Group #27181466 — subgroups

Subgroups — C₂×#20265541 #27181466

Order 12928.

488 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 149 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×#15692247	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265541	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15692176	C₂	1	4	yes	maximal, complemented
4	C₂×#15692166	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11717579	C₂²	1	2	yes	—
7	#15692247	—	2	2	no	—
8	#15692247	C₂²	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
10	C₂×#13108053	—	2	2	no	—
11	#15692176	C₂²	1	16	yes	complemented
12	#15692166	C₂²	1	4	yes	complemented
13	C₂²×D₄×C₁₀₁	—	2	2	no	—
14	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
15	#127×C₁₀₁	—	2	2	no	—
16	#127×C₁₀₁	C₂²	1	4	yes	—
17	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₀₁	—	2	2	no	—
19	C₂×#11717579	—	2	4	no	—
20	C₂×#11717579	C₂³	1	4	yes	—
21	C₄⋊Dic₁₀₁	—	2	2	no	—
22	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
23	#13108053	—	2	8	no	—
24	C₂⁴×C₁₀₁	—	2	2	no	—
25	C₂×D₄×C₁₀₁	—	2	2	no	—
26	C₂×D₄×C₁₀₁	—	2	8	no	—
27	C₂×D₄×C₁₀₁	—	2	8	no	—
28	C₂×D₄×C₁₀₁	—	4	8	no	—
29	C₂×D₄×C₁₀₁	C₂³	1	4	yes	—
30	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
31	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
32	C₄²×C₁₀₁	D₄	1	2	yes	—
33	C₄²×C₁₀₁	C₂³	1	2	yes	—
34	C₂³×C₁₀₁	—	2	4	no	—
35	C₂³×C₁₀₁	—	4	4	no	—
36	C₂³×C₁₀₁	—	4	8	no	—
37	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
38	D₄×C₁₀₁	—	2	8	no	—
39	D₄×C₁₀₁	—	4	16	no	—
40	D₄×C₁₀₁	—	4	4	no	—
41	C₂×C₄×C₁₀₁	—	2	4	no	—
42	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₁	#41	1	4	yes	—
45	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₁₀₁	—	2	4	no	—
47	C₂×Dic₁₀₁	—	4	4	no	—
48	#11717579	—	2	4	no	—
49	C₂²×C₁₀₁	—	4	4	no	—
50	C₂²×C₁₀₁	—	4	8	no	—
51	C₂²×C₁₀₁	—	8	8	no	—
52	C₂²×C₁₀₁	#124	1	2	yes	—
53	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	4	yes	—
54	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₁₀₁	—	2	4	no	—
56	C₄×C₁₀₁	#124	1	2	yes	—
57	C₄×C₁₀₁	C₂×#41	1	2	yes	—
58	Dic₁₀₁	—	4	4	no	—
59	C₂×C₁₀₁	—	8	4	no	—
60	C₂×C₁₀₁	#490	1	4	yes	complemented
61	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
62	C₂×C₁₀₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
63	C₂×#490	—	101	1	no	maximal, Sylow
64	C₁₀₁	C₂×#490	1	1	yes	characteristic, complemented, Sylow
65	C₂×#127	D₁₀₁	1	1	yes	characteristic
66	C₂×#125	—	101	1	no	—
67	#490	—	101	8	no	—
68	C₂×#102	—	101	4	no	—
69	C₂×#97	—	101	1	no	—
70	C₂²×D₄	—	2	2	no	—
71	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
72	C₂²×C₈	—	101	2	no	—
73	#127	—	2	2	no	—
74	#127	C₂×D₁₀₁	1	4	yes	—
75	#125	—	101	4	no	—
76	#125	—	202	2	no	—
77	C₂×C₄⋊C₄	—	101	2	no	—
78	C₂×C₄⋊C₄	—	202	2	no	—
79	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
80	#102	—	101	16	no	—
81	#97	—	101	4	no	—
82	C₂⁴	—	2	2	no	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	2	8	no	—
85	C₂×D₄	—	2	8	no	—
86	C₂×D₄	—	4	8	no	—
87	C₂×D₄	C₂²×D₁₀₁	1	4	yes	—
88	C₂²×C₄	—	202	2	no	—
89	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
91	C₂×C₈	—	101	4	no	—
92	C₂×C₈	—	202	4	no	—
93	C₄⋊C₄	—	101	4	no	—
94	C₄⋊C₄	—	202	8	no	—
95	C₄⋊C₄	—	202	2	no	—
96	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
97	C₄²	C₂²×D₁₀₁	1	2	yes	—
98	C₂³	—	2	4	no	—
99	C₂³	—	4	4	no	—
100	C₂³	—	4	8	no	—
101	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
102	D₄	—	2	8	no	—
103	D₄	—	4	16	no	—
104	D₄	—	4	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	404	4	no	—
107	C₂×C₄	—	202	4	no	—
108	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
110	C₂×C₄	#13108073	1	4	yes	—
111	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
112	C₈	—	202	4	no	—
113	C₂²	—	4	4	no	—
114	C₂²	—	4	8	no	—
115	C₂²	—	8	8	no	—
116	C₂²	#15692242	1	2	yes	—
117	C₂²	#15692281	1	4	yes	—
118	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
119	C₄	—	2	4	no	—
120	C₄	—	404	4	no	—
121	C₄	#15692242	1	2	yes	—
122	C₄	C₂×#13108073	1	2	yes	—
123	C₂	—	8	4	no	—
124	C₂	#20265541	1	4	yes	direct product factor
125	C₂	C₂×#15692242	1	1	yes	characteristic
126	C₂	C₂×#15692281	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692247

C₂

yes

maximal, characteristic, complemented

#20265541

C₂

yes

maximal, direct product factor

C₂×#15692176

C₂

yes

maximal, complemented

C₂×#15692166

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

#15692247

—

#15692247

C₂²

yes

complemented

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×#13108053

—

#15692176

C₂²

yes

complemented

#15692166

C₂²

yes

complemented

C₂²×D₄×C₁₀₁

—

C₂²×D₄×C₁₀₁

C₂²

yes

—

#127×C₁₀₁

—

#127×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₂×#11717579

C₂³

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂³

yes

—

#13108053

—

C₂⁴×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#124

yes

—

C₂²×C₁₀₁

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

#124

yes

—

C₄×C₁₀₁

C₂×#41

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×#490

—

101

maximal, Sylow

C₁₀₁

C₂×#490

yes

characteristic, complemented, Sylow

C₂×#127

D₁₀₁

yes

characteristic

C₂×#125

—

101

—

#490

—

101

—

C₂×#102

—

101

—

C₂×#97

—

101

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₀₁

yes

—

C₂²×C₈

—

101

—

#127

—

#127

C₂×D₁₀₁

yes

—

#125

—

101

—

#125

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#102

—

101

—

#97

—

101

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×D₁₀₁

yes

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₄²

C₂²⋊D₁₀₁

yes

—

C₄²

C₂²×D₁₀₁

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

102

D₄

—

103

D₄

—

104

D₄

—

105

C₂×C₄

—

106

C₂×C₄

—

404

—

107

C₂×C₄

—

202

—

108

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

110

C₂×C₄

#13108073

yes

—

111

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

112

C₈

—

202

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

#15692242

yes

—

117

C₂²

#15692281

yes

—

118

C₂²

C₂²×C₂²⋊D₁₀₁

yes

characteristic

119

C₄

—

120

C₄

—

404

—

121

C₄

#15692242

yes

—

122

C₄

C₂×#13108073

yes

—

123

C₂

—

124

C₂

#20265541

yes

direct product factor

125

C₂

C₂×#15692242

yes

characteristic

126

C₂

C₂×#15692281

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting, prime radical	C₂×#127×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂×D₄
hypercenter	C₂×#127	D₁₀₁
hypocenter	C₁₀₁	C₂×#490
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator, cosocle

C₂×C₄×C₁₀₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting, prime radical

C₂×#127×C₁₀₁

C₂

socle

C₂³×C₁₀₁

C₂×D₄

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#15692247	C₂	semidirect
2. #20265541	C₂	direct (N × H)
3. C₂×#15692176	C₂	semidirect
4. C₂×#15692166	C₂	semidirect
8. #15692247	C₂²	semidirect
11. #15692176	C₂²	semidirect
12. #15692166	C₂²	semidirect
60. C₂×C₁₀₁	#490	semidirect
64. C₁₀₁	C₂×#490	semidirect
124. C₂	#20265541	direct (N × H)

H ≅ G/N

type

1. C₂×#15692247

C₂

semidirect

2. #20265541

C₂

direct (N × H)

3. C₂×#15692176

C₂

semidirect

4. C₂×#15692166

C₂

semidirect

8. #15692247

C₂²

semidirect

11. #15692176

C₂²

semidirect

12. #15692166

C₂²

semidirect

60. C₂×C₁₀₁

#490

semidirect

64. C₁₀₁

C₂×#490

semidirect

124. C₂

#20265541

direct (N × H)

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

subgroup	index
C₂×#15692247	2
#20265541	2
C₂×#15692176	2
C₂×#15692166	2
C₂×#127×C₁₀₁	2
C₂×#490	101

subgroup

index

C₂×#127×C₁₀₁

101

Subgroups — C₂×#20265541 #27181466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20265541 #27181466

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20265541 #27181466