Group #27179399 — subgroups

Subgroups — #27179399

Order 12928.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#20265119	C₂	1	3	yes	maximal
2	#20265111	C₂	1	3	yes	maximal
3	#20265109	C₂	1	3	yes	maximal
4	#20265098	C₂	1	3	yes	maximal
5	#20265047	C₂	1	1	yes	maximal, characteristic
6	#20265045	C₂	1	1	yes	maximal, characteristic
7	#383×C₁₀₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₀₁	C₂²	1	3	yes	—
10	C₂×C₄⋊Dic₁₀₁	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₁₀₁	C₂²	1	3	yes	—
12	C₂×#13108053	C₂²	1	3	yes	—
13	C₂×#13108053	C₂²	1	3	yes	—
14	#15692162	C₂²	1	3	yes	—
15	#15692162	C₂²	1	3	yes	—
16	#15692162	C₂²	1	3	yes	—
17	#15692162	C₂²	1	3	yes	—
18	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
19	#95×C₁₀₁	C₂²	1	3	yes	—
20	#95×C₁₀₁	C₂²	1	3	yes	—
21	C₂×Dic₂₀₂	—	4	3	no	—
22	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
23	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₀₁	C₂³	1	3	yes	—
25	C₂²×Dic₁₀₁	C₂³	1	3	yes	—
26	C₄⋊Dic₁₀₁	—	4	3	no	—
27	C₄⋊Dic₁₀₁	—	4	3	no	—
28	C₄⋊Dic₁₀₁	—	4	3	no	—
29	#13108053	—	4	3	no	—
30	#13108053	—	4	3	no	—
31	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
33	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
34	C₄²×C₁₀₁	Q₈	1	4	yes	complemented
35	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
36	C₂×C₄×C₁₀₁	—	2	3	no	—
37	C₂×C₄×C₁₀₁	—	2	3	no	—
38	C₂×C₄×C₁₀₁	—	2	3	no	—
39	C₂×C₄×C₁₀₁	—	2	3	no	—
40	C₂×C₄×C₁₀₁	—	2	6	no	—
41	C₂×C₄×C₁₀₁	C₂×Q₈	1	6	yes	—
42	C₂×Dic₁₀₁	—	2	3	no	—
43	C₂×Dic₁₀₁	—	2	3	no	—
44	C₂×Dic₁₀₁	—	2	3	no	—
45	C₂×Dic₁₀₁	—	2	3	no	—
46	C₂×Dic₁₀₁	—	2	3	no	—
47	C₂×Dic₁₀₁	—	2	3	no	—
48	C₂×Dic₁₀₁	—	2	6	no	—
49	Dic₂₀₂	—	4	4	no	—
50	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₁₀₁	H(5,𝔽₂)	1	3	yes	—
52	C₂²×C₁₀₁	2₋¹⁺⁴	1	3	yes	—
53	C₄×C₁₀₁	—	2	6	no	—
54	C₄×C₁₀₁	—	4	1	no	—
55	C₄×C₁₀₁	—	4	3	no	—
56	Dic₁₀₁	—	4	1	no	—
57	Dic₁₀₁	—	4	1	no	—
58	Dic₁₀₁	—	4	3	no	—
59	Dic₁₀₁	—	4	3	no	—
60	C₂×C₁₀₁	#544	1	3	yes	—
61	C₂×C₁₀₁	#560	1	1	yes	characteristic
62	C₂×C₁₀₁	#563	1	3	yes	—
63	#2843	—	101	1	no	maximal, Sylow
64	C₁₀₁	#2843	1	1	yes	characteristic, complemented, Sylow
65	#400	—	101	3	no	—
66	#398	—	101	3	no	—
67	#398	—	101	3	no	—
68	#395	—	101	3	no	—
69	#384	—	101	1	no	—
70	#383	—	101	1	no	—
71	#383	D₁₀₁	1	1	yes	characteristic
72	C₂²×Q₈	—	101	1	no	—
73	C₂×C₄⋊C₄	—	101	3	no	—
74	C₂×C₄⋊C₄	—	101	3	no	—
75	C₂×C₄⋊C₄	—	101	3	no	—
76	C₂×C₄⋊C₄	—	101	3	no	—
77	C₂×C₄⋊C₄	—	101	3	no	—
78	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
79	#95	—	101	3	no	—
80	#95	—	101	3	no	—
81	#95	—	101	3	no	—
82	#95	—	101	3	no	—
83	#95	C₂×D₁₀₁	1	3	yes	—
84	#95	C₂×D₁₀₁	1	3	yes	—
85	C₂×Q₈	—	404	3	no	—
86	C₂²×C₄	—	101	3	no	—
87	C₂²×C₄	—	101	1	no	—
88	C₂²×C₄	—	101	1	no	—
89	C₂²×C₄	—	101	3	no	—
90	C₂²×C₄	C₂²×D₁₀₁	1	3	yes	—
91	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₁₀₁	1	3	yes	—
93	C₄⋊C₄	—	404	3	no	—
94	C₄⋊C₄	—	404	3	no	—
95	C₄⋊C₄	—	404	3	no	—
96	C₄⋊C₄	—	404	3	no	—
97	C₄⋊C₄	—	404	3	no	—
98	C₄²	Dic₂₀₂	1	4	yes	complemented
99	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
100	Q₈	—	404	4	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	2	6	no	—
106	C₂×C₄	—	202	3	no	—
107	C₂×C₄	—	202	3	no	—
108	C₂×C₄	—	202	3	no	—
109	C₂×C₄	—	202	3	no	—
110	C₂×C₄	—	202	3	no	—
111	C₂×C₄	—	202	3	no	—
112	C₂×C₄	—	202	6	no	—
113	C₂×C₄	C₂×Dic₂₀₂	1	6	yes	—
114	C₂²	C₂²×Dic₂₀₂	1	1	yes	characteristic
115	C₂²	#15692301	1	3	yes	—
116	C₂²	#15692302	1	3	yes	—
117	C₄	—	2	6	no	—
118	C₄	—	4	1	no	—
119	C₄	—	4	3	no	—
120	C₄	—	404	1	no	—
121	C₄	—	404	1	no	—
122	C₄	—	404	3	no	—
123	C₄	—	404	3	no	—
124	C₂	#20265805	1	3	yes	—
125	C₂	#20265918	1	1	yes	characteristic
126	C₂	#20265929	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265119

C₂

yes

maximal

#20265111

C₂

yes

maximal

#20265109

C₂

yes

maximal

#20265098

C₂

yes

maximal

#20265047

C₂

yes

maximal, characteristic

#20265045

C₂

yes

maximal, characteristic

#383×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×#13108053

C₂²

yes

—

C₂×#13108053

C₂²

yes

—

#15692162

C₂²

yes

—

#15692162

C₂²

yes

—

#15692162

C₂²

yes

—

#15692162

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

#95×C₁₀₁

C₂²

yes

—

#95×C₁₀₁

C₂²

yes

—

C₂×Dic₂₀₂

—

C₂²×Dic₁₀₁

C₂³

yes

characteristic

C₂²×Dic₁₀₁

C₂³

yes

characteristic

C₂²×Dic₁₀₁

C₂³

yes

—

C₂²×Dic₁₀₁

C₂³

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

#13108053

—

#13108053

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₄²×C₁₀₁

Q₈

yes

complemented

C₂³×C₁₀₁

C₂⁴

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

C₂²×C₁₀₁

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₁

H(5,𝔽₂)

yes

—

C₂²×C₁₀₁

2₋¹⁺⁴

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

#544

yes

—

C₂×C₁₀₁

#560

yes

characteristic

C₂×C₁₀₁

#563

yes

—

#2843

—

101

maximal, Sylow

C₁₀₁

#2843

yes

characteristic, complemented, Sylow

#400

—

101

—

#398

—

101

—

#398

—

101

—

#395

—

101

—

#384

—

101

—

#383

—

101

—

#383

D₁₀₁

yes

characteristic

C₂²×Q₈

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#95

—

101

—

#95

—

101

—

#95

—

101

—

#95

—

101

—

#95

C₂×D₁₀₁

yes

—

#95

C₂×D₁₀₁

yes

—

C₂×Q₈

—

404

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂²×D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂²×C₄

C₂²×D₁₀₁

yes

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

404

—

C₄²

Dic₂₀₂

yes

complemented

C₂³

C₂³×D₁₀₁

yes

characteristic

100

Q₈

—

404

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

202

—

107

C₂×C₄

—

202

—

108

C₂×C₄

—

202

—

109

C₂×C₄

—

202

—

110

C₂×C₄

—

202

—

111

C₂×C₄

—

202

—

112

C₂×C₄

—

202

—

113

C₂×C₄

C₂×Dic₂₀₂

yes

—

114

C₂²

C₂²×Dic₂₀₂

yes

characteristic

115

C₂²

#15692301

yes

—

116

C₂²

#15692302

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

404

—

121

C₄

—

404

—

122

C₄

—

404

—

123

C₄

—

404

—

124

C₂

#20265805

yes

—

125

C₂

#20265918

yes

characteristic

126

C₂

#20265929

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator, socle, prime radical, cosocle	C₂³×C₁₀₁	C₂⁴
Fitting	#383×C₁₀₁	C₂
hypercenter	#383	D₁₀₁
hypocenter	C₁₀₁	#2843
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #383

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₁

commutator, socle, prime radical, cosocle

C₂³×C₁₀₁

C₂⁴

Fitting

#383×C₁₀₁

C₂

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ #383

N	H ≅ G/N	type
34. C₄²×C₁₀₁	Q₈	semidirect
64. C₁₀₁	#2843	semidirect
98. C₄²	Dic₂₀₂	semidirect

H ≅ G/N

type

34. C₄²×C₁₀₁

Q₈

semidirect

64. C₁₀₁

#2843

semidirect

98. C₄²

Dic₂₀₂

semidirect

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

subgroup	index
#20265119	2
#20265111	2
#20265109	2
#20265098	2
#20265047	2
#20265045	2
#383×C₁₀₁	2
#2843	101

subgroup

index

#383×C₁₀₁

101

Subgroups — #27179399

Conjugacy classes of subgroups

Complements

Maximal subgroups