Group #27171856 — subgroups

Subgroups — #27171856

Order 12928.

248 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 121 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×C₄⋊#11717579	C₂	1	2	yes	maximal
2	C₂×C₄⋊#11717579	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692166	C₂	1	1	yes	maximal, characteristic
4	#20264932	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11717579	C₂²	1	2	yes	—
7	C₂²×#11717579	C₂²	1	1	yes	characteristic
8	C₂²×#11717579	C₂²	1	1	yes	characteristic
9	C₄⋊#11717579	—	2	4	no	—
10	C₄⋊#11717579	—	2	2	no	—
11	#15692166	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₁	C₄	1	2	yes	—
13	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
15	#95×C₁₀₁	C₄	1	1	yes	characteristic
16	#95×C₁₀₁	C₄	1	1	yes	characteristic
17	C₂×#11717579	—	2	1	no	—
18	C₂×#11717579	—	2	1	no	—
19	C₂×#11717579	—	2	1	no	—
20	C₂×#11717579	—	2	1	no	—
21	C₂×#11717579	—	2	2	no	—
22	C₂×#11717579	—	2	2	no	—
23	C₂×#11717579	—	2	2	no	—
24	C₂×#11717579	D₄	1	2	yes	—
25	C₂×#11717579	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₁₀₁	—	2	4	no	—
32	C₄²×C₁₀₁	—	2	2	no	—
33	C₄²×C₁₀₁	—	2	2	no	—
34	C₄²×C₁₀₁	D₄	1	2	yes	—
35	C₄²×C₁₀₁	Q₈	1	2	yes	—
36	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₁	—	2	2	no	—
38	C₂×C₄×C₁₀₁	—	2	2	no	—
39	C₂×C₄×C₁₀₁	—	2	2	no	—
40	C₂×C₄×C₁₀₁	—	2	2	no	—
41	C₂×C₄×C₁₀₁	—	2	2	no	—
42	C₂×C₄×C₁₀₁	—	2	2	no	—
43	C₂×C₄×C₁₀₁	—	2	2	no	—
44	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₀₁	M₁₆	1	4	yes	—
47	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
52	C₂×C₄×C₁₀₁	#41	1	1	yes	characteristic
53	#11717579	—	2	2	no	—
54	#11717579	—	2	2	no	—
55	#11717579	—	4	2	no	—
56	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₀₁	C₄×D₄	1	1	yes	characteristic
58	C₂²×C₁₀₁	C₄×Q₈	1	1	yes	characteristic
59	C₂²×C₁₀₁	C₂×M₁₆	1	2	yes	—
60	C₂²×C₁₀₁	#131	1	2	yes	—
61	C₄×C₁₀₁	—	2	2	no	—
62	C₄×C₁₀₁	—	2	2	no	—
63	C₄×C₁₀₁	—	2	4	no	—
64	C₄×C₁₀₁	—	4	2	no	—
65	C₄×C₁₀₁	#121	1	1	yes	characteristic
66	C₄×C₁₀₁	C₂²⋊Q₈	1	1	yes	characteristic
67	C₄×C₁₀₁	#125	1	1	yes	characteristic
68	C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₁₀₁	#389	1	1	yes	characteristic
70	C₂×C₁₀₁	#423	1	1	yes	characteristic
71	C₂×C₁₀₁	#424	1	1	yes	characteristic
72	C₂×C₁₀₁	#435	1	2	yes	—
73	C₂×C₁₀₁	#446	1	2	yes	—
74	#1940	—	101	1	no	maximal, Sylow
75	C₁₀₁	#1940	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄⋊C₈	—	101	2	no	—
77	C₂×C₄⋊C₈	—	101	1	no	—
78	C₂×#97	—	101	1	no	—
79	C₄×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic
80	#336	—	101	2	no	—
81	C₂²×C₈	—	101	1	no	—
82	C₂²×C₈	—	101	1	no	—
83	C₂²×C₈	—	101	2	no	—
84	C₂×C₄⋊C₄	Dic₁₀₁	1	2	yes	—
85	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
86	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
87	C₄⋊C₈	—	202	4	no	—
88	C₄⋊C₈	—	202	2	no	—
89	#97	—	202	2	no	—
90	#95	Dic₁₀₁	1	1	yes	characteristic
91	#95	Dic₁₀₁	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
93	C₂²×C₄	C₂×Dic₁₀₁	1	1	yes	characteristic
94	C₂²×C₄	C₂×Dic₁₀₁	1	1	yes	characteristic
95	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
97	C₂×C₈	—	101	2	no	—
98	C₂×C₈	—	101	2	no	—
99	C₂×C₈	—	202	2	no	—
100	C₂×C₈	—	202	2	no	—
101	C₂×C₈	—	202	2	no	—
102	C₂×C₈	—	202	1	no	—
103	C₂×C₈	—	202	1	no	—
104	C₂×C₈	—	202	1	no	—
105	C₂×C₈	—	202	1	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄²	—	2	2	no	—
108	C₄²	—	2	2	no	—
109	C₄²	Dic₂₀₂	1	2	yes	—
110	C₄²	D₄₀₄	1	2	yes	—
111	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	C₄⋊Dic₁₀₁	1	2	yes	—
120	C₂×C₄	C₄⋊Dic₁₀₁	1	2	yes	—
121	C₂×C₄	#13108058	1	4	yes	—
122	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
123	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
124	C₂×C₄	Q₈×D₁₀₁	1	1	yes	characteristic
125	C₂×C₄	C₂×Dic₂₀₂	1	1	yes	characteristic
126	C₂×C₄	#13108073	1	1	yes	characteristic
127	C₂×C₄	#13108075	1	1	yes	characteristic
128	C₈	—	404	2	no	—
129	C₈	—	202	2	no	—
130	C₈	—	202	2	no	—
131	C₂²	C₂×C₄⋊Dic₁₀₁	1	1	yes	characteristic
132	C₂²	D₄×Dic₁₀₁	1	1	yes	characteristic
133	C₂²	Q₈×Dic₁₀₁	1	1	yes	characteristic
134	C₂²	C₂×#13108058	1	2	yes	—
135	C₂²	#15692262	1	2	yes	—
136	C₄	—	2	4	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	2	no	—
139	C₄	—	4	2	no	—
140	C₄	#15692230	1	1	yes	characteristic
141	C₄	#15692235	1	1	yes	characteristic
142	C₄	#15692246	1	1	yes	characteristic
143	C₄	#15692254	1	1	yes	characteristic
144	C₂	#20265076	1	1	yes	characteristic
145	C₂	#20265201	1	1	yes	characteristic
146	C₂	#20265204	1	1	yes	characteristic
147	C₂	#20265249	1	2	yes	—
148	C₂	#20265308	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11717579

C₂

yes

maximal

C₂×C₄⋊#11717579

C₂

yes

maximal, characteristic

C₂×#15692166

C₂

yes

maximal, characteristic

#20264932

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

C₂²×#11717579

C₂²

yes

characteristic

C₂²×#11717579

C₂²

yes

—

—

—

C₂×C₄⋊C₄×C₁₀₁

C₄

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

#95×C₁₀₁

C₄

yes

characteristic

#95×C₁₀₁

C₄

yes

—

—

—

—

—

—

—

D₄

yes

—

C₂×#11717579

Q₈

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

Q₈

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

M₁₆

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

C₂×C₄×C₁₀₁

yes

C₂×C₄×C₁₀₁

yes

—

—

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₄×D₄

yes

characteristic

C₂²×C₁₀₁

C₄×Q₈

yes

characteristic

C₂²×C₁₀₁

C₂×M₁₆

yes

—

C₂²×C₁₀₁

#131

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#121

yes

characteristic

C₄×C₁₀₁

C₂²⋊Q₈

yes

characteristic

C₄×C₁₀₁

#125

yes

characteristic

C₄×C₁₀₁

C₄⋊Q₈

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×C₁₀₁

#446

yes

—

#1940

—

101

maximal, Sylow

C₁₀₁

#1940

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

101

—

C₂×C₄⋊C₈

—

101

—

C₂×#97

—

101

—

C₄×C₄⋊C₄

D₁₀₁

yes

characteristic

#336

—

101

—

C₂²×C₈

—

101

—

C₂²×C₈

—

101

—

C₂²×C₈

—

101

—

C₂×C₄⋊C₄

Dic₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

202

—

C₄⋊C₈

—

202

—

#97

—

202

—

#95

Dic₁₀₁

yes

characteristic

#95

Dic₁₀₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂×Dic₁₀₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₂×C₈

—

202

—

100

C₂×C₈

—

202

—

101

C₂×C₈

—

202

—

102

C₂×C₈

—

202

—

103

C₂×C₈

—

202

—

104

C₂×C₈

—

202

—

105

C₂×C₈

—

202

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

Dic₂₀₂

yes

—

110

C₄²

D₄₀₄

yes

—

111

C₂³

C₂²×Dic₁₀₁

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₄⋊Dic₁₀₁

yes

—

120

C₂×C₄

C₄⋊Dic₁₀₁

yes

—

121

C₂×C₄

#13108058

yes

—

122

C₂×C₄

D₄×D₁₀₁

yes

characteristic

123

C₂×C₄

C₂×D₄₀₄

yes

characteristic

124

C₂×C₄

Q₈×D₁₀₁

yes

characteristic

125

C₂×C₄

C₂×Dic₂₀₂

yes

characteristic

126

C₂×C₄

#13108073

yes

characteristic

127

C₂×C₄

#13108075

yes

characteristic

128

C₈

—

404

—

129

C₈

—

202

—

130

C₈

—

202

—

131

C₂²

C₂×C₄⋊Dic₁₀₁

yes

characteristic

132

C₂²

D₄×Dic₁₀₁

yes

characteristic

133

C₂²

Q₈×Dic₁₀₁

yes

characteristic

134

C₂²

C₂×#13108058

yes

—

135

C₂²

#15692262

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#15692230

yes

characteristic

141

C₄

#15692235

yes

characteristic

142

C₄

#15692246

yes

characteristic

143

C₄

#15692254

yes

characteristic

144

C₂

#20265076

yes

characteristic

145

C₂

#20265201

yes

characteristic

146

C₂

#20265204

yes

characteristic

147

C₂

#20265249

yes

—

148

C₂

#20265308

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₁
commutator, socle, prime radical	C₂³×C₁₀₁	C₂²×C₄
Fitting	C₄×C₄⋊C₄×C₁₀₁	C₂
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#1940
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₁

commutator, socle, prime radical

C₂³×C₁₀₁

C₂²×C₄

Fitting

C₄×C₄⋊C₄×C₁₀₁

C₂

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
75. C₁₀₁	#1940	semidirect

H ≅ G/N

type

75. C₁₀₁

#1940

semidirect

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

subgroup	index
C₂×C₄⋊#11717579	2
C₂×#15692166	2
#20264932	2
C₄×C₄⋊C₄×C₁₀₁	2
#1940	101

subgroup

index

C₄×C₄⋊C₄×C₁₀₁

101

Subgroups — #27171856

Conjugacy classes of subgroups

Complements

Maximal subgroups