Group #27182586 — subgroups

Subgroups — #27182586

Order 12928.

322 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 117 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	#20265789	C₂	1	2	yes	maximal
2	#20265485	C₂	1	4	yes	maximal, complemented
3	#20265435	C₂	1	2	yes	maximal
4	#20265418	C₂	1	2	yes	maximal
5	#20265388	C₂	1	4	yes	maximal, complemented
6	#424×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₄₀₄	C₂²	1	2	yes	—
8	C₂×Dic₄₀₄	C₂²	1	2	yes	—
9	#15692255	C₂²	1	2	yes	—
10	#15692248	—	2	4	no	—
11	#15692247	C₂²	1	2	yes	—
12	#15692245	—	2	4	no	—
13	#15692234	C₂²	1	4	yes	—
14	C₄×Dic₂₀₂	C₂²	1	4	yes	—
15	#15692189	C₂²	1	2	yes	—
16	#15692187	C₂²	1	2	yes	—
17	#15692180	C₂²	1	4	yes	—
18	#15692180	C₂²	1	4	yes	—
19	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
21	#117×C₁₀₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₀₁	C₂²	1	4	yes	—
23	C₂×Dic₂₀₂	C₂³	1	4	yes	—
24	Dic₄₀₄	—	4	2	no	—
25	Dic₄₀₄	—	4	2	no	—
26	C₄⋊Dic₁₀₁	—	2	4	no	—
27	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
28	#13108053	—	2	4	no	—
29	#13108053	—	2	4	no	—
30	#13108053	—	2	4	no	—
31	C₂²⋊Dic₁₀₁	—	2	4	no	—
32	C₄×Dic₁₀₁	—	2	4	no	—
33	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
34	M₁₆×C₁₀₁	—	4	1	no	—
35	C₂×C₈×C₁₀₁	—	4	1	no	—
36	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
37	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
39	C₂²⋊C₄×C₁₀₁	D₄	1	2	yes	—
40	C₄²×C₁₀₁	C₂³	1	2	yes	—
41	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₁	—	2	1	no	—
43	C₂×C₄×C₁₀₁	—	2	1	no	—
44	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₁₀₁	—	2	4	no	—
48	C₂×Dic₁₀₁	—	2	4	no	—
49	Dic₂₀₂	—	2	4	no	—
50	Dic₂₀₂	—	2	4	no	—
51	C₈×C₁₀₁	—	2	2	no	—
52	C₈×C₁₀₁	—	2	2	no	—
53	C₂²×C₁₀₁	—	2	1	no	—
54	C₂²×C₁₀₁	—	2	2	no	—
55	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₀₁	—	2	1	no	—
57	C₄×C₁₀₁	—	2	4	no	—
58	C₄×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₁₀₁	—	4	4	no	—
61	Dic₁₀₁	—	4	4	no	—
62	C₂×C₁₀₁	—	4	1	no	—
63	C₂×C₁₀₁	#537	1	1	yes	characteristic
64	C₂×C₁₀₁	#578	1	2	yes	—
65	#3261	—	101	1	no	maximal, Sylow
66	C₁₀₁	#3261	1	1	yes	characteristic, complemented, Sylow
67	#541	—	101	2	no	—
68	#479	—	101	4	no	—
69	#470	—	101	2	no	—
70	#467	—	101	2	no	—
71	#462	—	101	4	no	—
72	#424	D₁₀₁	1	1	yes	characteristic
73	C₂×Q₁₆	—	101	2	no	—
74	C₂×Q₁₆	—	101	2	no	—
75	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
77	C₄⋊Q₈	—	101	2	no	—
78	#126	—	202	4	no	—
79	#125	—	101	2	no	—
80	#125	—	202	4	no	—
81	C₂²⋊Q₈	—	101	4	no	—
82	C₄×Q₈	—	101	4	no	—
83	#117	C₂×D₁₀₁	1	1	yes	characteristic
84	#107	—	101	2	no	—
85	#106	—	101	2	no	—
86	C₄⋊C₈	C₂×D₁₀₁	1	4	yes	—
87	#103	—	101	4	no	—
88	#103	—	101	4	no	—
89	C₂×Q₈	—	101	4	no	—
90	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
91	Q₁₆	—	404	2	no	—
92	Q₁₆	—	404	2	no	—
93	M₁₆	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
96	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
97	C₄⋊C₄	—	101	4	no	—
98	C₄⋊C₄	—	202	4	no	—
99	C₄⋊C₄	—	202	4	no	—
100	C₄⋊C₄	—	202	4	no	—
101	C₄⋊C₄	—	202	4	no	—
102	C₄⋊C₄	D₄₀₄	1	2	yes	—
103	C₂²⋊C₄	—	202	4	no	—
104	C₂²⋊C₄	D₄₀₄	1	2	yes	—
105	C₄²	—	202	4	no	—
106	C₄²	C₂²×D₁₀₁	1	2	yes	—
107	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
108	Q₈	—	202	4	no	—
109	Q₈	—	202	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	202	4	no	—
113	C₂×C₄	—	202	4	no	—
114	C₂×C₄	C₂×D₄₀₄	1	4	yes	—
115	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
116	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
122	C₄	—	2	4	no	—
123	C₄	—	2	1	no	—
124	C₄	—	404	4	no	—
125	C₄	—	404	4	no	—
126	C₄	#15692300	1	1	yes	characteristic
127	C₄	#15692303	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#20265758	1	1	yes	characteristic
130	C₂	#20265994	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265789

C₂

yes

maximal

#20265485

C₂

yes

maximal, complemented

#20265435

C₂

yes

maximal

#20265418

C₂

yes

maximal

#20265388

C₂

yes

maximal, complemented

#424×C₁₀₁

C₂

yes

maximal, characteristic

C₂×Dic₄₀₄

C₂²

yes

—

C₂×Dic₄₀₄

C₂²

yes

—

#15692255

C₂²

yes

—

#15692248

—

#15692247

C₂²

yes

—

#15692245

—

#15692234

C₂²

yes

—

C₄×Dic₂₀₂

C₂²

yes

—

#15692189

C₂²

yes

—

#15692187

C₂²

yes

—

#15692180

C₂²

yes

—

#15692180

C₂²

yes

—

C₂×M₁₆×C₁₀₁

C₂²

yes

characteristic

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

#117×C₁₀₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₁

C₂²

yes

—

C₂×Dic₂₀₂

C₂³

yes

—

Dic₄₀₄

—

Dic₄₀₄

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂³

yes

—

#13108053

—

#13108053

—

#13108053

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

C₂³

yes

—

C₂×C₈×C₁₀₁

C₂³

yes

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₂²⋊C₄×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

C₈×C₁₀₁

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

H(5,𝔽₂)

yes

characteristic

C₄×C₁₀₁

2₋¹⁺⁴

yes

characteristic

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#537

yes

characteristic

C₂×C₁₀₁

#578

yes

—

#3261

—

101

maximal, Sylow

C₁₀₁

#3261

yes

characteristic, complemented, Sylow

#541

—

101

—

#479

—

101

—

#470

—

101

—

#467

—

101

—

#462

—

101

—

#424

D₁₀₁

yes

characteristic

C₂×Q₁₆

—

101

—

C₂×Q₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

C₄⋊Q₈

—

101

—

#126

—

202

—

#125

—

101

—

#125

—

202

—

C₂²⋊Q₈

—

101

—

C₄×Q₈

—

101

—

#117

C₂×D₁₀₁

yes

characteristic

#107

—

101

—

#106

—

101

—

C₄⋊C₈

C₂×D₁₀₁

yes

—

#103

—

101

—

#103

—

101

—

C₂×Q₈

—

101

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

Q₁₆

—

404

—

Q₁₆

—

404

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²×D₁₀₁

yes

—

C₂×C₈

C₂²×D₁₀₁

yes

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

100

C₄⋊C₄

—

202

—

101

C₄⋊C₄

—

202

—

102

C₄⋊C₄

D₄₀₄

yes

—

103

C₂²⋊C₄

—

202

—

104

C₂²⋊C₄

D₄₀₄

yes

—

105

C₄²

—

202

—

106

C₄²

C₂²×D₁₀₁

yes

—

107

C₂³

C₂×D₄₀₄

yes

characteristic

108

Q₈

—

202

—

109

Q₈

—

202

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

202

—

113

C₂×C₄

—

202

—

114

C₂×C₄

C₂×D₄₀₄

yes

—

115

C₂×C₄

C₂×D₄₀₄

yes

characteristic

116

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂²×D₄₀₄

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

404

—

125

C₄

—

404

—

126

C₄

#15692300

yes

characteristic

127

C₄

#15692303

yes

characteristic

128

C₂

—

129

C₂

#20265758

yes

characteristic

130

C₂

#20265994

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₄₀₄
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#424×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂³×C₁₀₁	C₂×D₄
hypercenter	#424	D₁₀₁
hypocenter	C₁₀₁	#3261
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₄₀₄

commutator, cosocle

C₂×C₄×C₁₀₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting

#424×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂²×D₄

prime radical

C₂³×C₁₀₁

C₂×D₄

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
2. #20265485	C₂	semidirect
5. #20265388	C₂	semidirect
66. C₁₀₁	#3261	semidirect

H ≅ G/N

type

2. #20265485

C₂

semidirect

5. #20265388

C₂

semidirect

66. C₁₀₁

#3261

semidirect

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

subgroup	index
#20265789	2
#20265485	2
#20265435	2
#20265418	2
#20265388	2
#424×C₁₀₁	2
#3261	101

subgroup

index

#424×C₁₀₁

101

Subgroups — #27182586

Conjugacy classes of subgroups

Complements

Maximal subgroups