Group #27206948 — subgroups

Subgroups — #27206948

Order 13184.

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

normal subgroups only

This group has 444 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	#20268604	C₂	1	4	yes	maximal, complemented
2	#20268063	C₂	1	4	yes	maximal, complemented
3	#20267973	C₂	1	4	yes	maximal, complemented
4	#20267909	C₂	1	2	yes	maximal, complemented
5	#494×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#15693069	C₂²	1	4	yes	—
7	#15693053	—	2	8	no	—
8	#15693052	C₂²	1	8	yes	—
9	#15693044	—	2	4	no	—
10	#15693043	C₂²	1	2	yes	—
11	Q₁₆×D₁₀₃	—	2	4	no	—
12	#15693027	C₂²	1	4	yes	—
13	#15692957	C₂²	1	2	yes	—
14	#15692949	C₂²	1	4	yes	—
15	#15692940	C₂²	1	4	yes	—
16	C₂×Q₁₆×C₁₀₃	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₁₀₃	C₂²	1	2	yes	—
18	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
19	#13110106	—	2	8	no	—
20	#13110105	—	4	4	no	—
21	#13110105	C₂³	1	4	yes	—
22	Q₈×D₁₀₃	—	2	8	no	—
23	Dic₄₁₂	D₄	1	2	yes	—
24	#13110095	—	2	8	no	—
25	#13110094	D₄	1	4	yes	—
26	#13110092	—	2	8	no	—
27	D₈₂₄	D₄	1	2	yes	—
28	#13110089	C₂³	1	4	yes	—
29	#13110088	—	2	4	no	—
30	C₈×D₁₀₃	—	2	4	no	—
31	C₂×Q₈×C₁₀₃	C₂³	1	4	yes	—
32	Q₁₆×C₁₀₃	—	2	4	no	—
33	Q₁₆×C₁₀₃	—	2	4	no	—
34	C₂×C₈×C₁₀₃	C₂³	1	2	yes	—
35	C₄⋊C₄×C₁₀₃	—	4	2	no	—
36	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
37	Q₈×C₁₀₃	—	2	8	no	—
38	C₂×C₄×C₁₀₃	—	2	1	no	—
39	C₂×C₄×C₁₀₃	—	2	4	no	—
40	C₂×C₄×C₁₀₃	C₂⁴	1	1	yes	characteristic
41	C₂²⋊D₁₀₃	—	2	4	no	—
42	D₄₁₂	—	4	4	no	—
43	D₄₁₂	C₂×D₄	1	4	yes	—
44	C₄×D₁₀₃	—	2	4	no	—
45	C₄×D₁₀₃	—	4	8	no	—
46	Dic₂₀₆	—	4	4	no	—
47	Dic₂₀₆	C₂×D₄	1	4	yes	—
48	C₈×C₁₀₃	C₂×D₄	1	2	yes	—
49	C₈×C₁₀₃	C₂×D₄	1	2	yes	—
50	C₁₀₃⋊C₈	—	2	4	no	—
51	C₂²×C₁₀₃	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂×D₁₀₃	—	2	4	no	—
53	C₄×C₁₀₃	—	2	2	no	—
54	C₄×C₁₀₃	—	4	4	no	—
55	C₄×C₁₀₃	C₂²×D₄	1	2	yes	—
56	Dic₁₀₃	—	2	4	no	—
57	C₂×C₁₀₃	—	2	1	no	—
58	C₂×C₁₀₃	D₄²	1	1	yes	characteristic
59	D₁₀₃	—	4	4	no	—
60	#3293	—	103	1	no	maximal, Sylow
61	C₁₀₃	#3293	1	1	yes	characteristic, complemented, Sylow
62	#578	—	103	4	no	—
63	#494	D₁₀₃	1	1	yes	characteristic, complemented
64	#473	—	103	4	no	—
65	#456	—	103	4	no	—
66	#443	—	103	2	no	—
67	2₋¹⁺⁴	—	103	4	no	—
68	#137	—	103	8	no	—
69	#137	—	206	8	no	—
70	#135	—	103	2	no	—
71	#135	—	206	4	no	—
72	C₂×Q₁₆	—	206	4	no	—
73	C₂×Q₁₆	C₂×D₁₀₃	1	4	yes	—
74	#131	—	103	4	no	—
75	C₄⋊Q₈	C₂×D₁₀₃	1	2	yes	—
76	#108	—	103	2	no	—
77	#104	—	103	4	no	—
78	#101	—	103	4	no	—
79	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
80	#41	—	103	4	no	—
81	#41	—	412	4	no	—
82	#41	—	206	8	no	—
83	C₂×Q₈	—	206	8	no	—
84	C₂×Q₈	C₂²×D₁₀₃	1	4	yes	—
85	Q₁₆	—	2	4	no	—
86	Q₁₆	—	2	4	no	—
87	Q₁₆	—	103	2	no	—
88	Q₁₆	—	206	8	no	—
89	SD₁₆	—	103	4	no	—
90	SD₁₆	—	206	8	no	—
91	D₈	—	103	2	no	—
92	M₁₆	—	103	4	no	—
93	M₁₆	—	206	4	no	—
94	C₂×C₈	—	206	4	no	—
95	C₂×C₈	C₂²×D₁₀₃	1	2	yes	—
96	C₄⋊C₄	—	4	2	no	—
97	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
98	Q₈	—	2	8	no	—
99	Q₈	—	103	4	no	—
100	Q₈	—	412	4	no	—
101	D₄	—	103	4	no	—
102	D₄	—	412	4	no	—
103	D₄	—	206	4	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	412	8	no	—
107	C₂×C₄	—	206	4	no	—
108	C₂×C₄	C₂³×D₁₀₃	1	1	yes	characteristic
109	C₈	—	206	4	no	—
110	C₈	D₄×D₁₀₃	1	2	yes	—
111	C₈	D₄×D₁₀₃	1	2	yes	—
112	C₂²	—	206	4	no	—
113	C₂²	#15693066	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	—	206	4	no	—
117	C₄	C₂×D₄×D₁₀₃	1	2	yes	—
118	C₂	—	2	1	no	—
119	C₂	—	412	4	no	—
120	C₂	#20268426	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268604

C₂

yes

maximal, complemented

#20268063

C₂

yes

maximal, complemented

#20267973

C₂

yes

maximal, complemented

#20267909

C₂

yes

maximal, complemented

#494×C₁₀₃

C₂

yes

maximal, characteristic, complemented

#15693069

C₂²

yes

—

#15693053

—

#15693052

C₂²

yes

—

#15693044

—

#15693043

C₂²

yes

—

Q₁₆×D₁₀₃

—

#15693027

C₂²

yes

—

#15692957

C₂²

yes

—

#15692949

C₂²

yes

—

#15692940

C₂²

yes

—

C₂×Q₁₆×C₁₀₃

C₂²

yes

—

C₄⋊Q₈×C₁₀₃

C₂²

yes

—

C₄×C₈×C₁₀₃

C₂²

yes

—

—

C₂³

yes

—

Q₈×D₁₀₃

—

Dic₄₁₂

D₄

yes

—

#13110095

—

#13110094

D₄

yes

—

#13110092

—

D₈₂₄

D₄

yes

—

#13110089

C₂³

yes

—

#13110088

—

C₈×D₁₀₃

—

C₂×Q₈×C₁₀₃

C₂³

yes

—

Q₁₆×C₁₀₃

—

Q₁₆×C₁₀₃

—

C₂×C₈×C₁₀₃

C₂³

yes

—

C₄⋊C₄×C₁₀₃

—

C₄²×C₁₀₃

C₂³

yes

characteristic

Q₈×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂⁴

yes

characteristic

C₂²⋊D₁₀₃

—

D₄₁₂

—

D₄₁₂

C₂×D₄

yes

—

C₄×D₁₀₃

—

C₄×D₁₀₃

—

Dic₂₀₆

—

Dic₂₀₆

C₂×D₄

yes

—

C₈×C₁₀₃

C₂×D₄

yes

—

C₈×C₁₀₃

C₂×D₄

yes

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

H(5,𝔽₂)

yes

characteristic

C₂×D₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₂²×D₄

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

D₄²

yes

characteristic

D₁₀₃

—

#3293

—

103

maximal, Sylow

C₁₀₃

#3293

yes

characteristic, complemented, Sylow

#578

—

103

—

#494

D₁₀₃

yes

characteristic, complemented

#473

—

103

—

#456

—

103

—

#443

—

103

—

2₋¹⁺⁴

—

103

—

#137

—

103

—

#137

—

206

—

#135

—

103

—

#135

—

206

—

C₂×Q₁₆

—

206

—

C₂×Q₁₆

C₂×D₁₀₃

yes

—

#131

—

103

—

C₄⋊Q₈

C₂×D₁₀₃

yes

—

#108

—

103

—

#104

—

103

—

#101

—

103

—

C₄×C₈

C₂×D₁₀₃

yes

characteristic

#41

—

103

—

#41

—

412

—

#41

—

206

—

C₂×Q₈

—

206

—

C₂×Q₈

C₂²×D₁₀₃

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

103

—

Q₁₆

—

206

—

SD₁₆

—

103

—

SD₁₆

—

206

—

D₈

—

103

—

M₁₆

—

103

—

M₁₆

—

206

—

C₂×C₈

—

206

—

C₂×C₈

C₂²×D₁₀₃

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₀₃

yes

characteristic

Q₈

—

Q₈

—

103

—

100

Q₈

—

412

—

101

D₄

—

103

—

102

D₄

—

412

—

103

D₄

—

206

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

412

—

107

C₂×C₄

—

206

—

108

C₂×C₄

C₂³×D₁₀₃

yes

characteristic

109

C₈

—

206

—

110

C₈

D₄×D₁₀₃

yes

—

111

C₈

D₄×D₁₀₃

yes

—

112

C₂²

—

206

—

113

C₂²

#15693066

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

206

—

117

C₄

C₂×D₄×D₁₀₃

yes

—

118

C₂

—

119

C₂

—

412

—

120

C₂

#20268426

yes

characteristic

	subgroup	G/H
center	C₂	#20268426
commutator, cosocle	C₂×C₄×C₁₀₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₃
Fitting	#494×C₁₀₃	C₂
socle	C₂×C₁₀₃	D₄²
hypercenter	#494	D₁₀₃
hypocenter	C₁₀₃	#3293
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

subgroup

G/H

center

C₂

#20268426

commutator, cosocle

C₂×C₄×C₁₀₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₃

Fitting

#494×C₁₀₃

C₂

socle

C₂×C₁₀₃

D₄²

D₁₀₃

C₁₀₃

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #494

N	H ≅ G/N	type
1. #20268604	C₂	semidirect
2. #20268063	C₂	semidirect
3. #20267973	C₂	semidirect
4. #20267909	C₂	semidirect
5. #494×C₁₀₃	C₂	semidirect
61. C₁₀₃	#3293	semidirect
63. #494	D₁₀₃	semidirect

H ≅ G/N

type

1. #20268604

C₂

semidirect

2. #20268063

C₂

semidirect

3. #20267973

C₂

semidirect

4. #20267909

C₂

semidirect

5. #494×C₁₀₃

C₂

semidirect

61. C₁₀₃

#3293

semidirect

63. #494

D₁₀₃

semidirect

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

subgroup	index
#20268604	2
#20268063	2
#20267973	2
#20267909	2
#494×C₁₀₃	2
#3293	103

subgroup

index

#494×C₁₀₃

103

Subgroups — #27206948

Conjugacy classes of subgroups

Complements

Maximal subgroups