Group #27170186 — subgroups

Subgroups — #27170186

Order 12928.

336 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20265553	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265372	C₂	1	2	yes	maximal, complemented
3	#20264921	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊#11717579	C₂	1	2	yes	maximal, complemented
5	D₄²×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13108059	—	2	2	no	—
7	C₂×#13108059	—	2	2	no	—
8	#15692252	C₂²	1	1	yes	characteristic, complemented
9	C₄⋊#11717579	C₂²	1	2	yes	complemented
10	#15692175	—	2	2	no	—
11	C₄×#11717579	C₂²	1	1	yes	characteristic, complemented
12	C₂²×D₄×C₁₀₁	—	2	2	no	—
13	#127×C₁₀₁	C₂²	1	1	yes	characteristic
14	#121×C₁₀₁	—	2	2	no	—
15	#120×C₁₀₁	—	2	2	no	—
16	C₄×D₄×C₁₀₁	C₂²	1	2	yes	—
17	C₂×D₄₀₄	—	2	2	no	—
18	C₂×D₄₀₄	—	2	1	no	—
19	#13108059	—	4	4	no	—
20	#13108059	—	4	4	no	—
21	C₂×#11717579	—	2	1	no	—
22	C₂×#11717579	—	2	2	no	—
23	C₂⁴×C₁₀₁	—	2	2	no	—
24	C₂×D₄×C₁₀₁	—	2	4	no	—
25	C₂×D₄×C₁₀₁	—	2	2	no	—
26	C₂×D₄×C₁₀₁	—	4	4	no	—
27	C₂×D₄×C₁₀₁	—	4	4	no	—
28	C₂×D₄×C₁₀₁	D₄	1	2	yes	—
29	C₂×D₄×C₁₀₁	D₄	1	2	yes	—
30	C₂²×C₄×C₁₀₁	—	2	2	no	—
31	C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
32	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
33	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
34	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
35	C₂³×C₁₀₁	—	2	2	no	—
36	C₂³×C₁₀₁	—	2	2	no	—
37	C₂³×C₁₀₁	—	4	8	no	—
38	C₂³×C₁₀₁	—	8	2	no	—
39	C₂²×D₁₀₁	—	4	1	no	—
40	D₄×C₁₀₁	—	2	2	no	—
41	D₄×C₁₀₁	—	4	2	no	—
42	D₄×C₁₀₁	—	4	8	no	—
43	D₄×C₁₀₁	—	4	2	no	—
44	D₄×C₁₀₁	—	4	2	no	—
45	D₄×C₁₀₁	D₈	1	4	yes	complemented
46	C₂×C₄×C₁₀₁	—	2	2	no	—
47	C₂×C₄×C₁₀₁	—	4	2	no	—
48	C₂×C₄×C₁₀₁	—	4	2	no	—
49	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
51	D₄₀₄	—	4	2	no	—
52	D₄₀₄	—	4	2	no	—
53	D₄₀₄	—	4	2	no	—
54	#11717579	—	2	2	no	—
55	#11717579	—	4	2	no	—
56	C₂²×C₁₀₁	—	2	4	no	—
57	C₂²×C₁₀₁	—	4	2	no	—
58	C₂²×C₁₀₁	—	4	2	no	—
59	C₂²×C₁₀₁	—	4	4	no	—
60	C₂²×C₁₀₁	—	4	2	no	—
61	C₂²×C₁₀₁	—	8	8	no	—
62	C₂²×C₁₀₁	#120	1	1	yes	characteristic
63	C₂×D₁₀₁	—	8	2	no	—
64	C₂×D₁₀₁	—	8	1	no	—
65	C₄×C₁₀₁	—	2	1	no	—
66	C₄×C₁₀₁	—	4	2	no	—
67	C₄×C₁₀₁	C₂×D₈	1	2	yes	—
68	C₄×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₁₀₁	—	4	4	no	—
70	C₂×C₁₀₁	—	8	2	no	—
71	C₂×C₁₀₁	#447	1	2	yes	—
72	C₂×C₁₀₁	#453	1	1	yes	characteristic
73	D₁₀₁	—	16	1	no	—
74	#1619	—	101	1	no	maximal, Sylow
75	C₁₀₁	#1619	1	1	yes	characteristic, complemented, Sylow
76	D₄²	D₁₀₁	1	1	yes	characteristic, complemented
77	#493	—	101	1	no	—
78	#459	—	101	2	no	—
79	#331	—	101	1	no	—
80	D₄⋊C₈	—	101	2	no	—
81	C₂²×D₄	—	2	2	no	—
82	C₂×D₈	—	202	2	no	—
83	C₂×D₈	—	202	2	no	—
84	#127	—	101	1	no	—
85	#127	C₂×D₁₀₁	1	1	yes	characteristic
86	#121	—	2	2	no	—
87	#120	—	2	2	no	—
88	C₄×D₄	C₂×D₁₀₁	1	2	yes	—
89	C₄⋊C₈	—	101	2	no	—
90	#102	—	202	2	no	—
91	C₄×C₈	—	101	1	no	—
92	C₂⁴	—	2	2	no	—
93	C₂×D₄	—	2	4	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	—	4	4	no	—
96	C₂×D₄	—	4	4	no	—
97	C₂×D₄	—	202	2	no	—
98	C₂×D₄	—	202	1	no	—
99	C₂×D₄	C₂²⋊D₁₀₁	1	2	yes	—
100	C₂×D₄	C₂²⋊D₁₀₁	1	2	yes	—
101	C₂²×C₄	—	2	2	no	—
102	D₈	—	404	4	no	—
103	D₈	—	404	4	no	—
104	C₂×C₈	—	202	2	no	—
105	C₂×C₈	—	202	1	no	—
106	C₄⋊C₄	C₂²⋊D₁₀₁	1	2	yes	—
107	C₂²⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
110	C₂³	—	2	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	4	8	no	—
113	C₂³	—	8	2	no	—
114	C₂³	—	404	1	no	—
115	D₄	—	2	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	4	8	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	404	2	no	—
121	D₄	—	404	2	no	—
122	D₄	—	404	2	no	—
123	D₄	#13108059	1	4	yes	complemented
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
129	C₈	—	404	2	no	—
130	C₈	—	202	2	no	—
131	C₂²	—	2	4	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	4	2	no	—
134	C₂²	—	4	4	no	—
135	C₂²	—	4	2	no	—
136	C₂²	—	8	8	no	—
137	C₂²	—	808	2	no	—
138	C₂²	—	808	1	no	—
139	C₂²	#15692224	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	4	2	no	—
142	C₄	C₂×#13108059	1	2	yes	—
143	C₄	#15692281	1	2	yes	—
144	C₂	—	4	4	no	—
145	C₂	—	8	2	no	—
146	C₂	—	1616	1	no	—
147	C₂	#20265311	1	2	yes	—
148	C₂	#20265345	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265553

C₂

yes

maximal, characteristic, complemented

#20265372

C₂

yes

maximal, complemented

#20264921

C₂

yes

maximal, characteristic, complemented

D₄⋊#11717579

C₂

yes

maximal, complemented

D₄²×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#13108059

—

C₂×#13108059

—

#15692252

C₂²

yes

characteristic, complemented

C₄⋊#11717579

C₂²

yes

complemented

#15692175

—

C₄×#11717579

C₂²

yes

characteristic, complemented

C₂²×D₄×C₁₀₁

—

#127×C₁₀₁

C₂²

yes

characteristic

#121×C₁₀₁

—

#120×C₁₀₁

—

C₄×D₄×C₁₀₁

C₂²

yes

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

—

—

—

—

C₂⁴×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₂²⋊C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂²×D₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

D₈

yes

complemented

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#120

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×D₈

yes

—

C₄×C₁₀₁

AGL(1,ℤ/8)

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#447

yes

—

C₂×C₁₀₁

#453

yes

characteristic

D₁₀₁

—

#1619

—

101

maximal, Sylow

C₁₀₁

#1619

yes

characteristic, complemented, Sylow

D₄²

D₁₀₁

yes

characteristic, complemented

#493

—

101

—

#459

—

101

—

#331

—

101

—

D₄⋊C₈

—

101

—

C₂²×D₄

—

C₂×D₈

—

202

—

C₂×D₈

—

202

—

#127

—

101

—

#127

C₂×D₁₀₁

yes

characteristic

#121

—

#120

—

C₄×D₄

C₂×D₁₀₁

yes

—

C₄⋊C₈

—

101

—

#102

—

202

—

C₄×C₈

—

101

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

C₂²⋊D₁₀₁

yes

—

100

C₂×D₄

C₂²⋊D₁₀₁

yes

—

101

C₂²×C₄

—

102

D₈

—

404

—

103

D₈

—

404

—

104

C₂×C₈

—

202

—

105

C₂×C₈

—

202

—

106

C₄⋊C₄

C₂²⋊D₁₀₁

yes

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

C₂²×D₁₀₁

yes

characteristic

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

404

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

404

—

121

D₄

—

404

—

122

D₄

—

404

—

123

D₄

#13108059

yes

complemented

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

129

C₈

—

404

—

130

C₈

—

202

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

808

—

138

C₂²

—

808

—

139

C₂²

#15692224

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

C₂×#13108059

yes

—

143

C₄

#15692281

yes

—

144

C₂

—

145

C₂

—

146

C₂

—

1616

—

147

C₂

#20265311

yes

—

148

C₂

#20265345

yes

characteristic

	subgroup	G/H
center	C₂²	#15692224
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	D₄²×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
hypercenter	D₄²	D₁₀₁
hypocenter	C₁₀₁	#1619
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ D₄²

subgroup

G/H

center

C₂²

#15692224

commutator, cosocle

C₄²×C₁₀₁

C₂³

Frattini

C₄²

C₂²×D₁₀₁

Fitting

D₄²×C₁₀₁

C₂

socle

C₂²×C₁₀₁

#120

hypercenter

D₄²

D₁₀₁

hypocenter

C₁₀₁

#1619

LCS

G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ D₄²

N	H ≅ G/N	type
1. #20265553	C₂	semidirect
2. #20265372	C₂	semidirect
3. #20264921	C₂	semidirect
4. D₄⋊#11717579	C₂	semidirect
5. D₄²×C₁₀₁	C₂	semidirect
8. #15692252	C₂²	semidirect
9. C₄⋊#11717579	C₂²	semidirect
11. C₄×#11717579	C₂²	semidirect
45. D₄×C₁₀₁	D₈	semidirect
75. C₁₀₁	#1619	semidirect
76. D₄²	D₁₀₁	semidirect
123. D₄	#13108059	semidirect

H ≅ G/N

type

1. #20265553

C₂

semidirect

2. #20265372

C₂

semidirect

3. #20264921

C₂

semidirect

4. D₄⋊#11717579

C₂

semidirect

5. D₄²×C₁₀₁

C₂

semidirect

8. #15692252

C₂²

semidirect

9. C₄⋊#11717579

C₂²

semidirect

11. C₄×#11717579

C₂²

semidirect

45. D₄×C₁₀₁

D₈

semidirect

75. C₁₀₁

#1619

semidirect

76. D₄²

D₁₀₁

semidirect

123. D₄

#13108059

semidirect

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

subgroup	index
#20265553	2
#20265372	2
#20264921	2
D₄⋊#11717579	2
D₄²×C₁₀₁	2
#1619	101

subgroup

index

D₄²×C₁₀₁

101

Subgroups — #27170186

Conjugacy classes of subgroups

Complements

Maximal subgroups