Group #27179615 — subgroups

Subgroups — #27179615

Order 12928.

704 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 289 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	#20265944	C₂	1	2	yes	maximal, complemented
2	#20265161	C₂	1	6	yes	maximal, complemented
3	C₂×#15692174	C₂	1	6	yes	maximal, complemented
4	C₂×2₋¹⁺⁴×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	#15692262	—	2	8	no	—
6	C₂×#13108058	C₂²	1	12	yes	—
7	#15692174	C₂²	1	12	yes	—
8	#15692172	C₂²	1	4	yes	—
9	2₋¹⁺⁴×C₁₀₁	—	2	8	no	—
10	C₂×#41×C₁₀₁	C₄	1	6	yes	—
11	C₂×#41×C₁₀₁	C₂²	1	3	yes	—
12	C₂×#41×C₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₁₀₁	C₄	1	2	yes	—
14	C₂²×Q₈×C₁₀₁	C₂²	1	3	yes	—
15	#13108058	—	2	12	no	—
16	#13108058	C₂³	1	8	yes	—
17	C₂×#11717579	—	2	12	no	—
18	#41×C₁₀₁	—	2	24	no	—
19	#41×C₁₀₁	—	2	12	no	—
20	#41×C₁₀₁	D₄	1	8	yes	—
21	C₂×Q₈×C₁₀₁	—	2	8	no	—
22	C₂×Q₈×C₁₀₁	—	2	12	no	—
23	C₂×Q₈×C₁₀₁	C₂×C₄	1	6	yes	—
24	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
25	C₂×Q₈×C₁₀₁	C₂³	1	3	yes	—
26	C₂×D₄×C₁₀₁	C₂×C₄	1	6	yes	—
27	C₂×D₄×C₁₀₁	C₂×C₄	1	3	yes	—
28	C₂×D₄×C₁₀₁	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₁	C₂×C₄	1	6	yes	—
30	C₂²×C₄×C₁₀₁	C₂×C₄	1	6	yes	—
31	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
32	C₂³×C₁₀₁	C₂²×C₄	1	2	yes	—
33	C₂³×C₁₀₁	C₂²×C₄	1	3	yes	—
34	Q₈×C₁₀₁	—	2	12	no	—
35	Q₈×C₁₀₁	—	2	6	no	—
36	Q₈×C₁₀₁	C₂²⋊C₄	1	4	yes	—
37	D₄×C₁₀₁	—	2	12	no	—
38	D₄×C₁₀₁	—	2	2	no	—
39	D₄×C₁₀₁	C₂²⋊C₄	1	12	yes	—
40	C₂×C₄×C₁₀₁	—	2	12	no	—
41	C₂×C₄×C₁₀₁	—	2	12	no	—
42	C₂×C₄×C₁₀₁	C₂²×C₄	1	6	yes	—
43	C₂×C₄×C₁₀₁	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₁₀₁	C₂×D₄	1	12	yes	—
45	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
46	#11717579	—	2	8	no	—
47	C₂²×C₁₀₁	—	2	2	no	—
48	C₂²×C₁₀₁	—	4	3	no	—
49	C₂²×C₁₀₁	—	4	2	no	—
50	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	6	yes	—
51	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
52	C₄×C₁₀₁	—	2	6	no	—
53	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	6	yes	—
54	C₄×C₁₀₁	C₂²×D₄	1	2	yes	—
55	C₂×C₁₀₁	—	2	1	no	—
56	C₂×C₁₀₁	—	2	6	no	—
57	C₂×C₁₀₁	—	4	2	no	—
58	C₂×C₁₀₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
59	#2889	—	101	1	no	maximal, Sylow
60	C₁₀₁	#2889	1	1	yes	characteristic, complemented, Sylow
61	C₂×2₋¹⁺⁴	D₁₀₁	1	1	yes	characteristic
62	#568	—	101	2	no	—
63	#413	—	101	6	no	—
64	C₂×#101	—	101	6	no	—
65	2₋¹⁺⁴	—	2	8	no	—
66	C₂×#41	Dic₁₀₁	1	6	yes	—
67	C₂×#41	C₂×D₁₀₁	1	3	yes	—
68	C₂×#41	C₂×D₁₀₁	1	1	yes	characteristic
69	C₂²×Q₈	Dic₁₀₁	1	2	yes	—
70	C₂²×Q₈	C₂×D₁₀₁	1	3	yes	—
71	#131	—	202	8	no	—
72	C₂×M₁₆	—	101	12	no	—
73	#101	—	101	12	no	—
74	#100	—	101	4	no	—
75	#41	—	2	24	no	—
76	#41	—	2	12	no	—
77	#41	C₂²⋊D₁₀₁	1	8	yes	—
78	C₂×Q₈	—	2	8	no	—
79	C₂×Q₈	—	2	12	no	—
80	C₂×Q₈	C₂×Dic₁₀₁	1	6	yes	—
81	C₂×Q₈	C₂×Dic₁₀₁	1	1	yes	characteristic
82	C₂×Q₈	C₂²×D₁₀₁	1	3	yes	—
83	C₂×D₄	C₂×Dic₁₀₁	1	6	yes	—
84	C₂×D₄	C₂×Dic₁₀₁	1	3	yes	—
85	C₂×D₄	C₂²×D₁₀₁	1	1	yes	characteristic
86	C₂²×C₄	C₂×Dic₁₀₁	1	6	yes	—
87	C₂²×C₄	C₂×Dic₁₀₁	1	6	yes	—
88	C₂²×C₄	C₂²×D₁₀₁	1	3	yes	—
89	M₁₆	—	101	8	no	—
90	M₁₆	—	202	12	no	—
91	C₂×C₈	—	202	12	no	—
92	C₂³	C₂²×Dic₁₀₁	1	2	yes	—
93	C₂³	C₂²×Dic₁₀₁	1	3	yes	—
94	Q₈	—	2	6	no	—
95	Q₈	—	2	12	no	—
96	Q₈	C₂²⋊Dic₁₀₁	1	4	yes	—
97	D₄	—	2	12	no	—
98	D₄	—	2	2	no	—
99	D₄	C₂²⋊Dic₁₀₁	1	12	yes	—
100	C₂×C₄	—	2	12	no	—
101	C₂×C₄	—	2	12	no	—
102	C₂×C₄	C₂²×Dic₁₀₁	1	6	yes	—
103	C₂×C₄	C₂²×Dic₁₀₁	1	3	yes	—
104	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	12	yes	—
105	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
106	C₈	—	202	8	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	4	3	no	—
109	C₂²	—	4	2	no	—
110	C₂²	C₂×C₂²⋊Dic₁₀₁	1	6	yes	—
111	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
112	C₄	—	2	6	no	—
113	C₄	C₂×C₂²⋊Dic₁₀₁	1	6	yes	—
114	C₄	C₂²×C₂²⋊D₁₀₁	1	2	yes	—
115	C₂	—	2	1	no	—
116	C₂	—	2	6	no	—
117	C₂	—	4	2	no	—
118	C₂	C₂²×C₂²⋊Dic₁₀₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265944

C₂

yes

maximal, complemented

#20265161

C₂

yes

maximal, complemented

C₂×#15692174

C₂

yes

maximal, complemented

C₂×2₋¹⁺⁴×C₁₀₁

C₂

yes

maximal, characteristic

#15692262

—

C₂×#13108058

C₂²

yes

—

#15692174

C₂²

yes

—

#15692172

C₂²

yes

—

2₋¹⁺⁴×C₁₀₁

—

C₂×#41×C₁₀₁

C₄

yes

—

C₂×#41×C₁₀₁

C₂²

yes

—

C₂×#41×C₁₀₁

C₂²

yes

characteristic

C₂²×Q₈×C₁₀₁

C₄

yes

—

C₂²×Q₈×C₁₀₁

C₂²

yes

—

#13108058

—

#13108058

C₂³

yes

—

C₂×#11717579

—

#41×C₁₀₁

—

#41×C₁₀₁

—

#41×C₁₀₁

D₄

yes

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₁₀₁

C₂³

yes

—

C₂×D₄×C₁₀₁

C₂×C₄

yes

—

C₂×D₄×C₁₀₁

C₂×C₄

yes

—

C₂×D₄×C₁₀₁

C₂³

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

C₂²⋊C₄

yes

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₀₁

C₂³×C₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×C₂²⋊C₄

yes

—

C₄×C₁₀₁

C₂²×D₄

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂²×C₂²⋊C₄

yes

characteristic

#2889

—

101

maximal, Sylow

C₁₀₁

#2889

yes

characteristic, complemented, Sylow

C₂×2₋¹⁺⁴

D₁₀₁

yes

characteristic

#568

—

101

—

#413

—

101

—

C₂×#101

—

101

—

2₋¹⁺⁴

—

C₂×#41

Dic₁₀₁

yes

—

C₂×#41

C₂×D₁₀₁

yes

—

C₂×#41

C₂×D₁₀₁

yes

characteristic

C₂²×Q₈

Dic₁₀₁

yes

—

C₂²×Q₈

C₂×D₁₀₁

yes

—

#131

—

202

—

C₂×M₁₆

—

101

—

#101

—

101

—

#100

—

101

—

#41

—

#41

—

#41

C₂²⋊D₁₀₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₀₁

yes

—

C₂×Q₈

C₂×Dic₁₀₁

yes

characteristic

C₂×Q₈

C₂²×D₁₀₁

yes

—

C₂×D₄

C₂×Dic₁₀₁

yes

—

C₂×D₄

C₂×Dic₁₀₁

yes

—

C₂×D₄

C₂²×D₁₀₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

—

M₁₆

—

101

—

M₁₆

—

202

—

C₂×C₈

—

202

—

C₂³

C₂²×Dic₁₀₁

yes

—

C₂³

C₂²×Dic₁₀₁

yes

—

Q₈

—

Q₈

—

Q₈

C₂²⋊Dic₁₀₁

yes

—

D₄

—

D₄

—

D₄

C₂²⋊Dic₁₀₁

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂²×Dic₁₀₁

yes

—

103

C₂×C₄

C₂²×Dic₁₀₁

yes

—

104

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

105

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

106

C₈

—

202

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

—

111

C₂²

C₂³×Dic₁₀₁

yes

characteristic

112

C₄

—

113

C₄

C₂×C₂²⋊Dic₁₀₁

yes

—

114

C₄

C₂²×C₂²⋊D₁₀₁

yes

—

115

C₂

—

116

C₂

—

117

C₂

—

118

C₂

C₂²×C₂²⋊Dic₁₀₁

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊Dic₁₀₁
commutator	C₂²×C₁₀₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting, prime radical	C₂×2₋¹⁺⁴×C₁₀₁	C₂
socle	C₂×C₁₀₁	C₂²×C₂²⋊C₄
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	C₂×2₋¹⁺⁴	D₁₀₁
hypocenter	C₁₀₁	#2889
LCS	G ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

subgroup

G/H

center

C₂

C₂²×C₂²⋊Dic₁₀₁

commutator

C₂²×C₁₀₁

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting, prime radical

C₂×2₋¹⁺⁴×C₁₀₁

C₂

socle

C₂×C₁₀₁

C₂²×C₂²⋊C₄

cosocle

C₂×C₄×C₁₀₁

C₂⁴

hypercenter

C₂×2₋¹⁺⁴

D₁₀₁

hypocenter

C₁₀₁

#2889

LCS

G ▷ C₂²×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×2₋¹⁺⁴

N	H ≅ G/N	type
1. #20265944	C₂	semidirect
2. #20265161	C₂	semidirect
3. C₂×#15692174	C₂	semidirect
60. C₁₀₁	#2889	semidirect

H ≅ G/N

type

1. #20265944

C₂

semidirect

2. #20265161

C₂

semidirect

3. C₂×#15692174

C₂

semidirect

60. C₁₀₁

#2889

semidirect

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

subgroup	index
#20265944	2
#20265161	2
C₂×#15692174	2
C₂×2₋¹⁺⁴×C₁₀₁	2
#2889	101

subgroup

index

C₂×2₋¹⁺⁴×C₁₀₁

101

Subgroups — #27179615

Conjugacy classes of subgroups

Complements

Maximal subgroups