Group #28363781 — subgroups

Subgroups — #28363781

Order 13952.

208 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20295053	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	—	2	2	no	—
10	C₄⋊#11724784	—	2	4	no	—
11	C₈×Dic₁₀₉	—	2	2	no	—
12	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	#117×C₁₀₉	C₂²	1	2	yes	—
14	#107×C₁₀₉	C₄	1	2	yes	complemented
15	#106×C₁₀₉	C₄	1	2	yes	complemented
16	C₂²×Dic₁₀₉	—	2	1	no	—
17	#13182013	—	2	2	no	—
18	#13182013	—	2	2	no	—
19	C₂×#11724784	—	2	1	no	—
20	C₂×#11724784	—	2	1	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₂×#11724784	D₄	1	2	yes	—
23	C₄×Dic₁₀₉	—	2	1	no	—
24	C₄×Dic₁₀₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
29	C₂×C₈×C₁₀₉	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
32	C₄²×C₁₀₉	—	2	2	no	—
33	C₂³×C₁₀₉	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	—	2	4	no	—
35	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
41	C₂×Dic₁₀₉	—	2	4	no	—
42	C₂×Dic₁₀₉	—	2	2	no	—
43	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
44	C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
45	#11724784	—	2	2	no	—
46	#11724784	—	4	2	no	—
47	C₂²×C₁₀₉	—	2	2	no	—
48	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₀₉	#125	1	1	yes	characteristic
51	C₄×C₁₀₉	—	4	4	no	—
52	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₀₉	#121	1	1	yes	characteristic
55	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₀₉	—	2	4	no	—
57	C₂×C₁₀₉	—	2	1	no	—
58	C₂×C₁₀₉	—	2	1	no	—
59	C₂×C₁₀₉	#389	1	1	yes	characteristic
60	C₂×C₁₀₉	#443	1	2	yes	—
61	#1945	—	109	1	no	maximal, Sylow
62	C₁₀₉	#1945	1	1	yes	characteristic, complemented, Sylow
63	C₂×#108	—	109	1	no	—
64	#427	D₁₀₉	1	1	yes	characteristic
65	#424	—	109	2	no	—
66	C₂×C₄×C₈	—	109	1	no	—
67	#339	—	109	2	no	—
68	C₂×M₁₆	—	109	2	no	—
69	C₂²×C₈	—	109	1	no	—
70	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
71	#117	C₂×D₁₀₉	1	2	yes	—
72	C₂×C₄²	—	109	1	no	—
73	#108	—	218	2	no	—
74	#107	Dic₁₀₉	1	2	yes	complemented
75	#106	Dic₁₀₉	1	2	yes	complemented
76	C₄⋊C₈	—	218	4	no	—
77	C₄×C₈	—	218	2	no	—
78	C₂²×C₄	—	218	1	no	—
79	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
80	M₁₆	—	218	2	no	—
81	M₁₆	—	218	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	Dic₂₁₈	1	2	yes	—
87	C₂×C₈	D₄₃₆	1	2	yes	—
88	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
89	C₂×C₈	C₂×Dic₁₀₉	1	1	yes	characteristic
90	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	109	2	no	—
94	C₄²	—	218	1	no	—
95	C₂³	#13182028	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	218	4	no	—
98	C₂×C₄	—	218	2	no	—
99	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
101	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
104	C₂×C₄	#13182030	1	1	yes	characteristic
105	C₈	—	436	2	no	—
106	C₈	—	218	2	no	—
107	C₈	C₄⋊Dic₁₀₉	1	2	yes	—
108	C₈	C₄⋊Dic₁₀₉	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₁₀₉	1	1	yes	characteristic
111	C₂²	#16812679	1	1	yes	characteristic
112	C₂²	#16812690	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	218	4	no	—
115	C₄	C₂×C₄⋊Dic₁₀₉	1	1	yes	characteristic
116	C₄	Q₈×Dic₁₀₉	1	1	yes	characteristic
117	C₄	#16812674	1	1	yes	characteristic
118	C₄	#16812698	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#20294925	1	1	yes	characteristic
122	C₂	#20295143	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812635

C₂

yes

maximal, characteristic

#20295053

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⋊#11724784

—

C₈×Dic₁₀₉

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

—

#107×C₁₀₉

C₄

yes

complemented

#106×C₁₀₉

C₄

yes

complemented

C₂²×Dic₁₀₉

—

—

—

—

—

—

D₄

yes

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

Q₈

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

—

C₂×C₈×C₁₀₉

Q₈

yes

—

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

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₄⋊C₄

yes

—

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₀₉

#125

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

#121

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#389

yes

characteristic

C₂×C₁₀₉

#443

yes

—

#1945

—

109

maximal, Sylow

C₁₀₉

#1945

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

#427

D₁₀₉

yes

characteristic

#424

—

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

—

218

—

#107

Dic₁₀₉

yes

complemented

#106

Dic₁₀₉

yes

complemented

C₄⋊C₈

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

D₄₃₆

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

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

100

C₂×C₄

D₄×D₁₀₉

yes

characteristic

101

C₂×C₄

C₂×D₄₃₆

yes

characteristic

102

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

103

C₂×C₄

C₂×Dic₂₁₈

yes

characteristic

104

C₂×C₄

#13182030

yes

characteristic

105

C₈

—

436

—

106

C₈

—

218

—

107

C₈

C₄⋊Dic₁₀₉

yes

—

108

C₈

C₄⋊Dic₁₀₉

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₁₀₉

yes

characteristic

111

C₂²

#16812679

yes

characteristic

112

C₂²

#16812690

yes

characteristic

113

C₄

—

114

C₄

—

218

—

115

C₄

C₂×C₄⋊Dic₁₀₉

yes

characteristic

116

C₄

Q₈×Dic₁₀₉

yes

characteristic

117

C₄

#16812674

yes

characteristic

118

C₄

#16812698

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#20294925

yes

characteristic

122

C₂

#20295143

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
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	#1945
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

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 ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₁₀₉	C₄	semidirect
15. #106×C₁₀₉	C₄	semidirect
62. C₁₀₉	#1945	semidirect
74. #107	Dic₁₀₉	semidirect
75. #106	Dic₁₀₉	semidirect

H ≅ G/N

type

14. #107×C₁₀₉

C₄

semidirect

15. #106×C₁₀₉

C₄

semidirect

62. C₁₀₉

#1945

semidirect

74. #107

Dic₁₀₉

semidirect

75. #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
#20295053	2
C₂×C₈×Dic₁₀₉	2
#20294785	2
#427×C₁₀₉	2
#1945	109

subgroup

index

C₂×C₈×Dic₁₀₉

#427×C₁₀₉

109

Subgroups — #28363781

Conjugacy classes of subgroups

Complements

Maximal subgroups