Group #27182936 — subgroups

Subgroups — #27182936

Order 12928.

444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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	#20265990	C₂	1	4	yes	maximal, complemented
2	#20265449	C₂	1	4	yes	maximal, complemented
3	#20265359	C₂	1	4	yes	maximal, complemented
4	#20265295	C₂	1	2	yes	maximal, complemented
5	#494×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#15692303	C₂²	1	4	yes	—
7	#15692287	—	2	8	no	—
8	#15692286	C₂²	1	8	yes	—
9	#15692278	—	2	4	no	—
10	#15692277	C₂²	1	2	yes	—
11	Q₁₆×D₁₀₁	—	2	4	no	—
12	#15692261	C₂²	1	4	yes	—
13	#15692191	C₂²	1	2	yes	—
14	#15692183	C₂²	1	4	yes	—
15	#15692174	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₀₁	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₀₁	C₂²	1	2	yes	—
18	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
19	#13108075	—	2	8	no	—
20	#13108074	—	4	4	no	—
21	#13108074	C₂³	1	4	yes	—
22	Q₈×D₁₀₁	—	2	8	no	—
23	Dic₄₀₄	D₄	1	2	yes	—
24	#13108064	—	2	8	no	—
25	#13108063	D₄	1	4	yes	—
26	#13108061	—	2	8	no	—
27	D₈₀₈	D₄	1	2	yes	—
28	#13108058	C₂³	1	4	yes	—
29	#13108057	—	2	4	no	—
30	C₈×D₁₀₁	—	2	4	no	—
31	C₂×Q₈×C₁₀₁	C₂³	1	4	yes	—
32	Q₁₆×C₁₀₁	—	2	4	no	—
33	Q₁₆×C₁₀₁	—	2	4	no	—
34	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₀₁	—	4	2	no	—
36	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
37	Q₈×C₁₀₁	—	2	8	no	—
38	C₂×C₄×C₁₀₁	—	2	1	no	—
39	C₂×C₄×C₁₀₁	—	2	4	no	—
40	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₁₀₁	—	2	4	no	—
42	D₄₀₄	—	4	4	no	—
43	D₄₀₄	C₂×D₄	1	4	yes	—
44	C₄×D₁₀₁	—	2	4	no	—
45	C₄×D₁₀₁	—	4	8	no	—
46	Dic₂₀₂	—	4	4	no	—
47	Dic₂₀₂	C₂×D₄	1	4	yes	—
48	C₈×C₁₀₁	C₂×D₄	1	2	yes	—
49	C₈×C₁₀₁	C₂×D₄	1	2	yes	—
50	#11717579	—	2	4	no	—
51	C₂²×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₀₁	—	2	4	no	—
53	C₄×C₁₀₁	—	2	2	no	—
54	C₄×C₁₀₁	—	4	4	no	—
55	C₄×C₁₀₁	C₂²×D₄	1	2	yes	—
56	Dic₁₀₁	—	2	4	no	—
57	C₂×C₁₀₁	—	2	1	no	—
58	C₂×C₁₀₁	D₄²	1	1	yes	characteristic
59	D₁₀₁	—	4	4	no	—
60	#3293	—	101	1	no	maximal, Sylow
61	C₁₀₁	#3293	1	1	yes	characteristic, complemented, Sylow
62	#578	—	101	4	no	—
63	#494	D₁₀₁	1	1	yes	characteristic, complemented
64	#473	—	101	4	no	—
65	#456	—	101	4	no	—
66	#443	—	101	2	no	—
67	2₋¹⁺⁴	—	101	4	no	—
68	#137	—	101	8	no	—
69	#137	—	202	8	no	—
70	#135	—	101	2	no	—
71	#135	—	202	4	no	—
72	C₂×Q₁₆	—	202	4	no	—
73	C₂×Q₁₆	C₂×D₁₀₁	1	4	yes	—
74	#131	—	101	4	no	—
75	C₄⋊Q₈	C₂×D₁₀₁	1	2	yes	—
76	#108	—	101	2	no	—
77	#104	—	101	4	no	—
78	#101	—	101	4	no	—
79	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
80	#41	—	101	4	no	—
81	#41	—	404	4	no	—
82	#41	—	202	8	no	—
83	C₂×Q₈	—	202	8	no	—
84	C₂×Q₈	C₂²×D₁₀₁	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	101	2	no	—
88	Q₁₆	—	202	8	no	—
89	SD₁₆	—	101	4	no	—
90	SD₁₆	—	202	8	no	—
91	D₈	—	101	2	no	—
92	M₁₆	—	101	4	no	—
93	M₁₆	—	202	4	no	—
94	C₂×C₈	—	202	4	no	—
95	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	101	4	no	—
100	Q₈	—	404	4	no	—
101	D₄	—	101	4	no	—
102	D₄	—	404	4	no	—
103	D₄	—	202	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	404	8	no	—
107	C₂×C₄	—	202	4	no	—
108	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
109	C₈	—	202	4	no	—
110	C₈	D₄×D₁₀₁	1	2	yes	—
111	C₈	D₄×D₁₀₁	1	2	yes	—
112	C₂²	—	202	4	no	—
113	C₂²	#15692300	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	202	4	no	—
117	C₄	C₂×D₄×D₁₀₁	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	404	4	no	—
120	C₂	#20265812	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265990

C₂

yes

maximal, complemented

#20265449

C₂

yes

maximal, complemented

#20265359

C₂

yes

maximal, complemented

#20265295

C₂

yes

maximal, complemented

#494×C₁₀₁

C₂

yes

maximal, characteristic, complemented

#15692303

C₂²

yes

—

#15692287

—

#15692286

C₂²

yes

—

#15692278

—

#15692277

C₂²

yes

—

Q₁₆×D₁₀₁

—

#15692261

C₂²

yes

—

#15692191

C₂²

yes

—

#15692183

C₂²

yes

—

#15692174

C₂²

yes

—

C₂×Q₁₆×C₁₀₁

C₂²

yes

—

C₄⋊Q₈×C₁₀₁

C₂²

yes

—

C₄×C₈×C₁₀₁

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₀₁

—

Dic₄₀₄

D₄

yes

—

#13108064

—

#13108063

D₄

yes

—

#13108061

—

D₈₀₈

D₄

yes

—

#13108058

C₂³

yes

—

#13108057

—

C₈×D₁₀₁

—

C₂×Q₈×C₁₀₁

C₂³

yes

—

Q₁₆×C₁₀₁

—

Q₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

C₂³

yes

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂²⋊D₁₀₁

—

D₄₀₄

—

D₄₀₄

C₂×D₄

yes

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

C₂×D₄

yes

—

C₈×C₁₀₁

C₂×D₄

yes

—

C₈×C₁₀₁

C₂×D₄

yes

—

#11717579

—

C₂²×C₁₀₁

H(5,𝔽₂)

yes

characteristic

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂²×D₄

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

D₄²

yes

characteristic

D₁₀₁

—

#3293

—

101

maximal, Sylow

C₁₀₁

#3293

yes

characteristic, complemented, Sylow

#578

—

101

—

#494

D₁₀₁

yes

characteristic, complemented

#473

—

101

—

#456

—

101

—

#443

—

101

—

2₋¹⁺⁴

—

101

—

#137

—

101

—

#137

—

202

—

#135

—

101

—

#135

—

202

—

C₂×Q₁₆

—

202

—

C₂×Q₁₆

C₂×D₁₀₁

yes

—

#131

—

101

—

C₄⋊Q₈

C₂×D₁₀₁

yes

—

#108

—

101

—

#104

—

101

—

#101

—

101

—

C₄×C₈

C₂×D₁₀₁

yes

characteristic

#41

—

101

—

#41

—

404

—

#41

—

202

—

C₂×Q₈

—

202

—

C₂×Q₈

C₂²×D₁₀₁

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

101

—

Q₁₆

—

202

—

SD₁₆

—

101

—

SD₁₆

—

202

—

D₈

—

101

—

M₁₆

—

101

—

M₁₆

—

202

—

C₂×C₈

—

202

—

C₂×C₈

C₂²×D₁₀₁

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₀₁

yes

characteristic

Q₈

—

Q₈

—

101

—

100

Q₈

—

404

—

101

D₄

—

101

—

102

D₄

—

404

—

103

D₄

—

202

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

404

—

107

C₂×C₄

—

202

—

108

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

109

C₈

—

202

—

110

C₈

D₄×D₁₀₁

yes

—

111

C₈

D₄×D₁₀₁

yes

—

112

C₂²

—

202

—

113

C₂²

#15692300

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

202

—

117

C₄

C₂×D₄×D₁₀₁

yes

—

118

C₂

—

119

C₂

—

404

—

120

C₂

#20265812

yes

characteristic

	subgroup	G/H
center	C₂	#20265812
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#494×C₁₀₁	C₂
socle	C₂×C₁₀₁	D₄²
hypercenter	#494	D₁₀₁
hypocenter	C₁₀₁	#3293
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#20265812

commutator, cosocle

C₂×C₄×C₁₀₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting

#494×C₁₀₁

C₂

socle

C₂×C₁₀₁

D₄²

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #20265990	C₂	semidirect
2. #20265449	C₂	semidirect
3. #20265359	C₂	semidirect
4. #20265295	C₂	semidirect
5. #494×C₁₀₁	C₂	semidirect
61. C₁₀₁	#3293	semidirect
63. #494	D₁₀₁	semidirect

H ≅ G/N

type

1. #20265990

C₂

semidirect

2. #20265449

C₂

semidirect

3. #20265359

C₂

semidirect

4. #20265295

C₂

semidirect

5. #494×C₁₀₁

C₂

semidirect

61. C₁₀₁

#3293

semidirect

63. #494

D₁₀₁

semidirect

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

subgroup	index
#20265990	2
#20265449	2
#20265359	2
#20265295	2
#494×C₁₀₁	2
#3293	101

subgroup

index

#494×C₁₀₁

101

Subgroups — #27182936

Conjugacy classes of subgroups

Complements

Maximal subgroups