Group #27172113 — subgroups

Subgroups — #27172113

Order 12928.

244 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20265734	C₂	1	1	yes	maximal, characteristic
2	#20265201	C₂	1	1	yes	maximal, characteristic
3	#20264937	C₂	1	4	yes	maximal
4	#376×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#13108058	C₂²	1	2	yes	—
6	#15692255	C₄	1	2	yes	complemented
7	#15692247	C₄	1	2	yes	complemented
8	#15692234	—	2	2	no	—
9	C₄×Dic₂₀₂	—	2	2	no	—
10	#15692214	C₂²	1	2	yes	—
11	C₄⋊#11717579	—	2	2	no	—
12	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
14	#95×C₁₀₁	—	2	2	no	—
15	C₂×Dic₂₀₂	—	2	2	no	—
16	#13108058	—	2	4	no	—
17	C₂×#11717579	—	2	2	no	—
18	C₄⋊Dic₁₀₁	—	2	2	no	—
19	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
20	#13108053	—	2	4	no	—
21	C₂²⋊Dic₁₀₁	—	2	2	no	—
22	C₄×Dic₁₀₁	—	2	2	no	—
23	C₂²×C₄×C₁₀₁	—	2	2	no	—
24	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
26	C₄²×C₁₀₁	—	2	2	no	—
27	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₀₁	D₄	1	2	yes	—
30	C₄²×C₁₀₁	Q₈	1	4	yes	complemented
31	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	—	2	4	no	—
33	C₂×C₄×C₁₀₁	—	4	2	no	—
34	C₂×C₄×C₁₀₁	—	4	4	no	—
35	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₀₁	#41	1	2	yes	—
41	C₂×Dic₁₀₁	—	2	4	no	—
42	Dic₂₀₂	—	4	4	no	—
43	#11717579	—	4	2	no	—
44	C₂²×C₁₀₁	—	2	2	no	—
45	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₁₀₁	—	2	2	no	—
48	C₄×C₁₀₁	—	2	2	no	—
49	C₄×C₁₀₁	—	4	4	no	—
50	C₄×C₁₀₁	C₄×Q₈	1	2	yes	—
51	C₄×C₁₀₁	#124	1	1	yes	characteristic
52	C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₀₁	—	4	4	no	—
54	C₂×C₁₀₁	—	2	2	no	—
55	C₂×C₁₀₁	#391	1	1	yes	characteristic
56	C₂×C₁₀₁	#421	1	2	yes	—
57	#1995	—	101	1	no	maximal, Sylow
58	C₁₀₁	#1995	1	1	yes	characteristic, complemented, Sylow
59	#533	—	101	1	no	—
60	#423	—	101	1	no	—
61	#376	D₁₀₁	1	1	yes	characteristic
62	#339	—	101	4	no	—
63	C₂×M₁₆	—	101	2	no	—
64	C₄⋊Q₈	—	101	2	no	—
65	#125	—	101	2	no	—
66	C₂²⋊Q₈	—	202	2	no	—
67	C₄×Q₈	—	202	2	no	—
68	#117	—	101	2	no	—
69	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
70	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
71	C₄⋊C₈	—	202	2	no	—
72	#95	—	2	2	no	—
73	C₂×Q₈	—	202	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
77	M₁₆	—	202	4	no	—
78	C₂×C₈	—	202	2	no	—
79	C₄⋊C₄	—	101	4	no	—
80	C₄⋊C₄	—	202	4	no	—
81	C₄⋊C₄	—	202	2	no	—
82	C₂²⋊C₄	—	202	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	202	2	no	—
85	C₄²	Dic₂₀₂	1	4	yes	complemented
86	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
87	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
88	C₄²	D₄₀₄	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
90	Q₈	—	404	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	—	202	4	no	—
95	C₂×C₄	#13108051	1	2	yes	—
96	C₂×C₄	#13108051	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
98	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₂₀₂	1	2	yes	—
100	C₂×C₄	#13108074	1	2	yes	—
101	C₈	—	404	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×#13108051	1	1	yes	characteristic
104	C₂²	#15692234	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	404	4	no	—
109	C₄	C₄×Dic₂₀₂	1	2	yes	—
110	C₄	#15692243	1	1	yes	characteristic
111	C₄	#15692255	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#20265083	1	1	yes	characteristic
114	C₂	#20265195	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265734

C₂

yes

maximal, characteristic

#20265201

C₂

yes

maximal, characteristic

#20264937

C₂

yes

maximal

#376×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

#15692255

C₄

yes

complemented

#15692247

C₄

yes

complemented

#15692234

—

C₄×Dic₂₀₂

—

#15692214

C₂²

yes

—

C₄⋊#11717579

—

C₂×C₄²×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

#95×C₁₀₁

—

C₂×Dic₂₀₂

—

#13108058

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂×C₄

yes

—

#13108053

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄²×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

Q₈

yes

complemented

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₂²⋊Q₈

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×Q₈

yes

—

C₄×C₁₀₁

#124

yes

characteristic

C₄×C₁₀₁

C₄⋊Q₈

yes

characteristic

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#391

yes

characteristic

C₂×C₁₀₁

#421

yes

—

#1995

—

101

maximal, Sylow

C₁₀₁

#1995

yes

characteristic, complemented, Sylow

#533

—

101

—

#423

—

101

—

#376

D₁₀₁

yes

characteristic

#339

—

101

—

C₂×M₁₆

—

101

—

C₄⋊Q₈

—

101

—

#125

—

101

—

C₂²⋊Q₈

—

202

—

C₄×Q₈

—

202

—

#117

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

202

—

#95

—

C₂×Q₈

—

202

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₄²

—

C₄²

—

202

—

C₄²

Dic₂₀₂

yes

complemented

C₄²

C₄×D₁₀₁

yes

characteristic

C₄²

C₄×D₁₀₁

yes

characteristic

C₄²

D₄₀₄

yes

—

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

Q₈

—

404

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

202

—

C₂×C₄

#13108051

yes

—

C₂×C₄

#13108051

yes

—

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

C₂×C₄

C₂×D₄₀₄

yes

characteristic

C₂×C₄

C₂×Dic₂₀₂

yes

—

100

C₂×C₄

#13108074

yes

—

101

C₈

—

404

—

102

C₂²

—

103

C₂²

C₂×#13108051

yes

characteristic

104

C₂²

#15692234

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

404

—

109

C₄

C₄×Dic₂₀₂

yes

—

110

C₄

#15692243

yes

characteristic

111

C₄

#15692255

yes

characteristic

112

C₂

—

113

C₂

#20265083

yes

characteristic

114

C₂

#20265195

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₀₄
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#376×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#376	D₁₀₁
hypocenter	C₁₀₁	#1995
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₄₀₄

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#376×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₀₁

C₂×D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
6. #15692255	C₄	semidirect
7. #15692247	C₄	semidirect
30. C₄²×C₁₀₁	Q₈	semidirect
58. C₁₀₁	#1995	semidirect
85. C₄²	Dic₂₀₂	semidirect

H ≅ G/N

type

6. #15692255

C₄

semidirect

7. #15692247

C₄

semidirect

30. C₄²×C₁₀₁

Q₈

semidirect

58. C₁₀₁

#1995

semidirect

85. C₄²

Dic₂₀₂

semidirect

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

subgroup	index
#20265734	2
#20265201	2
#20264937	2
#376×C₁₀₁	2
#1995	101

subgroup

index

#376×C₁₀₁

101

Subgroups — #27172113

Conjugacy classes of subgroups

Complements

Maximal subgroups