Group #28365173 — subgroups

Subgroups — #28365173

Order 13952.

130 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 49 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	#20295430	C₂	1	1	yes	maximal, characteristic
2	#20295120	C₂	1	1	yes	maximal, characteristic
3	#20294853	C₂	1	1	yes	maximal, characteristic
4	#20294844	C₂	1	1	yes	maximal, characteristic
5	#20294831	C₂	1	1	yes	maximal, characteristic
6	#20294829	C₂	1	1	yes	maximal, characteristic
7	#367×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₃₆	C₂²	1	1	yes	characteristic
9	#16812698	—	2	1	no	—
10	#16812664	—	2	1	no	—
11	C₂×#13182004	C₂²	1	1	yes	characteristic
12	#16812633	C₂²	1	1	yes	characteristic
13	#16812632	—	2	1	no	—
14	#16812622	—	2	1	no	—
15	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
17	#107×C₁₀₉	C₂²	1	1	yes	characteristic
18	#107×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₁₈	—	2	1	no	—
20	C₂×Dic₂₁₈	—	4	1	no	—
21	Dic₄₃₆	—	2	1	no	—
22	Dic₄₃₆	Q₈	1	2	yes	complemented
23	C₂×#11724784	D₄	1	1	yes	characteristic
24	C₄⋊Dic₁₀₉	—	2	1	no	—
25	#13182008	—	4	1	no	—
26	#13182008	—	4	1	no	—
27	C₄×Dic₁₀₉	—	4	1	no	—
28	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
29	#13182004	—	2	1	no	—
30	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₁₀₉	—	2	1	no	—
32	C₄⋊C₄×C₁₀₉	—	2	1	no	—
33	C₁₆×C₁₀₉	—	2	1	no	—
34	C₂×C₄×C₁₀₉	—	4	1	no	—
35	C₂×C₄×C₁₀₉	—	4	1	no	—
36	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₁₀₉	—	4	1	no	—
38	C₂×Dic₁₀₉	—	4	1	no	—
39	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
40	C₂×Dic₁₀₉	D₈	1	1	yes	characteristic
41	Dic₂₁₈	—	2	2	no	—
42	Dic₂₁₈	—	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	—	2	1	no	—
47	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
48	C₄×C₁₀₉	—	8	1	no	—
49	C₄×C₁₀₉	—	8	1	no	—
50	C₄×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
51	C₄×C₁₀₉	#137	1	1	yes	characteristic
52	Dic₁₀₉	—	2	1	no	—
53	Dic₁₀₉	—	4	2	no	—
54	Dic₁₀₉	—	8	1	no	—
55	Dic₁₀₉	SD₃₂	1	2	yes	—
56	C₂×C₁₀₉	#474	1	1	yes	characteristic
57	C₂×C₁₀₉	C₂×SD₃₂	1	1	yes	characteristic
58	C₂×C₁₀₉	#510	1	1	yes	characteristic
59	#2225	—	109	1	no	maximal, Sylow
60	C₁₀₉	#2225	1	1	yes	characteristic, complemented, Sylow
61	#500	—	109	1	no	—
62	C₄×Q₁₆	—	109	1	no	—
63	#367	—	109	1	no	—
64	#367	D₁₀₉	1	1	yes	characteristic
65	C₄⋊C₁₆	—	109	1	no	—
66	#358	—	109	1	no	—
67	#358	—	109	1	no	—
68	C₂×Q₁₆	—	109	1	no	—
69	C₄⋊Q₈	—	218	1	no	—
70	C₄×Q₈	—	218	1	no	—
71	C₂×C₁₆	—	109	1	no	—
72	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
73	#107	—	109	1	no	—
74	#107	—	218	1	no	—
75	#107	C₂×D₁₀₉	1	1	yes	characteristic
76	#107	C₂×D₁₀₉	1	1	yes	characteristic
77	#103	—	218	1	no	—
78	C₄×C₈	—	109	1	no	—
79	C₂×Q₈	—	436	1	no	—
80	C₂×Q₈	—	218	1	no	—
81	Q₁₆	—	109	2	no	—
82	Q₁₆	—	218	1	no	—
83	C₂×C₈	—	109	1	no	—
84	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	2	1	no	—
87	C₄⋊C₄	—	436	1	no	—
88	C₄⋊C₄	—	436	1	no	—
89	C₄⋊C₄	—	218	1	no	—
90	C₄²	—	109	1	no	—
91	C₄²	—	436	1	no	—
92	C₁₆	—	2	1	no	—
93	C₁₆	—	218	1	no	—
94	Q₈	—	872	2	no	—
95	Q₈	—	436	1	no	—
96	Q₈	—	218	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	109	1	no	—
100	C₂×C₄	—	109	1	no	—
101	C₂×C₄	—	436	1	no	—
102	C₂×C₄	—	436	1	no	—
103	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
104	C₈	—	218	1	no	—
105	C₈	Q₈×D₁₀₉	1	1	yes	characteristic
106	C₈	#13182029	1	1	yes	characteristic
107	C₂²	D₈×D₁₀₉	1	1	yes	characteristic
108	C₄	—	8	1	no	—
109	C₄	—	8	1	no	—
110	C₄	—	872	1	no	—
111	C₄	—	109	2	no	—
112	C₄	—	436	2	no	—
113	C₄	—	218	1	no	—
114	C₄	#16812677	1	1	yes	characteristic
115	C₄	#16812731	1	1	yes	characteristic
116	C₂	#20295305	1	1	yes	characteristic
117	C₂	SD₃₂×D₁₀₉	1	1	yes	characteristic
118	C₂	#20295465	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295430

C₂

yes

maximal, characteristic

#20295120

C₂

yes

maximal, characteristic

#20294853

C₂

yes

maximal, characteristic

#20294844

C₂

yes

maximal, characteristic

#20294831

C₂

yes

maximal, characteristic

#20294829

C₂

yes

maximal, characteristic

#367×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₄₃₆

C₂²

yes

—

—

C₂²

yes

characteristic

#16812633

C₂²

yes

characteristic

#16812632

—

#16812622

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

Dic₄₃₆

—

Dic₄₃₆

Q₈

yes

complemented

C₂×#11724784

D₄

yes

characteristic

C₄⋊Dic₁₀₉

—

#13182008

—

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

characteristic

#13182004

—

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₁₆×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

—

Dic₂₁₈

—

C₈×C₁₀₉

C₂×Q₈

yes

C₈×C₁₀₉

yes

—

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₀₉

#137

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

SD₃₂

yes

—

C₂×C₁₀₉

#474

yes

characteristic

C₂×C₁₀₉

C₂×SD₃₂

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2225

yes

characteristic, complemented, Sylow

#500

—

109

—

C₄×Q₁₆

—

109

—

#367

—

109

—

#367

D₁₀₉

yes

characteristic

C₄⋊C₁₆

—

109

—

#358

—

109

—

#358

—

109

—

C₂×Q₁₆

—

109

—

C₄⋊Q₈

—

218

—

C₄×Q₈

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

#107

—

109

—

#107

—

218

—

#107

C₂×D₁₀₉

yes

characteristic

#107

C₂×D₁₀₉

yes

characteristic

#103

—

218

—

C₄×C₈

—

109

—

C₂×Q₈

—

436

—

C₂×Q₈

—

218

—

Q₁₆

—

109

—

Q₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

436

—

C₄⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

436

—

C₁₆

—

C₁₆

—

218

—

Q₈

—

872

—

Q₈

—

436

—

Q₈

—

218

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

436

—

102

C₂×C₄

—

436

—

103

C₂×C₄

D₄×D₁₀₉

yes

characteristic

104

C₈

—

218

—

105

C₈

Q₈×D₁₀₉

yes

characteristic

106

C₈

#13182029

yes

characteristic

107

C₂²

D₈×D₁₀₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

872

—

111

C₄

—

109

—

112

C₄

—

436

—

113

C₄

—

218

—

114

C₄

#16812677

yes

characteristic

115

C₄

#16812731

yes

characteristic

116

C₂

#20295305

yes

characteristic

117

C₂

SD₃₂×D₁₀₉

yes

characteristic

118

C₂

#20295465

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₀₉
commutator, cosocle	C₂×C₈×C₁₀₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₀₉
Fitting	#367×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₂×D₈
hypercenter	#367	D₁₀₉
hypocenter	C₁₀₉	#2225
LCS	G ▷ C₂×C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

D₈×D₁₀₉

commutator, cosocle

C₂×C₈×C₁₀₉

C₂³

Frattini

C₂×C₈

C₂²×D₁₀₉

Fitting

#367×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 ◁ #367

N	H ≅ G/N	type
22. Dic₄₃₆	Q₈	semidirect
60. C₁₀₉	#2225	semidirect

H ≅ G/N

type

22. Dic₄₃₆

Q₈

semidirect

60. C₁₀₉

#2225

semidirect

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

subgroup	index
#20295430	2
#20295120	2
#20294853	2
#20294844	2
#20294831	2
#20294829	2
#367×C₁₀₉	2
#2225	109

subgroup

index

#367×C₁₀₉

109

Subgroups — #28365173

Conjugacy classes of subgroups

Complements

Maximal subgroups