Group #27186010 — subgroups

Subgroups — #27186010

Order 12928.

188 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 76 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×#15692322	C₂	1	2	yes	maximal
2	#20265046	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11717581	—	2	2	no	—
4	#15692322	—	2	2	no	—
5	#15692322	C₄	1	4	yes	complemented
6	C₂×C₄⋊Dic₁₀₁	C₄	1	1	yes	characteristic
7	C₂×#13108053	—	2	2	no	—
8	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₁₀₁	C₄	1	1	yes	characteristic
10	C₂×#11717581	—	2	4	no	—
11	C₂×#11717581	—	4	2	no	—
12	C₂×#11717581	—	4	2	no	—
13	C₂²×Dic₁₀₁	—	2	1	no	—
14	C₂²×Dic₁₀₁	D₄	1	2	yes	—
15	C₄⋊Dic₁₀₁	—	4	1	no	—
16	#13108053	—	4	2	no	—
17	#13108053	—	4	2	no	—
18	C₄×Dic₁₀₁	C₂×C₄	1	2	yes	—
19	C₄×Dic₁₀₁	D₄	1	1	yes	characteristic
20	C₄×Dic₁₀₁	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₁	—	2	1	no	—
22	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₁₀₁	—	4	1	no	—
24	C₂³×C₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₁	—	2	1	no	—
26	C₂×C₄×C₁₀₁	—	2	1	no	—
27	C₂×C₄×C₁₀₁	—	4	1	no	—
28	C₂×C₄×C₁₀₁	—	4	2	no	—
29	C₂×C₄×C₁₀₁	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₁₀₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₁	—	4	1	no	—
32	C₂×Dic₁₀₁	—	4	2	no	—
33	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
34	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
35	C₂×Dic₁₀₁	D₈	1	2	yes	—
36	C₂×Dic₁₀₁	SD₁₆	1	2	yes	—
37	C₂×Dic₁₀₁	SD₁₆	1	2	yes	—
38	C₂×Dic₁₀₁	Q₁₆	1	2	yes	—
39	#11717581	—	4	4	no	—
40	C₂²×C₁₀₁	#95	1	1	yes	characteristic
41	C₂²×C₁₀₁	#100	1	1	yes	characteristic
42	C₂²×C₁₀₁	#101	1	1	yes	characteristic
43	C₂²×C₁₀₁	#102	1	2	yes	—
44	C₂²×C₁₀₁	#103	1	2	yes	—
45	C₄×C₁₀₁	—	2	1	no	—
46	C₄×C₁₀₁	—	2	1	no	—
47	C₄×C₁₀₁	—	8	1	no	—
48	Dic₁₀₁	—	8	1	no	—
49	Dic₁₀₁	#102	1	2	yes	—
50	Dic₁₀₁	#103	1	2	yes	—
51	Dic₁₀₁	#106	1	2	yes	—
52	Dic₁₀₁	#107	1	2	yes	—
53	C₂×C₁₀₁	#331	1	1	yes	characteristic
54	C₂×C₁₀₁	#332	1	2	yes	—
55	C₂×C₁₀₁	#333	1	1	yes	characteristic
56	C₂×C₁₀₁	#340	1	2	yes	—
57	C₂×C₁₀₁	#343	1	1	yes	characteristic
58	#1296	—	101	1	no	maximal, Sylow
59	C₁₀₁	#1296	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₄⋊C₈	—	101	2	no	—
61	#384	—	101	1	no	—
62	C₂²×C₈	—	202	2	no	—
63	C₂×C₄⋊C₄	—	101	1	no	—
64	C₂×C₄⋊C₄	—	202	2	no	—
65	C₂×C₄⋊C₄	#87938	1	1	yes	characteristic
66	C₂×C₄²	—	101	1	no	—
67	C₄⋊C₈	—	101	4	no	—
68	C₄⋊C₈	—	202	2	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	—	101	2	no	—
71	C₂²×C₄	—	202	1	no	—
72	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
73	C₂×C₈	—	404	2	no	—
74	C₂×C₈	—	404	2	no	—
75	C₂×C₈	—	202	4	no	—
76	C₄⋊C₄	—	4	1	no	—
77	C₄⋊C₄	—	404	1	no	—
78	C₄⋊C₄	—	404	2	no	—
79	C₄⋊C₄	—	404	2	no	—
80	C₄²	—	101	2	no	—
81	C₄²	—	101	1	no	—
82	C₄²	—	101	1	no	—
83	C₂³	C₂²⋊#87938	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	101	2	no	—
89	C₂×C₄	—	101	2	no	—
90	C₂×C₄	—	101	2	no	—
91	C₂×C₄	—	101	2	no	—
92	C₂×C₄	—	101	2	no	—
93	C₂×C₄	—	101	2	no	—
94	C₂×C₄	—	404	1	no	—
95	C₂×C₄	—	404	2	no	—
96	C₂×C₄	C₄×#87938	1	1	yes	characteristic
97	C₂×C₄	C₄⋊#87938	1	1	yes	characteristic
98	C₈	—	404	4	no	—
99	C₂²	(C₂×#87938)⋊C₄	1	1	yes	characteristic
100	C₂²	#15692315	1	1	yes	characteristic
101	C₂²	#15692316	1	1	yes	characteristic
102	C₂²	#15692317	1	2	yes	—
103	C₂²	#15692318	1	2	yes	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	8	1	no	—
107	C₄	—	808	1	no	—
108	C₄	—	101	2	no	—
109	C₄	—	101	2	no	—
110	C₄	—	101	2	no	—
111	C₄	—	101	2	no	—
112	C₂	#20266035	1	1	yes	characteristic
113	C₂	#20266036	1	2	yes	—
114	C₂	#20266037	1	1	yes	characteristic
115	C₂	#20266050	1	2	yes	—
116	C₂	#20266057	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692322

C₂

yes

maximal

#20265046

C₂

yes

maximal, characteristic

C₂²×#11717581

—

#15692322

—

#15692322

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₁

C₄

yes

characteristic

C₂×#13108053

—

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₁

C₄

yes

—

—

—

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

D₄

yes

—

C₄⋊Dic₁₀₁

—

#13108053

—

#13108053

—

C₄×Dic₁₀₁

C₂×C₄

yes

—

C₄×Dic₁₀₁

D₄

yes

characteristic

C₄×Dic₁₀₁

Q₈

yes

characteristic

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₂³×C₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄²

yes

characteristic

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

C₂×Dic₁₀₁

D₈

yes

—

C₂×Dic₁₀₁

SD₁₆

yes

—

C₂×Dic₁₀₁

SD₁₆

yes

—

C₂×Dic₁₀₁

Q₁₆

yes

—

#11717581

—

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₂²×C₁₀₁

#103

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

#102

yes

—

Dic₁₀₁

#103

yes

—

Dic₁₀₁

#106

yes

—

Dic₁₀₁

#107

yes

—

C₂×C₁₀₁

#331

yes

characteristic

C₂×C₁₀₁

#332

yes

—

C₂×C₁₀₁

#333

yes

characteristic

C₂×C₁₀₁

#340

yes

—

C₂×C₁₀₁

#343

yes

characteristic

#1296

—

101

maximal, Sylow

C₁₀₁

#1296

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

101

—

#384

—

101

—

C₂²×C₈

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

202

—

C₂×C₄⋊C₄

#87938

yes

characteristic

C₂×C₄²

—

101

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

202

—

C₂²×C₄

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂×#87938

yes

characteristic

C₂×C₈

—

404

—

C₂×C₈

—

404

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

404

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

—

101

—

C₂³

C₂²⋊#87938

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

404

—

C₂²

(C₂×#87938)⋊C₄

yes

characteristic

100

C₂²

#15692315

yes

characteristic

101

C₂²

#15692316

yes

characteristic

102

C₂²

#15692317

yes

—

103

C₂²

#15692318

yes

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

808

—

108

C₄

—

101

—

109

C₄

—

101

—

110

C₄

—

101

—

111

C₄

—

101

—

112

C₂

#20266035

yes

characteristic

113

C₂

#20266036

yes

—

114

C₂

#20266037

yes

characteristic

115

C₂

#20266050

yes

—

116

C₂

#20266057

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#87938
commutator	C₂×C₄×C₁₀₁	C₄²
Frattini	C₂²×C₄	C₂×#87938
Fitting	C₂×C₄⋊C₄×C₁₀₁	C₄
socle, prime radical	C₂³×C₁₀₁	C₂²⋊C₄
cosocle	C₂×C₄×Dic₁₀₁	C₂²
hypercenter	C₂×C₄⋊C₄	#87938
hypocenter	C₁₀₁	#1296
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#87938

commutator

C₂×C₄×C₁₀₁

C₄²

Frattini

C₂²×C₄

C₂×#87938

Fitting

C₂×C₄⋊C₄×C₁₀₁

C₄

socle, prime radical

C₂³×C₁₀₁

C₂²⋊C₄

cosocle

C₂×C₄×Dic₁₀₁

C₂²

C₂×C₄⋊C₄

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
5. #15692322	C₄	semidirect
59. C₁₀₁	#1296	semidirect

H ≅ G/N

type

5. #15692322

C₄

semidirect

59. C₁₀₁

#1296

semidirect

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

subgroup	index
C₂×#15692322	2
#20265046	2
#1296	101

subgroup

index

C₂×#15692322

#20265046

#1296

101

Subgroups — #27186010

Conjugacy classes of subgroups

Complements

Maximal subgroups