Group #28365276 — subgroups

Subgroups — #28365276

Order 13952.

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

normal subgroups only

This group has 132 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	#20295427	C₂	1	1	yes	maximal, characteristic
2	#20295404	C₂	1	1	yes	maximal, characteristic
3	#20294831	C₂	1	1	yes	maximal, characteristic
4	#20294830	C₂	1	1	yes	maximal, characteristic
5	#20294829	C₂	1	1	yes	maximal, characteristic
6	#20294801	C₂	1	1	yes	maximal, characteristic
7	#358×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
9	C₂×#13182019	—	2	1	no	—
10	#16812697	—	2	1	no	—
11	#16812690	—	2	1	no	—
12	C₂×#13182004	C₂²	1	1	yes	characteristic
13	#16812633	C₂²	1	1	yes	characteristic
14	#16812630	—	2	1	no	—
15	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
18	#107×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₁₈	—	2	1	no	—
20	Dic₄₃₆	—	2	1	no	—
21	Dic₄₃₆	—	2	1	no	—
22	#13182019	—	4	2	no	—
23	C₂×#11724784	D₄	1	1	yes	characteristic
24	C₄⋊Dic₁₀₉	—	2	1	no	—
25	C₄⋊Dic₁₀₉	—	4	1	no	—
26	#13182008	—	4	1	no	—
27	#13182008	—	4	1	no	—
28	#13182008	—	4	1	no	—
29	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
30	#13182004	—	2	1	no	—
31	C₂×Q₈×C₁₀₉	—	2	1	no	—
32	Q₁₆×C₁₀₉	—	2	1	no	—
33	Q₁₆×C₁₀₉	—	2	1	no	—
34	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₀₉	—	2	1	no	—
36	C₁₆×C₁₀₉	—	2	1	no	—
37	Q₈×C₁₀₉	—	4	1	no	—
38	Q₈×C₁₀₉	—	4	1	no	—
39	C₂×C₄×C₁₀₉	—	4	1	no	—
40	C₂×C₄×C₁₀₉	—	4	1	no	—
41	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₁₀₉	—	4	1	no	—
43	C₂×Dic₁₀₉	—	4	1	no	—
44	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
45	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
46	Dic₂₁₈	—	4	1	no	—
47	Dic₂₁₈	—	4	1	no	—
48	C₈×C₁₀₉	#41	1	1	yes	characteristic
49	C₈×C₁₀₉	#41	1	1	yes	characteristic
50	#11724784	SD₁₆	1	2	yes	—
51	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
52	C₄×C₁₀₉	—	8	1	no	—
53	C₄×C₁₀₉	—	8	1	no	—
54	C₄×C₁₀₉	#124	1	1	yes	characteristic
55	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
56	Dic₁₀₉	—	2	1	no	—
57	Dic₁₀₉	—	2	1	no	—
58	Dic₁₀₉	—	8	1	no	—
59	Dic₁₀₉	—	8	1	no	—
60	C₂×C₁₀₉	#486	1	1	yes	characteristic
61	C₂×C₁₀₉	#510	1	1	yes	characteristic
62	C₂×C₁₀₉	#510	1	1	yes	characteristic
63	#2245	—	109	1	no	maximal, Sylow
64	C₁₀₉	#2245	1	1	yes	characteristic, complemented, Sylow
65	#499	—	109	1	no	—
66	#494	—	109	1	no	—
67	#358	—	109	1	no	—
68	#358	—	109	1	no	—
69	#358	—	109	1	no	—
70	#358	D₁₀₉	1	1	yes	characteristic
71	#346	—	109	1	no	—
72	C₂×Q₁₆	—	109	1	no	—
73	C₂×Q₁₆	—	218	1	no	—
74	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
75	C₄⋊Q₈	—	218	1	no	—
76	#125	—	218	1	no	—
77	C₂×C₁₆	—	109	1	no	—
78	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
79	#107	—	109	1	no	—
80	#107	C₂×D₁₀₉	1	1	yes	characteristic
81	#106	—	218	1	no	—
82	C₄×C₈	—	109	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	218	1	no	—
85	Q₁₆	—	2	1	no	—
86	Q₁₆	—	2	1	no	—
87	Q₁₆	—	436	2	no	—
88	Q₁₆	—	218	1	no	—
89	Q₁₆	—	218	1	no	—
90	C₂×C₈	—	109	1	no	—
91	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	436	1	no	—
94	C₄⋊C₄	—	436	1	no	—
95	C₄⋊C₄	—	436	1	no	—
96	C₄⋊C₄	—	436	1	no	—
97	C₄⋊C₄	—	218	1	no	—
98	C₄²	—	109	1	no	—
99	C₁₆	—	2	1	no	—
100	C₁₆	—	218	1	no	—
101	Q₈	—	4	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	436	1	no	—
104	Q₈	—	436	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	109	1	no	—
108	C₂×C₄	—	109	1	no	—
109	C₂×C₄	—	436	1	no	—
110	C₂×C₄	—	436	1	no	—
111	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
112	C₈	—	109	2	no	—
113	C₈	#13182028	1	1	yes	characteristic
114	C₈	#13182029	1	1	yes	characteristic
115	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	8	1	no	—
118	C₄	—	872	1	no	—
119	C₄	—	872	1	no	—
120	C₄	—	218	1	no	—
121	C₄	—	218	1	no	—
122	C₄	#16812685	1	1	yes	characteristic
123	C₄	SD₁₆×D₁₀₉	1	1	yes	characteristic
124	C₂	#20295371	1	1	yes	characteristic
125	C₂	#20295465	1	1	yes	characteristic
126	C₂	#20295468	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295427

C₂

yes

maximal, characteristic

#20295404

C₂

yes

maximal, characteristic

#20294831

C₂

yes

maximal, characteristic

#20294830

C₂

yes

maximal, characteristic

#20294829

C₂

yes

maximal, characteristic

#20294801

C₂

yes

maximal, characteristic

#358×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₄₃₆

C₂²

yes

—

—

—

C₂²

yes

characteristic

#16812633

C₂²

yes

characteristic

#16812630

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

Dic₄₃₆

—

Dic₄₃₆

—

#13182019

—

C₂×#11724784

D₄

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

#13182008

—

#13182008

—

#13182008

—

C₄×Dic₁₀₉

D₄

yes

characteristic

#13182004

—

C₂×Q₈×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₁₆×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₈

yes

characteristic

C₂×Dic₁₀₉

D₈

yes

characteristic

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

yes

C₈×C₁₀₉

yes

SD₁₆

yes

—

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#124

yes

characteristic

C₄×C₁₀₉

C₂×SD₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2245

yes

characteristic, complemented, Sylow

#499

—

109

—

#494

—

109

—

#358

—

109

—

#358

—

109

—

#358

—

109

—

#358

D₁₀₉

yes

characteristic

#346

—

109

—

C₂×Q₁₆

—

109

—

C₂×Q₁₆

—

218

—

C₂×Q₁₆

C₂×D₁₀₉

yes

characteristic

C₄⋊Q₈

—

218

—

#125

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

#107

—

109

—

#107

C₂×D₁₀₉

yes

characteristic

#106

—

218

—

C₄×C₈

—

109

—

C₂×Q₈

—

C₂×Q₈

—

218

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

436

—

Q₁₆

—

218

—

Q₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

218

—

C₄²

—

109

—

C₁₆

—

100

C₁₆

—

218

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

436

—

104

Q₈

—

436

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

109

—

108

C₂×C₄

—

109

—

109

C₂×C₄

—

436

—

110

C₂×C₄

—

436

—

111

C₂×C₄

D₄×D₁₀₉

yes

characteristic

112

C₈

—

109

—

113

C₈

#13182028

yes

characteristic

114

C₈

#13182029

yes

characteristic

115

C₂²

D₈×D₁₀₉

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

872

—

119

C₄

—

872

—

120

C₄

—

218

—

121

C₄

—

218

—

122

C₄

#16812685

yes

characteristic

123

C₄

SD₁₆×D₁₀₉

yes

characteristic

124

C₂

#20295371

yes

characteristic

125

C₂

#20295465

yes

characteristic

126

C₂

#20295468

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₉
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#358×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂×D₈
hypercenter	#358	D₁₀₉
hypocenter	C₁₀₉	#2245
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₀₉

commutator, cosocle

C₂×C₈×C₁₀₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#358×C₁₀₉

C₂

socle, prime radical

C₂²×C₁₀₉

C₂×D₈

D₁₀₉

C₁₀₉

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
64. C₁₀₉	#2245	semidirect

H ≅ G/N

type

64. C₁₀₉

#2245

semidirect

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

subgroup	index
#20295427	2
#20295404	2
#20294831	2
#20294830	2
#20294829	2
#20294801	2
#358×C₁₀₉	2
#2245	109

subgroup

index

#358×C₁₀₉

109

Subgroups — #28365276

Conjugacy classes of subgroups

Complements

Maximal subgroups