Group #27172400 — subgroups

Subgroups — #27172400

Order 12928.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×Dic₄₀₄	C₂	1	1	yes	maximal, characteristic
2	#20265185	C₂	1	2	yes	maximal
3	C₂×#15692173	C₂	1	2	yes	maximal
4	#20264942	C₂	1	1	yes	maximal, characteristic
5	#424×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
7	C₂×Dic₄₀₄	—	2	4	no	—
8	C₂×Dic₄₀₄	—	2	2	no	—
9	C₂×#13108058	C₂²	1	2	yes	—
10	#15692178	—	2	4	no	—
11	#15692173	—	2	4	no	—
12	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
14	#117×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₀₁	—	2	2	no	—
16	C₂×Dic₂₀₂	—	2	2	no	—
17	C₂×Dic₂₀₂	—	2	2	no	—
18	C₂×Dic₂₀₂	—	2	2	no	—
19	C₂×Dic₂₀₂	—	2	2	no	—
20	C₂×Dic₂₀₂	—	2	2	no	—
21	C₂×Dic₂₀₂	D₄	1	4	yes	—
22	C₂²×Dic₁₀₁	—	2	2	no	—
23	Dic₄₀₄	—	2	8	no	—
24	#13108058	—	2	2	no	—
25	#13108058	—	2	2	no	—
26	C₂×#11717579	—	2	2	no	—
27	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₀₁	—	2	2	no	—
29	C₂×C₈×C₁₀₁	—	2	2	no	—
30	C₂×C₈×C₁₀₁	—	2	1	no	—
31	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
32	C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
33	C₂²⋊C₄×C₁₀₁	—	2	1	no	—
34	C₄²×C₁₀₁	—	2	1	no	—
35	C₂³×C₁₀₁	#41	1	1	yes	characteristic
36	C₂×C₄×C₁₀₁	—	2	2	no	—
37	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₁	#41	1	2	yes	—
41	C₂×Dic₁₀₁	—	2	4	no	—
42	C₂×Dic₁₀₁	—	4	2	no	—
43	C₂×Dic₁₀₁	—	4	2	no	—
44	Dic₂₀₂	—	2	4	no	—
45	Dic₂₀₂	—	2	4	no	—
46	Dic₂₀₂	—	4	2	no	—
47	Dic₂₀₂	—	4	2	no	—
48	C₈×C₁₀₁	—	2	2	no	—
49	C₈×C₁₀₁	—	4	1	no	—
50	#11717579	—	4	2	no	—
51	C₂²×C₁₀₁	—	2	2	no	—
52	C₂²×C₁₀₁	#121	1	1	yes	characteristic
53	C₂²×C₁₀₁	#123	1	2	yes	—
54	C₄×C₁₀₁	—	4	2	no	—
55	C₄×C₁₀₁	#120	1	1	yes	characteristic
56	C₄×C₁₀₁	#121	1	2	yes	—
57	C₄×C₁₀₁	#124	1	1	yes	characteristic
58	Dic₁₀₁	—	4	4	no	—
59	C₂×C₁₀₁	—	2	2	no	—
60	C₂×C₁₀₁	#394	1	1	yes	characteristic
61	C₂×C₁₀₁	#473	1	2	yes	—
62	#2047	—	101	1	no	maximal, Sylow
63	C₁₀₁	#2047	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₁₆	—	101	1	no	—
65	#424	D₁₀₁	1	1	yes	characteristic
66	#419	—	101	2	no	—
67	C₂×#101	—	101	2	no	—
68	#341	—	101	1	no	—
69	C₂²×Q₈	—	101	2	no	—
70	C₂×Q₁₆	—	202	4	no	—
71	C₂×Q₁₆	—	202	2	no	—
72	C₂×M₁₆	—	101	2	no	—
73	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
75	#117	C₂×D₁₀₁	1	1	yes	characteristic
76	C₄⋊C₈	—	2	2	no	—
77	#103	—	202	4	no	—
78	#101	—	202	4	no	—
79	C₂×Q₈	—	101	4	no	—
80	C₂×Q₈	—	202	2	no	—
81	C₂×Q₈	—	202	2	no	—
82	C₂×Q₈	—	202	2	no	—
83	C₂×Q₈	—	202	2	no	—
84	C₂×Q₈	—	202	2	no	—
85	C₂²×C₄	—	202	2	no	—
86	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
87	Q₁₆	—	202	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	202	2	no	—
90	M₁₆	—	202	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	202	2	no	—
94	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
95	C₄⋊C₄	D₄₀₄	1	2	yes	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	#13108074	1	1	yes	characteristic
99	Q₈	—	404	2	no	—
100	Q₈	—	404	2	no	—
101	Q₈	—	202	4	no	—
102	Q₈	—	202	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	404	2	no	—
105	C₂×C₄	—	404	2	no	—
106	C₂×C₄	—	202	4	no	—
107	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
109	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
110	C₂×C₄	#13108075	1	2	yes	—
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	404	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#15692229	1	1	yes	characteristic
116	C₂²	#15692240	1	2	yes	—
117	C₄	—	4	2	no	—
118	C₄	—	404	4	no	—
119	C₄	#15692223	1	1	yes	characteristic
120	C₄	#15692230	1	2	yes	—
121	C₄	#15692244	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#20265094	1	1	yes	characteristic
124	C₂	#20265452	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₄₀₄

C₂

yes

maximal, characteristic

#20265185

C₂

yes

maximal

C₂×#15692173

C₂

yes

maximal

#20264942

C₂

yes

maximal, characteristic

#424×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₂²

yes

—

C₂×Dic₄₀₄

—

C₂×Dic₄₀₄

—

C₂×#13108058

C₂²

yes

—

#15692178

—

#15692173

—

C₂×M₁₆×C₁₀₁

C₂²

yes

characteristic

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

#117×C₁₀₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₁

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

D₄

yes

—

C₂²×Dic₁₀₁

—

Dic₄₀₄

—

#13108058

—

#13108058

—

C₂×#11717579

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

#41

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

C₈×C₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#121

yes

characteristic

C₂²×C₁₀₁

#123

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#120

yes

characteristic

C₄×C₁₀₁

#121

yes

—

C₄×C₁₀₁

#124

yes

characteristic

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#394

yes

characteristic

C₂×C₁₀₁

#473

yes

—

#2047

—

101

maximal, Sylow

C₁₀₁

#2047

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

101

—

#424

D₁₀₁

yes

characteristic

#419

—

101

—

C₂×#101

—

101

—

#341

—

101

—

C₂²×Q₈

—

101

—

C₂×Q₁₆

—

202

—

C₂×Q₁₆

—

202

—

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

#117

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

#103

—

202

—

#101

—

202

—

C₂×Q₈

—

101

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

Q₁₆

—

202

—

M₁₆

—

M₁₆

—

202

—

M₁₆

—

202

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

202

—

C₂×C₈

C₂²⋊D₁₀₁

yes

—

C₄⋊C₄

D₄₀₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#13108074

yes

characteristic

Q₈

—

404

—

100

Q₈

—

404

—

101

Q₈

—

202

—

102

Q₈

—

202

—

103

C₂×C₄

—

104

C₂×C₄

—

404

—

105

C₂×C₄

—

404

—

106

C₂×C₄

—

202

—

107

C₂×C₄

D₄×D₁₀₁

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

109

C₂×C₄

C₂×D₄₀₄

yes

characteristic

110

C₂×C₄

#13108075

yes

—

111

C₈

—

112

C₈

—

113

C₈

—

404

—

114

C₂²

—

115

C₂²

#15692229

yes

characteristic

116

C₂²

#15692240

yes

—

117

C₄

—

118

C₄

—

404

—

119

C₄

#15692223

yes

characteristic

120

C₄

#15692230

yes

—

121

C₄

#15692244

yes

characteristic

122

C₂

—

123

C₂

#20265094

yes

characteristic

124

C₂

#20265452

yes

—

	subgroup	G/H
center	C₂²	#15692229
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#424×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#121
prime radical	C₂³×C₁₀₁	#41
hypercenter	#424	D₁₀₁
hypocenter	C₁₀₁	#2047
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#15692229

commutator, cosocle

C₂²×C₄×C₁₀₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#424×C₁₀₁

C₂

C₂²×C₁₀₁

C₂³×C₁₀₁

D₁₀₁

C₁₀₁

G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
63. C₁₀₁	#2047	semidirect

H ≅ G/N

type

63. C₁₀₁

#2047

semidirect

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

subgroup	index
C₂²×Dic₄₀₄	2
#20265185	2
C₂×#15692173	2
#20264942	2
#424×C₁₀₁	2
#2047	101

subgroup

index

C₂²×Dic₄₀₄

#424×C₁₀₁

101

Subgroups — #27172400

Conjugacy classes of subgroups

Complements

Maximal subgroups