Group #27187050 — subgroups

Subgroups — #27187050

Order 12928.

216 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 82 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	1	yes	maximal, characteristic
2	C₂×C₄⋊#11717581	C₂	1	1	yes	maximal, characteristic
3	#20266043	C₂	1	4	yes	maximal
4	#20265022	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11717581	C₂²	1	2	yes	—
6	C₂²×#11717581	C₂²	1	2	yes	—
7	#15692322	—	2	2	no	—
8	C₄⋊#11717581	—	2	2	no	—
9	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
11	#15692162	C₄	1	2	yes	—
12	#15692162	C₄	1	1	yes	characteristic
13	#95×C₁₀₁	C₄	1	1	yes	characteristic
14	C₂×#11717581	—	2	2	no	—
15	C₂×#11717581	—	2	2	no	—
16	C₂×#11717581	—	2	4	no	—
17	C₂×#11717581	—	2	4	no	—
18	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
19	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
21	C₄×Dic₁₀₁	—	2	2	no	—
22	C₄×Dic₁₀₁	—	2	2	no	—
23	C₄×Dic₁₀₁	D₄	1	2	yes	—
24	C₄×Dic₁₀₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₁	—	2	2	no	—
29	C₂×C₄×C₁₀₁	—	2	2	no	—
30	C₂×C₄×C₁₀₁	—	2	2	no	—
31	C₂×C₄×C₁₀₁	—	2	2	no	—
32	C₂×C₄×C₁₀₁	C₄⋊C₄	1	2	yes	—
33	C₂×Dic₁₀₁	—	2	1	no	—
34	C₂×Dic₁₀₁	—	2	1	no	—
35	C₂×Dic₁₀₁	—	2	2	no	—
36	C₂×Dic₁₀₁	—	2	2	no	—
37	C₂×Dic₁₀₁	—	2	2	no	—
38	C₂×Dic₁₀₁	C₄⋊C₄	1	2	yes	—
39	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
40	C₂×Dic₁₀₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₁₀₁	#41	1	2	yes	—
42	C₂×Dic₁₀₁	#41	1	2	yes	—
43	#11717581	—	4	2	no	—
44	#11717581	—	4	2	no	—
45	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₁	#117	1	2	yes	—
47	C₂²×C₁₀₁	#131	1	2	yes	—
48	C₂²×C₁₀₁	#131	1	1	yes	characteristic
49	C₂²×C₁₀₁	#131	1	1	yes	characteristic
50	C₄×C₁₀₁	—	2	2	no	—
51	C₄×C₁₀₁	—	4	2	no	—
52	Dic₁₀₁	—	2	2	no	—
53	Dic₁₀₁	—	4	2	no	—
54	Dic₁₀₁	#124	1	1	yes	characteristic
55	Dic₁₀₁	#124	1	1	yes	characteristic
56	Dic₁₀₁	#125	1	2	yes	—
57	C₂×C₁₀₁	#382	1	1	yes	characteristic
58	C₂×C₁₀₁	#424	1	1	yes	characteristic
59	C₂×C₁₀₁	#424	1	1	yes	characteristic
60	C₂×C₁₀₁	#433	1	2	yes	—
61	C₂×C₁₀₁	#433	1	2	yes	—
62	#1832	—	101	1	no	maximal, Sylow
63	C₁₀₁	#1832	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊C₈	—	101	1	no	—
65	C₂×C₄⋊C₈	—	101	1	no	—
66	#376	—	101	1	no	—
67	#336	—	101	4	no	—
68	C₂²×C₈	—	101	2	no	—
69	C₂²×C₈	—	101	2	no	—
70	C₂×C₄²	—	101	1	no	—
71	C₂×C₄²	—	101	2	no	—
72	C₄⋊C₈	—	202	2	no	—
73	C₄⋊C₈	—	202	2	no	—
74	#95	—	101	2	no	—
75	#95	—	101	1	no	—
76	#95	#87938	1	1	yes	characteristic
77	C₂²×C₄	—	101	1	no	—
78	C₂²×C₄	—	101	1	no	—
79	C₂²×C₄	—	101	2	no	—
80	C₂²×C₄	C₂×#87938	1	2	yes	—
81	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
82	C₂×C₈	—	202	2	no	—
83	C₂×C₈	—	202	2	no	—
84	C₂×C₈	—	202	4	no	—
85	C₂×C₈	—	202	4	no	—
86	C₄²	—	101	2	no	—
87	C₄²	—	101	2	no	—
88	C₄²	—	202	2	no	—
89	C₄²	—	202	2	no	—
90	C₂³	C₂²×#87938	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	101	1	no	—
96	C₂×C₄	—	101	1	no	—
97	C₂×C₄	—	101	2	no	—
98	C₂×C₄	—	101	2	no	—
99	C₂×C₄	—	101	2	no	—
100	C₂×C₄	—	202	2	no	—
101	C₂×C₄	—	202	2	no	—
102	C₂×C₄	—	202	2	no	—
103	C₂×C₄	—	202	1	no	—
104	C₂×C₄	—	202	1	no	—
105	C₂×C₄	C₄⋊#87938	1	2	yes	—
106	C₈	—	404	2	no	—
107	C₈	—	404	2	no	—
108	C₂²	C₂×C₄⋊#87938	1	1	yes	characteristic
109	C₂²	#15692334	1	2	yes	—
110	C₂²	#15692342	1	2	yes	—
111	C₂²	#15692342	1	1	yes	characteristic
112	C₂²	#15692343	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	101	1	no	—
116	C₄	—	101	1	no	—
117	C₄	—	101	2	no	—
118	C₄	—	404	2	no	—
119	C₄	—	202	2	no	—
120	C₂	#20266121	1	1	yes	characteristic
121	C₂	#20266183	1	1	yes	characteristic
122	C₂	#20266184	1	1	yes	characteristic
123	C₂	#20266201	1	2	yes	—
124	C₂	#20266202	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692322

C₂

yes

maximal, characteristic

C₂×C₄⋊#11717581

C₂

yes

maximal, characteristic

#20266043

C₂

yes

maximal

#20265022

C₂

yes

maximal, characteristic

C₂²×#11717581

C₂²

yes

—

C₂²×#11717581

C₂²

yes

—

#15692322

—

C₄⋊#11717581

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

#15692162

C₄

yes

—

#15692162

C₄

yes

characteristic

#95×C₁₀₁

C₄

yes

—

—

—

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₁

C₂³

yes

characteristic

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

D₄

yes

—

C₄×Dic₁₀₁

Q₈

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₄⋊C₄

yes

—

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₁

#41

yes

—

C₂×Dic₁₀₁

#41

yes

—

#11717581

—

#11717581

—

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

#117

yes

—

C₂²×C₁₀₁

#131

yes

—

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

yes

Dic₁₀₁

yes

Dic₁₀₁

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

C₂×C₁₀₁

#433

yes

—

#1832

—

101

maximal, Sylow

C₁₀₁

#1832

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

101

—

C₂×C₄⋊C₈

—

101

—

#376

—

101

—

#336

—

101

—

C₂²×C₈

—

101

—

C₂²×C₈

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

202

—

C₄⋊C₈

—

202

—

#95

—

101

—

#95

—

101

—

#95

#87938

yes

characteristic

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂×#87938

yes

—

C₂²×C₄

C₂×#87938

yes

characteristic

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

—

202

—

C₄²

—

202

—

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

—

100

C₂×C₄

—

202

—

101

C₂×C₄

—

202

—

102

C₂×C₄

—

202

—

103

C₂×C₄

—

202

—

104

C₂×C₄

—

202

—

105

C₂×C₄

C₄⋊#87938

yes

—

106

C₈

—

404

—

107

C₈

—

404

—

108

C₂²

C₂×C₄⋊#87938

yes

characteristic

109

C₂²

#15692334

yes

—

110

C₂²

#15692342

yes

—

111

C₂²

#15692342

yes

characteristic

112

C₂²

#15692343

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

101

—

116

C₄

—

101

—

117

C₄

—

101

—

118

C₄

—

404

—

119

C₄

—

202

—

120

C₂

#20266121

yes

characteristic

121

C₂

#20266183

yes

characteristic

122

C₂

#20266184

yes

characteristic

123

C₂

#20266201

yes

—

124

C₂

#20266202

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87938
commutator, socle, prime radical	C₂³×C₁₀₁	C₂²×C₄
Fitting	#95×C₁₀₁	C₄
cosocle	C₂²×Dic₁₀₁	C₂³
hypercenter	#95	#87938
hypocenter	C₁₀₁	#1832
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×#87938

commutator, socle, prime radical

C₂³×C₁₀₁

C₂²×C₄

Fitting

#95×C₁₀₁

C₄

cosocle

C₂²×Dic₁₀₁

C₂³

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
63. C₁₀₁	#1832	semidirect

H ≅ G/N

type

63. C₁₀₁

#1832

semidirect

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

subgroup	index
C₂×#15692322	2
C₂×C₄⋊#11717581	2
#20266043	2
#20265022	2
#1832	101

subgroup

index

101

Subgroups — #27187050

Conjugacy classes of subgroups

Complements

Maximal subgroups