Group #27171033 — subgroups

Subgroups — #27171033

Order 12928.

228 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 109 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×#15692190	C₂	1	2	yes	maximal
2	M₁₆×Dic₁₀₁	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#20264942	C₂	1	1	yes	maximal, characteristic
5	#341×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	C₂²	1	2	yes	—
7	C₂²×#11717579	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
9	#15692190	C₄	1	8	yes	complemented
10	#15692165	—	2	2	no	—
11	C₈×Dic₁₀₁	—	2	2	no	—
12	C₈×Dic₁₀₁	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₁	Q₈	1	2	yes	—
16	#13108058	—	2	2	no	—
17	#13108058	C₂×C₄	1	4	yes	—
18	C₂×#11717579	—	2	2	no	—
19	C₂×#11717579	C₂×C₄	1	2	yes	—
20	C₂×#11717579	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
23	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₀₁	—	2	2	no	—
26	M₁₆×C₁₀₁	C₂×C₄	1	4	yes	—
27	C₂×C₈×C₁₀₁	—	2	2	no	—
28	C₂×C₈×C₁₀₁	—	2	2	no	—
29	C₂×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₂³×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₁	#41	1	2	yes	—
35	C₂×Dic₁₀₁	—	2	4	no	—
36	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
37	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
38	C₈×C₁₀₁	—	2	2	no	—
39	C₈×C₁₀₁	—	2	4	no	—
40	#11717579	—	2	4	no	—
41	#11717579	C₄²	1	4	yes	—
42	C₂²×C₁₀₁	—	2	2	no	—
43	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
45	C₄×C₁₀₁	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₁₀₁	#117	1	1	yes	characteristic
47	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
48	Dic₁₀₁	—	2	2	no	—
49	Dic₁₀₁	#108	1	4	yes	complemented
50	C₂×C₁₀₁	—	2	2	no	—
51	C₂×C₁₀₁	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₁₀₁	C₂×#108	1	2	yes	—
53	C₄×#108	—	101	1	no	maximal, Sylow
54	C₁₀₁	C₄×#108	1	1	yes	characteristic, complemented, Sylow
55	C₂×#108	—	101	2	no	—
56	C₄×M₁₆	—	101	2	no	—
57	C₂×C₄×C₈	—	101	1	no	—
58	#341	—	101	1	no	—
59	#341	D₁₀₁	1	1	yes	characteristic
60	C₂×M₁₆	—	101	2	no	—
61	C₂×M₁₆	C₂×D₁₀₁	1	2	yes	—
62	C₂²×C₈	—	101	1	no	—
63	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
64	C₂×C₄²	—	101	1	no	—
65	#108	—	101	8	no	—
66	#97	—	202	2	no	—
67	C₄×C₈	—	101	4	no	—
68	C₄×C₈	—	202	2	no	—
69	C₂²×C₄	—	101	2	no	—
70	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	101	4	no	—
73	M₁₆	—	202	2	no	—
74	M₁₆	C₄×D₁₀₁	1	4	yes	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	101	2	no	—
78	C₂×C₈	—	101	4	no	—
79	C₂×C₈	—	202	2	no	—
80	C₂×C₈	C₄×D₁₀₁	1	2	yes	—
81	C₄²	—	101	2	no	—
82	C₄²	—	101	1	no	—
83	C₄²	—	101	1	no	—
84	C₂³	Q₈×D₁₀₁	1	1	yes	characteristic
85	C₂×C₄	—	101	2	no	—
86	C₂×C₄	—	101	2	no	—
87	C₂×C₄	—	202	4	no	—
88	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
89	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
90	C₂×C₄	D₄×D₁₀₁	1	1	yes	characteristic
91	C₂×C₄	#13108073	1	2	yes	—
92	C₈	—	2	2	no	—
93	C₈	—	2	4	no	—
94	C₈	—	101	4	no	—
95	C₈	—	202	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₄⋊C₄×D₁₀₁	1	1	yes	characteristic
98	C₂²	#15692220	1	2	yes	—
99	C₄	—	101	4	no	—
100	C₄	—	202	2	no	—
101	C₄	C₄²×D₁₀₁	1	1	yes	characteristic
102	C₄	#15692211	1	1	yes	characteristic
103	C₄	#15692215	1	2	yes	—
104	C₂	—	2	2	no	—
105	C₂	#20265028	1	1	yes	characteristic
106	C₂	#108×D₁₀₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692190

C₂

yes

maximal

M₁₆×Dic₁₀₁

C₂

yes

maximal

C₂×C₈×Dic₁₀₁

C₂

yes

maximal, characteristic

#20264942

C₂

yes

maximal, characteristic

#341×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

C₂²×#11717579

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

#15692190

C₄

yes

complemented

#15692165

—

C₈×Dic₁₀₁

—

C₈×Dic₁₀₁

C₄

yes

—

C₂×M₁₆×C₁₀₁

C₂²

yes

—

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

Q₈

yes

—

#13108058

—

#13108058

C₂×C₄

yes

—

C₂×#11717579

—

C₂×#11717579

C₂×C₄

yes

—

C₂×#11717579

C₂×C₄

yes

—

C₄×Dic₁₀₁

C₂×C₄

yes

—

C₄×Dic₁₀₁

D₄

yes

characteristic

C₄×Dic₁₀₁

D₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

C₂×C₄

yes

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

C₂×C₄

yes

—

C₂³×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

C₈×C₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

#11717579

C₄²

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₄×Q₈

yes

—

C₄×C₁₀₁

C₂×C₄²

yes

characteristic

C₄×C₁₀₁

#117

yes

characteristic

C₄×C₁₀₁

C₄×D₄

yes

—

Dic₁₀₁

—

Dic₁₀₁

#108

yes

complemented

C₂×C₁₀₁

—

C₂×C₁₀₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₁

C₂×#108

yes

—

C₄×#108

—

101

maximal, Sylow

C₁₀₁

C₄×#108

yes

characteristic, complemented, Sylow

C₂×#108

—

101

—

C₄×M₁₆

—

101

—

C₂×C₄×C₈

—

101

—

#341

—

101

—

#341

D₁₀₁

yes

characteristic

C₂×M₁₆

—

101

—

C₂×M₁₆

C₂×D₁₀₁

yes

—

C₂²×C₈

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

C₂×C₄²

—

101

—

#108

—

101

—

#97

—

202

—

C₄×C₈

—

101

—

C₄×C₈

—

202

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

M₁₆

—

101

—

M₁₆

—

202

—

M₁₆

C₄×D₁₀₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

C₄×D₁₀₁

yes

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

—

101

—

C₂³

Q₈×D₁₀₁

yes

characteristic

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

202

—

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₁

yes

—

C₂×C₄

D₄×D₁₀₁

yes

characteristic

C₂×C₄

#13108073

yes

—

C₈

—

C₈

—

C₈

—

101

—

C₈

—

202

—

C₂²

—

C₂²

C₄⋊C₄×D₁₀₁

yes

characteristic

C₂²

#15692220

yes

—

C₄

—

101

—

100

C₄

—

202

—

101

C₄

C₄²×D₁₀₁

yes

characteristic

102

C₄

#15692211

yes

characteristic

103

C₄

#15692215

yes

—

104

C₂

—

105

C₂

#20265028

yes

characteristic

106

C₂

#108×D₁₀₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₁
commutator	C₄×C₁₀₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#341×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₁	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#341	D₁₀₁
hypocenter	C₁₀₁	C₄×#108
LCS	G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₁

commutator

C₄×C₁₀₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#341×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₁

C₂×Q₈

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #15692190	C₄	semidirect
49. Dic₁₀₁	#108	semidirect
54. C₁₀₁	C₄×#108	semidirect

H ≅ G/N

type

9. #15692190

C₄

semidirect

49. Dic₁₀₁

#108

semidirect

54. C₁₀₁

C₄×#108

semidirect

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

subgroup	index
C₂×#15692190	2
M₁₆×Dic₁₀₁	2
C₂×C₈×Dic₁₀₁	2
#20264942	2
#341×C₁₀₁	2
C₄×#108	101

subgroup

index

C₂×#15692190

M₁₆×Dic₁₀₁

C₂×C₈×Dic₁₀₁

#20264942

#341×C₁₀₁

C₄×#108

101

Subgroups — #27171033

Conjugacy classes of subgroups

Complements

Maximal subgroups