Group #28362913 — subgroups

Subgroups — #28362913

Order 13952.

232 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 117 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	#20294982	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20294785	C₂	1	2	yes	maximal
5	#427×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
9	#16812635	C₄	1	4	yes	complemented
10	#16812610	—	2	2	no	—
11	C₄×#11724784	—	2	2	no	—
12	C₈×Dic₁₀₉	C₄	1	4	yes	complemented
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	#117×C₁₀₉	C₂²	1	2	yes	—
15	#107×C₁₀₉	C₄	1	2	yes	complemented
16	#106×C₁₀₉	C₄	1	2	yes	complemented
17	C₂²×Dic₁₀₉	—	2	1	no	—
18	#13182013	—	2	2	no	—
19	#13182013	C₂×C₄	1	4	yes	—
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	C₂×C₄	1	2	yes	—
22	C₂×#11724784	D₄	1	1	yes	characteristic
23	C₂×#11724784	D₄	1	1	yes	characteristic
24	C₂×#11724784	Q₈	1	2	yes	—
25	C₄×Dic₁₀₉	—	2	1	no	—
26	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
33	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
34	C₄²×C₁₀₉	—	2	2	no	—
35	C₂³×C₁₀₉	#41	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	—	2	4	no	—
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
43	C₂×Dic₁₀₉	—	2	4	no	—
44	C₂×Dic₁₀₉	—	2	2	no	—
45	C₈×C₁₀₉	C₄²	1	2	yes	—
46	C₈×C₁₀₉	C₄²	1	2	yes	—
47	#11724784	—	2	4	no	—
48	#11724784	C₄⋊C₄	1	2	yes	—
49	#11724784	C₄⋊C₄	1	2	yes	—
50	C₂²×C₁₀₉	—	2	2	no	—
51	C₂²×C₁₀₉	#117	1	1	yes	characteristic
52	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₀₉	—	4	4	no	—
55	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
56	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
58	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
59	Dic₁₀₉	—	2	4	no	—
60	C₂×C₁₀₉	—	2	1	no	—
61	C₂×C₁₀₉	—	2	1	no	—
62	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₁₀₉	#443	1	2	yes	—
64	#1772	—	109	1	no	maximal, Sylow
65	C₁₀₉	#1772	1	1	yes	characteristic, complemented, Sylow
66	C₂×#108	—	109	1	no	—
67	#427	D₁₀₉	1	1	yes	characteristic
68	#405	—	109	2	no	—
69	C₂×C₄×C₈	—	109	1	no	—
70	#339	—	109	2	no	—
71	C₂×M₁₆	—	109	2	no	—
72	C₂²×C₈	—	109	1	no	—
73	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
74	#117	C₂×D₁₀₉	1	2	yes	—
75	C₂×C₄²	—	109	1	no	—
76	#108	—	109	4	no	—
77	#107	Dic₁₀₉	1	2	yes	complemented
78	#106	Dic₁₀₉	1	2	yes	complemented
79	#97	—	218	2	no	—
80	C₄×C₈	—	109	4	no	—
81	C₄×C₈	—	218	2	no	—
82	C₂²×C₄	—	218	1	no	—
83	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
84	M₁₆	—	109	4	no	—
85	M₁₆	—	218	2	no	—
86	C₂×C₈	—	109	2	no	—
87	C₂×C₈	—	109	1	no	—
88	C₂×C₈	—	109	1	no	—
89	C₂×C₈	—	109	2	no	—
90	C₂×C₈	—	218	2	no	—
91	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
92	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
93	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
94	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
95	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	109	2	no	—
99	C₄²	—	218	1	no	—
100	C₂³	#13182028	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	218	4	no	—
103	C₂×C₄	—	218	2	no	—
104	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
105	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
107	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
108	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
109	C₂×C₄	#13182030	1	1	yes	characteristic
110	C₈	—	109	2	no	—
111	C₈	—	109	2	no	—
112	C₈	—	218	4	no	—
113	C₈	C₄×Dic₁₀₉	1	2	yes	—
114	C₈	C₄×Dic₁₀₉	1	2	yes	—
115	C₂²	—	2	2	no	—
116	C₂²	#16812657	1	1	yes	characteristic
117	C₂²	D₄×Dic₁₀₉	1	1	yes	characteristic
118	C₂²	#16812664	1	1	yes	characteristic
119	C₄	—	4	4	no	—
120	C₄	—	218	4	no	—
121	C₄	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
122	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
123	C₄	#16812661	1	1	yes	characteristic
124	C₄	Q₈×Dic₁₀₉	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₄⋊C₄×Dic₁₀₉	1	1	yes	characteristic
128	C₂	#20295143	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812635

C₂

yes

maximal, characteristic

#20294982

C₂

yes

maximal

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294785

C₂

yes

maximal

#427×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812635

C₄

yes

complemented

#16812610

—

C₄×#11724784

—

C₈×Dic₁₀₉

C₄

yes

complemented

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

—

#107×C₁₀₉

C₄

yes

complemented

#106×C₁₀₉

C₄

yes

complemented

C₂²×Dic₁₀₉

—

#13182013

—

#13182013

C₂×C₄

yes

—

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

D₄

yes

characteristic

C₂×#11724784

D₄

yes

characteristic

C₂×#11724784

Q₈

yes

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

#41

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

C₄²

yes

—

C₈×C₁₀₉

C₄²

yes

—

#11724784

—

#11724784

C₄⋊C₄

yes

—

#11724784

C₄⋊C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#117

yes

characteristic

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂²×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

#443

yes

—

#1772

—

109

maximal, Sylow

C₁₀₉

#1772

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

#427

D₁₀₉

yes

characteristic

#405

—

109

—

C₂×C₄×C₈

—

109

—

#339

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

#108

—

109

—

#107

Dic₁₀₉

yes

complemented

#106

Dic₁₀₉

yes

complemented

#97

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₂×Dic₁₀₉

yes

characteristic

C₂×C₈

C₂×Dic₁₀₉

yes

characteristic

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

109

—

C₄²

—

218

—

100

C₂³

#13182028

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

218

—

103

C₂×C₄

—

218

—

104

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

105

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

106

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

107

C₂×C₄

D₄×D₁₀₉

yes

characteristic

108

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

109

C₂×C₄

#13182030

yes

characteristic

110

C₈

—

109

—

111

C₈

—

109

—

112

C₈

—

218

—

113

C₈

C₄×Dic₁₀₉

yes

—

114

C₈

C₄×Dic₁₀₉

yes

—

115

C₂²

—

116

C₂²

#16812657

yes

characteristic

117

C₂²

D₄×Dic₁₀₉

yes

characteristic

118

C₂²

#16812664

yes

characteristic

119

C₄

—

120

C₄

—

218

—

121

C₄

C₂×C₄×Dic₁₀₉

yes

characteristic

122

C₄

C₄⋊C₄×D₁₀₉

yes

characteristic

123

C₄

#16812661

yes

characteristic

124

C₄

Q₈×Dic₁₀₉

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

C₄⋊C₄×Dic₁₀₉

yes

characteristic

128

C₂

#20295143

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#427×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₄×D₄
prime radical	C₂³×C₁₀₉	#41
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#427	D₁₀₉
hypocenter	C₁₀₉	#1772
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#427×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₄×D₄

prime radical

C₂³×C₁₀₉

#41

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
9. #16812635	C₄	semidirect
12. C₈×Dic₁₀₉	C₄	semidirect
15. #107×C₁₀₉	C₄	semidirect
16. #106×C₁₀₉	C₄	semidirect
65. C₁₀₉	#1772	semidirect
77. #107	Dic₁₀₉	semidirect
78. #106	Dic₁₀₉	semidirect

H ≅ G/N

type

9. #16812635

C₄

semidirect

12. C₈×Dic₁₀₉

C₄

semidirect

15. #107×C₁₀₉

C₄

semidirect

16. #106×C₁₀₉

C₄

semidirect

65. C₁₀₉

#1772

semidirect

77. #107

Dic₁₀₉

semidirect

78. #106

Dic₁₀₉

semidirect

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

subgroup	index
C₂×#16812635	2
#20294982	2
C₂×C₈×Dic₁₀₉	2
#20294785	2
#427×C₁₀₉	2
#1772	109

subgroup

index

C₂×C₈×Dic₁₀₉

#427×C₁₀₉

109

Subgroups — #28362913

Conjugacy classes of subgroups

Complements

Maximal subgroups