Group #38742830 — subgroups

Subgroups — #38742830

Order 25216.

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

normal subgroups only

This group has 216 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×#20263255	C₂	1	2	yes	maximal
2	#27160925	C₂	1	1	yes	maximal, characteristic
3	#27160834	C₂	1	1	yes	maximal, characteristic
4	#27160833	C₂	1	2	yes	maximal
5	#336×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13096802	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
9	#20263255	—	2	4	no	—
10	#20263233	C₄	1	2	yes	—
11	#20263233	C₄	1	2	yes	—
12	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₉₇	C₂×C₄	1	2	yes	—
17	C₂×#13096802	—	2	2	no	—
18	C₂×#13096802	—	2	2	no	—
19	C₂×#13096802	—	2	2	no	—
20	C₄×Dic₁₉₇	—	2	2	no	—
21	C₄×Dic₁₉₇	D₄	1	2	yes	—
22	C₄×Dic₁₉₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₉₇	—	2	2	no	—
26	C₂×C₈×C₁₉₇	—	2	2	no	—
27	C₂×C₈×C₁₉₇	—	2	2	no	—
28	C₄²×C₁₉₇	—	2	2	no	—
29	C₂×Dic₁₉₇	—	2	2	no	—
30	C₂×Dic₁₉₇	—	2	2	no	—
31	C₂×Dic₁₉₇	—	2	1	no	—
32	C₂×Dic₁₉₇	—	2	1	no	—
33	C₂×Dic₁₉₇	—	2	2	no	—
34	C₂×Dic₁₉₇	—	2	2	no	—
35	C₂×Dic₁₉₇	C₄⋊C₄	1	4	yes	—
36	#13096802	—	4	2	no	—
37	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₉₇	—	2	2	no	—
39	C₂×C₄×C₁₉₇	—	2	2	no	—
40	C₂×C₄×C₁₉₇	—	2	2	no	—
41	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₉₇	#41	1	2	yes	—
44	C₂×C₄×C₁₉₇	#41	1	2	yes	—
45	C₈×C₁₉₇	—	4	2	no	—
46	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₉₇	#117	1	1	yes	characteristic
48	C₂²×C₁₉₇	#117	1	1	yes	characteristic
49	C₂²×C₁₉₇	#131	1	2	yes	—
50	C₂²×C₁₉₇	#131	1	2	yes	—
51	C₄×C₁₉₇	—	4	2	no	—
52	C₄×C₁₉₇	#124	1	2	yes	—
53	C₄×C₁₉₇	#125	1	2	yes	—
54	Dic₁₉₇	—	2	4	no	—
55	Dic₁₉₇	—	4	1	no	—
56	Dic₁₉₇	—	4	1	no	—
57	C₂×C₁₉₇	#382	1	1	yes	characteristic
58	C₂×C₁₉₇	#424	1	2	yes	—
59	C₂×C₁₉₇	#433	1	2	yes	—
60	C₂×C₁₉₇	#433	1	1	yes	characteristic
61	C₂×C₁₉₇	#433	1	1	yes	characteristic
62	C₁₉₇	#1832	1	1	yes	characteristic, complemented, Sylow
63	#1832	—	197	1	no	maximal, Sylow
64	C₂×C₄⋊C₈	—	197	2	no	—
65	#376	—	197	1	no	—
66	#336	—	197	2	no	—
67	#336	—	197	1	no	—
68	#336	D₁₉₇	1	1	yes	characteristic
69	C₂²×C₈	—	197	2	no	—
70	C₂²×C₈	C₂×D₁₉₇	1	2	yes	—
71	C₂×C₄²	—	197	1	no	—
72	C₂×C₄²	—	197	1	no	—
73	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
74	C₄⋊C₈	—	394	4	no	—
75	#95	—	197	2	no	—
76	#95	—	197	2	no	—
77	C₂²×C₄	—	197	1	no	—
78	C₂²×C₄	—	197	1	no	—
79	C₂²×C₄	—	197	2	no	—
80	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	394	2	no	—
86	C₂×C₈	—	394	2	no	—
87	C₂×C₈	—	394	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	394	2	no	—
90	C₄²	—	197	2	no	—
91	C₄²	—	197	2	no	—
92	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	394	1	no	—
97	C₂×C₄	—	394	1	no	—
98	C₂×C₄	—	394	2	no	—
99	C₂×C₄	—	394	2	no	—
100	C₂×C₄	—	394	2	no	—
101	C₂×C₄	—	394	2	no	—
102	C₂×C₄	—	197	4	no	—
103	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
104	C₂×C₄	Q₈×D₁₉₇	1	1	yes	characteristic
105	C₂×C₄	#15691357	1	2	yes	—
106	C₂×C₄	#15691358	1	2	yes	—
107	C₈	—	4	2	no	—
108	C₈	—	788	2	no	—
109	C₂²	C₄⋊C₄×D₁₉₇	1	1	yes	characteristic
110	C₂²	#20263282	1	1	yes	characteristic
111	C₂²	#20263283	1	1	yes	characteristic
112	C₂²	#20263332	1	2	yes	—
113	C₂²	#20263334	1	2	yes	—
114	C₄	—	4	2	no	—
115	C₄	—	788	1	no	—
116	C₄	—	788	1	no	—
117	C₄	—	394	4	no	—
118	C₄	#20263312	1	2	yes	—
119	C₄	#20263316	1	2	yes	—
120	C₂	#27160943	1	1	yes	characteristic
121	C₂	#27161104	1	2	yes	—
122	C₂	#27161137	1	2	yes	—
123	C₂	#27161138	1	1	yes	characteristic
124	C₂	#27161141	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263255

C₂

yes

maximal

#27160925

C₂

yes

maximal, characteristic

#27160834

C₂

yes

maximal, characteristic

#27160833

C₂

yes

maximal

#336×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

—

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

#20263255

—

#20263233

C₄

yes

—

#20263233

C₄

yes

—

C₂²×C₈×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂²×Dic₁₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₇

C₂×C₄

yes

—

C₂×#13096802

—

C₂×#13096802

—

C₂×#13096802

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

D₄

yes

—

C₄×Dic₁₉₇

Q₈

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₄²×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

C₄⋊C₄

yes

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

C₂²×C₁₉₇

yes

—

C₂²×C₁₉₇

#131

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#124

yes

—

C₄×C₁₉₇

#125

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

#382

yes

characteristic

C₂×C₁₉₇

#424

yes

—

C₂×C₁₉₇

#433

yes

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#1832

—

197

maximal, Sylow

C₂×C₄⋊C₈

—

197

—

#376

—

197

—

#336

—

197

—

#336

—

197

—

#336

D₁₉₇

yes

characteristic

C₂²×C₈

—

197

—

C₂²×C₈

C₂×D₁₉₇

yes

—

C₂×C₄²

—

197

—

C₂×C₄²

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

394

—

#95

—

197

—

#95

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

C₄×D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₄²

—

C₄²

—

394

—

C₄²

—

197

—

C₄²

—

197

—

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

100

C₂×C₄

—

394

—

101

C₂×C₄

—

394

—

102

C₂×C₄

—

197

—

103

C₂×C₄

D₄×D₁₉₇

yes

characteristic

104

C₂×C₄

Q₈×D₁₉₇

yes

characteristic

105

C₂×C₄

#15691357

yes

—

106

C₂×C₄

#15691358

yes

—

107

C₈

—

108

C₈

—

788

—

109

C₂²

C₄⋊C₄×D₁₉₇

yes

characteristic

110

C₂²

#20263282

yes

characteristic

111

C₂²

#20263283

yes

characteristic

112

C₂²

#20263332

yes

—

113

C₂²

#20263334

yes

—

114

C₄

—

115

C₄

—

788

—

116

C₄

—

788

—

117

C₄

—

394

—

118

C₄

#20263312

yes

—

119

C₄

#20263316

yes

—

120

C₂

#27160943

yes

characteristic

121

C₂

#27161104

yes

—

122

C₂

#27161137

yes

—

123

C₂

#27161138

yes

characteristic

124

C₂

#27161141

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₇
commutator, socle, prime radical	C₂³×C₁₉₇	C₂²×C₄
Fitting	#336×C₁₉₇	C₂
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#336	D₁₉₇
hypocenter	C₁₉₇	#1832
LCS	G ▷ C₂³×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₇

commutator, socle, prime radical

C₂³×C₁₉₇

C₂²×C₄

Fitting

#336×C₁₉₇

C₂

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂³×C₁₉₇ ▷ C₁₉₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
62. C₁₉₇	#1832	semidirect

H ≅ G/N

type

62. C₁₉₇

#1832

semidirect

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

subgroup	index
C₂×#20263255	2
#27160925	2
#27160834	2
#27160833	2
#336×C₁₉₇	2
#1832	197

subgroup

index

#336×C₁₉₇

197

Subgroups — #38742830

Conjugacy classes of subgroups

Complements

Maximal subgroups