Group #27186666 — subgroups

Subgroups — #27186666

Order 12928.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#20266200	C₂	1	2	yes	maximal
2	#20266037	C₂	1	4	yes	maximal, complemented
3	C₄⋊Q₈×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#15692322	C₂²	1	4	yes	—
5	#15692312	—	2	2	no	—
6	#15692310	—	2	2	no	—
7	C₂×Q₈×D₁₀₁	C₄	1	2	yes	—
8	#15692255	C₄	1	1	yes	characteristic
9	#15692254	—	2	2	no	—
10	#15692253	C₂²	1	2	yes	—
11	C₄⋊C₄×D₁₀₁	—	2	2	no	—
12	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₁₀₁	C₄	1	1	yes	characteristic
14	C₂×#11717581	—	2	4	no	—
15	C₂×Dic₂₀₂	—	2	2	no	—
16	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	—
17	Q₈×D₁₀₁	—	4	4	no	—
18	C₂×C₄×D₁₀₁	—	2	2	no	—
19	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
21	C₄⋊Dic₁₀₁	—	2	2	no	—
22	#13108053	—	2	4	no	—
23	C₄×Dic₁₀₁	D₄	1	2	yes	—
24	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
25	C₂×Q₈×C₁₀₁	C₂×C₄	1	2	yes	—
26	C₄⋊C₄×C₁₀₁	—	2	2	no	—
27	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
29	Q₈×C₁₀₁	—	4	2	no	—
30	C₂×C₄×C₁₀₁	—	2	2	no	—
31	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₁₀₁	—	2	2	no	—
34	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
36	C₄×D₁₀₁	—	2	2	no	—
37	C₄×D₁₀₁	—	2	4	no	—
38	C₄×D₁₀₁	—	4	4	no	—
39	Dic₂₀₂	—	4	4	no	—
40	Dic₂₀₂	—	4	2	no	—
41	#11717581	—	4	4	no	—
42	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂×D₁₀₁	—	2	2	no	—
44	C₂×D₁₀₁	#100	1	2	yes	—
45	C₄×C₁₀₁	—	2	1	no	—
46	C₄×C₁₀₁	—	2	2	no	—
47	C₄×C₁₀₁	—	4	2	no	—
48	Dic₁₀₁	—	2	1	no	—
49	Dic₁₀₁	—	4	2	no	—
50	Dic₁₀₁	#137	1	4	yes	—
51	C₂×C₁₀₁	C₂×#100	1	1	yes	characteristic
52	C₂×C₁₀₁	#419	1	2	yes	—
53	D₁₀₁	—	2	2	no	—
54	#1558	—	101	1	no	maximal, Sylow
55	C₁₀₁	#1558	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₄⋊Q₈	—	101	1	no	—
57	#432	—	101	2	no	—
58	#333	—	101	4	no	—
59	C₂²×Q₈	—	101	2	no	—
60	C₄⋊Q₈	—	101	2	no	—
61	C₄⋊Q₈	—	101	1	no	—
62	C₄⋊Q₈	—	202	2	no	—
63	C₄⋊Q₈	#87938	1	1	yes	characteristic
64	C₂×C₄⋊C₄	—	202	2	no	—
65	C₂×C₄²	—	101	1	no	—
66	C₄⋊C₈	—	101	4	no	—
67	C₂²⋊C₈	—	202	2	no	—
68	#97	—	202	2	no	—
69	C₂×Q₈	—	101	2	no	—
70	C₂×Q₈	—	404	4	no	—
71	C₂×Q₈	—	202	2	no	—
72	C₂×Q₈	C₂×#87938	1	2	yes	—
73	C₂²×C₄	—	101	2	no	—
74	C₂²×C₄	—	101	1	no	—
75	C₂²×C₄	—	202	2	no	—
76	C₂×C₈	—	202	4	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	202	4	no	—
79	C₄⋊C₄	—	202	2	no	—
80	C₄²	—	101	2	no	—
81	C₄²	—	101	1	no	—
82	C₄²	C₂×#87938	1	1	yes	characteristic
83	C₂³	—	101	1	no	—
84	Q₈	—	4	2	no	—
85	Q₈	—	404	2	no	—
86	Q₈	—	404	4	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	101	2	no	—
89	C₂×C₄	—	101	1	no	—
90	C₂×C₄	—	404	4	no	—
91	C₂×C₄	—	202	4	no	—
92	C₂×C₄	—	202	2	no	—
93	C₂×C₄	—	202	2	no	—
94	C₂×C₄	C₂²⋊#87938	1	2	yes	—
95	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
96	C₈	—	404	4	no	—
97	C₂²	—	101	2	no	—
98	C₂²	—	202	2	no	—
99	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	2	no	—
102	C₄	—	4	2	no	—
103	C₄	—	101	4	no	—
104	C₄	—	404	2	no	—
105	C₄	—	202	1	no	—
106	C₂	—	202	2	no	—
107	C₂	D₁₀₁⋊#100	1	1	yes	characteristic
108	C₂	#20266170	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20266200

C₂

yes

maximal

#20266037

C₂

yes

maximal, complemented

C₄⋊Q₈×D₁₀₁

C₂

yes

maximal, characteristic

#15692322

C₂²

yes

—

#15692312

—

#15692310

—

C₂×Q₈×D₁₀₁

C₄

yes

—

#15692255

C₄

yes

characteristic

#15692254

—

#15692253

C₂²

yes

—

C₄⋊C₄×D₁₀₁

—

C₄²×D₁₀₁

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₁

C₄

yes

characteristic

C₂×#11717581

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

Q₈×D₁₀₁

—

C₂×C₄×D₁₀₁

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₁

D₄

yes

—

C₄⋊Dic₁₀₁

—

#13108053

—

C₄×Dic₁₀₁

D₄

yes

—

C₄×Dic₁₀₁

C₂³

yes

characteristic

C₂×Q₈×C₁₀₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×D₁₀₁

C₂²⋊C₄

yes

characteristic

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₁

C₂×D₄

yes

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

#11717581

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

#100

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

#137

yes

—

C₂×C₁₀₁

C₂×#100

yes

characteristic

C₂×C₁₀₁

#419

yes

—

D₁₀₁

—

#1558

—

101

maximal, Sylow

C₁₀₁

#1558

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

101

—

#432

—

101

—

#333

—

101

—

C₂²×Q₈

—

101

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

—

202

—

C₄⋊Q₈

#87938

yes

characteristic

C₂×C₄⋊C₄

—

202

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

101

—

C₂²⋊C₈

—

202

—

#97

—

202

—

C₂×Q₈

—

101

—

C₂×Q₈

—

404

—

C₂×Q₈

—

202

—

C₂×Q₈

C₂×#87938

yes

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

C₂×#87938

yes

characteristic

C₂³

—

101

—

Q₈

—

Q₈

—

404

—

Q₈

—

404

—

C₂×C₄

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

404

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

C₂²⋊#87938

yes

—

C₂×C₄

C₂²×#87938

yes

characteristic

C₈

—

404

—

C₂²

—

101

—

C₂²

—

202

—

C₂²

C₂×C₂²⋊#87938

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

101

—

104

C₄

—

404

—

105

C₄

—

202

—

106

C₂

—

202

—

107

C₂

D₁₀₁⋊#100

yes

characteristic

108

C₂

#20266170

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87938
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87938
Fitting	C₄⋊Q₈×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₁	C₂²⋊C₄
cosocle	C₄×Dic₁₀₁	C₂³
hypercenter	C₄⋊Q₈	#87938
hypocenter	C₁₀₁	#1558
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#87938

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87938

Fitting

C₄⋊Q₈×C₁₀₁

C₄

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₀₁

C₂²⋊C₄

cosocle

C₄×Dic₁₀₁

C₂³

C₄⋊Q₈

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #20266037	C₂	semidirect
55. C₁₀₁	#1558	semidirect

H ≅ G/N

type

2. #20266037

C₂

semidirect

55. C₁₀₁

#1558

semidirect

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

subgroup	index
#20266200	2
#20266037	2
C₄⋊Q₈×D₁₀₁	2
#1558	101

subgroup

index

#20266200

#20266037

C₄⋊Q₈×D₁₀₁

#1558

101

Subgroups — #27186666

Conjugacy classes of subgroups

Complements

Maximal subgroups