Group #27171298 — subgroups

Subgroups — #27171298

Order 12928.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×#15692188	C₂	1	2	yes	maximal
2	C₂×#15692186	C₂	1	2	yes	maximal
3	C₂×#15692166	C₂	1	1	yes	maximal, characteristic
4	#20265043	C₂	1	1	yes	maximal, characteristic
5	#384×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11717579	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
9	#15692188	—	2	4	no	—
10	#15692186	—	2	4	no	—
11	#15692166	C₄	1	4	yes	complemented
12	#15692162	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₀₁	—	2	1	no	—
17	C₂²×Dic₁₀₁	—	2	1	no	—
18	C₂×#11717579	C₂×C₄	1	4	yes	—
19	C₂×#11717579	D₄	1	4	yes	—
20	C₂×#11717579	Q₈	1	4	yes	—
21	C₄⋊Dic₁₀₁	—	2	2	no	—
22	C₄⋊Dic₁₀₁	—	2	2	no	—
23	C₂²×C₄×C₁₀₁	—	2	2	no	—
24	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₁₀₁	—	2	2	no	—
27	C₄⋊C₄×C₁₀₁	—	2	2	no	—
28	C₄⋊C₄×C₁₀₁	—	4	4	no	—
29	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₄²×C₁₀₁	Q₈	1	2	yes	—
31	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₁	—	2	2	no	—
33	C₂×C₄×C₁₀₁	—	2	2	no	—
34	C₂×C₄×C₁₀₁	—	4	2	no	—
35	C₂×C₄×C₁₀₁	—	4	4	no	—
36	C₂×C₄×C₁₀₁	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₀₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₁₀₁	—	4	1	no	—
42	C₂×Dic₁₀₁	—	4	1	no	—
43	C₂×Dic₁₀₁	—	4	2	no	—
44	C₂×Dic₁₀₁	—	4	2	no	—
45	#11717579	C₄⋊C₄	1	8	yes	complemented
46	C₂²×C₁₀₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₀₁	#127	1	1	yes	characteristic
48	C₂²×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₁₀₁	#137	1	2	yes	—
51	C₄×C₁₀₁	—	2	4	no	—
52	C₄×C₁₀₁	—	8	2	no	—
53	C₄×C₁₀₁	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₁₀₁	C₄⋊Q₈	1	2	yes	—
55	Dic₁₀₁	—	8	1	no	—
56	Dic₁₀₁	—	8	1	no	—
57	C₂×C₁₀₁	#384	1	1	yes	characteristic
58	C₂×C₁₀₁	#428	1	2	yes	—
59	C₂×C₁₀₁	#496	1	1	yes	characteristic
60	C₂×C₁₀₁	#497	1	1	yes	characteristic
61	C₂×C₁₀₁	#501	1	2	yes	—
62	#1847	—	101	1	no	maximal, Sylow
63	C₁₀₁	#1847	1	1	yes	characteristic, complemented, Sylow
64	C₂×#107	—	101	2	no	—
65	C₂×#106	—	101	2	no	—
66	C₂×#97	—	101	1	no	—
67	#384	D₁₀₁	1	1	yes	characteristic
68	#382	—	101	1	no	—
69	C₂²×C₈	—	101	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	101	1	no	—
72	C₂×C₄⋊C₄	—	101	1	no	—
73	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
74	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
75	#107	—	202	4	no	—
76	#106	—	202	4	no	—
77	#97	—	101	4	no	—
78	#95	—	202	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	202	1	no	—
81	C₂²×C₄	—	202	1	no	—
82	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
84	C₂×C₈	—	101	4	no	—
85	C₂×C₈	—	101	4	no	—
86	C₂×C₈	—	101	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	202	2	no	—
91	C₄⋊C₄	—	202	2	no	—
92	C₄²	Dic₂₀₂	1	2	yes	—
93	C₄²	C₄×D₁₀₁	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	404	1	no	—
100	C₂×C₄	—	404	1	no	—
101	C₂×C₄	—	404	2	no	—
102	C₂×C₄	—	404	2	no	—
103	C₂×C₄	#13108053	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
105	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
106	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
107	C₂×C₄	C₂×Dic₂₀₂	1	1	yes	characteristic
108	C₈	—	101	8	no	—
109	C₂²	C₂×#13108053	1	1	yes	characteristic
110	C₂²	#15692251	1	1	yes	characteristic
111	C₂²	#15692253	1	1	yes	characteristic
112	C₂²	#15692281	1	2	yes	—
113	C₂²	#15692286	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	808	1	no	—
117	C₄	—	808	1	no	—
118	C₄	C₄⋊C₄×D₁₀₁	1	2	yes	—
119	C₄	#15692254	1	2	yes	—
120	C₂	#20265046	1	1	yes	characteristic
121	C₂	#20265217	1	2	yes	—
122	C₂	#20265564	1	1	yes	characteristic
123	C₂	#20265569	1	1	yes	characteristic
124	C₂	#20265583	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692188

C₂

yes

maximal

C₂×#15692186

C₂

yes

maximal

C₂×#15692166

C₂

yes

maximal, characteristic

#20265043

C₂

yes

maximal, characteristic

#384×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

—

C₄

yes

complemented

#15692162

—

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

—

C₂×#11717579

C₂×C₄

yes

—

C₂×#11717579

D₄

yes

—

C₂×#11717579

Q₈

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

Q₈

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

C₂×Q₈

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

#11717579

C₄⋊C₄

yes

complemented

C₂²×C₁₀₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₁

#127

yes

characteristic

C₂²×C₁₀₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₀₁

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₁

#137

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₁

C₄⋊Q₈

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

#384

yes

characteristic

C₂×C₁₀₁

#428

yes

—

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

#1847

—

101

maximal, Sylow

C₁₀₁

#1847

yes

characteristic, complemented, Sylow

C₂×#107

—

101

—

C₂×#106

—

101

—

C₂×#97

—

101

—

#384

D₁₀₁

yes

characteristic

#382

—

101

—

C₂²×C₈

—

101

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#107

—

202

—

#106

—

202

—

#97

—

101

—

#95

—

202

—

C₂²×C₄

—

C₂²×C₄

—

202

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₄²

Dic₂₀₂

yes

—

C₄²

C₄×D₁₀₁

yes

—

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

404

—

100

C₂×C₄

—

404

—

101

C₂×C₄

—

404

—

102

C₂×C₄

—

404

—

103

C₂×C₄

#13108053

yes

—

104

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

105

C₂×C₄

D₄×D₁₀₁

yes

—

106

C₂×C₄

Q₈×D₁₀₁

yes

—

107

C₂×C₄

C₂×Dic₂₀₂

yes

characteristic

108

C₈

—

101

—

109

C₂²

C₂×#13108053

yes

characteristic

110

C₂²

#15692251

yes

characteristic

111

C₂²

#15692253

yes

characteristic

112

C₂²

#15692281

yes

—

113

C₂²

#15692286

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

808

—

117

C₄

—

808

—

118

C₄

C₄⋊C₄×D₁₀₁

yes

—

119

C₄

#15692254

yes

—

120

C₂

#20265046

yes

characteristic

121

C₂

#20265217

yes

—

122

C₂

#20265564

yes

characteristic

123

C₂

#20265569

yes

characteristic

124

C₂

#20265583

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	#384×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	#384	D₁₀₁
hypocenter	C₁₀₁	#1847
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

#384×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. #15692166	C₄	semidirect
45. #11717579	C₄⋊C₄	semidirect
63. C₁₀₁	#1847	semidirect

H ≅ G/N

type

11. #15692166

C₄

semidirect

45. #11717579

C₄⋊C₄

semidirect

63. C₁₀₁

#1847

semidirect

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

subgroup	index
C₂×#15692188	2
C₂×#15692186	2
C₂×#15692166	2
#20265043	2
#384×C₁₀₁	2
#1847	101

subgroup

index

#384×C₁₀₁

101

Subgroups — #27171298

Conjugacy classes of subgroups

Complements

Maximal subgroups