Group #27180072 — subgroups

Subgroups — #27180072

Order 12928.

348 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 189 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	#20265734	C₂	1	1	yes	maximal, characteristic
2	#20265305	C₂	1	8	yes	maximal, complemented
3	#20265203	C₂	1	2	yes	maximal
4	#20265201	C₂	1	1	yes	maximal, characteristic
5	#20265132	C₂	1	2	yes	maximal
6	C₂×#97×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#13108058	C₂²	1	2	yes	—
8	#15692255	C₄	1	2	yes	—
9	#15692247	C₄	1	2	yes	—
10	#15692234	C₄	1	4	yes	—
11	C₄×Dic₂₀₂	C₂²	1	4	yes	—
12	#15692214	C₂²	1	2	yes	—
13	C₄⋊#11717579	C₂²	1	4	yes	—
14	#15692184	C₂²	1	8	yes	—
15	#15692165	C₂²	1	4	yes	—
16	C₈×Dic₁₀₁	C₂²	1	4	yes	—
17	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
20	#97×C₁₀₁	C₂²	1	4	yes	—
21	C₂×Dic₂₀₂	C₂×C₄	1	4	yes	—
22	#13108058	—	2	4	no	—
23	C₂×#11717579	C₂³	1	4	yes	—
24	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
26	#13108053	C₂×C₄	1	8	yes	—
27	C₂²⋊Dic₁₀₁	C₂×C₄	1	4	yes	—
28	C₄×Dic₁₀₁	C₂³	1	4	yes	—
29	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₀₁	—	2	2	no	—
32	C₂×C₈×C₁₀₁	Q₈	1	4	yes	complemented
33	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
34	C₂×C₈×C₁₀₁	C₂³	1	2	yes	—
35	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₁₀₁	C₂³	1	2	yes	—
38	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	—	4	2	no	—
40	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₁₀₁	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₁₀₁	#41	1	2	yes	—
45	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₁₀₁	C₂²×C₄	1	8	yes	—
47	Dic₂₀₂	—	4	4	no	—
48	C₈×C₁₀₁	—	2	2	no	—
49	C₈×C₁₀₁	C₂×Q₈	1	4	yes	—
50	#11717579	—	2	4	no	—
51	C₂²×C₁₀₁	—	2	2	no	—
52	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
53	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₀₁	—	2	2	no	—
55	C₄×C₁₀₁	—	2	2	no	—
56	C₄×C₁₀₁	C₄×Q₈	1	2	yes	—
57	C₄×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
58	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
59	Dic₁₀₁	—	2	8	no	—
60	C₂×C₁₀₁	—	2	2	no	—
61	C₂×C₁₀₁	C₂×C₄×Q₈	1	1	yes	characteristic
62	C₂×C₁₀₁	#568	1	2	yes	—
63	#2961	—	101	1	no	maximal, Sylow
64	C₁₀₁	#2961	1	1	yes	characteristic, complemented, Sylow
65	#533	—	101	1	no	—
66	#446	—	101	8	no	—
67	#424	—	101	2	no	—
68	#423	—	101	1	no	—
69	#405	—	101	2	no	—
70	C₂×#97	D₁₀₁	1	1	yes	characteristic
71	C₂×M₁₆	—	101	2	no	—
72	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
74	C₄⋊Q₈	—	101	2	no	—
75	#125	—	101	2	no	—
76	C₂²⋊Q₈	—	101	4	no	—
77	C₄×Q₈	—	101	4	no	—
78	#117	—	101	2	no	—
79	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
80	C₄⋊C₈	—	101	8	no	—
81	C₄⋊C₈	—	101	4	no	—
82	#97	—	101	4	no	—
83	#97	C₂×D₁₀₁	1	4	yes	—
84	C₄×C₈	—	101	4	no	—
85	C₂×Q₈	—	101	4	no	—
86	C₂²×C₄	C₄×D₁₀₁	1	2	yes	—
87	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
88	M₁₆	—	202	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	101	4	no	—
91	C₂×C₈	Dic₂₀₂	1	4	yes	complemented
92	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
93	C₂×C₈	C₂²×D₁₀₁	1	2	yes	—
94	C₄⋊C₄	—	101	8	no	—
95	C₄⋊C₄	—	101	4	no	—
96	C₄⋊C₄	—	101	4	no	—
97	C₂²⋊C₄	—	101	4	no	—
98	C₄²	—	101	4	no	—
99	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
100	C₄²	C₄×D₁₀₁	1	1	yes	characteristic
101	C₄²	C₂²×D₁₀₁	1	2	yes	—
102	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
103	Q₈	—	404	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	101	8	no	—
106	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
107	C₂×C₄	C₂×C₄×D₁₀₁	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
109	C₂×C₄	C₂×Dic₂₀₂	1	2	yes	—
110	C₂×C₄	#13108074	1	2	yes	—
111	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	202	4	no	—
114	C₈	C₂×Dic₂₀₂	1	4	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	C₄×Dic₂₀₂	1	2	yes	—
117	C₂²	C₂²×C₄×D₁₀₁	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	202	8	no	—
121	C₄	C₄×Dic₂₀₂	1	2	yes	—
122	C₄	C₂²×Dic₂₀₂	1	1	yes	characteristic
123	C₄	C₂×#13108074	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₂×C₄×Dic₂₀₂	1	1	yes	characteristic
126	C₂	#20265943	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265734

C₂

yes

maximal, characteristic

#20265305

C₂

yes

maximal, complemented

#20265203

C₂

yes

maximal

#20265201

C₂

yes

maximal, characteristic

#20265132

C₂

yes

maximal

C₂×#97×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

#15692255

C₄

yes

—

#15692247

C₄

yes

—

#15692234

C₄

yes

—

C₄×Dic₂₀₂

C₂²

yes

—

#15692214

C₂²

yes

—

C₄⋊#11717579

C₂²

yes

—

#15692184

C₂²

yes

—

#15692165

C₂²

yes

—

C₈×Dic₁₀₁

C₂²

yes

—

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

#97×C₁₀₁

C₂²

yes

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

#13108058

—

C₂×#11717579

C₂³

yes

—

C₄⋊Dic₁₀₁

C₂×C₄

yes

—

C₄⋊Dic₁₀₁

C₂×C₄

yes

—

#13108053

C₂×C₄

yes

—

C₂²⋊Dic₁₀₁

C₂×C₄

yes

—

C₄×Dic₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

Q₈

yes

complemented

C₂×C₈×C₁₀₁

C₂³

yes

—

C₂×C₈×C₁₀₁

C₂³

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₄²×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂×Dic₁₀₁

C₂²×C₄

yes

—

Dic₂₀₂

—

C₈×C₁₀₁

—

C₈×C₁₀₁

C₂×Q₈

yes

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₄×Q₈

yes

—

C₂²×C₁₀₁

C₂³×C₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×Q₈

yes

—

C₄×C₁₀₁

C₂²×Q₈

yes

characteristic

C₄×C₁₀₁

C₂×#41

yes

characteristic

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₀₁

#568

yes

—

#2961

—

101

maximal, Sylow

C₁₀₁

#2961

yes

characteristic, complemented, Sylow

#533

—

101

—

#446

—

101

—

#424

—

101

—

#423

—

101

—

#405

—

101

—

C₂×#97

D₁₀₁

yes

characteristic

C₂×M₁₆

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

C₄⋊Q₈

—

101

—

#125

—

101

—

C₂²⋊Q₈

—

101

—

C₄×Q₈

—

101

—

#117

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

101

—

C₄⋊C₈

—

101

—

#97

—

101

—

#97

C₂×D₁₀₁

yes

—

C₄×C₈

—

101

—

C₂×Q₈

—

101

—

C₂²×C₄

C₄×D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

M₁₆

—

202

—

C₂×C₈

—

C₂×C₈

—

101

—

C₂×C₈

Dic₂₀₂

yes

complemented

C₂×C₈

C₂²×D₁₀₁

yes

—

C₂×C₈

C₂²×D₁₀₁

yes

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₄²

—

101

—

C₄²

C₄×D₁₀₁

yes

characteristic

100

C₄²

C₄×D₁₀₁

yes

characteristic

101

C₄²

C₂²×D₁₀₁

yes

—

102

C₂³

C₂×C₄×D₁₀₁

yes

characteristic

103

Q₈

—

404

—

104

C₂×C₄

—

105

C₂×C₄

—

101

—

106

C₂×C₄

C₂×C₄×D₁₀₁

yes

—

107

C₂×C₄

C₂×C₄×D₁₀₁

yes

—

108

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

109

C₂×C₄

C₂×Dic₂₀₂

yes

—

110

C₂×C₄

#13108074

yes

—

111

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

112

C₈

—

113

C₈

—

202

—

114

C₈

C₂×Dic₂₀₂

yes

—

115

C₂²

—

116

C₂²

C₄×Dic₂₀₂

yes

—

117

C₂²

C₂²×C₄×D₁₀₁

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

202

—

121

C₄

C₄×Dic₂₀₂

yes

—

122

C₄

C₂²×Dic₂₀₂

yes

characteristic

123

C₄

C₂×#13108074

yes

characteristic

124

C₂

—

125

C₂

C₂×C₄×Dic₂₀₂

yes

characteristic

126

C₂

#20265943

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₂×#97×C₁₀₁	C₂
prime radical	C₂³×C₁₀₁	C₂²×C₄
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	C₂×#97	D₁₀₁
hypocenter	C₁₀₁	#2961
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₀₁

commutator, socle

C₂²×C₁₀₁

C₂³×C₄

Fitting

C₂×#97×C₁₀₁

C₂

prime radical

C₂³×C₁₀₁

C₂²×C₄

cosocle

C₂×C₄×C₁₀₁

C₂⁴

hypercenter

C₂×#97

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
2. #20265305	C₂	semidirect
32. C₂×C₈×C₁₀₁	Q₈	semidirect
64. C₁₀₁	#2961	semidirect
91. C₂×C₈	Dic₂₀₂	semidirect

H ≅ G/N

type

2. #20265305

C₂

semidirect

32. C₂×C₈×C₁₀₁

Q₈

semidirect

64. C₁₀₁

#2961

semidirect

91. C₂×C₈

Dic₂₀₂

semidirect

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

subgroup	index
#20265734	2
#20265305	2
#20265203	2
#20265201	2
#20265132	2
C₂×#97×C₁₀₁	2
#2961	101

subgroup

index

C₂×#97×C₁₀₁

101

Subgroups — #27180072

Conjugacy classes of subgroups

Complements

Maximal subgroups