Group #27186014 — subgroups

Subgroups — #27186014

Order 12928.

190 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 48 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	D₁₀₁⋊(C₂²⋊C₈)	C₂	1	2	yes	maximal
2	#20265140	C₂	1	1	yes	maximal, characteristic
3	C₂×D₁₀₁⋊C₈	—	2	2	no	—
4	#15692312	C₄	1	4	yes	—
5	C₂²⋊#11717581	—	2	2	no	—
6	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
7	C₂×#13108058	C₄	1	1	yes	characteristic
8	#15692167	—	2	2	no	—
9	C₂×M₁₆×C₁₀₁	C₄	1	1	yes	characteristic
10	D₁₀₁⋊C₈	—	4	2	no	—
11	D₁₀₁⋊C₈	—	4	2	no	—
12	C₂×#11717581	—	2	4	no	—
13	C₂³×D₁₀₁	Q₈	1	1	yes	characteristic
14	C₂²×Dic₁₀₁	D₄	1	1	yes	characteristic
15	C₂×C₄×D₁₀₁	—	4	1	no	—
16	C₂×C₄×D₁₀₁	—	4	1	no	—
17	C₂×C₄×D₁₀₁	C₂×C₄	1	2	yes	—
18	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
19	#13108058	—	4	1	no	—
20	C₂×#11717579	—	2	1	no	—
21	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
22	M₁₆×C₁₀₁	—	4	1	no	—
23	C₂×C₈×C₁₀₁	—	2	1	no	—
24	C₂³×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×D₁₀₁	—	2	4	no	—
26	C₂²×D₁₀₁	—	4	1	no	—
27	C₂²×D₁₀₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₀₁	—	2	1	no	—
29	C₂×C₄×C₁₀₁	—	2	1	no	—
30	C₂×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₁	—	4	1	no	—
33	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
34	C₄×D₁₀₁	—	2	2	no	—
35	C₄×D₁₀₁	—	2	2	no	—
36	C₄×D₁₀₁	—	4	2	no	—
37	#11717581	—	4	4	no	—
38	C₈×C₁₀₁	—	4	1	no	—
39	#11717579	—	4	1	no	—
40	C₂²×C₁₀₁	—	2	2	no	—
41	C₂²×C₁₀₁	—	2	1	no	—
42	C₂²×C₁₀₁	#95	1	1	yes	characteristic
43	C₂×D₁₀₁	—	2	2	no	—
44	C₂×D₁₀₁	—	2	2	no	—
45	C₂×D₁₀₁	—	4	4	no	—
46	C₂×D₁₀₁	#108	1	4	yes	—
47	C₄×C₁₀₁	—	2	1	no	—
48	C₄×C₁₀₁	#99	1	1	yes	characteristic
49	C₄×C₁₀₁	#99	1	1	yes	characteristic
50	Dic₁₀₁	—	2	2	no	—
51	C₂×C₁₀₁	—	4	1	no	—
52	C₂×C₁₀₁	#341	1	2	yes	—
53	C₂×C₁₀₁	#342	1	1	yes	characteristic
54	D₁₀₁	—	2	4	no	—
55	#1298	—	101	1	no	maximal, Sylow
56	C₁₀₁	#1298	1	1	yes	characteristic, complemented, Sylow
57	#407	—	101	1	no	—
58	C₂×C₂²⋊C₈	—	101	2	no	—
59	C₂³×C₄	—	101	1	no	—
60	C₂×M₁₆	—	101	1	no	—
61	C₂×M₁₆	#87938	1	1	yes	characteristic
62	C₂²×C₈	—	202	2	no	—
63	C₂²⋊C₈	—	101	4	no	—
64	C₂²⋊C₈	—	202	2	no	—
65	C₂²⋊C₈	—	202	2	no	—
66	C₂⁴	—	101	1	no	—
67	C₂²×C₄	—	101	1	no	—
68	C₂²×C₄	—	101	2	no	—
69	C₂²×C₄	—	101	2	no	—
70	C₂²×C₄	—	404	1	no	—
71	C₂²×C₄	—	404	1	no	—
72	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
73	M₁₆	—	4	1	no	—
74	M₁₆	—	404	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	404	2	no	—
77	C₂×C₈	—	404	2	no	—
78	C₂×C₈	—	202	4	no	—
79	C₂×C₈	—	202	1	no	—
80	C₂³	—	101	2	no	—
81	C₂³	—	404	1	no	—
82	C₂³	—	202	4	no	—
83	C₂³	C₄⋊#87938	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	101	2	no	—
87	C₂×C₄	—	404	2	no	—
88	C₂×C₄	—	404	1	no	—
89	C₂×C₄	—	202	2	no	—
90	C₂×C₄	—	202	2	no	—
91	C₂×C₄	C₄×#87938	1	1	yes	characteristic
92	C₂×C₄	C₂²⋊#87938	1	1	yes	characteristic
93	C₈	—	4	1	no	—
94	C₈	—	404	4	no	—
95	C₈	—	404	1	no	—
96	C₂²	—	2	2	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	101	4	no	—
99	C₂²	—	404	4	no	—
100	C₂²	—	202	2	no	—
101	C₂²	—	202	2	no	—
102	C₂²	(C₂×#87938)⋊C₄	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	202	2	no	—
105	C₄	#15692313	1	1	yes	characteristic
106	C₄	#15692314	1	1	yes	characteristic
107	C₂	—	4	1	no	—
108	C₂	—	202	4	no	—
109	C₂	#20266052	1	2	yes	—
110	C₂	#20266055	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₀₁⋊(C₂²⋊C₈)

C₂

yes

maximal

#20265140

C₂

yes

maximal, characteristic

C₂×D₁₀₁⋊C₈

—

#15692312

C₄

yes

—

C₂²⋊#11717581

—

C₂²×C₄×D₁₀₁

C₂²

yes

characteristic

C₂×#13108058

C₄

yes

characteristic

#15692167

—

C₂×M₁₆×C₁₀₁

C₄

yes

characteristic

D₁₀₁⋊C₈

—

D₁₀₁⋊C₈

—

C₂×#11717581

—

C₂³×D₁₀₁

Q₈

yes

characteristic

C₂²×Dic₁₀₁

D₄

yes

characteristic

C₂×C₄×D₁₀₁

—

C₂×C₄×D₁₀₁

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

—

C₂×C₄×D₁₀₁

D₄

yes

—

#13108058

—

C₂×#11717579

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂³×C₁₀₁

C₄⋊C₄

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄²

yes

characteristic

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

#11717581

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#95

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

#108

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

yes

C₄×C₁₀₁

yes

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#341

yes

—

C₂×C₁₀₁

#342

yes

characteristic

D₁₀₁

—

#1298

—

101

maximal, Sylow

C₁₀₁

#1298

yes

characteristic, complemented, Sylow

#407

—

101

—

C₂×C₂²⋊C₈

—

101

—

C₂³×C₄

—

101

—

C₂×M₁₆

—

101

—

C₂×M₁₆

#87938

yes

characteristic

C₂²×C₈

—

202

—

C₂²⋊C₈

—

101

—

C₂²⋊C₈

—

202

—

C₂²⋊C₈

—

202

—

C₂⁴

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

404

—

C₂²×C₄

—

404

—

C₂²×C₄

C₂×#87938

yes

characteristic

M₁₆

—

M₁₆

—

404

—

C₂×C₈

—

C₂×C₈

—

404

—

C₂×C₈

—

404

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂³

—

101

—

C₂³

—

404

—

C₂³

—

202

—

C₂³

C₄⋊#87938

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

101

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

404

—

C₈

—

404

—

C₂²

—

C₂²

—

C₂²

—

101

—

C₂²

—

404

—

100

C₂²

—

202

—

101

C₂²

—

202

—

102

C₂²

(C₂×#87938)⋊C₄

yes

characteristic

103

C₄

—

104

C₄

—

202

—

105

C₄

#15692313

yes

characteristic

106

C₄

#15692314

yes

characteristic

107

C₂

—

108

C₂

—

202

—

109

C₂

#20266052

yes

—

110

C₂

#20266055

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#87938
commutator	C₂×C₄×C₁₀₁	C₄²
Frattini	C₂²×C₄	C₂×#87938
Fitting	C₂×M₁₆×C₁₀₁	C₄
socle	C₂²×C₁₀₁	#95
prime radical	C₂³×D₁₀₁	Q₈
cosocle	C₂²×C₄×D₁₀₁	C₂²
hypercenter	C₂×M₁₆	#87938
hypocenter	C₁₀₁	#1298
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₂×C₄

C₂²⋊#87938

commutator

C₂×C₄×C₁₀₁

C₄²

Frattini

C₂²×C₄

C₂×#87938

Fitting

C₂×M₁₆×C₁₀₁

C₄

socle

C₂²×C₁₀₁

#95

prime radical

C₂³×D₁₀₁

Q₈

cosocle

C₂²×C₄×D₁₀₁

C₂²

C₂×M₁₆

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
56. C₁₀₁	#1298	semidirect

H ≅ G/N

type

56. C₁₀₁

#1298

semidirect

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

subgroup	index
D₁₀₁⋊(C₂²⋊C₈)	2
#20265140	2
#1298	101

subgroup

index

D₁₀₁⋊(C₂²⋊C₈)

#20265140

#1298

101

Subgroups — #27186014

Conjugacy classes of subgroups

Complements

Maximal subgroups