Group #28363252 — subgroups

Subgroups — #28363252

Order 13952.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20295074	C₂	1	2	yes	maximal, complemented
2	#20295010	C₂	1	1	yes	maximal, characteristic
3	#20294993	C₂	1	1	yes	maximal, characteristic
4	#20294790	C₂	1	2	yes	maximal, complemented
5	#568×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	1	yes	characteristic
7	C₂×#13182013	C₂²	1	2	yes	—
8	C₂²×#11724784	C₂²	1	1	yes	characteristic
9	#16812634	—	2	4	no	—
10	#16812618	C₄	1	2	yes	—
11	#16812616	C₄	1	2	yes	—
12	C₂²⋊#11724784	—	2	2	no	—
13	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
14	#131×C₁₀₉	—	2	4	no	—
15	C₂×M₁₆×C₁₀₉	—	2	2	no	—
16	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
17	#13182013	—	2	2	no	—
18	#13182013	—	2	2	no	—
19	#13182013	C₂×C₄	1	4	yes	—
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	—	2	1	no	—
22	C₂×#11724784	—	2	1	no	—
23	C₂×#11724784	—	2	2	no	—
24	#41×C₁₀₉	—	2	2	no	—
25	#41×C₁₀₉	—	2	2	no	—
26	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×D₄×C₁₀₉	Q₈	1	2	yes	—
29	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
30	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	M₁₆×C₁₀₉	—	2	4	no	—
32	M₁₆×C₁₀₉	—	2	2	no	—
33	M₁₆×C₁₀₉	D₄	1	4	yes	—
34	C₂×C₈×C₁₀₉	—	2	4	no	—
35	C₂×C₈×C₁₀₉	—	2	2	no	—
36	C₂³×C₁₀₉	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₁₀₉	—	2	2	no	—
39	D₄×C₁₀₉	—	2	4	no	—
40	D₄×C₁₀₉	—	2	2	no	—
41	C₂×C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	#41	1	2	yes	—
48	C₈×C₁₀₉	—	2	4	no	—
49	#11724784	—	2	4	no	—
50	#11724784	—	4	1	no	—
51	#11724784	—	4	1	no	—
52	C₂²×C₁₀₉	—	2	2	no	—
53	C₂²×C₁₀₉	—	4	1	no	—
54	C₂²×C₁₀₉	—	4	2	no	—
55	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
57	C₄×C₁₀₉	—	2	2	no	—
58	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
59	C₄×C₁₀₉	#120	1	1	yes	characteristic
60	C₄×C₁₀₉	#123	1	1	yes	characteristic
61	C₂×C₁₀₉	—	2	1	no	—
62	C₂×C₁₀₉	—	2	2	no	—
63	C₂×C₁₀₉	—	4	2	no	—
64	C₂×C₁₀₉	#385	1	1	yes	characteristic
65	#1858	—	109	1	no	maximal, Sylow
66	C₁₀₉	#1858	1	1	yes	characteristic, complemented, Sylow
67	#568	D₁₀₉	1	1	yes	characteristic
68	#430	—	109	2	no	—
69	#413	—	109	1	no	—
70	#408	—	109	1	no	—
71	#341	—	109	2	no	—
72	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
73	#131	—	2	4	no	—
74	C₂×M₁₆	—	2	2	no	—
75	C₂×M₁₆	—	109	2	no	—
76	C₂×M₁₆	—	109	1	no	—
77	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
78	C₂²×C₈	—	109	1	no	—
79	#108	—	218	4	no	—
80	#101	—	109	2	no	—
81	#100	—	109	2	no	—
82	C₂²⋊C₈	—	218	2	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₁₀₉	1	1	yes	characteristic
86	C₂×D₄	Dic₂₁₈	1	2	yes	—
87	C₂×D₄	C₄×D₁₀₉	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
90	M₁₆	—	2	4	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	109	4	no	—
93	M₁₆	—	218	2	no	—
94	M₁₆	—	218	2	no	—
95	M₁₆	C₂²⋊D₁₀₉	1	4	yes	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	218	1	no	—
99	C₂×C₈	—	218	1	no	—
100	C₂×C₈	—	218	2	no	—
101	C₂×C₈	—	218	2	no	—
102	C₂³	#13182008	1	2	yes	—
103	C₂³	C₂×Dic₂₁₈	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	#13182008	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
113	C₂×C₄	#13182029	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	436	1	no	—
116	C₈	—	436	1	no	—
117	C₈	—	218	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×#13182008	1	1	yes	characteristic
122	C₂²	#16812678	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	C₄×C₂²⋊D₁₀₉	1	2	yes	—
125	C₄	#16812668	1	1	yes	characteristic
126	C₄	#16812682	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#20294899	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295074

C₂

yes

maximal, complemented

#20295010

C₂

yes

maximal, characteristic

#20294993

C₂

yes

maximal, characteristic

#20294790

C₂

yes

maximal, complemented

#568×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

characteristic

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

#16812634

—

#16812618

C₄

yes

—

#16812616

C₄

yes

—

C₂²⋊#11724784

—

C₂×#41×C₁₀₉

C₂²

yes

characteristic

#131×C₁₀₉

—

C₂×M₁₆×C₁₀₉

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

#13182013

—

#13182013

—

#13182013

C₂×C₄

yes

—

—

—

—

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂³×C₁₀₉

C₄⋊C₄

yes

—

C₂³×C₁₀₉

C₂×Q₈

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₈×C₁₀₉

—

#11724784

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#385

yes

characteristic

#1858

—

109

maximal, Sylow

C₁₀₉

#1858

yes

characteristic, complemented, Sylow

#568

D₁₀₉

yes

characteristic

#430

—

109

—

#413

—

109

—

#408

—

109

—

#341

—

109

—

C₂×#41

C₂×D₁₀₉

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

#108

—

218

—

#101

—

109

—

#100

—

109

—

C₂²⋊C₈

—

218

—

#41

—

#41

—

C₂×Q₈

C₄×D₁₀₉

yes

characteristic

C₂×D₄

Dic₂₁₈

yes

—

C₂×D₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

C₂²⋊D₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

100

C₂×C₈

—

218

—

101

C₂×C₈

—

218

—

102

C₂³

#13182008

yes

—

103

C₂³

C₂×Dic₂₁₈

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#13182008

yes

—

110

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

113

C₂×C₄

#13182029

yes

—

114

C₈

—

115

C₈

—

436

—

116

C₈

—

436

—

117

C₈

—

218

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×#13182008

yes

characteristic

122

C₂²

#16812678

yes

—

123

C₄

—

124

C₄

C₄×C₂²⋊D₁₀₉

yes

—

125

C₄

#16812668

yes

characteristic

126

C₄

#16812682

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#20294899

yes

characteristic

	subgroup	G/H
center	C₄	#16812668
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#568×C₁₀₉	C₂
socle	C₂×C₁₀₉	#385
prime radical	C₂×#41×C₁₀₉	C₂²
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#568	D₁₀₉
hypocenter	C₁₀₉	#1858
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₄

#16812668

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#568×C₁₀₉

C₂

socle

C₂×C₁₀₉

#385

prime radical

C₂×#41×C₁₀₉

C₂²

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
1. #20295074	C₂	semidirect
4. #20294790	C₂	semidirect
66. C₁₀₉	#1858	semidirect

H ≅ G/N

type

1. #20295074

C₂

semidirect

4. #20294790

C₂

semidirect

66. C₁₀₉

#1858

semidirect

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

subgroup	index
#20295074	2
#20295010	2
#20294993	2
#20294790	2
#568×C₁₀₉	2
#1858	109

subgroup

index

#568×C₁₀₉

109

Subgroups — #28363252

Conjugacy classes of subgroups

Complements

Maximal subgroups