Group #28364920 — subgroups

Subgroups — #28364920

Order 13952.

142 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 51 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	#20295466	C₂	1	1	yes	maximal, characteristic
2	#20295148	C₂	1	1	yes	maximal, characteristic
3	#20295132	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294848	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294843	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294803	C₂	1	1	yes	maximal, characteristic, complemented
7	#360×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
9	#16812706	—	2	1	no	—
10	#16812664	—	2	1	no	—
11	#16812645	C₄	1	2	yes	complemented
12	#16812643	C₄	1	2	yes	complemented
13	#16812639	C₂²	1	1	yes	characteristic
14	#16812635	C₂²	1	1	yes	characteristic
15	#16812626	—	2	1	no	—
16	#16812622	—	2	1	no	—
17	#16812609	C₂²	1	1	yes	characteristic
18	#135×C₁₀₉	C₂²	1	1	yes	characteristic
19	M₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
20	#106×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₁₈	—	2	1	no	—
22	Dic₄₃₆	—	2	1	no	—
23	Dic₄₃₆	C₂×C₄	1	2	yes	—
24	#13182013	—	2	1	no	—
25	#13182013	—	4	1	no	—
26	C₂×#11724784	—	4	1	no	—
27	C₂×#11724784	D₄	1	1	yes	characteristic
28	#13182008	—	4	1	no	—
29	C₄×Dic₁₀₉	—	4	1	no	—
30	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
31	#13182004	C₂×C₄	1	2	yes	—
32	#41×C₁₀₉	—	2	1	no	—
33	Q₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	SD₁₆×C₁₀₉	—	2	1	no	—
35	D₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₁₀₉	—	2	1	no	—
38	C₁₆×C₁₀₉	—	2	1	no	—
39	Q₈×C₁₀₉	—	2	1	no	—
40	D₄×C₁₀₉	—	2	1	no	—
41	D₄×C₁₀₉	—	4	1	no	—
42	C₂×C₄×C₁₀₉	—	4	1	no	—
43	C₂×C₄×C₁₀₉	—	4	1	no	—
44	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₁₀₉	—	2	1	no	—
46	C₂×Dic₁₀₉	—	4	1	no	—
47	Dic₂₁₈	—	2	2	no	—
48	Dic₂₁₈	—	4	1	no	—
49	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₈×C₁₀₉	#41	1	1	yes	characteristic
51	#11724784	—	2	1	no	—
52	#11724784	—	4	1	no	—
53	C₂²×C₁₀₉	—	4	1	no	—
54	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₄×C₁₀₉	—	4	1	no	—
56	C₄×C₁₀₉	—	8	1	no	—
57	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
58	C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₁₀₉	—	4	1	no	—
60	Dic₁₀₉	—	4	2	no	—
61	C₂×C₁₀₉	—	2	1	no	—
62	C₂×C₁₀₉	—	8	1	no	—
63	C₂×C₁₀₉	#442	1	1	yes	characteristic
64	#2180	—	109	1	no	maximal, Sylow
65	C₁₀₉	#2180	1	1	yes	characteristic, complemented, Sylow
66	#510	—	109	1	no	—
67	#444	—	109	1	no	—
68	#441	—	109	1	no	—
69	#365	—	109	1	no	—
70	#362	—	109	1	no	—
71	#360	D₁₀₉	1	1	yes	characteristic
72	#347	—	109	1	no	—
73	#135	C₂×D₁₀₉	1	1	yes	characteristic
74	C₂×Q₁₆	—	109	1	no	—
75	#131	—	218	1	no	—
76	C₄×Q₈	—	218	1	no	—
77	Q₃₂	—	109	2	no	—
78	SD₃₂	—	109	2	no	—
79	M₃₂	—	109	1	no	—
80	M₃₂	C₂×D₁₀₉	1	1	yes	characteristic
81	#108	—	109	1	no	—
82	#106	C₂×D₁₀₉	1	1	yes	characteristic
83	#104	—	218	1	no	—
84	#103	—	218	1	no	—
85	#97	—	109	1	no	—
86	#41	—	2	1	no	—
87	C₂×Q₈	—	218	1	no	—
88	Q₁₆	—	109	2	no	—
89	Q₁₆	—	218	1	no	—
90	Q₁₆	C₄×D₁₀₉	1	1	yes	characteristic
91	SD₁₆	—	2	1	no	—
92	D₈	C₄×D₁₀₉	1	1	yes	characteristic
93	M₁₆	—	436	1	no	—
94	M₁₆	—	218	1	no	—
95	C₂×C₈	—	109	1	no	—
96	C₂×C₈	—	436	1	no	—
97	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	436	1	no	—
100	C₄²	—	109	1	no	—
101	C₄²	—	436	1	no	—
102	C₁₆	—	2	1	no	—
103	C₁₆	—	109	2	no	—
104	Q₈	—	2	1	no	—
105	Q₈	—	436	1	no	—
106	Q₈	—	218	2	no	—
107	D₄	—	2	1	no	—
108	D₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	436	1	no	—
112	C₂×C₄	—	218	1	no	—
113	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
114	C₈	—	436	1	no	—
115	C₈	—	218	1	no	—
116	C₈	C₂×C₄×D₁₀₉	1	1	yes	characteristic
117	C₈	#13182028	1	1	yes	characteristic
118	C₂²	—	4	1	no	—
119	C₂²	#16812723	1	1	yes	characteristic
120	C₄	—	4	1	no	—
121	C₄	—	8	1	no	—
122	C₄	—	436	1	no	—
123	C₄	—	436	2	no	—
124	C₄	#16812659	1	1	yes	characteristic
125	C₄	#16812726	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	8	1	no	—
128	C₂	#20295135	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295466

C₂

yes

maximal, characteristic

#20295148

C₂

yes

maximal, characteristic

#20295132

C₂

yes

maximal, characteristic, complemented

#20294848

C₂

yes

maximal, characteristic, complemented

#20294843

C₂

yes

maximal, characteristic, complemented

#20294803

C₂

yes

maximal, characteristic, complemented

#360×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₄₃₆

C₂²

yes

—

—

C₄

yes

complemented

#16812643

C₄

yes

complemented

#16812639

C₂²

yes

characteristic

#16812635

C₂²

yes

—

—

C₂²

yes

characteristic

#135×C₁₀₉

C₂²

yes

characteristic

M₃₂×C₁₀₉

C₂²

yes

characteristic

#106×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

Dic₄₃₆

—

Dic₄₃₆

C₂×C₄

yes

—

—

—

—

D₄

yes

characteristic

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

#13182004

C₂×C₄

yes

—

#41×C₁₀₉

—

Q₁₆×C₁₀₉

C₂×C₄

yes

characteristic

SD₁₆×C₁₀₉

—

D₈×C₁₀₉

C₂×C₄

yes

characteristic

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

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂²×C₄

yes

C₈×C₁₀₉

yes

—

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#442

yes

characteristic

#2180

—

109

maximal, Sylow

C₁₀₉

#2180

yes

characteristic, complemented, Sylow

#510

—

109

—

#444

—

109

—

#441

—

109

—

#365

—

109

—

#362

—

109

—

#360

D₁₀₉

yes

characteristic

#347

—

109

—

#135

C₂×D₁₀₉

yes

characteristic

C₂×Q₁₆

—

109

—

#131

—

218

—

C₄×Q₈

—

218

—

Q₃₂

—

109

—

SD₃₂

—

109

—

M₃₂

—

109

—

M₃₂

C₂×D₁₀₉

yes

characteristic

#108

—

109

—

#106

C₂×D₁₀₉

yes

characteristic

#104

—

218

—

#103

—

218

—

#97

—

109

—

#41

—

C₂×Q₈

—

218

—

Q₁₆

—

109

—

Q₁₆

—

218

—

Q₁₆

C₄×D₁₀₉

yes

characteristic

SD₁₆

—

D₈

C₄×D₁₀₉

yes

characteristic

M₁₆

—

436

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

436

—

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

436

—

100

C₄²

—

109

—

101

C₄²

—

436

—

102

C₁₆

—

103

C₁₆

—

109

—

104

Q₈

—

105

Q₈

—

436

—

106

Q₈

—

218

—

107

D₄

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

436

—

112

C₂×C₄

—

218

—

113

C₂×C₄

D₄×D₁₀₉

yes

characteristic

114

C₈

—

436

—

115

C₈

—

218

—

116

C₈

C₂×C₄×D₁₀₉

yes

characteristic

117

C₈

#13182028

yes

characteristic

118

C₂²

—

119

C₂²

#16812723

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

436

—

123

C₄

—

436

—

124

C₄

#16812659

yes

characteristic

125

C₄

#16812726

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#20295135

yes

characteristic

	subgroup	G/H
center	C₂	#20295135
commutator	C₈×C₁₀₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#360×C₁₀₉	C₂
socle	C₂×C₁₀₉	#442
prime radical	#135×C₁₀₉	C₂²
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	#360	D₁₀₉
hypocenter	C₁₀₉	#2180
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

subgroup

G/H

center

C₂

#20295135

commutator

C₈×C₁₀₉

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#360×C₁₀₉

C₂

socle

C₂×C₁₀₉

#442

prime radical

#135×C₁₀₉

C₂²

cosocle

C₂×C₈×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #360

N	H ≅ G/N	type
3. #20295132	C₂	semidirect
4. #20294848	C₂	semidirect
5. #20294843	C₂	semidirect
6. #20294803	C₂	semidirect
11. #16812645	C₄	semidirect
12. #16812643	C₄	semidirect
65. C₁₀₉	#2180	semidirect

H ≅ G/N

type

3. #20295132

C₂

semidirect

4. #20294848

C₂

semidirect

5. #20294843

C₂

semidirect

6. #20294803

C₂

semidirect

11. #16812645

C₄

semidirect

12. #16812643

C₄

semidirect

65. C₁₀₉

#2180

semidirect

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

subgroup	index
#20295466	2
#20295148	2
#20295132	2
#20294848	2
#20294843	2
#20294803	2
#360×C₁₀₉	2
#2180	109

subgroup

index

#360×C₁₀₉

109

Subgroups — #28364920

Conjugacy classes of subgroups

Complements

Maximal subgroups