Group #27173183 — subgroups

Subgroups — #27173183

Order 12928.

154 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 55 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×Dic₈₀₈	C₂	1	2	yes	maximal
2	#20265557	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₄₀₄	C₂	1	1	yes	maximal, characteristic
4	#20264982	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₀₄	—	2	1	no	—
7	C₂×Dic₄₀₄	C₂²	1	2	yes	—
8	C₂×Dic₄₀₄	C₂²	1	1	yes	characteristic
9	#15692255	—	2	1	no	—
10	C₄×Dic₂₀₂	—	2	1	no	—
11	Dic₈₀₈	—	2	4	no	—
12	#15692189	C₂²	1	1	yes	characteristic
13	#15692180	—	2	1	no	—
14	C₂×C₁₆×C₁₀₁	C₂²	1	2	yes	—
15	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₀₂	—	2	1	no	—
17	C₂×Dic₂₀₂	—	2	2	no	—
18	Dic₄₀₄	—	2	2	no	—
19	Dic₄₀₄	—	2	1	no	—
20	Dic₄₀₄	—	2	2	no	—
21	Dic₄₀₄	—	4	2	no	—
22	Dic₄₀₄	D₄	1	2	yes	—
23	C₄⋊Dic₁₀₁	—	2	1	no	—
24	C₄⋊Dic₁₀₁	—	4	1	no	—
25	C₄⋊Dic₁₀₁	—	4	1	no	—
26	#13108053	—	4	1	no	—
27	C₄×Dic₁₀₁	—	4	1	no	—
28	C₂×C₈×C₁₀₁	D₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
30	C₄²×C₁₀₁	D₄	1	1	yes	characteristic
31	C₁₆×C₁₀₁	—	2	2	no	—
32	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₁	D₈	1	1	yes	characteristic
34	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₁₀₁	—	4	1	no	—
36	C₂×Dic₁₀₁	—	4	1	no	—
37	C₂×Dic₁₀₁	—	4	2	no	—
38	Dic₂₀₂	—	2	2	no	—
39	Dic₂₀₂	—	4	1	no	—
40	Dic₂₀₂	—	4	2	no	—
41	Dic₂₀₂	—	4	2	no	—
42	C₈×C₁₀₁	—	2	1	no	—
43	C₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₀₁	#41	1	1	yes	characteristic
45	C₂²×C₁₀₁	C₂×D₈	1	1	yes	characteristic
46	C₄×C₁₀₁	—	2	1	no	—
47	C₄×C₁₀₁	Q₃₂	1	2	yes	complemented
48	C₄×C₁₀₁	#121	1	1	yes	characteristic
49	C₄×C₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₁₀₁	—	4	2	no	—
51	Dic₁₀₁	—	8	2	no	—
52	Dic₁₀₁	—	8	1	no	—
53	C₂×C₁₀₁	#459	1	1	yes	characteristic
54	C₂×C₁₀₁	C₂×Q₃₂	1	1	yes	characteristic
55	C₂×C₁₀₁	#510	1	1	yes	characteristic
56	#2209	—	101	1	no	maximal, Sylow
57	C₁₀₁	#2209	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₃₂	—	101	2	no	—
59	#494	—	101	1	no	—
60	C₄×Q₁₆	—	101	1	no	—
61	C₄⋊C₁₆	D₁₀₁	1	1	yes	characteristic
62	#358	—	101	2	no	—
63	C₂×Q₁₆	—	101	1	no	—
64	C₂×Q₁₆	—	101	2	no	—
65	C₂×Q₁₆	—	202	1	no	—
66	C₄⋊Q₈	—	202	1	no	—
67	C₄×Q₈	—	202	1	no	—
68	Q₃₂	—	202	4	no	—
69	C₂×C₁₆	C₂×D₁₀₁	1	2	yes	—
70	#107	—	101	1	no	—
71	#103	—	202	1	no	—
72	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
73	C₂×Q₈	—	202	2	no	—
74	C₂×Q₈	—	202	1	no	—
75	Q₁₆	—	101	2	no	—
76	Q₁₆	—	404	2	no	—
77	Q₁₆	—	202	2	no	—
78	Q₁₆	—	202	2	no	—
79	Q₁₆	—	202	1	no	—
80	C₂×C₈	D₄₀₄	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
82	C₄⋊C₄	—	404	1	no	—
83	C₄⋊C₄	—	404	1	no	—
84	C₄⋊C₄	—	404	1	no	—
85	C₄⋊C₄	—	202	1	no	—
86	C₄²	—	404	1	no	—
87	C₄²	D₄₀₄	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	404	2	no	—
90	Q₈	—	404	2	no	—
91	Q₈	—	404	1	no	—
92	Q₈	—	202	2	no	—
93	C₂×C₄	—	404	1	no	—
94	C₂×C₄	—	404	2	no	—
95	C₂×C₄	—	404	1	no	—
96	C₂×C₄	D₈₀₈	1	1	yes	characteristic
97	C₂×C₄	D₈₀₈	1	1	yes	characteristic
98	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₁₀₁	1	1	yes	characteristic
101	C₈	#13108075	1	1	yes	characteristic
102	C₂²	C₂×D₈₀₈	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	808	1	no	—
105	C₄	—	808	2	no	—
106	C₄	—	404	2	no	—
107	C₄	Dic₈₀₈	1	2	yes	complemented
108	C₄	#15692230	1	1	yes	characteristic
109	C₄	#15692283	1	1	yes	characteristic
110	C₂	#20265373	1	1	yes	characteristic
111	C₂	C₂×Dic₈₀₈	1	1	yes	characteristic
112	C₂	#20265620	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₈₀₈

C₂

yes

maximal

#20265557

C₂

yes

maximal, characteristic

C₄×Dic₄₀₄

C₂

yes

maximal, characteristic

#20264982

C₂

yes

maximal

C₄⋊C₁₆×C₁₀₁

C₂

yes

maximal, characteristic

C₂×Dic₄₀₄

—

C₂×Dic₄₀₄

C₂²

yes

—

C₂×Dic₄₀₄

C₂²

yes

characteristic

#15692255

—

C₄×Dic₂₀₂

—

Dic₈₀₈

—

#15692189

C₂²

yes

characteristic

#15692180

—

C₂×C₁₆×C₁₀₁

C₂²

yes

—

C₄×C₈×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

Dic₄₀₄

—

Dic₄₀₄

—

Dic₄₀₄

—

Dic₄₀₄

—

Dic₄₀₄

D₄

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

#13108053

—

C₄×Dic₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

characteristic

C₂×C₈×C₁₀₁

C₂³

yes

characteristic

C₄²×C₁₀₁

D₄

yes

characteristic

C₁₆×C₁₀₁

—

C₂×C₄×C₁₀₁

D₈

yes

characteristic

C₂×C₄×C₁₀₁

D₈

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

Dic₂₀₂

—

C₈×C₁₀₁

—

C₈×C₁₀₁

C₂×D₄

yes

characteristic

C₈×C₁₀₁

#41

yes

characteristic

C₂²×C₁₀₁

C₂×D₈

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

Q₃₂

yes

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

#459

yes

characteristic

C₂×C₁₀₁

C₂×Q₃₂

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#2209

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

101

—

#494

—

101

—

C₄×Q₁₆

—

101

—

C₄⋊C₁₆

D₁₀₁

yes

characteristic

#358

—

101

—

C₂×Q₁₆

—

101

—

C₂×Q₁₆

—

101

—

C₂×Q₁₆

—

202

—

C₄⋊Q₈

—

202

—

C₄×Q₈

—

202

—

Q₃₂

—

202

—

C₂×C₁₆

C₂×D₁₀₁

yes

—

#107

—

101

—

#103

—

202

—

C₄×C₈

C₂×D₁₀₁

yes

characteristic

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

Q₁₆

—

101

—

Q₁₆

—

404

—

Q₁₆

—

202

—

Q₁₆

—

202

—

Q₁₆

—

202

—

C₂×C₈

D₄₀₄

yes

characteristic

C₂×C₈

C₂²×D₁₀₁

yes

characteristic

C₄⋊C₄

—

404

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

202

—

C₄²

—

404

—

C₄²

D₄₀₄

yes

characteristic

C₁₆

—

Q₈

—

404

—

Q₈

—

404

—

Q₈

—

404

—

Q₈

—

202

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

D₈₀₈

yes

characteristic

C₂×C₄

D₈₀₈

yes

characteristic

C₂×C₄

C₂×D₄₀₄

yes

characteristic

C₈

—

100

C₈

D₄×D₁₀₁

yes

characteristic

101

C₈

#13108075

yes

characteristic

102

C₂²

C₂×D₈₀₈

yes

characteristic

103

C₄

—

104

C₄

—

808

—

105

C₄

—

808

—

106

C₄

—

404

—

107

C₄

Dic₈₀₈

yes

complemented

108

C₄

#15692230

yes

characteristic

109

C₄

#15692283

yes

characteristic

110

C₂

#20265373

yes

characteristic

111

C₂

C₂×Dic₈₀₈

yes

characteristic

112

C₂

#20265620

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₀₈
commutator, cosocle	C₂×C₈×C₁₀₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₁
Fitting	C₄⋊C₁₆×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	C₂×D₈
hypercenter	C₄⋊C₁₆	D₁₀₁
hypocenter	C₁₀₁	#2209
LCS	G ▷ C₂×C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₈₀₈

commutator, cosocle

C₂×C₈×C₁₀₁

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₁

Fitting

C₄⋊C₁₆×C₁₀₁

C₂

socle, prime radical

C₂²×C₁₀₁

C₂×D₈

hypercenter

C₄⋊C₁₆

D₁₀₁

hypocenter

C₁₀₁

#2209

LCS

G ▷ C₂×C₈×C₁₀₁ ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
47. C₄×C₁₀₁	Q₃₂	semidirect
57. C₁₀₁	#2209	semidirect
107. C₄	Dic₈₀₈	semidirect

H ≅ G/N

type

47. C₄×C₁₀₁

Q₃₂

semidirect

57. C₁₀₁

#2209

semidirect

107. C₄

Dic₈₀₈

semidirect

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

subgroup	index
C₂×Dic₈₀₈	2
#20265557	2
C₄×Dic₄₀₄	2
#20264982	2
C₄⋊C₁₆×C₁₀₁	2
#2209	101

subgroup

index

C₂×Dic₈₀₈

#20265557

C₄×Dic₄₀₄

#20264982

C₄⋊C₁₆×C₁₀₁

#2209

101

Subgroups — #27173183

Conjugacy classes of subgroups

Complements

Maximal subgroups