Group #28374831 — subgroups

Subgroups — #28374831

Order 13952.

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

normal subgroups only

This group has 476 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	#20295827	C₂	1	4	yes	maximal, complemented
2	#20295293	C₂	1	4	yes	maximal, complemented
3	#20295193	C₂	1	4	yes	maximal, complemented
4	#20295144	C₂	1	2	yes	maximal, complemented
5	#493×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812744	C₂²	1	4	yes	complemented
7	#16812725	C₂²	1	8	yes	complemented
8	#16812723	—	2	8	no	—
9	#16812721	C₂²	1	2	yes	—
10	#16812718	—	2	4	no	—
11	D₈×D₁₀₉	—	2	4	no	—
12	#16812705	C₂²	1	4	yes	complemented
13	#16812635	C₂²	1	2	yes	complemented
14	#16812627	C₂²	1	4	yes	complemented
15	#16812616	C₂²	1	4	yes	complemented
16	C₂×D₈×C₁₀₉	C₂²	1	4	yes	complemented
17	#127×C₁₀₉	C₂²	1	2	yes	complemented
18	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
19	#13182029	C₂³	1	4	yes	—
20	#13182028	—	2	8	no	—
21	C₂×C₂²⋊D₁₀₉	—	4	4	no	—
22	D₄×D₁₀₉	—	2	8	no	—
23	Dic₄₃₆	D₄	1	2	yes	complemented
24	#13182018	D₄	1	4	yes	complemented
25	#13182017	—	2	8	no	—
26	D₈₇₂	D₄	1	2	yes	complemented
27	#13182014	—	2	8	no	—
28	#13182013	C₂³	1	4	yes	—
29	#13182012	—	2	4	no	—
30	C₈×D₁₀₉	—	2	4	no	—
31	C₂×D₄×C₁₀₉	—	4	2	no	—
32	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
33	D₈×C₁₀₉	—	2	4	no	—
34	D₈×C₁₀₉	—	2	4	no	—
35	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
36	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂³×C₁₀₉	—	2	4	no	—
38	C₂²×D₁₀₉	—	4	4	no	—
39	D₄×C₁₀₉	—	2	8	no	—
40	D₄×C₁₀₉	—	8	4	no	—
41	C₂×C₄×C₁₀₉	—	2	1	no	—
42	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
43	C₂²⋊D₁₀₉	—	2	4	no	—
44	C₂²⋊D₁₀₉	—	4	8	no	—
45	C₂×Dic₁₀₉	—	4	4	no	—
46	D₄₃₆	C₂×D₄	1	4	yes	—
47	C₄×D₁₀₉	—	2	4	no	—
48	Dic₂₁₈	C₂×D₄	1	4	yes	—
49	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
50	C₈×C₁₀₉	C₂×D₄	1	2	yes	—
51	#11724784	—	2	4	no	—
52	C₂²×C₁₀₉	—	4	4	no	—
53	C₂²×C₁₀₉	—	8	4	no	—
54	C₂²×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
55	C₂×D₁₀₉	—	2	4	no	—
56	C₂×D₁₀₉	—	16	4	no	—
57	C₄×C₁₀₉	—	2	2	no	—
58	C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
59	Dic₁₀₉	—	2	4	no	—
60	C₂×C₁₀₉	—	2	1	no	—
61	C₂×C₁₀₉	—	8	4	no	—
62	C₂×C₁₀₉	D₄²	1	1	yes	characteristic
63	D₁₀₉	—	4	4	no	—
64	#3292	—	109	1	no	maximal, Sylow
65	C₁₀₉	#3292	1	1	yes	characteristic, complemented, Sylow
66	#576	—	109	4	no	—
67	#493	D₁₀₉	1	1	yes	characteristic, complemented
68	#472	—	109	4	no	—
69	#453	—	109	4	no	—
70	#443	—	109	2	no	—
71	H(5,𝔽₂)	—	109	4	no	—
72	AGL(1,ℤ/8)	—	109	8	no	—
73	AGL(1,ℤ/8)	—	218	8	no	—
74	#135	—	109	2	no	—
75	#135	—	218	4	no	—
76	C₂×D₈	—	218	4	no	—
77	C₂×D₈	C₂×D₁₀₉	1	4	yes	complemented
78	#131	—	109	4	no	—
79	#127	C₂×D₁₀₉	1	2	yes	complemented
80	#108	—	109	2	no	—
81	#104	—	109	4	no	—
82	#100	—	109	4	no	—
83	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
84	#41	—	109	4	no	—
85	#41	—	218	8	no	—
86	C₂×D₄	—	4	2	no	—
87	C₂×D₄	—	436	4	no	—
88	C₂×D₄	—	218	8	no	—
89	C₂×D₄	C₂²×D₁₀₉	1	4	yes	—
90	Q₁₆	—	109	2	no	—
91	SD₁₆	—	109	4	no	—
92	SD₁₆	—	218	8	no	—
93	D₈	—	2	4	no	—
94	D₈	—	2	4	no	—
95	D₈	—	109	2	no	—
96	D₈	—	218	8	no	—
97	M₁₆	—	109	4	no	—
98	M₁₆	—	218	4	no	—
99	C₂×C₈	—	218	4	no	—
100	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
101	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
102	C₂³	—	2	4	no	—
103	C₂³	—	436	4	no	—
104	Q₈	—	109	4	no	—
105	D₄	—	2	8	no	—
106	D₄	—	8	4	no	—
107	D₄	—	109	4	no	—
108	D₄	—	436	8	no	—
109	D₄	—	218	4	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	436	4	no	—
112	C₂×C₄	—	218	4	no	—
113	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
114	C₈	—	218	4	no	—
115	C₈	D₄×D₁₀₉	1	2	yes	—
116	C₈	D₄×D₁₀₉	1	2	yes	—
117	C₂²	—	4	4	no	—
118	C₂²	—	8	4	no	—
119	C₂²	—	1744	4	no	—
120	C₂²	—	218	4	no	—
121	C₂²	#16812744	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	218	4	no	—
124	C₄	C₂×D₄×D₁₀₉	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	8	4	no	—
127	C₂	—	436	4	no	—
128	C₂	#20295661	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295827

C₂

yes

maximal, complemented

#20295293

C₂

yes

maximal, complemented

#20295193

C₂

yes

maximal, complemented

#20295144

C₂

yes

maximal, complemented

#493×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812744

C₂²

yes

complemented

#16812725

C₂²

yes

complemented

#16812723

—

#16812721

C₂²

yes

—

#16812718

—

D₈×D₁₀₉

—

#16812705

C₂²

yes

complemented

#16812635

C₂²

yes

complemented

#16812627

C₂²

yes

complemented

#16812616

C₂²

yes

complemented

C₂×D₈×C₁₀₉

C₂²

yes

complemented

#127×C₁₀₉

C₂²

yes

complemented

C₄×C₈×C₁₀₉

C₂²

yes

characteristic, complemented

#13182029

C₂³

yes

—

#13182028

—

C₂×C₂²⋊D₁₀₉

—

D₄×D₁₀₉

—

Dic₄₃₆

D₄

yes

complemented

#13182018

D₄

yes

complemented

#13182017

—

D₈₇₂

D₄

yes

complemented

#13182014

—

#13182013

C₂³

yes

—

#13182012

—

C₈×D₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

C₂³

yes

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

C₂²×D₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂²⋊D₁₀₉

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

C₂×D₄

yes

—

C₄×D₁₀₉

—

Dic₂₁₈

C₂×D₄

yes

—

C₈×C₁₀₉

C₂×D₄

yes

—

C₈×C₁₀₉

C₂×D₄

yes

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

H(5,𝔽₂)

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×D₄

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

D₄²

yes

characteristic

D₁₀₉

—

#3292

—

109

maximal, Sylow

C₁₀₉

#3292

yes

characteristic, complemented, Sylow

#576

—

109

—

#493

D₁₀₉

yes

characteristic, complemented

#472

—

109

—

#453

—

109

—

#443

—

109

—

H(5,𝔽₂)

—

109

—

AGL(1,ℤ/8)

—

109

—

AGL(1,ℤ/8)

—

218

—

#135

—

109

—

#135

—

218

—

C₂×D₈

—

218

—

C₂×D₈

C₂×D₁₀₉

yes

complemented

#131

—

109

—

#127

C₂×D₁₀₉

yes

complemented

#108

—

109

—

#104

—

109

—

#100

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic, complemented

#41

—

109

—

#41

—

218

—

C₂×D₄

—

C₂×D₄

—

436

—

C₂×D₄

—

218

—

C₂×D₄

C₂²×D₁₀₉

yes

—

Q₁₆

—

109

—

SD₁₆

—

109

—

SD₁₆

—

218

—

D₈

—

D₈

—

D₈

—

109

—

D₈

—

218

—

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

218

—

100

C₂×C₈

C₂²×D₁₀₉

yes

—

101

C₄²

C₂²×D₁₀₉

yes

characteristic

102

C₂³

—

103

C₂³

—

436

—

104

Q₈

—

109

—

105

D₄

—

106

D₄

—

107

D₄

—

109

—

108

D₄

—

436

—

109

D₄

—

218

—

110

C₂×C₄

—

111

C₂×C₄

—

436

—

112

C₂×C₄

—

218

—

113

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

114

C₈

—

218

—

115

C₈

D₄×D₁₀₉

yes

—

116

C₈

D₄×D₁₀₉

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

1744

—

120

C₂²

—

218

—

121

C₂²

#16812744

yes

characteristic

122

C₄

—

123

C₄

—

218

—

124

C₄

C₂×D₄×D₁₀₉

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

436

—

128

C₂

#20295661

yes

characteristic

	subgroup	G/H
center	C₂	#20295661
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#493×C₁₀₉	C₂
socle	C₂×C₁₀₉	D₄²
hypercenter	#493	D₁₀₉
hypocenter	C₁₀₉	#3292
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

subgroup

G/H

center

C₂

#20295661

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#493×C₁₀₉

C₂

socle

C₂×C₁₀₉

D₄²

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #493

N	H ≅ G/N	type
1. #20295827	C₂	semidirect
2. #20295293	C₂	semidirect
3. #20295193	C₂	semidirect
4. #20295144	C₂	semidirect
5. #493×C₁₀₉	C₂	semidirect
6. #16812744	C₂²	semidirect
7. #16812725	C₂²	semidirect
12. #16812705	C₂²	semidirect
13. #16812635	C₂²	semidirect
14. #16812627	C₂²	semidirect
15. #16812616	C₂²	semidirect
16. C₂×D₈×C₁₀₉	C₂²	semidirect
17. #127×C₁₀₉	C₂²	semidirect
18. C₄×C₈×C₁₀₉	C₂²	semidirect
23. Dic₄₃₆	D₄	semidirect
24. #13182018	D₄	semidirect
26. D₈₇₂	D₄	semidirect
65. C₁₀₉	#3292	semidirect
67. #493	D₁₀₉	semidirect
77. C₂×D₈	C₂×D₁₀₉	semidirect
79. #127	C₂×D₁₀₉	semidirect
83. C₄×C₈	C₂×D₁₀₉	semidirect

H ≅ G/N

type

1. #20295827

C₂

semidirect

2. #20295293

C₂

semidirect

3. #20295193

C₂

semidirect

4. #20295144

C₂

semidirect

5. #493×C₁₀₉

C₂

semidirect

6. #16812744

C₂²

semidirect

7. #16812725

C₂²

semidirect

12. #16812705

C₂²

semidirect

13. #16812635

C₂²

semidirect

14. #16812627

C₂²

semidirect

15. #16812616

C₂²

semidirect

16. C₂×D₈×C₁₀₉

C₂²

semidirect

17. #127×C₁₀₉

C₂²

semidirect

18. C₄×C₈×C₁₀₉

C₂²

semidirect

23. Dic₄₃₆

D₄

semidirect

24. #13182018

D₄

semidirect

26. D₈₇₂

D₄

semidirect

65. C₁₀₉

#3292

semidirect

67. #493

D₁₀₉

semidirect

77. C₂×D₈

C₂×D₁₀₉

semidirect

79. #127

C₂×D₁₀₉

semidirect

83. C₄×C₈

C₂×D₁₀₉

semidirect

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

subgroup	index
#20295827	2
#20295293	2
#20295193	2
#20295144	2
#493×C₁₀₉	2
#3292	109

subgroup

index

#493×C₁₀₉

109

Subgroups — #28374831

Conjugacy classes of subgroups

Complements

Maximal subgroups