Group #27187270 — subgroups

Subgroups — #27187270

Order 12928.

320 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 70 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	#20266182	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692319	C₂	1	4	yes	maximal, complemented
3	#20266113	C₂	1	1	yes	maximal, characteristic, complemented
4	#20265725	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108084	C₂²	1	2	yes	—
6	C₂×C₄×#87938	C₂²	1	2	yes	—
7	#15692322	—	2	2	no	—
8	#15692319	—	2	8	no	—
9	(C₂×#87938)⋊C₄	—	2	2	no	—
10	C₂×#13108073	C₂²	1	2	yes	—
11	#15692255	C₄	1	1	yes	characteristic, complemented
12	#15692242	C₄	1	2	yes	complemented
13	#15692226	—	2	2	no	—
14	D₄×Dic₁₀₁	—	2	2	no	—
15	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
16	#127×C₁₀₁	C₄	1	1	yes	characteristic, complemented
17	C₂²×#87938	—	2	2	no	—
18	#13108084	—	2	4	no	—
19	C₂×#11717581	—	2	2	no	—
20	C₄×#87938	—	2	2	no	—
21	C₄×#87938	D₄	1	4	yes	complemented
22	#13108073	—	2	4	no	—
23	#13108073	—	2	4	no	—
24	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	—
25	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
26	C₂²×Dic₁₀₁	—	2	2	no	—
27	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
28	C₂×C₄×D₁₀₁	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₀₁	—	2	2	no	—
30	C₂²⋊Dic₁₀₁	—	2	4	no	—
31	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
32	C₄×Dic₁₀₁	D₄	1	2	yes	—
33	C₂×D₄×C₁₀₁	—	2	2	no	—
34	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
35	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₁₀₁	—	2	2	no	—
37	C₂²×D₁₀₁	C₂×D₄	1	1	yes	characteristic
38	C₂×#87938	—	2	4	no	—
39	C₂×#87938	—	4	4	no	—
40	D₄×C₁₀₁	—	2	4	no	—
41	D₄×C₁₀₁	—	4	4	no	—
42	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
44	C₂²⋊D₁₀₁	—	4	4	no	—
45	C₂×Dic₁₀₁	—	2	2	no	—
46	C₂×Dic₁₀₁	—	4	2	no	—
47	C₂×Dic₁₀₁	—	4	2	no	—
48	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
49	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
50	C₄×D₁₀₁	—	4	2	no	—
51	C₄×D₁₀₁	C₂×D₄	1	2	yes	—
52	C₄×D₁₀₁	#41	1	2	yes	—
53	Dic₂₀₂	—	2	4	no	—
54	#11717581	—	4	2	no	—
55	C₂²×C₁₀₁	—	4	2	no	—
56	C₂²×C₁₀₁	—	4	4	no	—
57	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×D₁₀₁	—	2	2	no	—
59	C₂×D₁₀₁	#121	1	2	yes	—
60	#87938	—	4	4	no	—
61	C₄×C₁₀₁	—	2	2	no	—
62	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
63	Dic₁₀₁	—	2	2	no	—
64	Dic₁₀₁	—	4	2	no	—
65	Dic₁₀₁	#124	1	1	yes	characteristic
66	Dic₁₀₁	#127	1	1	yes	characteristic
67	C₂×C₁₀₁	—	8	2	no	—
68	C₂×C₁₀₁	#390	1	1	yes	characteristic
69	C₂×C₁₀₁	#421	1	2	yes	—
70	D₁₀₁	—	2	2	no	—
71	#1963	—	101	1	no	maximal, Sylow
72	C₁₀₁	#1963	1	1	yes	characteristic, complemented, Sylow
73	#532	—	101	1	no	—
74	#423	—	101	1	no	—
75	C₂×#104	—	101	4	no	—
76	#376	—	101	1	no	—
77	C₂×#41	—	101	2	no	—
78	C₂×M₁₆	—	101	2	no	—
79	C₄⋊Q₈	—	101	1	no	—
80	#127	#87938	1	1	yes	characteristic, complemented
81	#124	—	101	2	no	—
82	#121	—	202	2	no	—
83	C₄×D₄	—	202	2	no	—
84	C₂×C₄²	—	101	1	no	—
85	C₂×C₄²	—	101	2	no	—
86	C₄⋊C₈	—	202	2	no	—
87	#104	—	202	8	no	—
88	#95	—	202	2	no	—
89	#41	—	202	4	no	—
90	#41	—	202	4	no	—
91	C₂×Q₈	—	101	2	no	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	202	2	no	—
94	C₂×D₄	C₂×#87938	1	2	yes	—
95	C₂²×C₄	—	101	1	no	—
96	C₂²×C₄	—	101	2	no	—
97	C₂²×C₄	—	202	2	no	—
98	C₂²×C₄	—	202	2	no	—
99	M₁₆	—	202	4	no	—
100	C₂×C₈	—	202	2	no	—
101	C₄⋊C₄	—	202	2	no	—
102	C₂²⋊C₄	—	202	4	no	—
103	C₄²	—	101	2	no	—
104	C₄²	—	101	1	no	—
105	C₄²	—	101	4	no	—
106	C₄²	—	202	2	no	—
107	C₄²	C₂×#87938	1	1	yes	characteristic
108	C₂³	—	2	2	no	—
109	C₂³	—	101	1	no	—
110	Q₈	—	202	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	4	4	no	—
113	D₄	—	404	4	no	—
114	C₂×C₄	—	101	1	no	—
115	C₂×C₄	—	101	2	no	—
116	C₂×C₄	—	101	2	no	—
117	C₂×C₄	—	101	2	no	—
118	C₂×C₄	—	404	2	no	—
119	C₂×C₄	—	404	2	no	—
120	C₂×C₄	—	404	2	no	—
121	C₂×C₄	—	404	4	no	—
122	C₂×C₄	—	202	4	no	—
123	C₂×C₄	—	202	2	no	—
124	C₂×C₄	C₂²⋊#87938	1	2	yes	—
125	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
126	C₈	—	404	2	no	—
127	C₂²	—	4	2	no	—
128	C₂²	—	4	4	no	—
129	C₂²	—	101	2	no	—
130	C₂²	—	202	2	no	—
131	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	101	1	no	—
134	C₄	—	101	1	no	—
135	C₄	—	404	4	no	—
136	C₄	—	404	2	no	—
137	C₄	—	202	2	no	—
138	C₄	D₄×#87938	1	2	yes	—
139	C₂	—	8	2	no	—
140	C₂	—	202	2	no	—
141	C₂	#20266129	1	1	yes	characteristic
142	C₂	#20266175	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20266182

C₂

yes

maximal, characteristic, complemented

C₂×#15692319

C₂

yes

maximal, complemented

#20266113

C₂

yes

maximal, characteristic, complemented

#20265725

C₂

yes

maximal, characteristic

C₂×#13108084

C₂²

yes

—

C₂×C₄×#87938

C₂²

yes

—

—

—

—

C₂²

yes

—

#15692255

C₄

yes

characteristic, complemented

#15692242

C₄

yes

complemented

#15692226

—

D₄×Dic₁₀₁

—

C₄²×D₁₀₁

C₂²

yes

characteristic

#127×C₁₀₁

C₄

yes

characteristic, complemented

C₂²×#87938

—

—

—

—

D₄

yes

complemented

#13108073

—

#13108073

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

C₂×C₂²⋊D₁₀₁

—

C₂²×Dic₁₀₁

—

C₂×C₄×D₁₀₁

D₄

yes

—

C₂×C₄×D₁₀₁

C₂³

yes

characteristic

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

C₂×C₄

yes

characteristic

C₄×Dic₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂³×C₁₀₁

—

C₂²×D₁₀₁

C₂×D₄

yes

characteristic

C₂×#87938

—

C₂×#87938

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₄×D₁₀₁

—

C₄×D₁₀₁

C₂×D₄

yes

—

C₄×D₁₀₁

#41

yes

—

Dic₂₀₂

—

#11717581

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

#121

yes

—

#87938

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

yes

Dic₁₀₁

yes

C₂×C₁₀₁

—

C₂×C₁₀₁

#390

yes

characteristic

C₂×C₁₀₁

#421

yes

—

D₁₀₁

—

#1963

—

101

maximal, Sylow

C₁₀₁

#1963

yes

characteristic, complemented, Sylow

#532

—

101

—

#423

—

101

—

C₂×#104

—

101

—

#376

—

101

—

C₂×#41

—

101

—

C₂×M₁₆

—

101

—

C₄⋊Q₈

—

101

—

#127

#87938

yes

characteristic, complemented

#124

—

101

—

#121

—

202

—

C₄×D₄

—

202

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

202

—

#104

—

202

—

#95

—

202

—

#41

—

202

—

#41

—

202

—

C₂×Q₈

—

101

—

C₂×D₄

—

C₂×D₄

—

202

—

C₂×D₄

C₂×#87938

yes

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

C₂²×C₄

—

202

—

M₁₆

—

202

—

100

C₂×C₈

—

202

—

101

C₄⋊C₄

—

202

—

102

C₂²⋊C₄

—

202

—

103

C₄²

—

101

—

104

C₄²

—

101

—

105

C₄²

—

101

—

106

C₄²

—

202

—

107

C₄²

C₂×#87938

yes

characteristic

108

C₂³

—

109

C₂³

—

101

—

110

Q₈

—

202

—

111

D₄

—

112

D₄

—

113

D₄

—

404

—

114

C₂×C₄

—

101

—

115

C₂×C₄

—

101

—

116

C₂×C₄

—

101

—

117

C₂×C₄

—

101

—

118

C₂×C₄

—

404

—

119

C₂×C₄

—

404

—

120

C₂×C₄

—

404

—

121

C₂×C₄

—

404

—

122

C₂×C₄

—

202

—

123

C₂×C₄

—

202

—

124

C₂×C₄

C₂²⋊#87938

yes

—

125

C₂×C₄

C₂²×#87938

yes

characteristic

126

C₈

—

404

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

101

—

130

C₂²

—

202

—

131

C₂²

C₂×C₂²⋊#87938

yes

characteristic

132

C₄

—

133

C₄

—

101

—

134

C₄

—

101

—

135

C₄

—

404

—

136

C₄

—

404

—

137

C₄

—

202

—

138

C₄

D₄×#87938

yes

—

139

C₂

—

140

C₂

—

202

—

141

C₂

#20266129

yes

characteristic

142

C₂

#20266175

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87938
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87938
Fitting	#127×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	#20265725	C₂
cosocle	C₂×C₄×D₁₀₁	C₂³
hypercenter	#127	#87938
hypocenter	C₁₀₁	#1963
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×C₂²⋊#87938

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87938

Fitting

#127×C₁₀₁

C₄

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

#20265725

C₂

cosocle

C₂×C₄×D₁₀₁

C₂³

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
1. #20266182	C₂	semidirect
2. C₂×#15692319	C₂	semidirect
3. #20266113	C₂	semidirect
11. #15692255	C₄	semidirect
12. #15692242	C₄	semidirect
16. #127×C₁₀₁	C₄	semidirect
21. C₄×#87938	D₄	semidirect
72. C₁₀₁	#1963	semidirect
80. #127	#87938	semidirect

H ≅ G/N

type

1. #20266182

C₂

semidirect

2. C₂×#15692319

C₂

semidirect

3. #20266113

C₂

semidirect

11. #15692255

C₄

semidirect

12. #15692242

C₄

semidirect

16. #127×C₁₀₁

C₄

semidirect

21. C₄×#87938

D₄

semidirect

72. C₁₀₁

#1963

semidirect

80. #127

#87938

semidirect

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

subgroup	index
#20266182	2
C₂×#15692319	2
#20266113	2
#20265725	2
#1963	101

subgroup

index

101

Subgroups — #27187270

Conjugacy classes of subgroups

Complements

Maximal subgroups