Group #27170208 — subgroups

Subgroups — #27170208

Order 12928.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20265907	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265374	C₂	1	2	yes	maximal
3	#20264926	C₂	1	1	yes	maximal, characteristic, complemented
4	#20264909	C₂	1	2	yes	maximal, complemented
5	#492×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108062	—	2	2	no	—
7	#15692254	—	2	2	no	—
8	#15692253	—	2	2	no	—
9	C₄×Dic₂₀₂	C₂²	1	2	yes	—
10	#15692220	—	2	2	no	—
11	C₄⋊#11717579	C₂²	1	2	yes	—
12	#15692176	—	2	2	no	—
13	C₂×SD₁₆×C₁₀₁	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	#127×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₀₂	—	2	2	no	—
18	C₂×Dic₂₀₂	D₄	1	2	yes	—
19	#13108062	—	4	4	no	—
20	C₂×#11717579	—	2	2	no	—
21	C₄⋊Dic₁₀₁	D₄	1	2	yes	—
22	#13108053	—	2	4	no	—
23	#13108053	—	2	2	no	—
24	C₄×Dic₁₀₁	—	2	2	no	—
25	C₄×Dic₁₀₁	—	2	2	no	—
26	C₂×Q₈×C₁₀₁	D₄	1	2	yes	—
27	C₂×D₄×C₁₀₁	—	2	1	no	—
28	C₂×D₄×C₁₀₁	—	2	2	no	—
29	SD₁₆×C₁₀₁	—	4	4	no	—
30	C₂×C₈×C₁₀₁	—	2	1	no	—
31	C₄⋊C₄×C₁₀₁	—	2	2	no	—
32	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
33	C₂³×C₁₀₁	—	4	1	no	—
34	Q₈×C₁₀₁	—	4	2	no	—
35	D₄×C₁₀₁	—	4	2	no	—
36	D₄×C₁₀₁	—	4	2	no	—
37	D₄×C₁₀₁	—	4	2	no	—
38	C₂×C₄×C₁₀₁	—	2	2	no	—
39	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₁₀₁	—	2	2	no	—
42	C₂×Dic₁₀₁	—	2	2	no	—
43	Dic₂₀₂	—	2	2	no	—
44	Dic₂₀₂	—	8	2	no	—
45	Dic₂₀₂	SD₁₆	1	4	yes	complemented
46	C₈×C₁₀₁	—	2	2	no	—
47	#11717579	—	4	2	no	—
48	C₂²×C₁₀₁	—	8	2	no	—
49	C₂²×C₁₀₁	—	8	1	no	—
50	C₂²×C₁₀₁	#120	1	1	yes	characteristic
51	C₄×C₁₀₁	—	2	1	no	—
52	C₄×C₁₀₁	—	4	2	no	—
53	C₄×C₁₀₁	C₂×SD₁₆	1	2	yes	—
54	C₄×C₁₀₁	#137	1	2	yes	—
55	Dic₁₀₁	—	2	4	no	—
56	Dic₁₀₁	—	4	2	no	—
57	C₂×C₁₀₁	—	16	1	no	—
58	C₂×C₁₀₁	#448	1	2	yes	—
59	C₂×C₁₀₁	#453	1	1	yes	characteristic
60	#1623	—	101	1	no	maximal, Sylow
61	C₁₀₁	#1623	1	1	yes	characteristic, complemented, Sylow
62	Q₈²	—	101	1	no	—
63	#492	D₁₀₁	1	1	yes	characteristic
64	#460	—	101	2	no	—
65	#333	—	101	1	no	—
66	Q₈⋊C₈	—	101	2	no	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	202	2	no	—
69	C₄⋊Q₈	—	202	2	no	—
70	C₄⋊Q₈	—	202	2	no	—
71	C₄⋊Q₈	C₂×D₁₀₁	1	1	yes	characteristic
72	#127	C₂×D₁₀₁	1	1	yes	characteristic
73	C₄×Q₈	—	101	2	no	—
74	C₄×Q₈	—	202	2	no	—
75	C₄⋊C₈	—	101	2	no	—
76	#102	—	202	2	no	—
77	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
78	C₂×Q₈	—	101	2	no	—
79	C₂×Q₈	—	202	2	no	—
80	C₂×Q₈	C₂²⋊D₁₀₁	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	404	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	202	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	101	2	no	—
89	C₄⋊C₄	—	202	4	no	—
90	C₄⋊C₄	—	202	2	no	—
91	C₄²	—	202	2	no	—
92	C₄²	—	202	2	no	—
93	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	808	2	no	—
97	Q₈	—	101	4	no	—
98	Q₈	—	202	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	202	2	no	—
104	C₂×C₄	—	202	2	no	—
105	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	404	2	no	—
109	C₂²	—	8	2	no	—
110	C₂²	—	8	1	no	—
111	C₂²	#15692223	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	404	2	no	—
115	C₄	—	202	4	no	—
116	C₄	SD₁₆×D₁₀₁	1	2	yes	—
117	C₄	#15692284	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#20265314	1	2	yes	—
120	C₂	#20265344	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265907

C₂

yes

maximal, characteristic, complemented

#20265374

C₂

yes

maximal

#20264926

C₂

yes

maximal, characteristic, complemented

#20264909

C₂

yes

maximal, complemented

#492×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108062

—

#15692254

—

#15692253

—

C₄×Dic₂₀₂

C₂²

yes

—

#15692220

—

C₄⋊#11717579

C₂²

yes

—

#15692176

—

C₂×SD₁₆×C₁₀₁

—

C₄⋊Q₈×C₁₀₁

C₂²

yes

characteristic

#127×C₁₀₁

C₂²

yes

characteristic

C₄×C₈×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

D₄

yes

—

#13108062

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

D₄

yes

—

#13108053

—

#13108053

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

SD₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

—

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

SD₁₆

yes

complemented

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#120

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×SD₁₆

yes

—

C₄×C₁₀₁

#137

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#448

yes

—

C₂×C₁₀₁

#453

yes

characteristic

#1623

—

101

maximal, Sylow

C₁₀₁

#1623

yes

characteristic, complemented, Sylow

Q₈²

—

101

—

#492

D₁₀₁

yes

characteristic

#460

—

101

—

#333

—

101

—

Q₈⋊C₈

—

101

—

C₂×SD₁₆

—

C₂×SD₁₆

—

202

—

C₄⋊Q₈

—

202

—

C₄⋊Q₈

—

202

—

C₄⋊Q₈

C₂×D₁₀₁

yes

characteristic

#127

C₂×D₁₀₁

yes

characteristic

C₄×Q₈

—

101

—

C₄×Q₈

—

202

—

C₄⋊C₈

—

101

—

#102

—

202

—

C₄×C₈

C₂×D₁₀₁

yes

characteristic

C₂×Q₈

—

101

—

C₂×Q₈

—

202

—

C₂×Q₈

C₂²⋊D₁₀₁

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

404

—

C₂×C₈

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₄²

—

202

—

C₄²

—

202

—

C₄²

C₂²×D₁₀₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

808

—

Q₈

—

101

—

Q₈

—

202

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

202

—

104

C₂×C₄

—

202

—

105

C₂×C₄

D₄×D₁₀₁

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

107

C₈

—

108

C₈

—

404

—

109

C₂²

—

110

C₂²

—

111

C₂²

#15692223

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

404

—

115

C₄

—

202

—

116

C₄

SD₁₆×D₁₀₁

yes

—

117

C₄

#15692284

yes

—

118

C₂

—

119

C₂

#20265314

yes

—

120

C₂

#20265344

yes

characteristic

	subgroup	G/H
center	C₂²	#15692223
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#492×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
prime radical	#127×C₁₀₁	C₂²
hypercenter	#492	D₁₀₁
hypocenter	C₁₀₁	#1623
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#15692223

commutator, cosocle

C₄²×C₁₀₁

C₂³

Frattini

C₄²

C₂²×D₁₀₁

Fitting

#492×C₁₀₁

C₂

socle

C₂²×C₁₀₁

#120

prime radical

#127×C₁₀₁

C₂²

D₁₀₁

C₁₀₁

G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #20265907	C₂	semidirect
3. #20264926	C₂	semidirect
4. #20264909	C₂	semidirect
45. Dic₂₀₂	SD₁₆	semidirect
61. C₁₀₁	#1623	semidirect

H ≅ G/N

type

1. #20265907

C₂

semidirect

3. #20264926

C₂

semidirect

4. #20264909

C₂

semidirect

45. Dic₂₀₂

SD₁₆

semidirect

61. C₁₀₁

#1623

semidirect

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

subgroup	index
#20265907	2
#20265374	2
#20264926	2
#20264909	2
#492×C₁₀₁	2
#1623	101

subgroup

index

#492×C₁₀₁

101

Subgroups — #27170208

Conjugacy classes of subgroups

Complements

Maximal subgroups