Group #27172696 — subgroups

Subgroups — #27172696

Order 12928.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20265075	C₂	1	1	yes	maximal, characteristic
2	#20265047	C₂	1	1	yes	maximal, characteristic
3	#20264940	C₂	1	2	yes	maximal
4	#20264940	C₂	1	2	yes	maximal
5	#336×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
7	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
8	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
11	C₂×#13108053	—	2	1	no	—
12	#15692162	—	2	1	no	—
13	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₁	—	2	1	no	—
16	C₂²×Dic₁₀₁	—	2	1	no	—
17	C₂²×Dic₁₀₁	—	2	2	no	—
18	C₄⋊Dic₁₀₁	—	2	1	no	—
19	C₄⋊Dic₁₀₁	—	2	2	no	—
20	C₄⋊Dic₁₀₁	—	2	4	no	—
21	C₄⋊Dic₁₀₁	—	4	2	no	—
22	C₄⋊Dic₁₀₁	—	4	2	no	—
23	C₄⋊Dic₁₀₁	Q₈	1	2	yes	—
24	#13108053	—	4	2	no	—
25	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₀₁	—	2	2	no	—
29	C₂×C₈×C₁₀₁	—	2	2	no	—
30	C₂×C₈×C₁₀₁	—	2	2	no	—
31	C₄²×C₁₀₁	—	2	2	no	—
32	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₁	—	2	1	no	—
34	C₂×C₄×C₁₀₁	—	2	1	no	—
35	C₂×C₄×C₁₀₁	—	2	1	no	—
36	C₂×C₄×C₁₀₁	—	2	1	no	—
37	C₂×C₄×C₁₀₁	SD₁₆	1	2	yes	—
38	C₂×C₄×C₁₀₁	Q₁₆	1	2	yes	—
39	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₁	#41	1	2	yes	—
41	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₀₁	#41	1	2	yes	—
43	C₂×Dic₁₀₁	—	2	2	no	—
44	C₂×Dic₁₀₁	—	4	1	no	—
45	C₂×Dic₁₀₁	—	4	1	no	—
46	C₂×Dic₁₀₁	—	4	1	no	—
47	C₂×Dic₁₀₁	—	4	2	no	—
48	C₂×Dic₁₀₁	—	4	1	no	—
49	C₂×Dic₁₀₁	—	4	2	no	—
50	C₂×Dic₁₀₁	—	4	2	no	—
51	C₂×Dic₁₀₁	—	4	1	no	—
52	C₈×C₁₀₁	—	4	2	no	—
53	C₂²×C₁₀₁	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₁₀₁	#123	1	1	yes	characteristic
55	C₂²×C₁₀₁	#124	1	1	yes	characteristic
56	C₂²×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic
59	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₄×C₁₀₁	—	2	2	no	—
61	C₄×C₁₀₁	—	2	2	no	—
62	C₄×C₁₀₁	#125	1	2	yes	—
63	C₄×C₁₀₁	#126	1	2	yes	—
64	Dic₁₀₁	—	4	2	no	—
65	Dic₁₀₁	—	8	1	no	—
66	Dic₁₀₁	—	8	2	no	—
67	C₂×C₁₀₁	#400	1	1	yes	characteristic
68	C₂×C₁₀₁	#476	1	1	yes	characteristic
69	C₂×C₁₀₁	#477	1	1	yes	characteristic
70	C₂×C₁₀₁	#481	1	1	yes	characteristic
71	C₂×C₁₀₁	#484	1	1	yes	characteristic
72	C₂×C₁₀₁	#487	1	1	yes	characteristic
73	C₂×C₁₀₁	#490	1	1	yes	characteristic
74	#2100	—	101	1	no	maximal, Sylow
75	C₁₀₁	#2100	1	1	yes	characteristic, complemented, Sylow
76	#389	—	101	1	no	—
77	#384	—	101	1	no	—
78	#340	—	101	2	no	—
79	#340	—	101	2	no	—
80	#336	D₁₀₁	1	1	yes	characteristic
81	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
82	C₂×C₄⋊C₄	—	101	1	no	—
83	C₂×C₄⋊C₄	—	101	2	no	—
84	C₂×C₄⋊C₄	—	101	1	no	—
85	C₂×C₄⋊C₄	—	202	1	no	—
86	C₂×C₄⋊C₄	—	202	1	no	—
87	C₂×C₄⋊C₄	—	202	1	no	—
88	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
89	#95	—	202	1	no	—
90	C₂²×C₄	—	202	1	no	—
91	C₂²×C₄	—	202	2	no	—
92	C₂²×C₄	—	202	1	no	—
93	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
94	C₂²×C₄	D₄₀₄	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₄⋊C₄	—	101	2	no	—
100	C₄⋊C₄	—	404	2	no	—
101	C₄⋊C₄	—	404	2	no	—
102	C₄⋊C₄	—	404	2	no	—
103	C₄⋊C₄	—	202	2	no	—
104	C₄⋊C₄	—	202	4	no	—
105	C₄⋊C₄	—	202	1	no	—
106	C₄²	—	2	2	no	—
107	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	404	1	no	—
113	C₂×C₄	—	404	2	no	—
114	C₂×C₄	—	404	1	no	—
115	C₂×C₄	—	404	2	no	—
116	C₂×C₄	—	404	2	no	—
117	C₂×C₄	—	404	1	no	—
118	C₂×C₄	—	404	1	no	—
119	C₂×C₄	—	404	1	no	—
120	C₂×C₄	—	202	2	no	—
121	C₂×C₄	#13108063	1	2	yes	—
122	C₂×C₄	Dic₄₀₄	1	2	yes	—
123	C₂×C₄	Q₈×D₁₀₁	1	1	yes	characteristic
124	C₂×C₄	#13108073	1	2	yes	—
125	C₂×C₄	#13108073	1	1	yes	characteristic
126	C₂×C₄	#13108074	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₂²	#15692235	1	1	yes	characteristic
129	C₂²	#15692239	1	1	yes	characteristic
130	C₂²	#15692243	1	1	yes	characteristic
131	C₂²	C₂×#13108063	1	1	yes	characteristic
132	C₂²	C₂×Dic₄₀₄	1	1	yes	characteristic
133	C₂²	#15692283	1	1	yes	characteristic
134	C₂²	#15692283	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	808	1	no	—
138	C₄	—	808	2	no	—
139	C₄	—	404	2	no	—
140	C₄	#15692245	1	2	yes	—
141	C₄	#15692248	1	2	yes	—
142	C₂	#20265117	1	1	yes	characteristic
143	C₂	#20265468	1	1	yes	characteristic
144	C₂	#20265473	1	1	yes	characteristic
145	C₂	#20265496	1	1	yes	characteristic
146	C₂	#20265513	1	1	yes	characteristic
147	C₂	#20265528	1	1	yes	characteristic
148	C₂	#20265542	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265075

C₂

yes

maximal, characteristic

#20265047

C₂

yes

maximal, characteristic

#20264940

C₂

yes

maximal

#20264940

C₂

yes

maximal

#336×C₁₀₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₂×#13108053

—

#15692162

—

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

Q₈

yes

—

#13108053

—

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄²×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

SD₁₆

yes

—

C₂×C₄×C₁₀₁

Q₁₆

yes

—

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

C₂²⋊Q₈

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₁

C₂×Q₁₆

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#125

yes

—

C₄×C₁₀₁

#126

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#2100

yes

characteristic, complemented, Sylow

#389

—

101

—

#384

—

101

—

#340

—

101

—

#340

—

101

—

#336

D₁₀₁

yes

characteristic

C₂²×C₈

C₂×D₁₀₁

yes

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#95

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

D₄₀₄

yes

characteristic

C₂²×C₄

D₄₀₄

yes

characteristic

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

101

—

100

C₄⋊C₄

—

404

—

101

C₄⋊C₄

—

404

—

102

C₄⋊C₄

—

404

—

103

C₄⋊C₄

—

202

—

104

C₄⋊C₄

—

202

—

105

C₄⋊C₄

—

202

—

106

C₄²

—

107

C₂³

C₂×D₄₀₄

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

404

—

113

C₂×C₄

—

404

—

114

C₂×C₄

—

404

—

115

C₂×C₄

—

404

—

116

C₂×C₄

—

404

—

117

C₂×C₄

—

404

—

118

C₂×C₄

—

404

—

119

C₂×C₄

—

404

—

120

C₂×C₄

—

202

—

121

C₂×C₄

#13108063

yes

—

122

C₂×C₄

Dic₄₀₄

yes

—

123

C₂×C₄

Q₈×D₁₀₁

yes

characteristic

124

C₂×C₄

#13108073

yes

—

125

C₂×C₄

#13108073

yes

characteristic

126

C₂×C₄

#13108074

yes

—

127

C₈

—

128

C₂²

#15692235

yes

characteristic

129

C₂²

#15692239

yes

characteristic

130

C₂²

#15692243

yes

characteristic

131

C₂²

C₂×#13108063

yes

characteristic

132

C₂²

C₂×Dic₄₀₄

yes

characteristic

133

C₂²

#15692283

yes

characteristic

134

C₂²

#15692283

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

808

—

138

C₄

—

808

—

139

C₄

—

404

—

140

C₄

#15692245

yes

—

141

C₄

#15692248

yes

—

142

C₂

#20265117

yes

characteristic

143

C₂

#20265468

yes

characteristic

144

C₂

#20265473

yes

characteristic

145

C₂

#20265496

yes

characteristic

146

C₂

#20265513

yes

characteristic

147

C₂

#20265528

yes

characteristic

148

C₂

#20265542

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₀₄
commutator, cosocle	C₂²×C₄×C₁₀₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#336×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
hypercenter	#336	D₁₀₁
hypocenter	C₁₀₁	#2100
LCS	G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₄₀₄

commutator, cosocle

C₂²×C₄×C₁₀₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#336×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

D₁₀₁

C₁₀₁

G ▷ C₂²×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
75. C₁₀₁	#2100	semidirect

H ≅ G/N

type

75. C₁₀₁

#2100

semidirect

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

subgroup	index
#20265075	2
#20265047	2
#20264940	2
#336×C₁₀₁	2
#2100	101

subgroup

index

#336×C₁₀₁

101

Subgroups — #27172696

Conjugacy classes of subgroups

Complements

Maximal subgroups