Group #28362918 — subgroups

Subgroups — #28362918

Order 13952.

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

normal subgroups only

This group has 224 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×#16812609	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	#16812609	C₄	1	4	yes	complemented
12	C₄×#11724784	—	2	2	no	—
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	2	yes	—
23	C₂×#11724784	Q₈	1	2	yes	—
24	C₄×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	2	yes	—
28	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
32	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
33	C₄²×C₁₀₉	—	2	2	no	—
34	C₂³×C₁₀₉	#41	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	—	2	4	no	—
36	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
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×Dic₁₀₉	—	2	2	no	—
43	C₂×Dic₁₀₉	—	4	2	no	—
44	C₈×C₁₀₉	C₄²	1	2	yes	—
45	C₈×C₁₀₉	C₄²	1	2	yes	—
46	#11724784	—	2	4	no	—
47	#11724784	C₄⋊C₄	1	4	yes	—
48	C₂²×C₁₀₉	—	2	2	no	—
49	C₂²×C₁₀₉	#117	1	1	yes	characteristic
50	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
51	C₂²×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
52	C₄×C₁₀₉	—	4	4	no	—
53	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
54	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
56	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
57	Dic₁₀₉	—	4	2	no	—
58	C₂×C₁₀₉	—	2	1	no	—
59	C₂×C₁₀₉	—	2	1	no	—
60	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₁₀₉	#444	1	2	yes	—
62	#1773	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1773	1	1	yes	characteristic, complemented, Sylow
64	C₂×#108	—	109	1	no	—
65	#427	D₁₀₉	1	1	yes	characteristic
66	#405	—	109	2	no	—
67	C₂×#97	—	109	1	no	—
68	#339	—	109	2	no	—
69	C₂×M₁₆	—	109	2	no	—
70	C₂²×C₈	—	109	1	no	—
71	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
72	#117	C₂×D₁₀₉	1	2	yes	—
73	C₂×C₄²	—	109	1	no	—
74	#108	—	109	4	no	—
75	#107	Dic₁₀₉	1	2	yes	complemented
76	#106	Dic₁₀₉	1	2	yes	complemented
77	#97	—	109	4	no	—
78	#97	—	218	2	no	—
79	C₄×C₈	—	218	2	no	—
80	C₂²×C₄	—	218	1	no	—
81	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
82	M₁₆	—	109	4	no	—
83	M₁₆	—	218	2	no	—
84	C₂×C₈	—	109	2	no	—
85	C₂×C₈	—	109	2	no	—
86	C₂×C₈	—	109	2	no	—
87	C₂×C₈	—	218	2	no	—
88	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
89	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
90	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
91	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
92	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	—	2	2	no	—
95	C₄²	—	109	2	no	—
96	C₄²	—	218	1	no	—
97	C₂³	#13182028	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	436	2	no	—
100	C₂×C₄	—	218	2	no	—
101	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
102	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
103	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
104	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
106	C₂×C₄	#13182030	1	1	yes	characteristic
107	C₈	—	109	4	no	—
108	C₈	—	218	4	no	—
109	C₈	C₄×Dic₁₀₉	1	2	yes	—
110	C₈	C₄×Dic₁₀₉	1	2	yes	—
111	C₂²	—	2	2	no	—
112	C₂²	#16812657	1	1	yes	characteristic
113	C₂²	D₄×Dic₁₀₉	1	1	yes	characteristic
114	C₂²	#16812664	1	1	yes	characteristic
115	C₄	—	4	4	no	—
116	C₄	—	436	2	no	—
117	C₄	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
118	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
119	C₄	#16812661	1	1	yes	characteristic
120	C₄	Q₈×Dic₁₀₉	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	2	1	no	—
123	C₂	C₄⋊C₄×Dic₁₀₉	1	1	yes	characteristic
124	C₂	#20295148	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812635

C₂

yes

maximal, characteristic

#20294982

C₂

yes

maximal

C₂×#16812609

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

—

#16812609

C₄

yes

complemented

C₄×#11724784

—

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

—

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

—

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

#444

yes

—

#1773

—

109

maximal, Sylow

C₁₀₉

#1773

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

#427

D₁₀₉

yes

characteristic

#405

—

109

—

C₂×#97

—

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

—

109

—

#97

—

218

—

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

—

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

—

C₂³

#13182028

yes

characteristic

C₂×C₄

—

C₂×C₄

—

436

—

100

C₂×C₄

—

218

—

101

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

102

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

103

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

104

C₂×C₄

D₄×D₁₀₉

yes

characteristic

105

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

106

C₂×C₄

#13182030

yes

characteristic

107

C₈

—

109

—

108

C₈

—

218

—

109

C₈

C₄×Dic₁₀₉

yes

—

110

C₈

C₄×Dic₁₀₉

yes

—

111

C₂²

—

112

C₂²

#16812657

yes

characteristic

113

C₂²

D₄×Dic₁₀₉

yes

characteristic

114

C₂²

#16812664

yes

characteristic

115

C₄

—

116

C₄

—

436

—

117

C₄

C₂×C₄×Dic₁₀₉

yes

characteristic

118

C₄

C₄⋊C₄×D₁₀₉

yes

characteristic

119

C₄

#16812661

yes

characteristic

120

C₄

Q₈×Dic₁₀₉

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

C₄⋊C₄×Dic₁₀₉

yes

characteristic

124

C₂

#20295148

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	#1773
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
11. #16812609	C₄	semidirect
15. #107×C₁₀₉	C₄	semidirect
16. #106×C₁₀₉	C₄	semidirect
63. C₁₀₉	#1773	semidirect
75. #107	Dic₁₀₉	semidirect
76. #106	Dic₁₀₉	semidirect

H ≅ G/N

type

9. #16812635

C₄

semidirect

11. #16812609

C₄

semidirect

15. #107×C₁₀₉

C₄

semidirect

16. #106×C₁₀₉

C₄

semidirect

63. C₁₀₉

#1773

semidirect

75. #107

Dic₁₀₉

semidirect

76. #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×#16812609	2
#20294785	2
#427×C₁₀₉	2
#1773	109

subgroup

index

#427×C₁₀₉

109

Subgroups — #28362918

Conjugacy classes of subgroups

Complements

Maximal subgroups