Group #27195965 — subgroups

Subgroups — #27195965

Order 13184.

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×#15692945	C₂	1	4	yes	maximal
2	C₂×#15692932	C₂	1	1	yes	maximal, characteristic
3	#20267657	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₀₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
7	#15692945	—	2	8	no	—
8	#15692932	—	2	2	no	—
9	C₁₀₃⋊#95	—	2	2	no	—
10	C₂²×Q₈×C₁₀₃	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₀₃	—	2	2	no	—
12	C₄⋊Q₈×C₁₀₃	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₀₃	—	2	2	no	—
14	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₃	—	2	2	no	—
16	C₂×C₁₀₃⋊C₈	—	2	2	no	—
17	C₂×C₁₀₃⋊C₈	—	2	2	no	—
18	C₂×C₁₀₃⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₁₀₃	—	2	4	no	—
20	C₂×Q₈×C₁₀₃	—	2	2	no	—
21	C₂×Q₈×C₁₀₃	—	2	4	no	—
22	C₂×Q₈×C₁₀₃	—	2	4	no	—
23	C₂×Q₈×C₁₀₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₀₃	—	2	2	no	—
25	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₀₃	—	2	8	no	—
28	C₄²×C₁₀₃	C₂×C₄	1	2	yes	—
29	C₄²×C₁₀₃	D₄	1	2	yes	—
30	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₁₀₃	—	2	8	no	—
32	Q₈×C₁₀₃	—	4	4	no	—
33	C₂×C₄×C₁₀₃	—	2	4	no	—
34	C₂×C₄×C₁₀₃	—	2	4	no	—
35	C₂×C₄×C₁₀₃	—	4	2	no	—
36	C₂×C₄×C₁₀₃	—	4	2	no	—
37	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₃	#41	1	2	yes	—
42	C₂×Dic₁₀₃	—	4	2	no	—
43	C₂×Dic₁₀₃	—	4	4	no	—
44	C₁₀₃⋊C₈	—	2	4	no	—
45	C₂²×C₁₀₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₃	#124	1	1	yes	characteristic
47	C₂²×C₁₀₃	#127	1	1	yes	characteristic
48	C₂²×C₁₀₃	#137	1	4	yes	—
49	C₄×C₁₀₃	—	2	4	no	—
50	C₄×C₁₀₃	—	4	4	no	—
51	C₄×C₁₀₃	C₄×D₄	1	2	yes	—
52	C₄×C₁₀₃	#121	1	2	yes	—
53	Dic₁₀₃	—	8	2	no	—
54	C₂×C₁₀₃	#390	1	1	yes	characteristic
55	C₂×C₁₀₃	#419	1	2	yes	—
56	C₂×C₁₀₃	#491	1	2	yes	—
57	C₂×C₁₀₃	#497	1	2	yes	—
58	#1960	—	103	1	no	maximal, Sylow
59	C₁₀₃	#1960	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₄⋊Q₈	D₁₀₃	1	1	yes	characteristic
61	C₂×#103	—	103	4	no	—
62	C₂×#97	—	103	1	no	—
63	#382	—	103	1	no	—
64	C₂²×Q₈	C₂×D₁₀₃	1	2	yes	—
65	C₂²×C₈	—	103	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₁₀₃	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	103	2	no	—
70	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
71	#103	—	206	8	no	—
72	#97	—	206	2	no	—
73	#95	—	206	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₁₀₃	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	206	2	no	—
80	C₂²×C₄	C₂²⋊D₁₀₃	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
82	C₂×C₈	—	103	4	no	—
83	C₂×C₈	—	206	2	no	—
84	C₂×C₈	—	206	2	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	206	4	no	—
87	C₄²	C₂×Dic₁₀₃	1	2	yes	—
88	C₄²	C₂²⋊D₁₀₃	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₀₃	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	—	412	2	no	—
97	C₂×C₄	—	412	4	no	—
98	C₂×C₄	C₂²⋊Dic₁₀₃	1	4	yes	—
99	C₂×C₄	C₂²×Dic₁₀₃	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₀₃	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
102	C₂×C₄	#13110104	1	2	yes	—
103	C₈	—	206	4	no	—
104	C₂²	C₂×C₂²⋊Dic₁₀₃	1	1	yes	characteristic
105	C₂²	#15693008	1	1	yes	characteristic
106	C₂²	#15693017	1	1	yes	characteristic
107	C₂²	#15693052	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	824	2	no	—
111	C₄	D₄×Dic₁₀₃	1	2	yes	—
112	C₄	#15692992	1	2	yes	—
113	C₂	#20267694	1	1	yes	characteristic
114	C₂	#20267800	1	2	yes	—
115	C₂	#20268159	1	2	yes	—
116	C₂	#20268183	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692945

C₂

yes

maximal

C₂×#15692932

C₂

yes

maximal, characteristic

#20267657

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×C₁₀₃⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

#15692945

—

#15692932

—

C₁₀₃⋊#95

—

C₂²×Q₈×C₁₀₃

C₂²

yes

—

C₄⋊Q₈×C₁₀₃

—

C₄⋊Q₈×C₁₀₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₃

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₂²×Dic₁₀₃

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

D₄

yes

—

C₄⋊Dic₁₀₃

—

C₂×Q₈×C₁₀₃

—

C₂×Q₈×C₁₀₃

—

C₂×Q₈×C₁₀₃

—

C₂×Q₈×C₁₀₃

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

—

C₂²×C₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

D₄

yes

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

Q₈×C₁₀₃

—

Q₈×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₄×D₄

yes

—

C₄×C₁₀₃

#121

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

#390

yes

characteristic

C₂×C₁₀₃

#419

yes

—

C₂×C₁₀₃

#491

yes

—

C₂×C₁₀₃

#497

yes

—

#1960

—

103

maximal, Sylow

C₁₀₃

#1960

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₀₃

yes

characteristic

C₂×#103

—

103

—

C₂×#97

—

103

—

#382

—

103

—

C₂²×Q₈

C₂×D₁₀₃

yes

—

C₂²×C₈

—

103

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₀₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#103

—

206

—

#97

—

206

—

#95

—

206

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₀₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

206

—

C₂²×C₄

C₂²⋊D₁₀₃

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

103

—

C₂×C₈

—

206

—

C₂×C₈

—

206

—

C₄⋊C₄

—

C₄⋊C₄

—

206

—

C₄²

C₂×Dic₁₀₃

yes

—

C₄²

C₂²⋊D₁₀₃

yes

—

C₂³

C₂×C₂²⋊D₁₀₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

412

—

C₂×C₄

—

412

—

C₂×C₄

C₂²⋊Dic₁₀₃

yes

—

C₂×C₄

C₂²×Dic₁₀₃

yes

characteristic

100

C₂×C₄

D₄×D₁₀₃

yes

—

101

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

102

C₂×C₄

#13110104

yes

—

103

C₈

—

206

—

104

C₂²

C₂×C₂²⋊Dic₁₀₃

yes

characteristic

105

C₂²

#15693008

yes

characteristic

106

C₂²

#15693017

yes

characteristic

107

C₂²

#15693052

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

824

—

111

C₄

D₄×Dic₁₀₃

yes

—

112

C₄

#15692992

yes

—

113

C₂

#20267694

yes

characteristic

114

C₂

#20267800

yes

—

115

C₂

#20268159

yes

—

116

C₂

#20268183

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₃
commutator	C₂×C₄×C₁₀₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	C₂×C₄⋊Q₈×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂×D₄
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₀₃
hypocenter	C₁₀₃	#1960
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₃

commutator

C₂×C₄×C₁₀₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

C₂×C₄⋊Q₈×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂×D₄

cosocle

C₂²×C₄×C₁₀₃

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₀₃

hypocenter

C₁₀₃

#1960

LCS

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₀₃	C₄	semidirect
59. C₁₀₃	#1960	semidirect
67. C₄⋊Q₈	Dic₁₀₃	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₀₃

C₄

semidirect

59. C₁₀₃

#1960

semidirect

67. C₄⋊Q₈

Dic₁₀₃

semidirect

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

subgroup	index
C₂×#15692945	2
C₂×#15692932	2
#20267657	2
C₂×C₄⋊Q₈×C₁₀₃	2
#1960	103

subgroup

index

C₂×C₄⋊Q₈×C₁₀₃

103

Subgroups — #27195965

Conjugacy classes of subgroups

Complements

Maximal subgroups