Group #27171212 — subgroups

Subgroups — #27171212

Order 12928.

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

normal subgroups only

This group has 316 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	#20265141	C₂	1	2	yes	maximal
2	#20264946	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#13108058	C₂²	1	4	yes	—
5	C₂²⋊#11717579	—	2	2	no	—
6	C₂²⋊#11717579	—	2	2	no	—
7	C₂³×C₄×C₁₀₁	C₄	1	2	yes	—
8	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
11	C₂×C₄⋊C₄×C₁₀₁	C₄	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
13	#13108058	—	2	8	no	—
14	C₂×#11717579	—	2	4	no	—
15	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₀₁	—	2	4	no	—
17	C₂²×C₄×C₁₀₁	—	2	2	no	—
18	C₂²×C₄×C₁₀₁	—	2	2	no	—
19	C₂²×C₄×C₁₀₁	—	4	4	no	—
20	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₁	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
23	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
24	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₀₁	—	2	4	no	—
26	C₄⋊C₄×C₁₀₁	—	2	4	no	—
27	C₂³×C₁₀₁	—	2	2	no	—
28	C₂³×C₁₀₁	—	2	2	no	—
29	C₂³×C₁₀₁	—	2	2	no	—
30	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₁	—	2	2	no	—
34	C₂×C₄×C₁₀₁	—	2	4	no	—
35	C₂×C₄×C₁₀₁	—	2	4	no	—
36	C₂×C₄×C₁₀₁	—	2	4	no	—
37	C₂×C₄×C₁₀₁	—	4	4	no	—
38	C₂×C₄×C₁₀₁	—	4	4	no	—
39	C₂×C₄×C₁₀₁	—	4	4	no	—
40	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₁₀₁	#41	1	4	yes	—
43	#11717579	—	4	4	no	—
44	C₂²×C₁₀₁	—	2	8	no	—
45	C₂²×C₁₀₁	—	2	1	no	—
46	C₂²×C₁₀₁	—	2	1	no	—
47	C₂²×C₁₀₁	—	2	1	no	—
48	C₂²×C₁₀₁	—	2	1	no	—
49	C₂²×C₁₀₁	—	2	1	no	—
50	C₂²×C₁₀₁	—	2	1	no	—
51	C₂²×C₁₀₁	#100	1	2	yes	—
52	C₂²×C₁₀₁	#101	1	2	yes	—
53	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₁₀₁	#117	1	2	yes	—
55	C₄×C₁₀₁	—	2	2	no	—
56	C₄×C₁₀₁	—	4	4	no	—
57	C₄×C₁₀₁	#123	1	4	yes	—
58	C₂×C₁₀₁	—	2	2	no	—
59	C₂×C₁₀₁	—	2	2	no	—
60	C₂×C₁₀₁	—	2	2	no	—
61	C₂×C₁₀₁	#381	1	1	yes	characteristic
62	C₂×C₁₀₁	C₂×#100	1	1	yes	characteristic
63	C₂×C₁₀₁	C₂×#101	1	1	yes	characteristic
64	#1821	—	101	1	no	maximal, Sylow
65	C₁₀₁	#1821	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic
67	#407	—	101	2	no	—
68	#343	—	101	4	no	—
69	C₂³×C₄	Dic₁₀₁	1	2	yes	—
70	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
71	C₂×M₁₆	—	101	4	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	2	2	no	—
74	C₂×C₄⋊C₄	Dic₁₀₁	1	2	yes	—
75	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
76	C₂²⋊C₈	—	202	2	no	—
77	C₂²⋊C₈	—	202	2	no	—
78	C₂⁴	C₂×Dic₁₀₁	1	1	yes	characteristic
79	C₂²×C₄	—	2	4	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	—	4	4	no	—
83	C₂²×C₄	C₂×Dic₁₀₁	1	1	yes	characteristic
84	C₂²×C₄	C₂×Dic₁₀₁	1	4	yes	—
85	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
86	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
88	M₁₆	—	202	8	no	—
89	C₂×C₈	—	202	4	no	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₂³	—	2	2	no	—
93	C₂³	—	2	2	no	—
94	C₂³	—	2	2	no	—
95	C₂³	C₂²⋊Dic₁₀₁	1	1	yes	characteristic
96	C₂³	C₂²⋊Dic₁₀₁	1	1	yes	characteristic
97	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	4	4	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₂²⋊Dic₁₀₁	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
107	C₂×C₄	#13108074	1	4	yes	—
108	C₈	—	404	4	no	—
109	C₂²	—	2	8	no	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#15692172	1	2	yes	—
117	C₂²	#15692174	1	2	yes	—
118	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
119	C₂²	#15692214	1	2	yes	—
120	C₄	—	2	2	no	—
121	C₄	—	4	4	no	—
122	C₄	#15692238	1	4	yes	—
123	C₂	—	2	2	no	—
124	C₂	—	2	2	no	—
125	C₂	—	2	2	no	—
126	C₂	#20265040	1	1	yes	characteristic
127	C₂	C₂×#15692172	1	1	yes	characteristic
128	C₂	C₂×#15692174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265141

C₂

yes

maximal

#20264946

C₂

yes

maximal, complemented

C₂²×C₄⋊C₄×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

C₂²⋊#11717579

—

C₂²⋊#11717579

—

C₂³×C₄×C₁₀₁

C₄

yes

—

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₄

yes

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

#13108058

—

C₂×#11717579

—

C₂⁴×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂³×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#100

yes

—

C₂²×C₁₀₁

#101

yes

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

#117

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#123

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1821

yes

characteristic, complemented, Sylow

C₂²×C₄⋊C₄

D₁₀₁

yes

characteristic

#407

—

101

—

#343

—

101

—

C₂³×C₄

Dic₁₀₁

yes

—

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

C₂×M₁₆

—

101

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₁₀₁

yes

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂²⋊C₈

—

202

—

C₂²⋊C₈

—

202

—

C₂⁴

C₂×Dic₁₀₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₀₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊Dic₁₀₁

yes

characteristic

C₂³

C₂²⋊Dic₁₀₁

yes

characteristic

C₂³

C₂²×Dic₁₀₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

107

C₂×C₄

#13108074

yes

—

108

C₈

—

404

—

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#15692172

yes

—

117

C₂²

#15692174

yes

—

118

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

characteristic

119

C₂²

#15692214

yes

—

120

C₄

—

121

C₄

—

122

C₄

#15692238

yes

—

123

C₂

—

124

C₂

—

125

C₂

—

126

C₂

#20265040

yes

characteristic

127

C₂

C₂×#15692172

yes

characteristic

128

C₂

C₂×#15692174

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₀₁
commutator	C₂³×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂²×C₄⋊C₄×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₀₁	C₂×C₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂²×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#1821
LCS	G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₀₁

commutator

C₂³×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂²×C₄⋊C₄×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₁₀₁

C₂×C₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂²×C₄⋊C₄

D₁₀₁

hypocenter

C₁₀₁

#1821

LCS

G ▷ C₂³×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
2. #20264946	C₂	semidirect
65. C₁₀₁	#1821	semidirect

H ≅ G/N

type

2. #20264946

C₂

semidirect

65. C₁₀₁

#1821

semidirect

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

subgroup	index
#20265141	2
#20264946	2
C₂²×C₄⋊C₄×C₁₀₁	2
#1821	101

subgroup

index

#20265141

#20264946

C₂²×C₄⋊C₄×C₁₀₁

#1821

101

Subgroups — #27171212

Conjugacy classes of subgroups

Complements

Maximal subgroups