Group #28363155 — subgroups

Subgroups — #28363155

Order 13952.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×#16812635	C₂	1	1	yes	maximal, characteristic
2	#20294981	C₂	1	1	yes	maximal, characteristic
3	#20294796	C₂	1	2	yes	maximal
4	#20294782	C₂	1	2	yes	maximal
5	#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂×#13182013	C₂²	1	1	yes	characteristic
8	#16812658	C₂²	1	1	yes	characteristic
9	#16812635	—	2	1	no	—
10	#16812635	—	2	1	no	—
11	#16812609	C₄	1	2	yes	complemented
12	C₈×Dic₁₀₉	C₄	1	2	yes	complemented
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	#117×C₁₀₉	C₂²	1	2	yes	—
15	#107×C₁₀₉	—	2	1	no	—
16	#106×C₁₀₉	—	2	1	no	—
17	#13182013	—	2	1	no	—
18	#13182013	—	2	1	no	—
19	#13182013	—	2	2	no	—
20	#13182013	—	2	2	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₂×#11724784	C₂×C₄	1	2	yes	—
23	C₄⋊Dic₁₀₉	—	2	1	no	—
24	C₂²⋊Dic₁₀₉	—	2	1	no	—
25	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
32	C₂×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
33	C₄⋊C₄×C₁₀₉	—	2	2	no	—
34	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
35	C₄²×C₁₀₉	—	2	2	no	—
36	C₂³×C₁₀₉	#41	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	—	4	2	no	—
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
44	C₂×Dic₁₀₉	—	2	2	no	—
45	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
46	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
49	#11724784	—	2	2	no	—
50	#11724784	—	4	2	no	—
51	C₂²×C₁₀₉	—	4	1	no	—
52	C₂²×C₁₀₉	#117	1	1	yes	characteristic
53	C₂²×C₁₀₉	#124	1	1	yes	characteristic
54	C₂²×C₁₀₉	#125	1	1	yes	characteristic
55	C₄×C₁₀₉	—	8	2	no	—
56	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₁₀₉	#117	1	1	yes	characteristic
58	C₄×C₁₀₉	#124	1	1	yes	characteristic
59	C₄×C₁₀₉	#125	1	1	yes	characteristic
60	Dic₁₀₉	—	4	2	no	—
61	C₂×C₁₀₉	—	2	1	no	—
62	C₂×C₁₀₉	—	2	1	no	—
63	C₂×C₁₀₉	—	2	1	no	—
64	C₂×C₁₀₉	#382	1	1	yes	characteristic
65	#1833	—	109	1	no	maximal, Sylow
66	C₁₀₉	#1833	1	1	yes	characteristic, complemented, Sylow
67	C₂×#108	—	109	1	no	—
68	#427	D₁₀₉	1	1	yes	characteristic
69	#405	—	109	1	no	—
70	#344	—	109	2	no	—
71	H(3,ℤ/4)	—	109	2	no	—
72	C₂×M₁₆	—	109	1	no	—
73	C₂×M₁₆	—	109	2	no	—
74	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
75	#117	—	109	1	no	—
76	#117	C₂×D₁₀₉	1	2	yes	—
77	#108	—	218	1	no	—
78	#108	—	218	1	no	—
79	#107	—	2	1	no	—
80	#106	—	2	1	no	—
81	#97	—	109	2	no	—
82	C₄×C₈	—	109	2	no	—
83	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
84	M₁₆	—	218	2	no	—
85	M₁₆	—	218	2	no	—
86	M₁₆	—	218	1	no	—
87	M₁₆	—	218	1	no	—
88	C₂×C₈	—	109	2	no	—
89	C₂×C₈	—	218	2	no	—
90	C₂×C₈	Dic₂₁₈	1	1	yes	characteristic
91	C₂×C₈	Dic₂₁₈	1	1	yes	characteristic
92	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
93	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
94	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	218	1	no	—
98	C₂²⋊C₄	—	2	2	no	—
99	C₂²⋊C₄	—	218	1	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	109	2	no	—
102	C₂³	#13182028	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	218	2	no	—
105	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
108	C₂×C₄	#13182028	1	1	yes	characteristic
109	C₂×C₄	#13182030	1	1	yes	characteristic
110	C₂×C₄	#13182030	1	1	yes	characteristic
111	C₈	—	436	2	no	—
112	C₈	—	218	2	no	—
113	C₈	#13182008	1	1	yes	characteristic
114	C₈	#13182008	1	1	yes	characteristic
115	C₈	#13182008	1	1	yes	characteristic
116	C₈	#13182008	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#16812657	1	1	yes	characteristic
119	C₂²	#16812686	1	1	yes	characteristic
120	C₂²	#16812690	1	1	yes	characteristic
121	C₄	—	8	2	no	—
122	C₄	—	436	2	no	—
123	C₄	C₂×#13182008	1	1	yes	characteristic
124	C₄	#16812657	1	1	yes	characteristic
125	C₄	#16812688	1	1	yes	characteristic
126	C₄	#16812690	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#20294891	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812635

C₂

yes

maximal, characteristic

#20294981

C₂

yes

maximal, characteristic

#20294796

C₂

yes

maximal

#20294782

C₂

yes

maximal

#427×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂×#13182013

C₂²

yes

characteristic

#16812658

C₂²

yes

—

—

C₄

yes

complemented

C₈×Dic₁₀₉

C₄

yes

complemented

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

—

#107×C₁₀₉

—

#106×C₁₀₉

—

—

—

—

—

—

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

#41

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×Dic₁₀₉

—

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#382

yes

characteristic

#1833

—

109

maximal, Sylow

C₁₀₉

#1833

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

#427

D₁₀₉

yes

characteristic

#405

—

109

—

#344

—

109

—

H(3,ℤ/4)

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

—

109

—

#117

C₂×D₁₀₉

yes

—

#108

—

218

—

#108

—

218

—

#107

—

#106

—

#97

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

Dic₂₁₈

yes

characteristic

C₂×C₈

Dic₂₁₈

yes

characteristic

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

C₂²⋊C₄

—

218

—

100

C₄²

—

101

C₄²

—

109

—

102

C₂³

#13182028

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

218

—

105

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

107

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

108

C₂×C₄

#13182028

yes

characteristic

109

C₂×C₄

#13182030

yes

characteristic

110

C₂×C₄

#13182030

yes

characteristic

111

C₈

—

436

—

112

C₈

—

218

—

113

C₈

#13182008

yes

characteristic

114

C₈

#13182008

yes

characteristic

115

C₈

#13182008

yes

characteristic

116

C₈

#13182008

yes

characteristic

117

C₂²

—

118

C₂²

#16812657

yes

characteristic

119

C₂²

#16812686

yes

characteristic

120

C₂²

#16812690

yes

characteristic

121

C₄

—

122

C₄

—

436

—

123

C₄

C₂×#13182008

yes

characteristic

124

C₄

#16812657

yes

characteristic

125

C₄

#16812688

yes

characteristic

126

C₄

#16812690

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#20294891

yes

characteristic

	subgroup	G/H
center	C₄	#16812657
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#427×C₁₀₉	C₂
socle	C₂×C₁₀₉	#382
prime radical	C₂³×C₁₀₉	#41
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#427	D₁₀₉
hypocenter	C₁₀₉	#1833
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₄

#16812657

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#427×C₁₀₉

C₂

C₂×C₁₀₉

C₂³×C₁₀₉

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
11. #16812609	C₄	semidirect
12. C₈×Dic₁₀₉	C₄	semidirect
66. C₁₀₉	#1833	semidirect

H ≅ G/N

type

11. #16812609

C₄

semidirect

12. C₈×Dic₁₀₉

C₄

semidirect

66. C₁₀₉

#1833

semidirect

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

subgroup	index
C₂×#16812635	2
#20294981	2
#20294796	2
#20294782	2
#427×C₁₀₉	2
#1833	109

subgroup

index

#427×C₁₀₉

109

Subgroups — #28363155

Conjugacy classes of subgroups

Complements

Maximal subgroups