Group #27171267 — subgroups

Subgroups — #27171267

Order 12928.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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×#15692166	C₂	1	1	yes	maximal, characteristic
2	#20265043	C₂	1	1	yes	maximal, characteristic
3	#20264939	C₂	1	4	yes	maximal
4	#384×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11717579	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
7	#15692166	C₄	1	4	yes	complemented
8	#15692162	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₀₁	—	2	2	no	—
13	C₂×#11717579	—	2	2	no	—
14	C₂×#11717579	—	2	2	no	—
15	C₂×#11717579	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₁₀₁	—	2	4	no	—
17	C₂²×C₄×C₁₀₁	—	2	2	no	—
18	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
19	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₁	—	2	4	no	—
21	C₄⋊C₄×C₁₀₁	—	4	4	no	—
22	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
23	C₄²×C₁₀₁	Q₈	1	2	yes	—
24	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₁	—	2	2	no	—
26	C₂×C₄×C₁₀₁	—	2	2	no	—
27	C₂×C₄×C₁₀₁	—	4	2	no	—
28	C₂×C₄×C₁₀₁	—	4	4	no	—
29	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
30	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	#41	1	2	yes	—
33	C₂×C₄×C₁₀₁	#41	1	2	yes	—
34	C₂×Dic₁₀₁	—	4	2	no	—
35	C₂×Dic₁₀₁	—	4	2	no	—
36	C₂×Dic₁₀₁	—	4	2	no	—
37	#11717579	—	2	4	no	—
38	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₀₁	#124	1	1	yes	characteristic
40	C₂²×C₁₀₁	#124	1	1	yes	characteristic
41	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₁₀₁	#137	1	2	yes	—
43	C₄×C₁₀₁	—	2	4	no	—
44	C₄×C₁₀₁	—	8	2	no	—
45	C₄×C₁₀₁	#117	1	2	yes	—
46	C₄×C₁₀₁	#125	1	2	yes	—
47	Dic₁₀₁	—	8	2	no	—
48	C₂×C₁₀₁	#382	1	1	yes	characteristic
49	C₂×C₁₀₁	#428	1	2	yes	—
50	C₂×C₁₀₁	#489	1	2	yes	—
51	C₂×C₁₀₁	#490	1	1	yes	characteristic
52	C₂×C₁₀₁	#491	1	1	yes	characteristic
53	#1836	—	101	1	no	maximal, Sylow
54	C₁₀₁	#1836	1	1	yes	characteristic, complemented, Sylow
55	C₂×#97	—	101	1	no	—
56	#384	D₁₀₁	1	1	yes	characteristic
57	#382	—	101	1	no	—
58	#340	—	101	4	no	—
59	C₂²×C₈	—	101	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	101	2	no	—
62	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
63	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
64	#97	—	101	4	no	—
65	#95	—	202	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	202	2	no	—
68	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
70	C₂×C₈	—	101	4	no	—
71	C₂×C₈	—	202	2	no	—
72	C₂×C₈	—	202	2	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	—	202	4	no	—
76	C₄²	Dic₂₀₂	1	2	yes	—
77	C₄²	C₄×D₁₀₁	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	404	2	no	—
84	C₂×C₄	—	404	2	no	—
85	C₂×C₄	—	404	2	no	—
86	C₂×C₄	#13108053	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₂₀₂	1	1	yes	characteristic
89	C₂×C₄	#13108073	1	2	yes	—
90	C₂×C₄	#13108075	1	2	yes	—
91	C₈	—	202	4	no	—
92	C₂²	C₂×#13108053	1	1	yes	characteristic
93	C₂²	#15692242	1	1	yes	characteristic
94	C₂²	#15692244	1	1	yes	characteristic
95	C₂²	#15692281	1	2	yes	—
96	C₂²	#15692286	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	808	2	no	—
100	C₄	#15692213	1	2	yes	—
101	C₄	#15692246	1	2	yes	—
102	C₂	#20265042	1	1	yes	characteristic
103	C₂	#20265217	1	2	yes	—
104	C₂	#20265539	1	2	yes	—
105	C₂	#20265541	1	1	yes	characteristic
106	C₂	#20265545	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692166

C₂

yes

maximal, characteristic

#20265043

C₂

yes

maximal, characteristic

#20264939

C₂

yes

maximal

#384×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

#15692166

C₄

yes

complemented

#15692162

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

C₂×C₄

yes

—

C₄⋊Dic₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

Q₈

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

#11717579

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂²×C₁₀₁

#137

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#117

yes

—

C₄×C₁₀₁

#125

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

#382

yes

characteristic

C₂×C₁₀₁

#428

yes

—

C₂×C₁₀₁

#489

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#1836

yes

characteristic, complemented, Sylow

C₂×#97

—

101

—

#384

D₁₀₁

yes

characteristic

#382

—

101

—

#340

—

101

—

C₂²×C₈

—

101

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#97

—

101

—

#95

—

202

—

C₂²×C₄

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

202

—

C₄²

Dic₂₀₂

yes

—

C₄²

C₄×D₁₀₁

yes

—

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

#13108053

yes

—

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

C₂×C₄

C₂×Dic₂₀₂

yes

characteristic

C₂×C₄

#13108073

yes

—

C₂×C₄

#13108075

yes

—

C₈

—

202

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#15692286

yes

—

C₄

—

C₄

—

C₄

—

808

—

100

C₄

#15692213

yes

—

101

C₄

#15692246

yes

—

102

C₂

#20265042

yes

characteristic

103

C₂

#20265217

yes

—

104

C₂

#20265539

yes

—

105

C₂

#20265541

yes

characteristic

106

C₂

#20265545

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#384×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#384	D₁₀₁
hypocenter	C₁₀₁	#1836
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#384×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
7. #15692166	C₄	semidirect
54. C₁₀₁	#1836	semidirect

H ≅ G/N

type

7. #15692166

C₄

semidirect

54. C₁₀₁

#1836

semidirect

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

subgroup	index
C₂×#15692166	2
#20265043	2
#20264939	2
#384×C₁₀₁	2
#1836	101

subgroup

index

#384×C₁₀₁

101

Subgroups — #27171267

Conjugacy classes of subgroups

Complements

Maximal subgroups