Group #27171956 — subgroups

Subgroups — #27171956

Order 12928.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#15692177	C₂	1	4	yes	maximal, complemented
3	#20265043	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#97×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄₀₄	—	2	2	no	—
6	C₂²×D₄₀₄	C₂²	1	2	yes	—
7	#15692252	—	2	2	no	—
8	#15692252	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
10	#15692177	—	2	8	no	—
11	#15692162	—	2	2	no	—
12	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
14	#97×C₁₀₁	—	2	2	no	—
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	8	no	—
21	C₂×D₄₀₄	C₂×C₄	1	4	yes	—
22	C₂²×Dic₁₀₁	—	2	2	no	—
23	C₄⋊Dic₁₀₁	—	2	4	no	—
24	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₀₁	—	2	2	no	—
27	C₂×C₈×C₁₀₁	—	2	2	no	—
28	C₂×C₈×C₁₀₁	D₄	1	4	yes	complemented
29	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₁	D₄	1	2	yes	—
31	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
32	C₂²×D₁₀₁	—	2	4	no	—
33	C₂²×D₁₀₁	—	4	2	no	—
34	C₂²×D₁₀₁	—	4	2	no	—
35	C₂²×D₁₀₁	—	4	8	no	—
36	C₂×C₄×C₁₀₁	—	2	4	no	—
37	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₁₀₁	#41	1	2	yes	—
42	C₂×Dic₁₀₁	—	4	2	no	—
43	C₂×Dic₁₀₁	—	4	4	no	—
44	D₄₀₄	—	2	8	no	—
45	D₄₀₄	—	4	16	no	—
46	D₄₀₄	—	4	4	no	—
47	C₈×C₁₀₁	—	2	4	no	—
48	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₀₁	#124	1	1	yes	characteristic
50	C₂²×C₁₀₁	#127	1	1	yes	characteristic
51	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	4	yes	—
52	C₂×D₁₀₁	—	4	4	no	—
53	C₂×D₁₀₁	—	4	8	no	—
54	C₂×D₁₀₁	—	8	4	no	—
55	C₂×D₁₀₁	—	8	4	no	—
56	C₄×C₁₀₁	—	2	4	no	—
57	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
58	C₄×C₁₀₁	#121	1	2	yes	—
59	Dic₁₀₁	—	8	2	no	—
60	C₂×C₁₀₁	#390	1	1	yes	characteristic
61	C₂×C₁₀₁	#418	1	2	yes	—
62	C₂×C₁₀₁	#490	1	2	yes	—
63	C₂×C₁₀₁	#496	1	2	yes	—
64	D₁₀₁	—	8	4	no	—
65	#1961	—	101	1	no	maximal, Sylow
66	C₁₀₁	#1961	1	1	yes	characteristic, complemented, Sylow
67	C₂×#127	—	101	1	no	—
68	C₂×#102	—	101	4	no	—
69	C₂×#97	D₁₀₁	1	1	yes	characteristic, complemented
70	#382	—	101	1	no	—
71	C₂²×D₄	—	101	2	no	—
72	C₂²×D₄	—	202	2	no	—
73	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
74	#127	—	101	4	no	—
75	#127	—	202	2	no	—
76	C₂×C₄⋊C₄	—	101	2	no	—
77	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
78	#102	—	202	8	no	—
79	#97	—	2	2	no	—
80	#95	—	202	2	no	—
81	C₂⁴	—	202	2	no	—
82	C₂×D₄	—	101	4	no	—
83	C₂×D₄	—	404	8	no	—
84	C₂×D₄	—	202	8	no	—
85	C₂×D₄	—	202	2	no	—
86	C₂×D₄	—	202	4	no	—
87	C₂×D₄	—	202	4	no	—
88	C₂²×C₄	—	202	2	no	—
89	C₂²×C₄	D₄₀₄	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	D₄₀₄	1	4	yes	complemented
94	C₄⋊C₄	—	202	4	no	—
95	C₄²	C₄×D₁₀₁	1	2	yes	—
96	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
97	C₂³	—	404	2	no	—
98	C₂³	—	404	2	no	—
99	C₂³	—	404	8	no	—
100	C₂³	—	202	4	no	—
101	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
102	D₄	—	404	16	no	—
103	D₄	—	404	4	no	—
104	D₄	—	202	8	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	404	2	no	—
107	C₂×C₄	—	404	4	no	—
108	C₂×C₄	#13108051	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
111	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
112	C₂×C₄	#13108074	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₂²	—	808	4	no	—
115	C₂²	—	808	4	no	—
116	C₂²	—	404	4	no	—
117	C₂²	—	404	8	no	—
118	C₂²	C₂×#13108051	1	1	yes	characteristic
119	C₂²	#15692243	1	1	yes	characteristic
120	C₂²	#15692252	1	1	yes	characteristic
121	C₂²	#15692283	1	4	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	808	2	no	—
124	C₄	C₄×D₄₀₄	1	2	yes	—
125	C₄	#15692229	1	2	yes	—
126	C₂	—	808	4	no	—
127	C₂	#20265079	1	1	yes	characteristic
128	C₂	#20265183	1	2	yes	—
129	C₂	#20265542	1	2	yes	—
130	C₂	#20265566	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692252

C₂

yes

maximal, characteristic

C₂×#15692177

C₂

yes

maximal, complemented

#20265043

C₂

yes

maximal, characteristic, complemented

C₂×#97×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₀₄

—

C₂²×D₄₀₄

C₂²

yes

—

#15692252

—

#15692252

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

#15692177

—

#15692162

—

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

#97×C₁₀₁

—

C₂³×D₁₀₁

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

C₂×C₄

yes

—

C₂²×Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

complemented

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

C₈×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

#121

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

#390

yes

characteristic

C₂×C₁₀₁

#418

yes

—

C₂×C₁₀₁

#490

yes

—

C₂×C₁₀₁

#496

yes

—

D₁₀₁

—

#1961

—

101

maximal, Sylow

C₁₀₁

#1961

yes

characteristic, complemented, Sylow

C₂×#127

—

101

—

C₂×#102

—

101

—

C₂×#97

D₁₀₁

yes

characteristic, complemented

#382

—

101

—

C₂²×D₄

—

101

—

C₂²×D₄

—

202

—

C₂²×C₈

C₂×D₁₀₁

yes

—

#127

—

101

—

#127

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#102

—

202

—

#97

—

#95

—

202

—

C₂⁴

—

202

—

C₂×D₄

—

101

—

C₂×D₄

—

404

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

D₄₀₄

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄₀₄

yes

complemented

C₄⋊C₄

—

202

—

C₄²

C₄×D₁₀₁

yes

—

C₄²

C₂²⋊D₁₀₁

yes

—

C₂³

—

404

—

C₂³

—

404

—

C₂³

—

404

—

100

C₂³

—

202

—

101

C₂³

C₂×D₄₀₄

yes

characteristic

102

D₄

—

404

—

103

D₄

—

404

—

104

D₄

—

202

—

105

C₂×C₄

—

106

C₂×C₄

—

404

—

107

C₂×C₄

—

404

—

108

C₂×C₄

#13108051

yes

—

109

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

111

C₂×C₄

C₂×D₄₀₄

yes

—

112

C₂×C₄

#13108074

yes

—

113

C₈

—

114

C₂²

—

808

—

115

C₂²

—

808

—

116

C₂²

—

404

—

117

C₂²

—

404

—

118

C₂²

C₂×#13108051

yes

characteristic

119

C₂²

#15692243

yes

characteristic

120

C₂²

#15692252

yes

characteristic

121

C₂²

#15692283

yes

—

122

C₄

—

123

C₄

—

808

—

124

C₄

C₄×D₄₀₄

yes

—

125

C₄

#15692229

yes

—

126

C₂

—

808

—

127

C₂

#20265079

yes

characteristic

128

C₂

#20265183

yes

—

129

C₂

#20265542

yes

—

130

C₂

#20265566

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₀₄
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×#97×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂×D₄
prime radical	C₂×#15692252	C₂
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×#97	D₁₀₁
hypocenter	C₁₀₁	#1961
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₄₀₄

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×#97×C₁₀₁

C₂

socle

C₂³×C₁₀₁

C₂×D₄

prime radical

C₂×#15692252

C₂

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂×#97

D₁₀₁

hypocenter

C₁₀₁

#1961

LCS

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
2. C₂×#15692177	C₂	semidirect
3. #20265043	C₂	semidirect
4. C₂×#97×C₁₀₁	C₂	semidirect
8. #15692252	C₄	semidirect
28. C₂×C₈×C₁₀₁	D₄	semidirect
66. C₁₀₁	#1961	semidirect
69. C₂×#97	D₁₀₁	semidirect
93. C₂×C₈	D₄₀₄	semidirect

H ≅ G/N

type

2. C₂×#15692177

C₂

semidirect

3. #20265043

C₂

semidirect

4. C₂×#97×C₁₀₁

C₂

semidirect

8. #15692252

C₄

semidirect

28. C₂×C₈×C₁₀₁

D₄

semidirect

66. C₁₀₁

#1961

semidirect

69. C₂×#97

D₁₀₁

semidirect

93. C₂×C₈

D₄₀₄

semidirect

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

subgroup	index
C₂×#15692252	2
C₂×#15692177	2
#20265043	2
C₂×#97×C₁₀₁	2
#1961	101

subgroup

index

C₂×#97×C₁₀₁

101

Subgroups — #27171956

Conjugacy classes of subgroups

Complements

Maximal subgroups