Group #27171201 — subgroups

Subgroups — #27171201

Order 12928.

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

normal subgroups only

This group has 292 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	C₂×#15692214	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#11717579	C₂	1	1	yes	maximal, characteristic
3	#20264931	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11717579	C₂²	1	2	yes	—
6	#15692214	—	2	2	no	—
7	#15692214	—	2	2	no	—
8	C₂×C₄⋊Dic₁₀₁	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₀₁	C₄	1	1	yes	characteristic
10	C₂×C₂²⋊Dic₁₀₁	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
12	C₂²⋊#11717579	—	2	2	no	—
13	C₂³×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₀₁	—	2	2	no	—
16	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
17	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
18	C₂×#11717579	—	2	2	no	—
19	C₂×#11717579	—	2	4	no	—
20	C₄⋊Dic₁₀₁	—	2	2	no	—
21	C₄⋊Dic₁₀₁	—	2	2	no	—
22	C₂²⋊Dic₁₀₁	—	2	4	no	—
23	C₄×Dic₁₀₁	—	2	2	no	—
24	C₄×Dic₁₀₁	—	2	2	no	—
25	C₂⁴×C₁₀₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₁	—	2	4	no	—
27	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
29	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
30	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₀₁	—	2	2	no	—
32	C₂×C₈×C₁₀₁	—	2	4	no	—
33	C₂³×C₁₀₁	—	2	2	no	—
34	C₂³×C₁₀₁	—	2	2	no	—
35	C₂³×C₁₀₁	—	2	2	no	—
36	C₂³×C₁₀₁	C₂²⋊C₄	1	2	yes	—
37	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₁	—	2	2	no	—
39	C₂×C₄×C₁₀₁	—	2	4	no	—
40	C₂×C₄×C₁₀₁	—	2	4	no	—
41	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₁	#41	1	4	yes	—
45	C₂×Dic₁₀₁	—	2	2	no	—
46	C₂×Dic₁₀₁	—	2	2	no	—
47	C₂×Dic₁₀₁	—	2	4	no	—
48	C₂×Dic₁₀₁	—	2	4	no	—
49	C₈×C₁₀₁	—	4	2	no	—
50	#11717579	—	4	2	no	—
51	C₂²×C₁₀₁	—	2	2	no	—
52	C₂²×C₁₀₁	—	2	2	no	—
53	C₂²×C₁₀₁	—	2	2	no	—
54	C₂²×C₁₀₁	—	4	4	no	—
55	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₀₁	#117	1	2	yes	—
57	C₂²×C₁₀₁	#131	1	2	yes	—
58	C₂²×C₁₀₁	#131	1	2	yes	—
59	C₄×C₁₀₁	—	2	2	no	—
60	C₄×C₁₀₁	#123	1	2	yes	—
61	C₄×C₁₀₁	#123	1	2	yes	—
62	Dic₁₀₁	—	4	2	no	—
63	Dic₁₀₁	—	4	2	no	—
64	C₂×C₁₀₁	—	4	2	no	—
65	C₂×C₁₀₁	#381	1	1	yes	characteristic
66	C₂×C₁₀₁	#408	1	1	yes	characteristic
67	C₂×C₁₀₁	#408	1	1	yes	characteristic
68	C₂×C₁₀₁	#433	1	4	yes	—
69	#1818	—	101	1	no	maximal, Sylow
70	C₁₀₁	#1818	1	1	yes	characteristic, complemented, Sylow
71	C₂×#117	—	101	1	no	—
72	C₂×C₂²⋊C₈	—	101	1	no	—
73	C₂×C₂²⋊C₈	D₁₀₁	1	1	yes	characteristic
74	#336	—	101	4	no	—
75	C₂³×C₄	C₂×D₁₀₁	1	1	yes	characteristic
76	C₂²×C₈	—	101	2	no	—
77	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
78	#117	—	202	2	no	—
79	#117	—	202	2	no	—
80	C₂×C₄⋊C₄	—	101	1	no	—
81	C₂×C₄⋊C₄	—	101	1	no	—
82	C₂×C₂²⋊C₄	—	101	2	no	—
83	C₂×C₄²	—	101	2	no	—
84	C₂²⋊C₈	—	2	2	no	—
85	C₂²⋊C₈	—	202	2	no	—
86	C₂⁴	C₄×D₁₀₁	1	1	yes	characteristic
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	—	101	2	no	—
89	C₂²×C₄	—	101	2	no	—
90	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
91	C₂²×C₄	D₄₀₄	1	2	yes	—
92	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	202	2	no	—
97	C₂×C₈	—	202	4	no	—
98	C₄⋊C₄	—	202	2	no	—
99	C₄⋊C₄	—	202	2	no	—
100	C₂²⋊C₄	—	202	4	no	—
101	C₄²	—	202	2	no	—
102	C₄²	—	202	2	no	—
103	C₂³	—	2	2	no	—
104	C₂³	—	2	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	#13108051	1	2	yes	—
107	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	202	2	no	—
112	C₂×C₄	—	202	2	no	—
113	C₂×C₄	—	202	4	no	—
114	C₂×C₄	—	202	4	no	—
115	C₂×C₄	#13108051	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
117	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
118	C₂×C₄	#13108073	1	4	yes	—
119	C₈	—	4	2	no	—
120	C₈	—	404	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	4	4	no	—
125	C₂²	C₂×#13108051	1	1	yes	characteristic
126	C₂²	#15692211	1	2	yes	—
127	C₂²	#15692261	1	2	yes	—
128	C₂²	#15692263	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	404	2	no	—
131	C₄	—	404	2	no	—
132	C₄	#15692236	1	2	yes	—
133	C₄	#15692239	1	2	yes	—
134	C₂	—	4	2	no	—
135	C₂	#20265038	1	1	yes	characteristic
136	C₂	#20265143	1	1	yes	characteristic
137	C₂	#20265145	1	1	yes	characteristic
138	C₂	#20265235	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692214

C₂

yes

maximal, characteristic

C₂×C₂²⋊#11717579

C₂

yes

maximal, characteristic

#20264931

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

#15692214

—

#15692214

—

C₂×C₄⋊Dic₁₀₁

C₄

yes

characteristic

C₂×C₄⋊Dic₁₀₁

C₄

yes

characteristic

C₂×C₂²⋊Dic₁₀₁

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

C₂²⋊#11717579

—

C₂³×C₄×C₁₀₁

C₂²

yes

characteristic

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂²⋊C₈×C₁₀₁

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂×#11717579

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂⁴×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

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

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

#117

yes

—

C₂²×C₁₀₁

#131

yes

—

C₂²×C₁₀₁

#131

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#123

yes

—

C₄×C₁₀₁

#123

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

#1818

—

101

maximal, Sylow

C₁₀₁

#1818

yes

characteristic, complemented, Sylow

C₂×#117

—

101

—

C₂×C₂²⋊C₈

—

101

—

C₂×C₂²⋊C₈

D₁₀₁

yes

characteristic

#336

—

101

—

C₂³×C₄

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

—

#117

—

202

—

#117

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₄²

—

101

—

C₂²⋊C₈

—

C₂²⋊C₈

—

202

—

C₂⁴

C₄×D₁₀₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₄×D₁₀₁

yes

characteristic

C₂²×C₄

D₄₀₄

yes

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

100

C₂²⋊C₄

—

202

—

101

C₄²

—

202

—

102

C₄²

—

202

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

#13108051

yes

—

107

C₂³

C₂×C₄×D₁₀₁

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

202

—

112

C₂×C₄

—

202

—

113

C₂×C₄

—

202

—

114

C₂×C₄

—

202

—

115

C₂×C₄

#13108051

yes

—

116

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

117

C₂×C₄

C₂×D₄₀₄

yes

characteristic

118

C₂×C₄

#13108073

yes

—

119

C₈

—

120

C₈

—

404

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×#13108051

yes

characteristic

126

C₂²

#15692211

yes

—

127

C₂²

#15692261

yes

—

128

C₂²

#15692263

yes

—

129

C₄

—

130

C₄

—

404

—

131

C₄

—

404

—

132

C₄

#15692236

yes

—

133

C₄

#15692239

yes

—

134

C₂

—

135

C₂

#20265038

yes

characteristic

136

C₂

#20265143

yes

characteristic

137

C₂

#20265145

yes

characteristic

138

C₂

#20265235

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₁
commutator, socle	C₂³×C₁₀₁	C₂²×C₄
Fitting	C₂×C₂²⋊C₈×C₁₀₁	C₂
prime radical	C₂⁴×C₁₀₁	C₂×C₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×C₂²⋊C₈	D₁₀₁
hypocenter	C₁₀₁	#1818
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₁

commutator, socle

C₂³×C₁₀₁

C₂²×C₄

Fitting

C₂×C₂²⋊C₈×C₁₀₁

C₂

prime radical

C₂⁴×C₁₀₁

C₂×C₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂×C₂²⋊C₈

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #20264931	C₂	semidirect
70. C₁₀₁	#1818	semidirect

H ≅ G/N

type

3. #20264931

C₂

semidirect

70. C₁₀₁

#1818

semidirect

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

subgroup	index
C₂×#15692214	2
C₂×C₂²⋊#11717579	2
#20264931	2
C₂×C₂²⋊C₈×C₁₀₁	2
#1818	101

subgroup

index

C₂×#15692214

C₂×C₂²⋊#11717579

#20264931

C₂×C₂²⋊C₈×C₁₀₁

#1818

101

Subgroups — #27171201

Conjugacy classes of subgroups

Complements

Maximal subgroups