Group #28365105 — subgroups

Subgroups — #28365105

Order 13952.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#20295462	C₂	1	2	yes	maximal, complemented
2	#20295144	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294847	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294839	C₂	1	2	yes	maximal, complemented
5	#493×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812721	C₂²	1	1	yes	characteristic
7	#16812705	—	2	1	no	—
8	#16812643	—	2	2	no	—
9	#16812640	—	2	2	no	—
10	#16812639	C₂²	1	2	yes	—
11	#16812635	C₂²	1	1	yes	characteristic
12	#16812627	—	2	1	no	—
13	C₂×D₈×C₁₀₉	—	2	1	no	—
14	C₂×D₈×C₁₀₉	C₂²	1	2	yes	—
15	#127×C₁₀₉	—	2	1	no	—
16	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	#13182029	—	2	1	no	—
18	Dic₄₃₆	D₄	1	1	yes	characteristic, complemented
19	#13182018	—	2	1	no	—
20	D₈₇₂	D₄	1	1	yes	characteristic, complemented
21	#13182013	—	2	1	no	—
22	#13182012	—	4	1	no	—
23	C₈×D₁₀₉	—	4	1	no	—
24	#13182004	—	2	2	no	—
25	C₂×D₄×C₁₀₉	—	2	2	no	—
26	C₂×D₄×C₁₀₉	—	4	2	no	—
27	D₈×C₁₀₉	—	2	2	no	—
28	D₈×C₁₀₉	—	2	2	no	—
29	D₈×C₁₀₉	—	4	1	no	—
30	D₈×C₁₀₉	—	4	1	no	—
31	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂³×C₁₀₉	—	4	2	no	—
35	D₄×C₁₀₉	—	4	2	no	—
36	D₄×C₁₀₉	—	4	2	no	—
37	D₄×C₁₀₉	—	8	4	no	—
38	C₂×C₄×C₁₀₉	—	2	1	no	—
39	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂²⋊D₁₀₉	—	4	1	no	—
41	D₄₃₆	—	2	1	no	—
42	C₄×D₁₀₉	—	4	1	no	—
43	Dic₂₁₈	—	2	1	no	—
44	C₈×C₁₀₉	D₈	1	1	yes	characteristic
45	C₈×C₁₀₉	D₈	1	1	yes	characteristic
46	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₈×C₁₀₉	#41	1	1	yes	characteristic
48	#11724784	—	4	1	no	—
49	C₂²×C₁₀₉	—	8	4	no	—
50	C₂²×C₁₀₉	—	8	2	no	—
51	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
52	C₂×D₁₀₉	—	4	1	no	—
53	C₄×C₁₀₉	—	2	2	no	—
54	C₄×C₁₀₉	#121	1	1	yes	characteristic
55	C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
56	Dic₁₀₉	—	4	1	no	—
57	C₂×C₁₀₉	—	2	1	no	—
58	C₂×C₁₀₉	—	16	2	no	—
59	C₂×C₁₀₉	#459	1	1	yes	characteristic
60	D₁₀₉	—	8	1	no	—
61	#2213	—	109	1	no	maximal, Sylow
62	C₁₀₉	#2213	1	1	yes	characteristic, complemented, Sylow
63	#509	—	109	2	no	—
64	#493	D₁₀₉	1	1	yes	characteristic, complemented
65	#443	—	109	1	no	—
66	#364	—	109	1	no	—
67	#361	—	109	2	no	—
68	#135	—	109	1	no	—
69	C₂×D₈	—	2	1	no	—
70	C₂×D₈	C₂×D₁₀₉	1	2	yes	—
71	#131	—	218	1	no	—
72	#127	—	2	1	no	—
73	SD₃₂	—	218	2	no	—
74	D₁₆	—	218	2	no	—
75	M₃₂	—	109	2	no	—
76	#108	—	109	1	no	—
77	#104	—	218	1	no	—
78	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
79	#41	—	218	1	no	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	—	4	2	no	—
82	Q₁₆	—	109	1	no	—
83	SD₁₆	—	218	1	no	—
84	D₈	—	2	2	no	—
85	D₈	—	2	2	no	—
86	D₈	—	4	1	no	—
87	D₈	—	4	1	no	—
88	D₈	—	109	1	no	—
89	M₁₆	—	436	1	no	—
90	M₁₆	—	218	1	no	—
91	C₂×C₈	—	436	1	no	—
92	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
94	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
95	C₁₆	—	218	2	no	—
96	C₂³	—	4	2	no	—
97	Q₈	—	218	1	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	8	4	no	—
101	D₄	—	436	1	no	—
102	D₄	—	218	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	436	1	no	—
105	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
106	C₈	—	436	1	no	—
107	C₈	#13182014	1	1	yes	characteristic
108	C₈	#13182014	1	1	yes	characteristic
109	C₈	D₄×D₁₀₉	1	1	yes	characteristic
110	C₈	#13182028	1	1	yes	characteristic
111	C₂²	—	8	4	no	—
112	C₂²	—	8	2	no	—
113	C₂²	—	436	1	no	—
114	C₂²	#16812725	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	436	1	no	—
117	C₄	#16812670	1	1	yes	characteristic
118	C₄	C₂×#13182014	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	16	2	no	—
121	C₂	—	872	1	no	—
122	C₂	#20295218	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295462

C₂

yes

maximal, complemented

#20295144

C₂

yes

maximal, characteristic, complemented

#20294847

C₂

yes

maximal, characteristic, complemented

#20294839

C₂

yes

maximal, complemented

#493×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812721

C₂²

yes

—

—

—

C₂²

yes

—

#16812635

C₂²

yes

characteristic

#16812627

—

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

C₂²

yes

—

#127×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

#13182029

—

Dic₄₃₆

D₄

yes

characteristic, complemented

#13182018

—

D₈₇₂

D₄

yes

characteristic, complemented

#13182013

—

#13182012

—

C₈×D₁₀₉

—

#13182004

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₂³×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂²⋊D₁₀₉

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

D₈

yes

characteristic

C₈×C₁₀₉

D₈

yes

characteristic

C₈×C₁₀₉

C₂×D₄

yes

C₈×C₁₀₉

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#121

yes

characteristic

C₄×C₁₀₉

C₂×D₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#459

yes

characteristic

D₁₀₉

—

#2213

—

109

maximal, Sylow

C₁₀₉

#2213

yes

characteristic, complemented, Sylow

#509

—

109

—

#493

D₁₀₉

yes

characteristic, complemented

#443

—

109

—

#364

—

109

—

#361

—

109

—

#135

—

109

—

C₂×D₈

—

C₂×D₈

C₂×D₁₀₉

yes

—

#131

—

218

—

#127

—

SD₃₂

—

218

—

D₁₆

—

218

—

M₃₂

—

109

—

#108

—

109

—

#104

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

#41

—

218

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

109

—

SD₁₆

—

218

—

D₈

—

D₈

—

D₈

—

D₈

—

D₈

—

109

—

M₁₆

—

436

—

M₁₆

—

218

—

C₂×C₈

—

436

—

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

characteristic

C₁₆

—

218

—

C₂³

—

Q₈

—

218

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

436

—

102

D₄

—

218

—

103

C₂×C₄

—

104

C₂×C₄

—

436

—

105

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

106

C₈

—

436

—

107

C₈

#13182014

yes

characteristic

108

C₈

#13182014

yes

characteristic

109

C₈

D₄×D₁₀₉

yes

characteristic

110

C₈

#13182028

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

436

—

114

C₂²

#16812725

yes

characteristic

115

C₄

—

116

C₄

—

436

—

117

C₄

#16812670

yes

characteristic

118

C₄

C₂×#13182014

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

872

—

122

C₂

#20295218

yes

characteristic

	subgroup	G/H
center	C₂	#20295218
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#493×C₁₀₉	C₂
socle	C₂×C₁₀₉	#459
hypercenter	#493	D₁₀₉
hypocenter	C₁₀₉	#2213
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂

#20295218

commutator, cosocle

C₂×C₈×C₁₀₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#493×C₁₀₉

C₂

C₂×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #20295462	C₂	semidirect
2. #20295144	C₂	semidirect
3. #20294847	C₂	semidirect
4. #20294839	C₂	semidirect
5. #493×C₁₀₉	C₂	semidirect
18. Dic₄₃₆	D₄	semidirect
20. D₈₇₂	D₄	semidirect
62. C₁₀₉	#2213	semidirect
64. #493	D₁₀₉	semidirect

H ≅ G/N

type

1. #20295462

C₂

semidirect

2. #20295144

C₂

semidirect

3. #20294847

C₂

semidirect

4. #20294839

C₂

semidirect

5. #493×C₁₀₉

C₂

semidirect

18. Dic₄₃₆

D₄

semidirect

20. D₈₇₂

D₄

semidirect

62. C₁₀₉

#2213

semidirect

64. #493

D₁₀₉

semidirect

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

subgroup	index
#20295462	2
#20295144	2
#20294847	2
#20294839	2
#493×C₁₀₉	2
#2213	109

subgroup

index

#493×C₁₀₉

109

Subgroups — #28365105

Conjugacy classes of subgroups

Complements

Maximal subgroups