Group #27169028 — subgroups

Subgroups — #27169028

Order 12928.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#20265201	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11717579	C₂	1	1	yes	maximal, characteristic
3	#431×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#13108058	—	2	1	no	—
5	C₂²×#11717579	—	2	1	no	—
6	C₄⋊#11717579	—	2	1	no	—
7	C₄⋊#11717579	—	2	1	no	—
8	C₄⋊#11717579	C₄	1	2	yes	—
9	C₄⋊#11717579	C₄	1	2	yes	—
10	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₀₁	—	2	1	no	—
12	C₂²⋊C₈×C₁₀₁	C₄	1	2	yes	—
13	C₄×C₈×C₁₀₁	—	2	1	no	—
14	#13108058	—	4	1	no	—
15	#13108058	—	4	1	no	—
16	C₂×#11717579	—	2	2	no	—
17	C₂×#11717579	—	2	2	no	—
18	C₂×#11717579	—	4	1	no	—
19	C₂×#11717579	—	4	1	no	—
20	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₁	Q₈	1	1	yes	characteristic
23	C₂×C₈×C₁₀₁	—	2	2	no	—
24	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
27	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
28	C₂³×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₁	—	2	2	no	—
30	C₂×C₄×C₁₀₁	—	4	1	no	—
31	C₂×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
38	C₂×C₄×C₁₀₁	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₁₀₁	Q₁₆	1	1	yes	characteristic
41	C₈×C₁₀₁	—	4	2	no	—
42	#11717579	—	4	2	no	—
43	#11717579	—	4	2	no	—
44	C₂²×C₁₀₁	—	2	1	no	—
45	C₂²×C₁₀₁	—	2	1	no	—
46	C₂²×C₁₀₁	#95	1	1	yes	characteristic
47	C₂²×C₁₀₁	#106	1	1	yes	characteristic
48	C₂²×C₁₀₁	#107	1	1	yes	characteristic
49	C₄×C₁₀₁	—	2	1	no	—
50	C₄×C₁₀₁	—	2	1	no	—
51	C₄×C₁₀₁	—	2	1	no	—
52	C₄×C₁₀₁	#102	1	1	yes	characteristic
53	C₄×C₁₀₁	#102	1	1	yes	characteristic
54	C₄×C₁₀₁	#103	1	1	yes	characteristic
55	C₄×C₁₀₁	#103	1	1	yes	characteristic
56	C₄×C₁₀₁	#108	1	2	yes	—
57	C₂×C₁₀₁	—	2	1	no	—
58	C₂×C₁₀₁	—	2	1	no	—
59	C₂×C₁₀₁	#340	1	1	yes	characteristic
60	C₂×C₁₀₁	#341	1	1	yes	characteristic
61	C₂×C₁₀₁	#344	1	1	yes	characteristic
62	#1301	—	101	1	no	maximal, Sylow
63	C₁₀₁	#1301	1	1	yes	characteristic, complemented, Sylow
64	#431	D₁₀₁	1	1	yes	characteristic
65	#423	—	101	1	no	—
66	C₂×C₄⋊C₈	—	101	1	no	—
67	C₂×M₁₆	—	202	1	no	—
68	C₂²×C₈	—	202	1	no	—
69	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	101	2	no	—
72	C₄⋊C₈	—	101	2	no	—
73	C₄⋊C₈	—	202	1	no	—
74	C₄⋊C₈	—	202	1	no	—
75	C₂²⋊C₈	Dic₁₀₁	1	2	yes	—
76	C₄×C₈	—	2	1	no	—
77	C₂²×C₄	Dic₂₀₂	1	1	yes	characteristic
78	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
79	C₂²×C₄	C₂×Dic₁₀₁	1	1	yes	characteristic
80	M₁₆	—	404	1	no	—
81	M₁₆	—	404	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	404	1	no	—
84	C₂×C₈	—	404	1	no	—
85	C₂×C₈	—	202	2	no	—
86	C₂×C₈	—	202	2	no	—
87	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
88	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
89	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
90	C₄²	C₂²⋊D₁₀₁	1	1	yes	characteristic
91	C₂³	C₄⋊Dic₁₀₁	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₁₀₁	1	1	yes	characteristic
95	C₂×C₄	#13108051	1	1	yes	characteristic
96	C₂×C₄	#13108051	1	1	yes	characteristic
97	C₂×C₄	C₂²⋊Dic₁₀₁	1	1	yes	characteristic
98	C₂×C₄	#13108053	1	1	yes	characteristic
99	C₂×C₄	#13108053	1	1	yes	characteristic
100	C₂×C₄	D₈₀₈	1	1	yes	characteristic
101	C₂×C₄	#13108063	1	1	yes	characteristic
102	C₂×C₄	#13108063	1	1	yes	characteristic
103	C₂×C₄	Dic₄₀₄	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	404	2	no	—
106	C₈	—	404	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#15692162	1	1	yes	characteristic
110	C₂²	#15692187	1	1	yes	characteristic
111	C₂²	#15692189	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#15692177	1	1	yes	characteristic
116	C₄	#15692177	1	1	yes	characteristic
117	C₄	#15692180	1	1	yes	characteristic
118	C₄	#15692180	1	1	yes	characteristic
119	C₄	#15692190	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#20264940	1	1	yes	characteristic
123	C₂	#20264941	1	1	yes	characteristic
124	C₂	#20264947	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265201

C₂

yes

maximal, characteristic

C₂×C₄⋊#11717579

C₂

yes

maximal, characteristic

#431×C₁₀₁

C₂

yes

maximal, characteristic

—

—

—

—

C₄

yes

—

C₄⋊#11717579

C₄

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₁

—

C₂²⋊C₈×C₁₀₁

C₄

yes

—

C₄×C₈×C₁₀₁

—

—

—

—

—

—

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

Q₈

yes

characteristic

C₂×C₈×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

D₄

yes

characteristic

C₄²×C₁₀₁

D₄

yes

characteristic

C₂³×C₁₀₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄²

yes

characteristic

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

D₈

yes

characteristic

C₂×C₄×C₁₀₁

SD₁₆

yes

characteristic

C₂×C₄×C₁₀₁

SD₁₆

yes

characteristic

C₂×C₄×C₁₀₁

Q₁₆

yes

characteristic

C₈×C₁₀₁

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1301

yes

characteristic, complemented, Sylow

#431

D₁₀₁

yes

characteristic

#423

—

101

—

C₂×C₄⋊C₈

—

101

—

C₂×M₁₆

—

202

—

C₂²×C₈

—

202

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

202

—

C₄⋊C₈

—

202

—

C₂²⋊C₈

Dic₁₀₁

yes

—

C₄×C₈

—

C₂²×C₄

Dic₂₀₂

yes

characteristic

C₂²×C₄

D₄₀₄

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₁

yes

characteristic

M₁₆

—

404

—

M₁₆

—

404

—

C₂×C₈

—

C₂×C₈

—

404

—

C₂×C₈

—

404

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄²

C₄×D₁₀₁

yes

characteristic

C₄²

C₄×D₁₀₁

yes

characteristic

C₄²

C₂²⋊D₁₀₁

yes

characteristic

C₄²

C₂²⋊D₁₀₁

yes

characteristic

C₂³

C₄⋊Dic₁₀₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₀₁

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₁₀₁

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

D₈₀₈

yes

characteristic

101

C₂×C₄

#13108063

yes

characteristic

102

C₂×C₄

#13108063

yes

characteristic

103

C₂×C₄

Dic₄₀₄

yes

characteristic

104

C₈

—

105

C₈

—

404

—

106

C₈

—

404

—

107

C₂²

—

108

C₂²

—

109

C₂²

#15692162

yes

characteristic

110

C₂²

#15692187

yes

characteristic

111

C₂²

#15692189

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#15692177

yes

characteristic

116

C₄

#15692177

yes

characteristic

117

C₄

#15692180

yes

characteristic

118

C₄

#15692180

yes

characteristic

119

C₄

#15692190

yes

—

120

C₂

—

121

C₂

—

122

C₂

#20264940

yes

characteristic

123

C₂

#20264941

yes

characteristic

124

C₂

#20264947

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13108051
commutator	C₂×C₄×C₁₀₁	C₄²
Frattini	C₂×C₄²	C₂×D₁₀₁
Fitting	#431×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#95
prime radical	C₂³×C₁₀₁	C₄⋊C₄
cosocle	C₂×C₄²×C₁₀₁	C₂²
hypercenter	#431	D₁₀₁
hypocenter	C₁₀₁	#1301
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

#13108051

commutator

C₂×C₄×C₁₀₁

C₄²

Frattini

C₂×C₄²

C₂×D₁₀₁

Fitting

#431×C₁₀₁

C₂

socle

C₂²×C₁₀₁

#95

prime radical

C₂³×C₁₀₁

C₄⋊C₄

cosocle

C₂×C₄²×C₁₀₁

C₂²

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
63. C₁₀₁	#1301	semidirect

H ≅ G/N

type

63. C₁₀₁

#1301

semidirect

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

subgroup	index
#20265201	2
C₂×C₄⋊#11717579	2
#431×C₁₀₁	2
#1301	101

subgroup

index

#20265201

C₂×C₄⋊#11717579

#431×C₁₀₁

#1301

101

Subgroups — #27169028

Conjugacy classes of subgroups

Complements

Maximal subgroups