Group #28362513 — subgroups

Subgroups — #28362513

Order 13952.

184 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 55 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	#20295223	C₂	1	2	yes	maximal, complemented
2	#20295156	C₂	1	1	yes	maximal, characteristic, complemented
3	#20294777	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294770	C₂	1	2	yes	maximal, complemented
5	#493×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182017	—	2	2	no	—
7	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
8	C₄⋊#11724784	C₂²	1	2	yes	—
9	C₄⋊#11724784	C₂²	1	1	yes	characteristic
10	#16812628	—	2	1	no	—
11	#16812620	—	2	2	no	—
12	#16812609	—	2	1	no	—
13	C₂×D₈×C₁₀₉	—	2	1	no	—
14	C₂×D₈×C₁₀₉	—	2	1	no	—
15	#127×C₁₀₉	C₂²	1	2	yes	—
16	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
18	#13182017	—	4	4	no	—
19	C₂×#11724784	—	2	1	no	—
20	C₂×#11724784	—	2	2	no	—
21	C₄⋊Dic₁₀₉	D₄	1	1	yes	characteristic
22	#13182008	—	2	1	no	—
23	C₄×Dic₁₀₉	—	2	1	no	—
24	C₂×D₄×C₁₀₉	—	2	2	no	—
25	C₂×D₄×C₁₀₉	—	2	2	no	—
26	C₂×D₄×C₁₀₉	—	2	2	no	—
27	D₈×C₁₀₉	—	4	2	no	—
28	D₈×C₁₀₉	—	4	2	no	—
29	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
31	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂³×C₁₀₉	—	4	2	no	—
33	D₄×C₁₀₉	—	4	2	no	—
34	D₄×C₁₀₉	—	4	2	no	—
35	D₄×C₁₀₉	—	4	2	no	—
36	D₄×C₁₀₉	—	4	2	no	—
37	D₄×C₁₀₉	—	8	2	no	—
38	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
41	C₂×Dic₁₀₉	—	2	1	no	—
42	C₂×Dic₁₀₉	—	2	1	no	—
43	Dic₂₁₈	—	2	2	no	—
44	C₈×C₁₀₉	—	2	1	no	—
45	C₈×C₁₀₉	SD₁₆	1	2	yes	complemented
46	#11724784	—	4	1	no	—
47	#11724784	—	4	2	no	—
48	C₂²×C₁₀₉	—	8	2	no	—
49	C₂²×C₁₀₉	—	8	2	no	—
50	C₂²×C₁₀₉	—	8	2	no	—
51	C₂²×C₁₀₉	#121	1	1	yes	characteristic
52	C₄×C₁₀₉	—	2	1	no	—
53	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
54	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
56	Dic₁₀₉	—	4	1	no	—
57	Dic₁₀₉	—	4	1	no	—
58	C₂×C₁₀₉	—	16	2	no	—
59	C₂×C₁₀₉	#460	1	1	yes	characteristic
60	C₂×C₁₀₉	#469	1	1	yes	characteristic
61	C₂×C₁₀₉	#472	1	1	yes	characteristic
62	#1688	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1688	1	1	yes	characteristic, complemented, Sylow
64	#493	D₁₀₉	1	1	yes	characteristic
65	#460	—	109	2	no	—
66	#446	—	109	1	no	—
67	#334	—	109	1	no	—
68	#331	—	109	2	no	—
69	C₂×SD₁₆	—	218	2	no	—
70	C₂×D₈	—	2	1	no	—
71	C₂×D₈	—	2	1	no	—
72	#127	C₂×D₁₀₉	1	2	yes	—
73	C₄×Q₈	—	109	1	no	—
74	C₄⋊C₈	—	109	1	no	—
75	C₄⋊C₈	—	109	2	no	—
76	C₄⋊C₈	—	218	1	no	—
77	#102	—	218	2	no	—
78	#97	—	218	1	no	—
79	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
80	C₂×Q₈	—	109	1	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	2	no	—
83	C₂×D₄	—	2	2	no	—
84	SD₁₆	—	436	4	no	—
85	D₈	—	4	2	no	—
86	D₈	—	4	2	no	—
87	C₂×C₈	—	218	2	no	—
88	C₂×C₈	—	218	1	no	—
89	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
91	C₄⋊C₄	—	109	1	no	—
92	C₄⋊C₄	—	218	1	no	—
93	C₄²	—	218	1	no	—
94	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
95	C₂³	—	4	2	no	—
96	Q₈	—	218	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	8	2	no	—
102	C₂×C₄	—	218	1	no	—
103	C₂×C₄	—	218	1	no	—
104	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
106	C₂×C₄	#13182028	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	436	2	no	—
109	C₈	—	436	1	no	—
110	C₈	#13182017	1	2	yes	complemented
111	C₂²	—	8	2	no	—
112	C₂²	—	8	2	no	—
113	C₂²	—	8	2	no	—
114	C₂²	#16812670	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	436	1	no	—
117	C₄	—	436	1	no	—
118	C₄	C₂×#13182017	1	1	yes	characteristic
119	C₄	#16812723	1	2	yes	—
120	C₄	#16812725	1	1	yes	characteristic
121	C₂	—	16	2	no	—
122	C₂	#20295223	1	1	yes	characteristic
123	C₂	#20295276	1	1	yes	characteristic
124	C₂	#20295293	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295223

C₂

yes

maximal, complemented

#20295156

C₂

yes

maximal, characteristic, complemented

#20294777

C₂

yes

maximal, characteristic, complemented

#20294770

C₂

yes

maximal, complemented

#493×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182017

—

C₄×Dic₂₁₈

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

—

—

—

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

—

#127×C₁₀₉

C₂²

yes

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

D₄

yes

—

—

—

C₄⋊Dic₁₀₉

D₄

yes

characteristic

#13182008

—

C₄×Dic₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

SD₁₆

yes

complemented

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#121

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1688

yes

characteristic, complemented, Sylow

#493

D₁₀₉

yes

characteristic

#460

—

109

—

#446

—

109

—

#334

—

109

—

#331

—

109

—

C₂×SD₁₆

—

218

—

C₂×D₈

—

C₂×D₈

—

#127

C₂×D₁₀₉

yes

—

C₄×Q₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

#102

—

218

—

#97

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

436

—

D₈

—

D₈

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄²

—

218

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

—

Q₈

—

218

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

218

—

103

C₂×C₄

—

218

—

104

C₂×C₄

D₄×D₁₀₉

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

106

C₂×C₄

#13182028

yes

characteristic

107

C₈

—

108

C₈

—

436

—

109

C₈

—

436

—

110

C₈

#13182017

yes

complemented

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#16812670

yes

characteristic

115

C₄

—

116

C₄

—

436

—

117

C₄

—

436

—

118

C₄

C₂×#13182017

yes

characteristic

119

C₄

#16812723

yes

—

120

C₄

#16812725

yes

characteristic

121

C₂

—

122

C₂

#20295223

yes

characteristic

123

C₂

#20295276

yes

characteristic

124

C₂

#20295293

yes

characteristic

	subgroup	G/H
center	C₂²	#16812670
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting, prime radical	#493×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#121
hypercenter	#493	D₁₀₉
hypocenter	C₁₀₉	#1688
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#16812670

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting, prime radical

#493×C₁₀₉

C₂

C₂²×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #20295223	C₂	semidirect
2. #20295156	C₂	semidirect
3. #20294777	C₂	semidirect
4. #20294770	C₂	semidirect
45. C₈×C₁₀₉	SD₁₆	semidirect
63. C₁₀₉	#1688	semidirect
110. C₈	#13182017	semidirect

H ≅ G/N

type

1. #20295223

C₂

semidirect

2. #20295156

C₂

semidirect

3. #20294777

C₂

semidirect

4. #20294770

C₂

semidirect

45. C₈×C₁₀₉

SD₁₆

semidirect

63. C₁₀₉

#1688

semidirect

110. C₈

#13182017

semidirect

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

subgroup	index
#20295223	2
#20295156	2
#20294777	2
#20294770	2
#493×C₁₀₉	2
#1688	109

subgroup

index

#493×C₁₀₉

109

Subgroups — #28362513

Conjugacy classes of subgroups

Complements

Maximal subgroups