Group #27170272 — subgroups

Subgroups — #27170272

Order 12928.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20265556	C₂	1	1	yes	maximal, characteristic
2	#20265472	C₂	1	2	yes	maximal
3	#20264926	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#11717579	C₂	1	2	yes	maximal
5	Q₈²×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108064	—	2	2	no	—
7	#15692255	C₂²	1	1	yes	characteristic
8	#15692188	—	2	2	no	—
9	C₄⋊#11717579	C₂²	1	2	yes	—
10	#15692179	—	2	2	no	—
11	C₄×#11717579	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₀₁	—	2	2	no	—
13	C₄⋊Q₈×C₁₀₁	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₁₀₁	—	2	2	no	—
16	C₄×Q₈×C₁₀₁	C₂²	1	2	yes	—
17	C₂×Dic₂₀₂	—	2	1	no	—
18	#13108064	—	4	4	no	—
19	C₂×#11717579	—	2	1	no	—
20	C₂×#11717579	—	2	2	no	—
21	C₄⋊Dic₁₀₁	—	2	2	no	—
22	C₂×Q₈×C₁₀₁	—	2	2	no	—
23	C₂×Q₈×C₁₀₁	D₄	1	2	yes	—
24	C₂×Q₈×C₁₀₁	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₀₁	—	2	4	no	—
26	C₄⋊C₄×C₁₀₁	—	2	2	no	—
27	C₄⋊C₄×C₁₀₁	—	2	2	no	—
28	C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
29	C₄²×C₁₀₁	—	2	2	no	—
30	C₄²×C₁₀₁	—	2	2	no	—
31	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
32	Q₈×C₁₀₁	—	2	2	no	—
33	Q₈×C₁₀₁	—	4	2	no	—
34	Q₈×C₁₀₁	—	8	2	no	—
35	Q₈×C₁₀₁	Q₁₆	1	4	yes	complemented
36	C₂×C₄×C₁₀₁	—	2	2	no	—
37	C₂×C₄×C₁₀₁	—	2	2	no	—
38	C₂×C₄×C₁₀₁	—	2	2	no	—
39	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
41	C₂×Dic₁₀₁	—	4	1	no	—
42	Dic₂₀₂	—	4	2	no	—
43	#11717579	—	2	2	no	—
44	#11717579	—	4	2	no	—
45	C₂²×C₁₀₁	#120	1	1	yes	characteristic
46	C₄×C₁₀₁	—	2	4	no	—
47	C₄×C₁₀₁	—	2	1	no	—
48	C₄×C₁₀₁	—	4	2	no	—
49	C₄×C₁₀₁	—	4	2	no	—
50	C₄×C₁₀₁	C₂×Q₁₆	1	2	yes	—
51	C₄×C₁₀₁	#137	1	2	yes	—
52	Dic₁₀₁	—	8	1	no	—
53	C₂×C₁₀₁	#451	1	2	yes	—
54	C₂×C₁₀₁	#453	1	1	yes	characteristic
55	#1633	—	101	1	no	maximal, Sylow
56	C₁₀₁	#1633	1	1	yes	characteristic, complemented, Sylow
57	Q₈²	D₁₀₁	1	1	yes	characteristic
58	#494	—	101	1	no	—
59	#477	—	101	2	no	—
60	#333	—	101	1	no	—
61	Q₈⋊C₈	—	101	2	no	—
62	C₂×Q₁₆	—	202	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	101	1	no	—
66	C₄⋊Q₈	C₂×D₁₀₁	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₁₀₁	1	2	yes	—
69	#107	—	202	2	no	—
70	C₄⋊C₈	—	101	2	no	—
71	#103	—	202	2	no	—
72	C₄×C₈	—	101	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	202	1	no	—
75	C₂×Q₈	C₂²⋊D₁₀₁	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₁₀₁	1	2	yes	—
77	Q₁₆	—	404	4	no	—
78	C₂×C₈	—	202	2	no	—
79	C₂×C₈	—	202	1	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	202	2	no	—
84	C₄⋊C₄	C₂²⋊D₁₀₁	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	404	2	no	—
92	Q₈	#13108064	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	404	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
99	C₈	—	404	2	no	—
100	C₈	—	202	2	no	—
101	C₂²	#15692224	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	808	1	no	—
107	C₄	C₂×#13108064	1	2	yes	—
108	C₄	#15692286	1	2	yes	—
109	C₂	#20265333	1	2	yes	—
110	C₂	#20265345	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265556

C₂

yes

maximal, characteristic

#20265472

C₂

yes

maximal

#20264926

C₂

yes

maximal, characteristic

Q₈⋊#11717579

C₂

yes

maximal

Q₈²×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108064

—

#15692255

C₂²

yes

characteristic

#15692188

—

C₄⋊#11717579

C₂²

yes

—

#15692179

—

C₄×#11717579

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₁

—

C₄⋊Q₈×C₁₀₁

—

C₄⋊Q₈×C₁₀₁

C₂²

yes

characteristic

C₄×Q₈×C₁₀₁

—

C₄×Q₈×C₁₀₁

C₂²

yes

—

C₂×Dic₂₀₂

—

#13108064

—

C₂×#11717579

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

—

C₂×Q₈×C₁₀₁

D₄

yes

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

Q₁₆

yes

complemented

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

#120

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×Q₁₆

yes

—

C₄×C₁₀₁

#137

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

#451

yes

—

C₂×C₁₀₁

#453

yes

characteristic

#1633

—

101

maximal, Sylow

C₁₀₁

#1633

yes

characteristic, complemented, Sylow

Q₈²

D₁₀₁

yes

characteristic

#494

—

101

—

#477

—

101

—

#333

—

101

—

Q₈⋊C₈

—

101

—

C₂×Q₁₆

—

202

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

C₂×D₁₀₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₁₀₁

yes

—

#107

—

202

—

C₄⋊C₈

—

101

—

#103

—

202

—

C₄×C₈

—

101

—

C₂×Q₈

—

C₂×Q₈

—

202

—

C₂×Q₈

C₂²⋊D₁₀₁

yes

—

C₂×Q₈

C₂²⋊D₁₀₁

yes

—

Q₁₆

—

404

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

202

—

C₄⋊C₄

C₂²⋊D₁₀₁

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₁₀₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

404

—

Q₈

#13108064

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

404

—

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

C₈

—

404

—

100

C₈

—

202

—

101

C₂²

#15692224

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

808

—

107

C₄

C₂×#13108064

yes

—

108

C₄

#15692286

yes

—

109

C₂

#20265333

yes

—

110

C₂

#20265345

yes

characteristic

	subgroup	G/H
center	C₂²	#15692224
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	Q₈²×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	#120
hypercenter	Q₈²	D₁₀₁
hypocenter	C₁₀₁	#1633
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#15692224

commutator, cosocle

C₄²×C₁₀₁

C₂³

Frattini

C₄²

C₂²×D₁₀₁

Fitting

Q₈²×C₁₀₁

C₂

socle, prime radical

C₂²×C₁₀₁

#120

hypercenter

Q₈²

D₁₀₁

hypocenter

C₁₀₁

#1633

LCS

G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
35. Q₈×C₁₀₁	Q₁₆	semidirect
56. C₁₀₁	#1633	semidirect
92. Q₈	#13108064	semidirect

H ≅ G/N

type

35. Q₈×C₁₀₁

Q₁₆

semidirect

56. C₁₀₁

#1633

semidirect

92. Q₈

#13108064

semidirect

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

subgroup	index
#20265556	2
#20265472	2
#20264926	2
Q₈⋊#11717579	2
Q₈²×C₁₀₁	2
#1633	101

subgroup

index

Q₈²×C₁₀₁

101

Subgroups — #27170272

Conjugacy classes of subgroups

Complements

Maximal subgroups