Group #27171953 — subgroups

Subgroups — #27171953

Order 12928.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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×#15692179	C₂	1	4	yes	maximal
2	C₂×#15692166	C₂	1	1	yes	maximal, characteristic
3	#20265043	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11717579	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
7	#15692179	—	2	8	no	—
8	#15692166	—	2	2	no	—
9	#15692162	—	2	2	no	—
10	C₂²×Q₈×C₁₀₁	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₀₁	—	2	2	no	—
12	C₄⋊Q₈×C₁₀₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₀₁	—	2	2	no	—
14	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₁	—	2	2	no	—
16	C₂×#11717579	—	2	2	no	—
17	C₂×#11717579	—	2	2	no	—
18	C₂×#11717579	D₄	1	4	yes	—
19	C₄⋊Dic₁₀₁	—	2	4	no	—
20	C₂×Q₈×C₁₀₁	—	2	2	no	—
21	C₂×Q₈×C₁₀₁	—	2	4	no	—
22	C₂×Q₈×C₁₀₁	—	2	4	no	—
23	C₂×Q₈×C₁₀₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₀₁	—	2	2	no	—
25	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₀₁	—	2	8	no	—
28	C₄²×C₁₀₁	C₂×C₄	1	2	yes	—
29	C₄²×C₁₀₁	D₄	1	2	yes	—
30	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₁₀₁	—	2	8	no	—
32	Q₈×C₁₀₁	—	4	4	no	—
33	C₂×C₄×C₁₀₁	—	2	4	no	—
34	C₂×C₄×C₁₀₁	—	2	4	no	—
35	C₂×C₄×C₁₀₁	—	4	2	no	—
36	C₂×C₄×C₁₀₁	—	4	2	no	—
37	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₁	#41	1	2	yes	—
42	C₂×Dic₁₀₁	—	4	2	no	—
43	C₂×Dic₁₀₁	—	4	4	no	—
44	#11717579	—	2	4	no	—
45	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₁	#124	1	1	yes	characteristic
47	C₂²×C₁₀₁	#127	1	1	yes	characteristic
48	C₂²×C₁₀₁	#137	1	4	yes	—
49	C₄×C₁₀₁	—	2	4	no	—
50	C₄×C₁₀₁	—	4	4	no	—
51	C₄×C₁₀₁	C₄×D₄	1	2	yes	—
52	C₄×C₁₀₁	#121	1	2	yes	—
53	Dic₁₀₁	—	8	2	no	—
54	C₂×C₁₀₁	#390	1	1	yes	characteristic
55	C₂×C₁₀₁	#419	1	2	yes	—
56	C₂×C₁₀₁	#491	1	2	yes	—
57	C₂×C₁₀₁	#497	1	2	yes	—
58	#1960	—	101	1	no	maximal, Sylow
59	C₁₀₁	#1960	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₄⋊Q₈	D₁₀₁	1	1	yes	characteristic
61	C₂×#103	—	101	4	no	—
62	C₂×#97	—	101	1	no	—
63	#382	—	101	1	no	—
64	C₂²×Q₈	C₂×D₁₀₁	1	2	yes	—
65	C₂²×C₈	—	101	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₁₀₁	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	101	2	no	—
70	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
71	#103	—	202	8	no	—
72	#97	—	202	2	no	—
73	#95	—	202	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₁₀₁	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	202	2	no	—
80	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
82	C₂×C₈	—	101	4	no	—
83	C₂×C₈	—	202	2	no	—
84	C₂×C₈	—	202	2	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	202	4	no	—
87	C₄²	C₂×Dic₁₀₁	1	2	yes	—
88	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	404	2	no	—
97	C₂×C₄	—	404	4	no	—
98	C₂×C₄	C₂²⋊Dic₁₀₁	1	4	yes	—
99	C₂×C₄	C₂²×Dic₁₀₁	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
102	C₂×C₄	#13108073	1	2	yes	—
103	C₈	—	202	4	no	—
104	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
105	C₂²	#15692242	1	1	yes	characteristic
106	C₂²	#15692251	1	1	yes	characteristic
107	C₂²	#15692286	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	808	2	no	—
111	C₄	D₄×Dic₁₀₁	1	2	yes	—
112	C₄	#15692226	1	2	yes	—
113	C₂	#20265080	1	1	yes	characteristic
114	C₂	#20265186	1	2	yes	—
115	C₂	#20265545	1	2	yes	—
116	C₂	#20265569	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692179

C₂

yes

maximal

C₂×#15692166

C₂

yes

maximal, characteristic

#20265043

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

#15692179

—

#15692166

—

#15692162

—

C₂²×Q₈×C₁₀₁

C₂²

yes

—

C₄⋊Q₈×C₁₀₁

—

C₄⋊Q₈×C₁₀₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₁

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

D₄

yes

—

C₄⋊Dic₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

#11717579

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

C₂²×C₁₀₁

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄×D₄

yes

—

C₄×C₁₀₁

#121

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

#390

yes

characteristic

C₂×C₁₀₁

#419

yes

—

C₂×C₁₀₁

#491

yes

—

C₂×C₁₀₁

#497

yes

—

#1960

—

101

maximal, Sylow

C₁₀₁

#1960

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₀₁

yes

characteristic

C₂×#103

—

101

—

C₂×#97

—

101

—

#382

—

101

—

C₂²×Q₈

C₂×D₁₀₁

yes

—

C₂²×C₈

—

101

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₀₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#103

—

202

—

#97

—

202

—

#95

—

202

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₀₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

202

—

C₄²

C₂×Dic₁₀₁

yes

—

C₄²

C₂²⋊D₁₀₁

yes

—

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

C₂×C₄

C₂²×Dic₁₀₁

yes

characteristic

100

C₂×C₄

D₄×D₁₀₁

yes

—

101

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

102

C₂×C₄

#13108073

yes

—

103

C₈

—

202

—

104

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

characteristic

105

C₂²

#15692242

yes

characteristic

106

C₂²

#15692251

yes

characteristic

107

C₂²

#15692286

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

808

—

111

C₄

D₄×Dic₁₀₁

yes

—

112

C₄

#15692226

yes

—

113

C₂

#20265080

yes

characteristic

114

C₂

#20265186

yes

—

115

C₂

#20265545

yes

—

116

C₂

#20265569

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₁
Fitting	C₂×C₄⋊Q₈×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂×D₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₀₁
hypocenter	C₁₀₁	#1960
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₁

Fitting

C₂×C₄⋊Q₈×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂×D₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₀₁

hypocenter

C₁₀₁

#1960

LCS

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₀₁	C₄	semidirect
59. C₁₀₁	#1960	semidirect
67. C₄⋊Q₈	Dic₁₀₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₀₁

C₄

semidirect

59. C₁₀₁

#1960

semidirect

67. C₄⋊Q₈

Dic₁₀₁

semidirect

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

subgroup	index
C₂×#15692179	2
C₂×#15692166	2
#20265043	2
C₂×C₄⋊Q₈×C₁₀₁	2
#1960	101

subgroup

index

C₂×C₄⋊Q₈×C₁₀₁

101

Subgroups — #27171953

Conjugacy classes of subgroups

Complements

Maximal subgroups