	4	no	—
76	#104	—	394	4	no	—
77	C₂²⋊C₈	—	394	2	no	—
78	#41	—	394	2	no	—
79	#41	—	394	2	no	—
80	C₂×Q₈	—	197	1	no	—
81	C₂×D₄	—	394	1	no	—
82	C₂×D₄	—	197	1	no	—
83	C₂²×C₄	—	394	1	no	—
84	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	2	2	no	—
87	M₁₆	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	394	2	no	—
90	C₂×C₈	—	394	1	no	—
91	C₂×C₈	C₂²⋊D₁₉₇	1	2	yes	—
92	C₄⋊C₄	—	394	2	no	—
93	C₂²⋊C₄	—	394	2	no	—
94	C₄²	—	197	4	no	—
95	C₂³	—	394	1	no	—
96	C₂³	#15691359	1	1	yes	characteristic
97	Q₈	—	394	2	no	—
98	D₄	—	788	2	no	—
99	D₄	—	394	2	no	—
100	C₂×C₄	—	788	1	no	—
101	C₂×C₄	—	788	1	no	—
102	C₂×C₄	—	394	1	no	—
103	C₂×C₄	—	394	4	no	—
104	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₉₇	1	1	yes	characteristic
107	C₂×C₄	#15691358	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	4	2	no	—
110	C₈	—	788	1	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	788	2	no	—
113	C₂²	—	788	1	no	—
114	C₂²	#20263302	1	1	yes	characteristic
115	C₂²	#20263311	1	2	yes	—
116	C₄	—	788	4	no	—
117	C₄	—	788	1	no	—
118	C₄	#20263304	1	2	yes	—
119	C₄	#20263309	1	1	yes	characteristic
120	C₄	#20263322	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	1576	1	no	—
124	C₂	#27161014	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161107

C₂

yes

maximal, complemented

#27161095

C₂

yes

maximal

#27161047

C₂

yes

maximal, characteristic

#27160836

C₂

yes

maximal, characteristic, complemented

#430×C₁₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#15691358

C₂²

yes

characteristic

C₂²×#13096802

C₂²

yes

characteristic

#20263285

C₂²

yes

—

#20263255

—

#20263252

—

#20263238

—

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂×M₁₆×C₁₉₇

C₂²

yes

characteristic

#108×C₁₉₇

—

#15691358

—

#15691358

—

C₂×Dic₃₉₄

Q₈

yes

characteristic

C₂×D₇₈₈

Q₈

yes

characteristic

C₂×C₂²⋊D₁₉₇

—

C₂×C₄×D₁₉₇

—

C₂×#13096802

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

—

C₂²⋊Dic₁₉₇

—

C₄×Dic₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

D₄

yes

—

C₂²×D₁₉₇

—

Dic₃₉₄

—

D₇₈₈

—

C₂²⋊D₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₄×D₁₉₇

—

C₄×D₁₉₇

—

#13096802

—

C₂³×C₁₉₇

#41

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₉₇

#123

yes

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₄×C₁₉₇

C₂²⋊Q₈

yes

—

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#399

yes

characteristic

D₁₉₇

—

C₁₉₇

#2098

yes

characteristic, complemented, Sylow

#2098

—

197

maximal, Sylow

#430

D₁₉₇

yes

characteristic, complemented

#424

—

197

—

#421

—

197

—

#408

—

197

—

H(3,ℤ/4)

—

197

—

C₂×#41

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

C₂×M₁₆

C₂×D₁₉₇

yes

characteristic

C₂²×C₈

—

197

—

#117

—

197

—

#108

—

C₄⋊C₈

—

394

—

#104

—

394

—

C₂²⋊C₈

—

394

—

#41

—

394

—

#41

—

394

—

C₂×Q₈

—

197

—

C₂×D₄

—

394

—

C₂×D₄

—

197

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

C₂²⋊D₁₉₇

yes

—

C₄⋊C₄

—

394

—

C₂²⋊C₄

—

394

—

C₄²

—

197

—

C₂³

—

394

—

C₂³

#15691359

yes

characteristic

Q₈

—

394

—

D₄

—

788

—

D₄

—

394

—

100

C₂×C₄

—

788

—

101

C₂×C₄

—

788

—

102

C₂×C₄

—

394

—

103

C₂×C₄

—

394

—

104

C₂×C₄

D₄×D₁₉₇

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

106

C₂×C₄

Q₈×D₁₉₇

yes

characteristic

107

C₂×C₄

#15691358

yes

—

108

C₈

—

109

C₈

—

110

C₈

—

788

—

111

C₂²

—

112

C₂²

—

788

—

113

C₂²

—

788

—

114

C₂²

#20263302

yes

characteristic

115

C₂²

#20263311

yes

—

116

C₄

—

788

—

117

C₄

—

788

—

118

C₄

#20263304

yes

—

119

C₄

#20263309

yes

characteristic

120

C₄

#20263322

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

1576

—

124

C₂

#27161014

yes

characteristic

	subgroup	G/H
center	C₄	#20263322
commutator, cosocle	C₂²×C₄×C₁₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	#430×C₁₉₇	C₂
socle	C₂×C₁₉₇	#399
prime radical	C₂×#15691358	C₂²
hypercenter	#430	D₁₉₇
hypocenter	C₁₉₇	#2098
LCS	G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₄

#20263322

commutator, cosocle

C₂²×C₄×C₁₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

#430×C₁₉₇

C₂

C₂×C₁₉₇

C₂²

D₁₉₇

C₁₉₇

G ▷ C₂²×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
1. #27161107	C₂	semidirect
4. #27160836	C₂	semidirect
5. #430×C₁₉₇	C₂	semidirect
62. C₁₉₇	#2098	semidirect
64. #430	D₁₉₇	semidirect

H ≅ G/N

type

1. #27161107

C₂

semidirect

4. #27160836

C₂

semidirect

5. #430×C₁₉₇

C₂

semidirect

62. C₁₉₇

#2098

semidirect

64. #430

D₁₉₇

semidirect

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

subgroup	index
#27161107	2
#27161095	2
#27161047	2
#27160836	2
#430×C₁₉₇	2
#2098	197

subgroup

index

#430×C₁₉₇

197

Subgroups — #38744264

Conjugacy classes of subgroups

Complements

Maximal subgroups