Group #28362940 — subgroups

Subgroups — #28362940

Order 13952.

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

normal subgroups only

This group has 220 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	M₁₆×Dic₁₀₉	C₂	1	2	yes	maximal
3	C₂×#16812609	C₂	1	1	yes	maximal, characteristic
4	#20294790	C₂	1	2	yes	maximal
5	C₂×#108×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	#16812609	—	2	2	no	—
11	#16812609	C₄	1	4	yes	—
12	C₈×Dic₁₀₉	—	2	2	no	—
13	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	#108×C₁₀₉	C₄	1	4	yes	complemented
16	C₂²×Dic₁₀₉	Q₈	1	2	yes	—
17	#13182013	—	2	2	no	—
18	#13182013	C₂×C₄	1	4	yes	—
19	C₂×#11724784	—	2	2	no	—
20	C₂×#11724784	—	2	1	no	—
21	C₂×#11724784	—	2	1	no	—
22	C₂×#11724784	C₂×C₄	1	2	yes	—
23	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₀₉	D₄	1	2	yes	—
25	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₀₉	—	2	2	no	—
27	M₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₁₀₉	—	2	2	no	—
29	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
32	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
39	C₂×Dic₁₀₉	—	4	2	no	—
40	C₂×Dic₁₀₉	C₄⋊C₄	1	4	yes	—
41	C₈×C₁₀₉	—	2	4	no	—
42	C₈×C₁₀₉	C₄²	1	4	yes	—
43	#11724784	—	2	2	no	—
44	#11724784	—	2	4	no	—
45	C₂²×C₁₀₉	—	2	2	no	—
46	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
48	C₂²×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
49	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₁₀₉	#117	1	1	yes	characteristic
51	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
52	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
53	Dic₁₀₉	—	2	4	no	—
54	C₂×C₁₀₉	—	2	1	no	—
55	C₂×C₁₀₉	—	2	1	no	—
56	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₁₀₉	#430	1	2	yes	—
58	#1778	—	109	1	no	maximal, Sylow
59	C₁₀₉	#1778	1	1	yes	characteristic, complemented, Sylow
60	C₂×#108	—	109	1	no	—
61	C₂×#108	D₁₀₉	1	1	yes	characteristic
62	C₄×M₁₆	—	109	2	no	—
63	C₂×#97	—	109	1	no	—
64	#341	—	109	2	no	—
65	C₂×M₁₆	—	109	2	no	—
66	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
67	C₂²×C₈	—	109	1	no	—
68	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
69	C₂×C₄²	—	109	1	no	—
70	#108	—	109	4	no	—
71	#108	Dic₁₀₉	1	4	yes	complemented
72	#97	—	109	4	no	—
73	#97	—	218	2	no	—
74	C₄×C₈	—	218	2	no	—
75	C₂²×C₄	—	109	2	no	—
76	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	109	4	no	—
79	M₁₆	—	218	2	no	—
80	M₁₆	C₂×Dic₁₀₉	1	4	yes	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	109	2	no	—
83	C₂×C₈	—	218	2	no	—
84	C₂×C₈	—	218	1	no	—
85	C₂×C₈	—	218	1	no	—
86	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
87	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
88	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
89	C₄²	—	109	2	no	—
90	C₄²	—	109	2	no	—
91	C₂³	Q₈×D₁₀₉	1	1	yes	characteristic
92	C₂×C₄	—	109	4	no	—
93	C₂×C₄	—	436	2	no	—
94	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
95	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
96	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
97	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
98	C₂×C₄	#13182028	1	1	yes	characteristic
99	C₂×C₄	#13182030	1	1	yes	characteristic
100	C₈	—	2	4	no	—
101	C₈	—	218	2	no	—
102	C₈	—	218	4	no	—
103	C₈	C₄×Dic₁₀₉	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
106	C₂²	#16812664	1	1	yes	characteristic
107	C₂²	Q₈×Dic₁₀₉	1	1	yes	characteristic
108	C₄	—	218	4	no	—
109	C₄	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
110	C₄	#16812657	1	1	yes	characteristic
111	C₄	#16812661	1	1	yes	characteristic
112	C₄	D₄×Dic₁₀₉	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	C₄⋊C₄×Dic₁₀₉	1	1	yes	characteristic
116	C₂	#20295073	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812635

C₂

yes

maximal, characteristic

M₁₆×Dic₁₀₉

C₂

yes

maximal

C₂×#16812609

C₂

yes

maximal, characteristic

#20294790

C₂

yes

maximal

C₂×#108×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812635

C₄

yes

complemented

#16812609

—

#16812609

C₄

yes

—

C₈×Dic₁₀₉

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#108×C₁₀₉

C₄

yes

complemented

C₂²×Dic₁₀₉

Q₈

yes

—

#13182013

—

#13182013

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₂×Q₈

yes

characteristic

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

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₄²

yes

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×Q₈

yes

characteristic

C₂²×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

#430

yes

—

#1778

—

109

maximal, Sylow

C₁₀₉

#1778

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

C₂×#108

D₁₀₉

yes

characteristic

C₄×M₁₆

—

109

—

C₂×#97

—

109

—

#341

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄²

—

109

—

#108

—

109

—

#108

Dic₁₀₉

yes

complemented

#97

—

109

—

#97

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

109

—

M₁₆

—

218

—

M₁₆

C₂×Dic₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

Q₈×D₁₀₉

yes

characteristic

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

C₂×C₄

D₄×D₁₀₉

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₈

—

101

C₈

—

218

—

102

C₈

—

218

—

103

C₈

C₄×Dic₁₀₉

yes

—

104

C₂²

—

105

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

106

C₂²

#16812664

yes

characteristic

107

C₂²

Q₈×Dic₁₀₉

yes

characteristic

108

C₄

—

218

—

109

C₄

C₂×C₄×Dic₁₀₉

yes

characteristic

110

C₄

#16812657

yes

characteristic

111

C₄

#16812661

yes

characteristic

112

C₄

D₄×Dic₁₀₉

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

C₄⋊C₄×Dic₁₀₉

yes

characteristic

116

C₂

#20295073

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#108×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂³×C₁₀₉	C₂×Q₈
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#108	D₁₀₉
hypocenter	C₁₀₉	#1778
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#108×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂³×C₁₀₉

C₂×Q₈

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
9. #16812635	C₄	semidirect
15. #108×C₁₀₉	C₄	semidirect
59. C₁₀₉	#1778	semidirect
71. #108	Dic₁₀₉	semidirect

H ≅ G/N

type

9. #16812635

C₄

semidirect

15. #108×C₁₀₉

C₄

semidirect

59. C₁₀₉

#1778

semidirect

71. #108

Dic₁₀₉

semidirect

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

subgroup	index
C₂×#16812635	2
M₁₆×Dic₁₀₉	2
C₂×#16812609	2
#20294790	2
C₂×#108×C₁₀₉	2
#1778	109

subgroup

index

M₁₆×Dic₁₀₉

C₂×#108×C₁₀₉

109

Subgroups — #28362940

Conjugacy classes of subgroups

Complements

Maximal subgroups