Group #27172055 — subgroups

Subgroups — #27172055

Order 12928.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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×#15692255	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11717579	C₂	1	1	yes	maximal, characteristic
3	C₂×#15692178	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
6	C₂²×#11717579	C₂²	1	2	yes	—
7	#15692255	—	2	2	no	—
8	#15692255	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₁	—	2	1	no	—
11	C₄⋊#11717579	—	2	2	no	—
12	#15692178	—	2	8	no	—
13	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₁	—	2	1	no	—
15	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
16	#95×C₁₀₁	—	2	1	no	—
17	C₂×Dic₂₀₂	—	2	4	no	—
18	C₂×Dic₂₀₂	—	2	4	no	—
19	C₂×Dic₂₀₂	—	2	2	no	—
20	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	—
21	C₂²×Dic₁₀₁	—	2	2	no	—
22	C₂×#11717579	—	2	4	no	—
23	C₂×#11717579	—	2	2	no	—
24	C₄⋊Dic₁₀₁	—	2	4	no	—
25	C₄⋊Dic₁₀₁	—	2	4	no	—
26	C₂²×C₄×C₁₀₁	—	2	2	no	—
27	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₁	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₀₁	—	2	2	no	—
31	C₄⋊C₄×C₁₀₁	D₄	1	4	yes	—
32	C₄²×C₁₀₁	—	4	2	no	—
33	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
34	C₄²×C₁₀₁	D₄	1	2	yes	—
35	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₁	—	2	2	no	—
37	C₂×C₄×C₁₀₁	—	2	4	no	—
38	C₂×C₄×C₁₀₁	—	4	2	no	—
39	C₂×C₄×C₁₀₁	—	4	2	no	—
40	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₀₁	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₀₁	Q₁₆	1	2	yes	—
44	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₁	#41	1	2	yes	—
48	C₂×Dic₁₀₁	—	2	4	no	—
49	C₂×Dic₁₀₁	—	4	4	no	—
50	Dic₂₀₂	—	2	8	no	—
51	Dic₂₀₂	—	4	2	no	—
52	Dic₂₀₂	—	4	2	no	—
53	#11717579	—	4	2	no	—
54	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₁₀₁	#121	1	2	yes	—
56	C₂²×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
57	C₂²×C₁₀₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₂²×C₁₀₁	#137	1	2	yes	—
59	C₄×C₁₀₁	—	2	2	no	—
60	C₄×C₁₀₁	—	4	4	no	—
61	C₄×C₁₀₁	#103	1	4	yes	complemented
62	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
63	C₄×C₁₀₁	#124	1	1	yes	characteristic
64	C₄×C₁₀₁	#127	1	1	yes	characteristic
65	Dic₁₀₁	—	4	4	no	—
66	C₂×C₁₀₁	#390	1	1	yes	characteristic
67	C₂×C₁₀₁	C₂×#103	1	1	yes	characteristic
68	C₂×C₁₀₁	#419	1	1	yes	characteristic
69	C₂×C₁₀₁	#461	1	2	yes	—
70	C₂×C₁₀₁	#462	1	2	yes	—
71	#1981	—	101	1	no	maximal, Sylow
72	C₁₀₁	#1981	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₄⋊Q₈	—	101	1	no	—
74	C₂×C₄⋊C₈	—	101	1	no	—
75	C₂×#103	—	101	4	no	—
76	C₄×C₄⋊C₄	D₁₀₁	1	1	yes	characteristic
77	C₂²×Q₈	—	101	2	no	—
78	C₂²×C₈	—	101	2	no	—
79	C₄⋊Q₈	—	101	4	no	—
80	C₄⋊Q₈	—	202	2	no	—
81	C₂×C₄⋊C₄	—	202	1	no	—
82	C₂×C₄⋊C₄	—	202	1	no	—
83	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
84	C₂×C₄²	—	2	1	no	—
85	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
86	C₄⋊C₈	—	202	2	no	—
87	#103	—	202	8	no	—
88	#95	—	2	1	no	—
89	C₂×Q₈	—	101	4	no	—
90	C₂×Q₈	—	202	4	no	—
91	C₂×Q₈	—	202	4	no	—
92	C₂×Q₈	—	202	2	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	202	2	no	—
95	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₀₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
98	C₂×C₈	—	202	4	no	—
99	C₂×C₈	—	202	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	202	4	no	—
102	C₄⋊C₄	—	202	4	no	—
103	C₄⋊C₄	D₄₀₄	1	4	yes	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄×D₁₀₁	1	2	yes	—
106	C₄²	D₄₀₄	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
108	Q₈	—	404	2	no	—
109	Q₈	—	404	2	no	—
110	Q₈	—	202	8	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	404	4	no	—
116	C₂×C₄	—	202	4	no	—
117	C₂×C₄	#13108051	1	2	yes	—
118	C₂×C₄	#13108051	1	2	yes	—
119	C₂×C₄	#13108062	1	2	yes	—
120	C₂×C₄	#13108064	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
122	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
123	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
124	C₂×C₄	#13108074	1	2	yes	—
125	C₈	—	404	2	no	—
126	C₂²	C₂×#13108051	1	1	yes	characteristic
127	C₂²	#15692229	1	2	yes	—
128	C₂²	C₂×#13108062	1	1	yes	characteristic
129	C₂²	C₂×#13108064	1	1	yes	characteristic
130	C₂²	#15692285	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	404	4	no	—
134	C₄	#15692178	1	4	yes	complemented
135	C₄	C₄×D₄₀₄	1	2	yes	—
136	C₄	#15692243	1	1	yes	characteristic
137	C₄	#15692252	1	1	yes	characteristic
138	C₂	#20265079	1	1	yes	characteristic
139	C₂	C₂×#15692178	1	1	yes	characteristic
140	C₂	#20265185	1	1	yes	characteristic
141	C₂	#20265382	1	2	yes	—
142	C₂	#20265387	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692255

C₂

yes

maximal, characteristic

C₂×C₄⋊#11717579

C₂

yes

maximal, characteristic

C₂×#15692178

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₂²

yes

—

C₂²×#11717579

C₂²

yes

—

#15692255

—

#15692255

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

—

C₄⋊#11717579

—

#15692178

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

#95×C₁₀₁

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

SD₁₆

yes

—

C₂×C₄×C₁₀₁

Q₁₆

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

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

#11717579

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

#121

yes

—

C₂²×C₁₀₁

C₂×SD₁₆

yes

characteristic

C₂²×C₁₀₁

C₂×Q₁₆

yes

characteristic

C₂²×C₁₀₁

#137

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#103

yes

complemented

C₄×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

Dic₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×C₁₀₁

#462

yes

—

#1981

—

101

maximal, Sylow

C₁₀₁

#1981

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

101

—

C₂×C₄⋊C₈

—

101

—

C₂×#103

—

101

—

C₄×C₄⋊C₄

D₁₀₁

yes

characteristic

C₂²×Q₈

—

101

—

C₂²×C₈

—

101

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

—

202

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

202

—

#103

—

202

—

#95

—

C₂×Q₈

—

101

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂²×C₄

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₁

yes

characteristic

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

202

—

C₂×C₈

—

202

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

202

—

102

C₄⋊C₄

—

202

—

103

C₄⋊C₄

D₄₀₄

yes

—

104

C₄²

—

105

C₄²

C₄×D₁₀₁

yes

—

106

C₄²

D₄₀₄

yes

—

107

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

108

Q₈

—

404

—

109

Q₈

—

404

—

110

Q₈

—

202

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

404

—

116

C₂×C₄

—

202

—

117

C₂×C₄

#13108051

yes

—

118

C₂×C₄

#13108051

yes

—

119

C₂×C₄

#13108062

yes

—

120

C₂×C₄

#13108064

yes

—

121

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

122

C₂×C₄

C₂×D₄₀₄

yes

—

123

C₂×C₄

C₂×D₄₀₄

yes

characteristic

124

C₂×C₄

#13108074

yes

—

125

C₈

—

404

—

126

C₂²

C₂×#13108051

yes

characteristic

127

C₂²

#15692229

yes

—

128

C₂²

C₂×#13108062

yes

characteristic

129

C₂²

C₂×#13108064

yes

characteristic

130

C₂²

#15692285

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

404

—

134

C₄

#15692178

yes

complemented

135

C₄

C₄×D₄₀₄

yes

—

136

C₄

#15692243

yes

characteristic

137

C₄

#15692252

yes

characteristic

138

C₂

#20265079

yes

characteristic

139

C₂

C₂×#15692178

yes

characteristic

140

C₂

#20265185

yes

characteristic

141

C₂

#20265382

yes

—

142

C₂

#20265387

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₄×C₄⋊C₄×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₀₁
hypocenter	C₁₀₁	#1981
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₄×C₄⋊C₄×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₀₁

hypocenter

C₁₀₁

#1981

LCS

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #15692255	C₄	semidirect
61. C₄×C₁₀₁	#103	semidirect
72. C₁₀₁	#1981	semidirect
134. C₄	#15692178	semidirect

H ≅ G/N

type

8. #15692255

C₄

semidirect

61. C₄×C₁₀₁

#103

semidirect

72. C₁₀₁

#1981

semidirect

134. C₄

#15692178

semidirect

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

subgroup	index
C₂×#15692255	2
C₂×C₄⋊#11717579	2
C₂×#15692178	2
C₄×C₄⋊C₄×C₁₀₁	2
#1981	101

subgroup

index

C₄×C₄⋊C₄×C₁₀₁

101

Subgroups — #27172055

Conjugacy classes of subgroups

Complements

Maximal subgroups