Group #27196124 — subgroups

Subgroups — #27196124

Order 13184.

244 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20268348	C₂	1	1	yes	maximal, characteristic
2	#20267637	C₂	1	1	yes	maximal, characteristic
3	#20267552	C₂	1	4	yes	maximal
4	#423×C₁₀₃	C₂	1	1	yes	maximal, characteristic
5	#15693021	C₄	1	2	yes	complemented
6	#15693013	C₄	1	2	yes	complemented
7	#15693000	—	2	2	no	—
8	C₄×Dic₂₀₆	—	2	2	no	—
9	#15692980	C₂²	1	2	yes	—
10	C₂×C₄×Dic₁₀₃	C₂²	1	2	yes	—
11	C₁₀₃⋊#95	—	2	2	no	—
12	C₂×M₁₆×C₁₀₃	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₀₃	—	2	2	no	—
15	C₂×Dic₂₀₆	—	2	2	no	—
16	C₂²×Dic₁₀₃	—	2	2	no	—
17	C₄⋊Dic₁₀₃	—	2	2	no	—
18	C₄⋊Dic₁₀₃	C₂×C₄	1	4	yes	—
19	Dic₁₀₃⋊C₄	—	2	4	no	—
20	C₂²⋊Dic₁₀₃	—	2	2	no	—
21	C₄×Dic₁₀₃	—	2	2	no	—
22	C₄×Dic₁₀₃	—	2	2	no	—
23	C₄×Dic₁₀₃	Q₈	1	4	yes	complemented
24	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₀₃	—	2	4	no	—
27	C₂×C₈×C₁₀₃	—	2	2	no	—
28	C₄²×C₁₀₃	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₀₃	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₀₃	D₄	1	2	yes	—
31	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₃	—	4	2	no	—
33	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₀₃	#41	1	2	yes	—
39	C₂×Dic₁₀₃	—	2	4	no	—
40	C₂×Dic₁₀₃	—	2	4	no	—
41	C₂×Dic₁₀₃	—	4	4	no	—
42	Dic₂₀₆	—	4	4	no	—
43	C₈×C₁₀₃	—	4	2	no	—
44	C₂²×C₁₀₃	—	2	2	no	—
45	C₂²×C₁₀₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₃	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₁₀₃	—	2	2	no	—
48	C₄×C₁₀₃	—	2	2	no	—
49	C₄×C₁₀₃	C₄×Q₈	1	2	yes	—
50	C₄×C₁₀₃	#124	1	1	yes	characteristic
51	C₄×C₁₀₃	C₄⋊Q₈	1	1	yes	characteristic
52	Dic₁₀₃	—	4	4	no	—
53	Dic₁₀₃	—	4	4	no	—
54	C₂×C₁₀₃	—	2	2	no	—
55	C₂×C₁₀₃	#391	1	1	yes	characteristic
56	C₂×C₁₀₃	#421	1	2	yes	—
57	#1995	—	103	1	no	maximal, Sylow
58	C₁₀₃	#1995	1	1	yes	characteristic, complemented, Sylow
59	#533	—	103	1	no	—
60	#423	D₁₀₃	1	1	yes	characteristic
61	#376	—	103	1	no	—
62	#339	—	103	4	no	—
63	C₂×M₁₆	C₂×D₁₀₃	1	2	yes	—
64	C₄⋊Q₈	—	103	2	no	—
65	#125	—	103	2	no	—
66	C₂²⋊Q₈	—	206	2	no	—
67	C₄×Q₈	—	206	2	no	—
68	#117	—	103	2	no	—
69	C₂×C₄²	—	103	2	no	—
70	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
71	C₄⋊C₈	—	2	2	no	—
72	#95	—	206	2	no	—
73	C₂×Q₈	—	206	2	no	—
74	C₂²×C₄	—	206	2	no	—
75	C₂²×C₄	D₄₁₂	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₄⋊C₄	—	103	4	no	—
80	C₄⋊C₄	—	206	4	no	—
81	C₄⋊C₄	—	206	2	no	—
82	C₂²⋊C₄	—	206	2	no	—
83	C₄²	—	103	4	no	—
84	C₄²	—	206	2	no	—
85	C₄²	—	206	2	no	—
86	C₄²	C₄×D₁₀₃	1	1	yes	characteristic
87	C₄²	C₄×D₁₀₃	1	1	yes	characteristic
88	C₄²	C₂²⋊D₁₀₃	1	2	yes	—
89	C₂³	C₂×D₄₁₂	1	1	yes	characteristic
90	Q₈	—	412	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	412	4	no	—
93	C₂×C₄	—	206	4	no	—
94	C₂×C₄	—	206	4	no	—
95	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	2	yes	—
96	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₁₀₃	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
99	C₂×C₄	Q₈×D₁₀₃	1	2	yes	—
100	C₂×C₄	#13110104	1	2	yes	—
101	C₈	—	4	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×(C₂×D₁₀₃)⋊C₄	1	1	yes	characteristic
104	C₂²	#15693001	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	412	4	no	—
108	C₄	—	412	4	no	—
109	C₄	#15692986	1	2	yes	—
110	C₄	#15693008	1	1	yes	characteristic
111	C₄	#15693019	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#20267696	1	1	yes	characteristic
114	C₂	#20267807	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20268348

C₂

yes

maximal, characteristic

#20267637

C₂

yes

maximal, characteristic

#20267552

C₂

yes

maximal

#423×C₁₀₃

C₂

yes

maximal, characteristic

#15693021

C₄

yes

complemented

#15693013

C₄

yes

complemented

#15693000

—

C₄×Dic₂₀₆

—

#15692980

C₂²

yes

—

C₂×C₄×Dic₁₀₃

C₂²

yes

—

C₁₀₃⋊#95

—

C₂×M₁₆×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₃

—

C₂×Dic₂₀₆

—

C₂²×Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

C₂×C₄

yes

—

Dic₁₀₃⋊C₄

—

C₂²⋊Dic₁₀₃

—

C₄×Dic₁₀₃

—

C₄×Dic₁₀₃

—

C₄×Dic₁₀₃

Q₈

yes

complemented

C₂²×C₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

M₁₆×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₄²×C₁₀₃

C₂×C₄

yes

characteristic

C₄²×C₁₀₃

C₂×C₄

yes

characteristic

C₄²×C₁₀₃

D₄

yes

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

—

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

Dic₂₀₆

—

C₈×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₃

C₂²⋊Q₈

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₄×Q₈

yes

—

C₄×C₁₀₃

#124

yes

characteristic

C₄×C₁₀₃

C₄⋊Q₈

yes

characteristic

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#391

yes

characteristic

C₂×C₁₀₃

#421

yes

—

#1995

—

103

maximal, Sylow

C₁₀₃

#1995

yes

characteristic, complemented, Sylow

#533

—

103

—

#423

D₁₀₃

yes

characteristic

#376

—

103

—

#339

—

103

—

C₂×M₁₆

C₂×D₁₀₃

yes

—

C₄⋊Q₈

—

103

—

#125

—

103

—

C₂²⋊Q₈

—

206

—

C₄×Q₈

—

206

—

#117

—

103

—

C₂×C₄²

—

103

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

C₄⋊C₈

—

#95

—

206

—

C₂×Q₈

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

D₄₁₂

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₂²⋊C₄

—

206

—

C₄²

—

103

—

C₄²

—

206

—

C₄²

—

206

—

C₄²

C₄×D₁₀₃

yes

characteristic

C₄²

C₄×D₁₀₃

yes

characteristic

C₄²

C₂²⋊D₁₀₃

yes

—

C₂³

C₂×D₄₁₂

yes

characteristic

Q₈

—

412

—

C₂×C₄

—

C₂×C₄

—

412

—

C₂×C₄

—

206

—

C₂×C₄

—

206

—

C₂×C₄

(C₂×D₁₀₃)⋊C₄

yes

—

C₂×C₄

(C₂×D₁₀₃)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₀₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

C₂×C₄

Q₈×D₁₀₃

yes

—

100

C₂×C₄

#13110104

yes

—

101

C₈

—

102

C₂²

—

103

C₂²

C₂×(C₂×D₁₀₃)⋊C₄

yes

characteristic

104

C₂²

#15693001

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

412

—

108

C₄

—

412

—

109

C₄

#15692986

yes

—

110

C₄

#15693008

yes

characteristic

111

C₄

#15693019

yes

characteristic

112

C₂

—

113

C₂

#20267696

yes

characteristic

114

C₂

#20267807

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₃
commutator	C₂×C₄×C₁₀₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#423×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₀₃	C₂×D₄
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	#423	D₁₀₃
hypocenter	C₁₀₃	#1995
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₃

commutator

C₂×C₄×C₁₀₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#423×C₁₀₃

C₂

socle

C₂²×C₁₀₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₀₃

C₂×D₄

cosocle

C₂²×C₄×C₁₀₃

C₂³

D₁₀₃

C₁₀₃

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #15693021	C₄	semidirect
6. #15693013	C₄	semidirect
23. C₄×Dic₁₀₃	Q₈	semidirect
58. C₁₀₃	#1995	semidirect

H ≅ G/N

type

5. #15693021

C₄

semidirect

6. #15693013

C₄

semidirect

23. C₄×Dic₁₀₃

Q₈

semidirect

58. C₁₀₃

#1995

semidirect

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

subgroup	index
#20268348	2
#20267637	2
#20267552	2
#423×C₁₀₃	2
#1995	103

subgroup

index

#423×C₁₀₃

103

Subgroups — #27196124

Conjugacy classes of subgroups

Complements

Maximal subgroups