Group #28365201 — subgroups

Subgroups — #28365201

Order 13952.

126 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 45 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	#20295423	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295143	C₂	1	1	yes	maximal, characteristic
3	#20294849	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294848	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294846	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294836	C₂	1	1	yes	maximal, characteristic
7	#360×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#16812706	—	2	1	no	—
9	#16812698	—	2	1	no	—
10	#16812639	C₂²	1	1	yes	characteristic
11	#16812635	C₂²	1	1	yes	characteristic
12	#16812631	C₂²	1	1	yes	characteristic
13	#16812630	—	2	1	no	—
14	#16812626	—	2	1	no	—
15	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
16	#135×C₁₀₉	C₂²	1	1	yes	characteristic
17	M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
18	#106×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₁₈	—	4	1	no	—
20	#13182013	—	2	1	no	—
21	#13182013	—	4	1	no	—
22	C₂×#11724784	—	4	1	no	—
23	C₂×#11724784	D₄	1	1	yes	characteristic
24	C₄⋊Dic₁₀₉	—	2	1	no	—
25	#13182008	—	4	1	no	—
26	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
27	#13182004	—	2	1	no	—
28	#41×C₁₀₉	—	2	1	no	—
29	Q₁₆×C₁₀₉	Q₈	1	1	yes	characteristic, complemented
30	SD₁₆×C₁₀₉	—	2	1	no	—
31	D₈×C₁₀₉	Q₈	1	1	yes	characteristic, complemented
32	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₀₉	—	2	1	no	—
34	C₁₆×C₁₀₉	—	2	1	no	—
35	Q₈×C₁₀₉	—	2	1	no	—
36	D₄×C₁₀₉	—	2	1	no	—
37	D₄×C₁₀₉	—	4	1	no	—
38	C₂×C₄×C₁₀₉	—	4	1	no	—
39	C₂×C₄×C₁₀₉	—	4	1	no	—
40	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₁₀₉	—	2	1	no	—
42	C₂×Dic₁₀₉	—	4	1	no	—
43	Dic₂₁₈	—	8	2	no	—
44	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₁₀₉	#41	1	1	yes	characteristic
46	#11724784	—	4	1	no	—
47	#11724784	SD₁₆	1	1	yes	characteristic
48	#11724784	SD₁₆	1	1	yes	characteristic
49	C₂²×C₁₀₉	—	4	1	no	—
50	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₄×C₁₀₉	—	4	1	no	—
52	C₄×C₁₀₉	—	8	1	no	—
53	C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₁₀₉	—	2	2	no	—
56	Dic₁₀₉	—	8	1	no	—
57	C₂×C₁₀₉	—	2	1	no	—
58	C₂×C₁₀₉	—	8	1	no	—
59	C₂×C₁₀₉	#476	1	1	yes	characteristic
60	#2230	—	109	1	no	maximal, Sylow
61	C₁₀₉	#2230	1	1	yes	characteristic, complemented, Sylow
62	#498	—	109	1	no	—
63	#443	—	109	1	no	—
64	#365	—	109	1	no	—
65	#365	—	109	1	no	—
66	#364	—	109	1	no	—
67	#360	—	109	1	no	—
68	#360	D₁₀₉	1	1	yes	characteristic
69	#135	C₂×D₁₀₉	1	1	yes	characteristic
70	#131	—	218	1	no	—
71	C₄⋊Q₈	—	218	1	no	—
72	M₃₂	—	109	1	no	—
73	M₃₂	C₂×D₁₀₉	1	1	yes	characteristic
74	#108	—	109	1	no	—
75	#106	—	109	1	no	—
76	#106	—	218	1	no	—
77	#106	C₂×D₁₀₉	1	1	yes	characteristic
78	#104	—	218	1	no	—
79	C₄×C₈	—	109	1	no	—
80	#41	—	2	1	no	—
81	C₂×Q₈	—	436	1	no	—
82	Q₁₆	Dic₂₁₈	1	1	yes	characteristic, complemented
83	SD₁₆	—	2	1	no	—
84	D₈	Dic₂₁₈	1	1	yes	characteristic, complemented
85	M₁₆	—	436	1	no	—
86	M₁₆	—	218	1	no	—
87	C₂×C₈	—	109	1	no	—
88	C₂×C₈	—	436	1	no	—
89	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	436	1	no	—
92	C₄⋊C₄	—	218	1	no	—
93	C₄²	—	109	1	no	—
94	C₁₆	—	2	1	no	—
95	C₁₆	—	218	1	no	—
96	Q₈	—	2	1	no	—
97	Q₈	—	872	2	no	—
98	D₄	—	2	1	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	436	1	no	—
103	C₂×C₄	—	218	1	no	—
104	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
105	C₈	—	109	1	no	—
106	C₈	—	109	1	no	—
107	C₈	—	436	1	no	—
108	C₈	C₂×Dic₂₁₈	1	1	yes	characteristic
109	C₈	#13182028	1	1	yes	characteristic
110	C₂²	—	4	1	no	—
111	C₂²	#16812723	1	1	yes	characteristic
112	C₄	—	4	1	no	—
113	C₄	—	8	1	no	—
114	C₄	—	872	1	no	—
115	C₄	—	218	2	no	—
116	C₄	#16812676	1	1	yes	characteristic
117	C₄	SD₁₆×D₁₀₉	1	1	yes	characteristic
118	C₂	—	2	1	no	—
119	C₂	—	8	1	no	—
120	C₂	#20295314	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295423

C₂

yes

maximal, characteristic, complemented

#20295143

C₂

yes

maximal, characteristic

#20294849

C₂

yes

maximal, characteristic, complemented

#20294848

C₂

yes

maximal, characteristic, complemented

#20294846

C₂

yes

maximal, characteristic, complemented

#20294836

C₂

yes

maximal, characteristic

#360×C₁₀₉

C₂

yes

maximal, characteristic

#16812706

—

#16812698

—

#16812639

C₂²

yes

characteristic

#16812635

C₂²

yes

characteristic

#16812631

C₂²

yes

characteristic

#16812630

—

#16812626

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

#135×C₁₀₉

C₂²

yes

characteristic

M₃₂×C₁₀₉

C₂²

yes

characteristic

#106×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

—

—

—

D₄

yes

characteristic

C₄⋊Dic₁₀₉

—

#13182008

—

C₄×Dic₁₀₉

D₄

yes

characteristic

#13182004

—

#41×C₁₀₉

—

Q₁₆×C₁₀₉

Q₈

yes

characteristic, complemented

SD₁₆×C₁₀₉

—

D₈×C₁₀₉

Q₈

yes

characteristic, complemented

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₁₆×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂×Q₈

yes

C₈×C₁₀₉

yes

—

SD₁₆

yes

characteristic

#11724784

SD₁₆

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₀₉

C₂×SD₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#476

yes

characteristic

#2230

—

109

maximal, Sylow

C₁₀₉

#2230

yes

characteristic, complemented, Sylow

#498

—

109

—

#443

—

109

—

#365

—

109

—

#365

—

109

—

#364

—

109

—

#360

—

109

—

#360

D₁₀₉

yes

characteristic

#135

C₂×D₁₀₉

yes

characteristic

#131

—

218

—

C₄⋊Q₈

—

218

—

M₃₂

—

109

—

M₃₂

C₂×D₁₀₉

yes

characteristic

#108

—

109

—

#106

—

109

—

#106

—

218

—

#106

C₂×D₁₀₉

yes

characteristic

#104

—

218

—

C₄×C₈

—

109

—

#41

—

C₂×Q₈

—

436

—

Q₁₆

Dic₂₁₈

yes

characteristic, complemented

SD₁₆

—

D₈

Dic₂₁₈

yes

characteristic, complemented

M₁₆

—

436

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

436

—

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

218

—

C₄²

—

109

—

C₁₆

—

C₁₆

—

218

—

Q₈

—

Q₈

—

872

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

436

—

103

C₂×C₄

—

218

—

104

C₂×C₄

D₄×D₁₀₉

yes

characteristic

105

C₈

—

109

—

106

C₈

—

109

—

107

C₈

—

436

—

108

C₈

C₂×Dic₂₁₈

yes

characteristic

109

C₈

#13182028

yes

characteristic

110

C₂²

—

111

C₂²

#16812723

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

872

—

115

C₄

—

218

—

116

C₄

#16812676

yes

characteristic

117

C₄

SD₁₆×D₁₀₉

yes

characteristic

118

C₂

—

119

C₂

—

120

C₂

#20295314

yes

characteristic

	subgroup	G/H
center	C₂	#20295314
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#360×C₁₀₉	C₂
socle	C₂×C₁₀₉	#476
prime radical	#135×C₁₀₉	C₂²
hypercenter	#360	D₁₀₉
hypocenter	C₁₀₉	#2230
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

subgroup

G/H

center

C₂

#20295314

commutator, cosocle

C₂×C₈×C₁₀₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#360×C₁₀₉

C₂

socle

C₂×C₁₀₉

#476

prime radical

#135×C₁₀₉

C₂²

D₁₀₉

C₁₀₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

N	H ≅ G/N	type
1. #20295423	C₂	semidirect
3. #20294849	C₂	semidirect
4. #20294848	C₂	semidirect
5. #20294846	C₂	semidirect
29. Q₁₆×C₁₀₉	Q₈	semidirect
31. D₈×C₁₀₉	Q₈	semidirect
61. C₁₀₉	#2230	semidirect
82. Q₁₆	Dic₂₁₈	semidirect
84. D₈	Dic₂₁₈	semidirect

H ≅ G/N

type

1. #20295423

C₂

semidirect

3. #20294849

C₂

semidirect

4. #20294848

C₂

semidirect

5. #20294846

C₂

semidirect

29. Q₁₆×C₁₀₉

Q₈

semidirect

31. D₈×C₁₀₉

Q₈

semidirect

61. C₁₀₉

#2230

semidirect

82. Q₁₆

Dic₂₁₈

semidirect

84. D₈

Dic₂₁₈

semidirect

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

subgroup	index
#20295423	2
#20295143	2
#20294849	2
#20294848	2
#20294846	2
#20294836	2
#360×C₁₀₉	2
#2230	109

subgroup

index

#360×C₁₀₉

109

Subgroups — #28365201

Conjugacy classes of subgroups

Complements

Maximal subgroups