Group #27171181 — subgroups

Subgroups — #27171181

Order 12928.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×#15692263	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692190	C₂	1	2	yes	maximal, complemented
3	#20265159	C₂	1	2	yes	maximal
4	#20264935	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13108074	C₂²	1	1	yes	characteristic
7	#15692263	—	2	4	no	—
8	C₂×#13108058	C₂²	1	2	yes	—
9	C₂×#13108057	C₄	1	2	yes	—
10	C₂²×#11717579	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₀₁	C₄	1	2	yes	—
12	#15692190	—	2	4	no	—
13	#15692173	—	2	2	no	—
14	#15692171	—	2	2	no	—
15	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₀₁	C₂²	1	1	yes	characteristic
17	#13108074	—	2	2	no	—
18	#13108074	—	2	2	no	—
19	C₂×Dic₂₀₂	D₄	1	1	yes	characteristic
20	C₂×D₄₀₄	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₁₀₁	Q₈	1	2	yes	—
22	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
23	#13108058	—	2	2	no	—
24	#13108058	—	2	2	no	—
25	#13108057	—	2	4	no	—
26	C₂×#11717579	—	2	2	no	—
27	C₂×#11717579	C₂×C₄	1	1	yes	characteristic
28	C₂×#11717579	C₂×C₄	1	1	yes	characteristic
29	C₂×#11717579	D₄	1	4	yes	—
30	C₈×D₁₀₁	—	2	4	no	—
31	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₀₁	—	2	2	no	—
33	M₁₆×C₁₀₁	—	2	2	no	—
34	M₁₆×C₁₀₁	—	2	2	no	—
35	C₂×C₈×C₁₀₁	—	2	2	no	—
36	C₂×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
37	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₁₀₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₁	#41	1	2	yes	—
43	C₂²⋊D₁₀₁	—	2	4	no	—
44	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
45	D₄₀₄	—	2	2	no	—
46	C₄×D₁₀₁	—	2	2	no	—
47	C₄×D₁₀₁	—	2	2	no	—
48	Dic₂₀₂	—	2	2	no	—
49	C₈×C₁₀₁	—	2	2	no	—
50	C₈×C₁₀₁	—	4	2	no	—
51	#11717579	—	4	2	no	—
52	#11717579	C₂²⋊C₄	1	2	yes	—
53	#11717579	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₁₀₁	—	4	1	no	—
55	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₁₀₁	—	2	2	no	—
58	C₂×D₁₀₁	—	4	2	no	—
59	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₀₁	#117	1	1	yes	characteristic
61	C₄×C₁₀₁	#121	1	2	yes	—
62	Dic₁₀₁	—	2	2	no	—
63	C₂×C₁₀₁	—	2	1	no	—
64	C₂×C₁₀₁	—	2	2	no	—
65	C₂×C₁₀₁	#380	1	1	yes	characteristic
66	D₁₀₁	—	4	2	no	—
67	#1813	—	101	1	no	maximal, Sylow
68	C₁₀₁	#1813	1	1	yes	characteristic, complemented, Sylow
69	C₂×#131	—	101	1	no	—
70	C₂×#108	—	101	2	no	—
71	#413	—	101	2	no	—
72	#338	—	101	1	no	—
73	#338	D₁₀₁	1	1	yes	characteristic, complemented
74	C₂×#41	—	101	1	no	—
75	#131	—	202	4	no	—
76	C₂×M₁₆	—	101	2	no	—
77	C₂×M₁₆	—	101	2	no	—
78	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
79	C₂×M₁₆	C₂×D₁₀₁	1	1	yes	characteristic
80	C₂²×C₈	—	101	1	no	—
81	C₂²×C₈	—	101	2	no	—
82	#108	—	202	4	no	—
83	#101	—	202	2	no	—
84	#100	—	202	2	no	—
85	#41	—	202	2	no	—
86	#41	—	202	2	no	—
87	C₂×Q₈	—	101	1	no	—
88	C₂×D₄	—	101	2	no	—
89	C₂×D₄	—	101	1	no	—
90	C₂²×C₄	—	101	2	no	—
91	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	202	4	no	—
96	M₁₆	—	202	2	no	—
97	M₁₆	—	202	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	101	1	no	—
100	C₂×C₈	—	101	1	no	—
101	C₂×C₈	—	101	4	no	—
102	C₂×C₈	—	202	4	no	—
103	C₂×C₈	—	202	2	no	—
104	C₂×C₈	C₄×D₁₀₁	1	2	yes	—
105	C₂³	—	101	2	no	—
106	C₂³	Q₈×D₁₀₁	1	1	yes	characteristic
107	Q₈	—	202	2	no	—
108	D₄	—	202	4	no	—
109	D₄	—	202	2	no	—
110	C₂×C₄	—	101	2	no	—
111	C₂×C₄	—	202	2	no	—
112	C₂×C₄	—	202	2	no	—
113	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
114	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
115	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
116	C₂×C₄	#13108074	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	101	2	no	—
120	C₈	—	101	2	no	—
121	C₈	—	404	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	404	2	no	—
124	C₂²	—	202	2	no	—
125	C₂²	C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
126	C₂²	#15692233	1	2	yes	—
127	C₄	—	202	2	no	—
128	C₄	C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
129	C₄	#15692212	1	1	yes	characteristic
130	C₄	#15692228	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	404	2	no	—
134	C₂	#20265036	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692263

C₂

yes

maximal, characteristic

C₂×#15692190

C₂

yes

maximal, complemented

#20265159

C₂

yes

maximal

#20264935

C₂

yes

maximal, characteristic, complemented

#338×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#13108074

C₂²

yes

characteristic

#15692263

—

C₂×#13108058

C₂²

yes

—

C₂×#13108057

C₄

yes

—

C₂²×#11717579

C₂²

yes

characteristic

C₂×C₈×D₁₀₁

C₄

yes

—

#15692190

—

#15692173

—

#15692171

—

C₂×M₁₆×C₁₀₁

C₂²

yes

—

C₂×M₁₆×C₁₀₁

C₂²

yes

characteristic

#13108074

—

#13108074

—

C₂×Dic₂₀₂

D₄

yes

characteristic

C₂×D₄₀₄

D₄

yes

characteristic

C₂×C₂²⋊D₁₀₁

Q₈

yes

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11717579

C₂×C₄

yes

characteristic

C₂×#11717579

D₄

yes

—

C₈×D₁₀₁

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

C₂×C₄

yes

—

C₂³×C₁₀₁

C₂×Q₈

yes

characteristic

C₂²×D₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

D₄₀₄

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

Dic₂₀₂

—

C₈×C₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

#11717579

C₂²⋊C₄

yes

—

#11717579

C₂²⋊C₄

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₂²⋊Q₈

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

C₂×C₂²⋊C₄

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#380

yes

characteristic

D₁₀₁

—

#1813

—

101

maximal, Sylow

C₁₀₁

#1813

yes

characteristic, complemented, Sylow

C₂×#131

—

101

—

C₂×#108

—

101

—

#413

—

101

—

#338

—

101

—

#338

D₁₀₁

yes

characteristic, complemented

C₂×#41

—

101

—

#131

—

202

—

C₂×M₁₆

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₂×M₁₆

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

—

101

—

C₂²×C₈

—

101

—

#108

—

202

—

#101

—

202

—

#100

—

202

—

#41

—

202

—

#41

—

202

—

C₂×Q₈

—

101

—

C₂×D₄

—

101

—

C₂×D₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

202

—

M₁₆

—

202

—

M₁₆

—

202

—

C₂×C₈

—

C₂×C₈

—

101

—

100

C₂×C₈

—

101

—

101

C₂×C₈

—

101

—

102

C₂×C₈

—

202

—

103

C₂×C₈

—

202

—

104

C₂×C₈

C₄×D₁₀₁

yes

—

105

C₂³

—

101

—

106

C₂³

Q₈×D₁₀₁

yes

characteristic

107

Q₈

—

202

—

108

D₄

—

202

—

109

D₄

—

202

—

110

C₂×C₄

—

101

—

111

C₂×C₄

—

202

—

112

C₂×C₄

—

202

—

113

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

114

C₂×C₄

D₄×D₁₀₁

yes

—

115

C₂×C₄

D₄×D₁₀₁

yes

characteristic

116

C₂×C₄

#13108074

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

101

—

120

C₈

—

101

—

121

C₈

—

404

—

122

C₂²

—

123

C₂²

—

404

—

124

C₂²

—

202

—

125

C₂²

C₄⋊C₄×D₁₀₁

yes

characteristic

126

C₂²

#15692233

yes

—

127

C₄

—

202

—

128

C₄

C₂²⋊C₄×D₁₀₁

yes

characteristic

129

C₄

#15692212

yes

characteristic

130

C₄

#15692228

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

404

—

134

C₂

#20265036

yes

characteristic

	subgroup	G/H
center	C₄	#15692212
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#338×C₁₀₁	C₂
socle	C₂×C₁₀₁	#380
prime radical	C₂×#13108074	C₂²
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#338	D₁₀₁
hypocenter	C₁₀₁	#1813
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#15692212

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#338×C₁₀₁

C₂

C₂×C₁₀₁

C₂²

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#15692190	C₂	semidirect
4. #20264935	C₂	semidirect
5. #338×C₁₀₁	C₂	semidirect
68. C₁₀₁	#1813	semidirect
73. #338	D₁₀₁	semidirect

H ≅ G/N

type

2. C₂×#15692190

C₂

semidirect

4. #20264935

C₂

semidirect

5. #338×C₁₀₁

C₂

semidirect

68. C₁₀₁

#1813

semidirect

73. #338

D₁₀₁

semidirect

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

subgroup	index
C₂×#15692263	2
C₂×#15692190	2
#20265159	2
#20264935	2
#338×C₁₀₁	2
#1813	101

subgroup

index

#338×C₁₀₁

101

Subgroups — #27171181

Conjugacy classes of subgroups

Complements

Maximal subgroups