Group #27171937 — subgroups

Subgroups — #27171937

Order 12928.

436 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 137 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×#15692252	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692175	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×#11717579	C₂	1	1	yes	maximal, characteristic, complemented
4	#384×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₀₄	—	2	2	no	—
6	C₂²×D₄₀₄	C₂²	1	2	yes	—
7	C₂²×#11717579	C₂²	1	2	yes	—
8	#15692252	—	2	2	no	—
9	#15692252	C₄	1	4	yes	complemented
10	#15692175	—	2	8	no	—
11	C₄×#11717579	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂³×D₁₀₁	—	2	2	no	—
16	C₂×D₄₀₄	—	2	4	no	—
17	C₂×D₄₀₄	—	2	4	no	—
18	C₂×D₄₀₄	—	2	8	no	—
19	C₂×D₄₀₄	—	2	2	no	—
20	C₂×D₄₀₄	—	4	4	no	—
21	C₂×D₄₀₄	—	4	4	no	—
22	C₂×D₄₀₄	C₂×C₄	1	4	yes	—
23	C₂×#11717579	—	2	2	no	—
24	C₂×#11717579	—	2	2	no	—
25	C₂×#11717579	D₄	1	4	yes	complemented
26	C₂²×C₄×C₁₀₁	—	2	2	no	—
27	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
28	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₀₁	—	2	4	no	—
30	C₄⋊C₄×C₁₀₁	—	4	4	no	—
31	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
32	C₄²×C₁₀₁	D₄	1	2	yes	—
33	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₁₀₁	—	2	4	no	—
35	C₂²×D₁₀₁	—	4	2	no	—
36	C₂²×D₁₀₁	—	4	2	no	—
37	C₂²×D₁₀₁	—	4	4	no	—
38	C₂²×D₁₀₁	—	4	4	no	—
39	C₂×C₄×C₁₀₁	—	4	2	no	—
40	C₂×C₄×C₁₀₁	—	4	4	no	—
41	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
42	C₂×C₄×C₁₀₁	D₈	1	4	yes	—
43	C₂×C₄×C₁₀₁	SD₁₆	1	4	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	D₄₀₄	—	2	8	no	—
49	D₄₀₄	—	4	16	no	—
50	D₄₀₄	—	4	4	no	—
51	#11717579	—	2	4	no	—
52	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₁₀₁	#124	1	1	yes	characteristic
54	C₂²×C₁₀₁	#127	1	1	yes	characteristic
55	C₂²×C₁₀₁	C₂×D₈	1	2	yes	—
56	C₂²×C₁₀₁	C₂×SD₁₆	1	2	yes	—
57	C₂×D₁₀₁	—	4	4	no	—
58	C₂×D₁₀₁	—	4	8	no	—
59	C₂×D₁₀₁	—	8	4	no	—
60	C₂×D₁₀₁	—	8	2	no	—
61	C₂×D₁₀₁	—	8	2	no	—
62	C₄×C₁₀₁	—	8	2	no	—
63	C₄×C₁₀₁	#102	1	8	yes	complemented
64	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
65	C₄×C₁₀₁	#121	1	2	yes	—
66	C₂×C₁₀₁	#390	1	1	yes	characteristic
67	C₂×C₁₀₁	C₂×#102	1	2	yes	—
68	C₂×C₁₀₁	#486	1	2	yes	—
69	C₂×C₁₀₁	#492	1	1	yes	characteristic
70	C₂×C₁₀₁	#493	1	1	yes	characteristic
71	D₁₀₁	—	8	4	no	—
72	#1957	—	101	1	no	maximal, Sylow
73	C₁₀₁	#1957	1	1	yes	characteristic, complemented, Sylow
74	C₂×#127	—	101	1	no	—
75	C₂×#102	—	101	4	no	—
76	C₂×C₄×C₈	—	101	1	no	—
77	#384	D₁₀₁	1	1	yes	characteristic, complemented
78	C₂²×D₄	—	101	2	no	—
79	C₂²×D₄	—	202	2	no	—
80	C₂²×C₈	—	101	2	no	—
81	#127	—	101	4	no	—
82	#127	—	202	2	no	—
83	C₂×C₄⋊C₄	—	2	2	no	—
84	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
85	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
86	#102	—	202	8	no	—
87	C₄×C₈	—	202	2	no	—
88	C₂⁴	—	202	2	no	—
89	C₂×D₄	—	101	4	no	—
90	C₂×D₄	—	404	4	no	—
91	C₂×D₄	—	404	4	no	—
92	C₂×D₄	—	202	8	no	—
93	C₂×D₄	—	202	4	no	—
94	C₂×D₄	—	202	2	no	—
95	C₂×D₄	—	202	4	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
98	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
99	C₂×C₈	—	101	4	no	—
100	C₂×C₈	—	202	2	no	—
101	C₂×C₈	—	202	2	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	4	4	no	—
104	C₄²	C₄×D₁₀₁	1	2	yes	—
105	C₄²	D₄₀₄	1	2	yes	—
106	C₂³	—	404	2	no	—
107	C₂³	—	404	2	no	—
108	C₂³	—	404	4	no	—
109	C₂³	—	404	4	no	—
110	C₂³	—	202	4	no	—
111	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
112	D₄	—	404	16	no	—
113	D₄	—	404	4	no	—
114	D₄	—	202	8	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	#13108051	1	4	yes	—
118	C₂×C₄	#13108059	1	4	yes	—
119	C₂×C₄	#13108061	1	4	yes	—
120	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
121	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
122	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
123	C₂×C₄	#13108075	1	2	yes	—
124	C₈	—	202	4	no	—
125	C₂²	—	808	4	no	—
126	C₂²	—	808	2	no	—
127	C₂²	—	808	2	no	—
128	C₂²	—	404	4	no	—
129	C₂²	—	404	8	no	—
130	C₂²	C₂×#13108051	1	1	yes	characteristic
131	C₂²	#15692244	1	1	yes	characteristic
132	C₂²	#15692251	1	1	yes	characteristic
133	C₂²	C₂×#13108059	1	2	yes	—
134	C₂²	C₂×#13108061	1	2	yes	—
135	C₄	—	8	2	no	—
136	C₄	#15692175	1	8	yes	complemented
137	C₄	#15692217	1	2	yes	—
138	C₄	#15692230	1	2	yes	—
139	C₂	—	808	4	no	—
140	C₂	#20265078	1	1	yes	characteristic
141	C₂	C₂×#15692175	1	2	yes	—
142	C₂	#20265525	1	2	yes	—
143	C₂	#20265551	1	1	yes	characteristic
144	C₂	#20265553	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692252

C₂

yes

maximal, characteristic

C₂×#15692175

C₂

yes

maximal, complemented

C₂×C₄×#11717579

C₂

yes

maximal, characteristic, complemented

#384×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₀₄

—

C₂²×D₄₀₄

C₂²

yes

—

C₂²×#11717579

C₂²

yes

—

#15692252

—

#15692252

C₄

yes

complemented

#15692175

—

C₄×#11717579

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂³×D₁₀₁

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

C₂×C₄

yes

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

D₄

yes

complemented

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

D₈

yes

—

C₂×C₄×C₁₀₁

SD₁₆

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

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

#11717579

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

C₂×D₈

yes

—

C₂²×C₁₀₁

C₂×SD₁₆

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#102

yes

complemented

C₄×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

#121

yes

—

C₂×C₁₀₁

#390

yes

characteristic

C₂×C₁₀₁

C₂×#102

yes

—

C₂×C₁₀₁

#486

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

D₁₀₁

—

#1957

—

101

maximal, Sylow

C₁₀₁

#1957

yes

characteristic, complemented, Sylow

C₂×#127

—

101

—

C₂×#102

—

101

—

C₂×C₄×C₈

—

101

—

#384

D₁₀₁

yes

characteristic, complemented

C₂²×D₄

—

101

—

C₂²×D₄

—

202

—

C₂²×C₈

—

101

—

#127

—

101

—

#127

—

202

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#102

—

202

—

C₄×C₈

—

202

—

C₂⁴

—

202

—

C₂×D₄

—

101

—

C₂×D₄

—

404

—

C₂×D₄

—

404

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

100

C₂×C₈

—

202

—

101

C₂×C₈

—

202

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₄×D₁₀₁

yes

—

105

C₄²

D₄₀₄

yes

—

106

C₂³

—

404

—

107

C₂³

—

404

—

108

C₂³

—

404

—

109

C₂³

—

404

—

110

C₂³

—

202

—

111

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

112

D₄

—

404

—

113

D₄

—

404

—

114

D₄

—

202

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#13108051

yes

—

118

C₂×C₄

#13108059

yes

—

119

C₂×C₄

#13108061

yes

—

120

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

121

C₂×C₄

D₄×D₁₀₁

yes

—

122

C₂×C₄

C₂×D₄₀₄

yes

characteristic

123

C₂×C₄

#13108075

yes

—

124

C₈

—

202

—

125

C₂²

—

808

—

126

C₂²

—

808

—

127

C₂²

—

808

—

128

C₂²

—

404

—

129

C₂²

—

404

—

130

C₂²

C₂×#13108051

yes

characteristic

131

C₂²

#15692244

yes

characteristic

132

C₂²

#15692251

yes

characteristic

133

C₂²

C₂×#13108059

yes

—

134

C₂²

C₂×#13108061

yes

—

135

C₄

—

136

C₄

#15692175

yes

complemented

137

C₄

#15692217

yes

—

138

C₄

#15692230

yes

—

139

C₂

—

808

—

140

C₂

#20265078

yes

characteristic

141

C₂

C₂×#15692175

yes

—

142

C₂

#20265525

yes

—

143

C₂

#20265551

yes

characteristic

144

C₂

#20265553

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#384×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂×D₄
prime radical	C₂×#15692252	C₂
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#384	D₁₀₁
hypocenter	C₁₀₁	#1957
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#384×C₁₀₁

C₂

socle

C₂³×C₁₀₁

C₂×D₄

prime radical

C₂×#15692252

C₂

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
2. C₂×#15692175	C₂	semidirect
3. C₂×C₄×#11717579	C₂	semidirect
4. #384×C₁₀₁	C₂	semidirect
9. #15692252	C₄	semidirect
25. C₂×#11717579	D₄	semidirect
63. C₄×C₁₀₁	#102	semidirect
73. C₁₀₁	#1957	semidirect
77. #384	D₁₀₁	semidirect
136. C₄	#15692175	semidirect

H ≅ G/N

type

2. C₂×#15692175

C₂

semidirect

3. C₂×C₄×#11717579

C₂

semidirect

4. #384×C₁₀₁

C₂

semidirect

9. #15692252

C₄

semidirect

25. C₂×#11717579

D₄

semidirect

63. C₄×C₁₀₁

#102

semidirect

73. C₁₀₁

#1957

semidirect

77. #384

D₁₀₁

semidirect

136. C₄

#15692175

semidirect

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

subgroup	index
C₂×#15692252	2
C₂×#15692175	2
C₂×C₄×#11717579	2
#384×C₁₀₁	2
#1957	101

subgroup

index

#384×C₁₀₁

101

Subgroups — #27171937

Conjugacy classes of subgroups

Complements

Maximal subgroups