Group #27181646 — subgroups

Subgroups — #27181646

Order 12928.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 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	2	yes	maximal
2	#20265582	C₂	1	2	yes	maximal, complemented
3	#20265579	C₂	1	4	yes	maximal, complemented
4	#20265575	C₂	1	2	yes	maximal, complemented
5	#20265215	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	#15692255	C₂²	1	4	yes	—
8	#15692247	C₂²	1	4	yes	—
9	#15692234	—	2	4	no	—
10	C₄×Dic₂₀₂	—	2	4	no	—
11	#15692214	C₂²	1	4	yes	—
12	#15692189	C₂²	1	8	yes	—
13	#15692187	C₂²	1	8	yes	—
14	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₁₀₁	C₂²	1	2	yes	—
18	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₀₂	—	2	4	no	—
20	C₄⋊Dic₁₀₁	—	2	4	no	—
21	C₄⋊Dic₁₀₁	C₂³	1	8	yes	—
22	#13108053	—	2	8	no	—
23	C₂²⋊Dic₁₀₁	—	2	4	no	—
24	C₄×Dic₁₀₁	—	2	4	no	—
25	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₀₁	Q₈	1	8	yes	complemented
28	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
29	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
30	C₄²×C₁₀₁	D₄	1	2	yes	—
31	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
32	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
33	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₁	—	2	4	no	—
35	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
39	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₁₀₁	—	2	8	no	—
41	Dic₂₀₂	—	4	8	no	—
42	C₈×C₁₀₁	C₂×Q₈	1	4	yes	—
43	C₈×C₁₀₁	C₂×Q₈	1	4	yes	—
44	C₂²×C₁₀₁	—	2	2	no	—
45	C₂²×C₁₀₁	C₄⋊Q₈	1	2	yes	—
46	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₁₀₁	—	2	2	no	—
48	C₄×C₁₀₁	C₄⋊Q₈	1	2	yes	—
49	C₄×C₁₀₁	#135	1	4	yes	—
50	C₄×C₁₀₁	C₂²×Q₈	1	2	yes	—
51	Dic₁₀₁	—	4	8	no	—
52	C₂×C₁₀₁	—	2	2	no	—
53	C₂×C₁₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₁₀₁	C₂×#135	1	2	yes	—
55	#3160	—	101	1	no	maximal, Sylow
56	C₁₀₁	#3160	1	1	yes	characteristic, complemented, Sylow
57	#533	—	101	2	no	—
58	#500	—	101	2	no	—
59	#499	—	101	4	no	—
60	#498	—	101	2	no	—
61	#427	—	101	4	no	—
62	C₂×C₄×C₈	D₁₀₁	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
64	C₄⋊Q₈	—	101	4	no	—
65	#125	—	101	4	no	—
66	C₂²⋊Q₈	—	202	4	no	—
67	C₄×Q₈	—	202	4	no	—
68	#117	—	101	4	no	—
69	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
70	#107	—	101	8	no	—
71	#106	—	101	8	no	—
72	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
73	C₄×C₈	C₂×D₁₀₁	1	2	yes	—
74	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
75	C₂×Q₈	—	202	4	no	—
76	C₂²×C₄	D₄₀₄	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
78	C₂×C₈	Dic₂₀₂	1	8	yes	complemented
79	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
80	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
81	C₄⋊C₄	—	101	8	no	—
82	C₄⋊C₄	—	202	8	no	—
83	C₄⋊C₄	—	202	4	no	—
84	C₂²⋊C₄	—	202	4	no	—
85	C₄²	—	202	4	no	—
86	C₄²	D₄₀₄	1	2	yes	—
87	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
88	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
89	C₂³	C₂×D₄₀₄	1	1	yes	characteristic
90	Q₈	—	404	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	202	8	no	—
93	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
94	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
95	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
96	C₂×C₄	C₂×Dic₂₀₂	1	4	yes	—
97	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
98	C₈	C₂×Dic₂₀₂	1	4	yes	—
99	C₈	C₂×Dic₂₀₂	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#15692255	1	2	yes	—
102	C₂²	C₂²×D₄₀₄	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	404	8	no	—
105	C₄	#15692255	1	2	yes	—
106	C₄	#15692277	1	4	yes	—
107	C₄	C₂²×Dic₂₀₂	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#15692255	1	1	yes	characteristic
110	C₂	C₂×#15692277	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265734

C₂

yes

maximal

#20265582

C₂

yes

maximal, complemented

#20265579

C₂

yes

maximal, complemented

#20265575

C₂

yes

maximal, complemented

#20265215

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₁

C₂

yes

maximal, characteristic

#15692255

C₂²

yes

—

#15692247

C₂²

yes

—

#15692234

—

C₄×Dic₂₀₂

—

#15692214

C₂²

yes

—

#15692189

C₂²

yes

—

#15692187

C₂²

yes

—

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₄×C₈×C₁₀₁

C₂²

yes

characteristic

C₄×C₈×C₁₀₁

C₂²

yes

—

C₄×C₈×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂³

yes

—

#13108053

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

Q₈

yes

complemented

C₂×C₈×C₁₀₁

C₂³

yes

—

C₂×C₈×C₁₀₁

C₂³

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂×Dic₁₀₁

—

Dic₂₀₂

—

C₈×C₁₀₁

C₂×Q₈

yes

—

C₈×C₁₀₁

C₂×Q₈

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₄⋊Q₈

yes

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄⋊Q₈

yes

—

C₄×C₁₀₁

#135

yes

—

C₄×C₁₀₁

C₂²×Q₈

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₁

C₂×#135

yes

—

#3160

—

101

maximal, Sylow

C₁₀₁

#3160

yes

characteristic, complemented, Sylow

#533

—

101

—

#500

—

101

—

#499

—

101

—

#498

—

101

—

#427

—

101

—

C₂×C₄×C₈

D₁₀₁

yes

characteristic

C₂²×C₈

C₂×D₁₀₁

yes

—

C₄⋊Q₈

—

101

—

#125

—

101

—

C₂²⋊Q₈

—

202

—

C₄×Q₈

—

202

—

#117

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#107

—

101

—

#106

—

101

—

C₄×C₈

C₂×D₁₀₁

yes

characteristic

C₄×C₈

C₂×D₁₀₁

yes

—

C₄×C₈

C₂×D₁₀₁

yes

characteristic

C₂×Q₈

—

202

—

C₂²×C₄

D₄₀₄

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

Dic₂₀₂

yes

complemented

C₂×C₈

C₂²×D₁₀₁

yes

—

C₂×C₈

C₂²×D₁₀₁

yes

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₄²

—

202

—

C₄²

D₄₀₄

yes

—

C₄²

C₂²×D₁₀₁

yes

characteristic

C₄²

C₂²×D₁₀₁

yes

characteristic

C₂³

C₂×D₄₀₄

yes

characteristic

Q₈

—

404

—

C₂×C₄

—

C₂×C₄

—

202

—

C₂×C₄

C₂×D₄₀₄

yes

—

C₂×C₄

C₂×D₄₀₄

yes

—

C₂×C₄

C₂×D₄₀₄

yes

characteristic

C₂×C₄

C₂×Dic₂₀₂

yes

—

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

C₈

C₂×Dic₂₀₂

yes

—

C₈

C₂×Dic₂₀₂

yes

—

100

C₂²

—

101

C₂²

#15692255

yes

—

102

C₂²

C₂²×D₄₀₄

yes

characteristic

103

C₄

—

104

C₄

—

404

—

105

C₄

#15692255

yes

—

106

C₄

#15692277

yes

—

107

C₄

C₂²×Dic₂₀₂

yes

—

108

C₂

—

109

C₂

C₂×#15692255

yes

characteristic

110

C₂

C₂×#15692277

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₀₄
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	C₂×C₄×C₈×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂³×C₁₀₁	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₀₁
hypocenter	C₁₀₁	#3160
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₄₀₄

commutator, cosocle

C₂×C₄×C₁₀₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting

C₂×C₄×C₈×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂²×D₄

prime radical

C₂³×C₁₀₁

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₀₁

hypocenter

C₁₀₁

#3160

LCS

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #20265582	C₂	semidirect
3. #20265579	C₂	semidirect
4. #20265575	C₂	semidirect
27. C₂×C₈×C₁₀₁	Q₈	semidirect
56. C₁₀₁	#3160	semidirect
78. C₂×C₈	Dic₂₀₂	semidirect

H ≅ G/N

type

2. #20265582

C₂

semidirect

3. #20265579

C₂

semidirect

4. #20265575

C₂

semidirect

27. C₂×C₈×C₁₀₁

Q₈

semidirect

56. C₁₀₁

#3160

semidirect

78. C₂×C₈

Dic₂₀₂

semidirect

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

subgroup	index
#20265734	2
#20265582	2
#20265579	2
#20265575	2
#20265215	2
C₂×C₄×C₈×C₁₀₁	2
#3160	101

subgroup

index

C₂×C₄×C₈×C₁₀₁

101

Subgroups — #27181646

Conjugacy classes of subgroups

Complements

Maximal subgroups