Group #27196136 — subgroups

Subgroups — #27196136

Order 13184.

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

normal subgroups only

This group has 228 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	C₂×#15693013	C₂	1	1	yes	maximal, characteristic
2	C₄×#13110089	C₂	1	1	yes	maximal, characteristic
3	#20267559	C₂	1	4	yes	maximal
4	#423×C₁₀₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13110089	C₂²	1	2	yes	—
6	#15693013	—	2	1	no	—
7	#15693013	—	2	1	no	—
8	#15693013	—	2	2	no	—
9	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
10	C₂×Dic₁₀₃⋊C₄	C₄	1	4	yes	—
11	#15692932	—	2	1	no	—
12	C₄×C₁₀₃⋊C₈	—	2	1	no	—
13	C₂×M₁₆×C₁₀₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₀₃	—	2	2	no	—
16	C₂²×Dic₁₀₃	C₂×C₄	1	4	yes	—
17	#13110089	—	2	2	no	—
18	#13110089	Q₈	1	4	yes	—
19	C₂×C₁₀₃⋊C₈	—	2	2	no	—
20	C₄⋊Dic₁₀₃	—	2	2	no	—
21	C₄⋊Dic₁₀₃	—	2	2	no	—
22	Dic₁₀₃⋊C₄	—	2	4	no	—
23	Dic₁₀₃⋊C₄	—	2	4	no	—
24	C₂²×C₄×C₁₀₃	C₂×C₄	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	D₄	1	1	yes	characteristic
29	C₄²×C₁₀₃	D₄	1	1	yes	characteristic
30	C₄²×C₁₀₃	D₄	1	2	yes	—
31	C₂³×C₁₀₃	C₂²×C₄	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	C₂×D₄	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	—	4	4	no	—
41	C₈×C₁₀₃	—	4	2	no	—
42	C₁₀₃⋊C₈	—	2	4	no	—
43	C₂²×C₁₀₃	—	2	2	no	—
44	C₂²×C₁₀₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₀₃	C₄×Q₈	1	2	yes	—
46	C₄×C₁₀₃	—	2	2	no	—
47	C₄×C₁₀₃	—	2	2	no	—
48	C₄×C₁₀₃	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₁₀₃	#124	1	1	yes	characteristic
50	C₄×C₁₀₃	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₁₀₃	—	4	4	no	—
52	C₂×C₁₀₃	—	2	2	no	—
53	C₂×C₁₀₃	#391	1	1	yes	characteristic
54	C₂×C₁₀₃	#413	1	2	yes	—
55	#1998	—	103	1	no	maximal, Sylow
56	C₁₀₃	#1998	1	1	yes	characteristic, complemented, Sylow
57	C₂×#125	—	103	1	no	—
58	#423	D₁₀₃	1	1	yes	characteristic
59	C₄×M₁₆	—	103	1	no	—
60	#343	—	103	4	no	—
61	C₂×M₁₆	—	103	2	no	—
62	C₂×M₁₆	C₂×D₁₀₃	1	2	yes	—
63	#125	—	206	1	no	—
64	#125	—	206	1	no	—
65	#125	—	206	2	no	—
66	C₂×C₄⋊C₄	—	103	4	no	—
67	C₂×C₄⋊C₄	—	103	2	no	—
68	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
69	C₄⋊C₈	—	2	2	no	—
70	#97	—	206	1	no	—
71	C₄×C₈	—	206	1	no	—
72	C₂²×C₄	—	103	4	no	—
73	C₂²×C₄	C₄×D₁₀₃	1	2	yes	—
74	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
75	M₁₆	—	2	4	no	—
76	M₁₆	—	103	4	no	—
77	M₁₆	—	206	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	206	2	no	—
80	C₄⋊C₄	—	206	4	no	—
81	C₄⋊C₄	—	206	4	no	—
82	C₄⋊C₄	—	206	2	no	—
83	C₄⋊C₄	—	206	2	no	—
84	C₄²	D₄₁₂	1	1	yes	characteristic
85	C₄²	D₄₁₂	1	1	yes	characteristic
86	C₄²	C₂²⋊D₁₀₃	1	2	yes	—
87	C₂³	C₂×C₄×D₁₀₃	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	412	4	no	—
90	C₂×C₄	—	206	4	no	—
91	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	2	yes	—
92	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
94	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
95	C₂×C₄	Q₈×D₁₀₃	1	2	yes	—
96	C₂×C₄	#13110104	1	2	yes	—
97	C₈	—	4	2	no	—
98	C₈	—	206	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×(C₂×D₁₀₃)⋊C₄	1	1	yes	characteristic
101	C₂²	#15692986	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	412	4	no	—
105	C₄	#15693001	1	2	yes	—
106	C₄	#15693008	1	1	yes	characteristic
107	C₄	#15693019	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#20267696	1	1	yes	characteristic
110	C₂	#20267773	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15693013

C₂

yes

maximal, characteristic

C₄×#13110089

C₂

yes

maximal, characteristic

#20267559

C₂

yes

maximal

#423×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110089

C₂²

yes

—

#15693013

—

#15693013

—

#15693013

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

C₂×Dic₁₀₃⋊C₄

C₄

yes

—

#15692932

—

C₄×C₁₀₃⋊C₈

—

C₂×M₁₆×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₃

—

C₂²×Dic₁₀₃

C₂×C₄

yes

—

#13110089

—

#13110089

Q₈

yes

—

C₂×C₁₀₃⋊C₈

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

Dic₁₀₃⋊C₄

—

Dic₁₀₃⋊C₄

—

C₂²×C₄×C₁₀₃

C₂×C₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

M₁₆×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₄²×C₁₀₃

D₄

yes

characteristic

C₄²×C₁₀₃

D₄

yes

characteristic

C₄²×C₁₀₃

D₄

yes

—

C₂³×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

—

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₃

C₄×Q₈

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂²⋊Q₈

yes

—

C₄×C₁₀₃

#124

yes

characteristic

C₄×C₁₀₃

C₄⋊Q₈

yes

characteristic

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

#391

yes

characteristic

C₂×C₁₀₃

#413

yes

—

#1998

—

103

maximal, Sylow

C₁₀₃

#1998

yes

characteristic, complemented, Sylow

C₂×#125

—

103

—

#423

D₁₀₃

yes

characteristic

C₄×M₁₆

—

103

—

#343

—

103

—

C₂×M₁₆

—

103

—

C₂×M₁₆

C₂×D₁₀₃

yes

—

#125

—

206

—

#125

—

206

—

#125

—

206

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

C₄⋊C₈

—

#97

—

206

—

C₄×C₈

—

206

—

C₂²×C₄

—

103

—

C₂²×C₄

C₄×D₁₀₃

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

M₁₆

—

M₁₆

—

103

—

M₁₆

—

206

—

C₂×C₈

—

C₂×C₈

—

206

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄²

D₄₁₂

yes

characteristic

C₄²

D₄₁₂

yes

characteristic

C₄²

C₂²⋊D₁₀₃

yes

—

C₂³

C₂×C₄×D₁₀₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

412

—

C₂×C₄

—

206

—

C₂×C₄

(C₂×D₁₀₃)⋊C₄

yes

—

C₂×C₄

(C₂×D₁₀₃)⋊C₄

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

C₂×C₄

C₂×D₄₁₂

yes

characteristic

C₂×C₄

Q₈×D₁₀₃

yes

—

C₂×C₄

#13110104

yes

—

C₈

—

C₈

—

206

—

C₂²

—

100

C₂²

C₂×(C₂×D₁₀₃)⋊C₄

yes

characteristic

101

C₂²

#15692986

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

412

—

105

C₄

#15693001

yes

—

106

C₄

#15693008

yes

characteristic

107

C₄

#15693019

yes

characteristic

108

C₂

—

109

C₂

#20267696

yes

characteristic

110

C₂

#20267773

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₁₀₃)⋊C₄
commutator, prime radical	C₂³×C₁₀₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	#423×C₁₀₃	C₂
socle	C₂²×C₁₀₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	#423	D₁₀₃
hypocenter	C₁₀₃	#1998
LCS	G ▷ C₂³×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₁₀₃)⋊C₄

commutator, prime radical

C₂³×C₁₀₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

#423×C₁₀₃

C₂

socle

C₂²×C₁₀₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₃

C₂³

D₁₀₃

C₁₀₃

G ▷ C₂³×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
56. C₁₀₃	#1998	semidirect

H ≅ G/N

type

56. C₁₀₃

#1998

semidirect

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

subgroup	index
C₂×#15693013	2
C₄×#13110089	2
#20267559	2
#423×C₁₀₃	2
#1998	103

subgroup

index

#423×C₁₀₃

103

Subgroups — #27196136

Conjugacy classes of subgroups

Complements

Maximal subgroups