Group #28378097 — subgroups

Subgroups — #28378097

Order 13952.

136 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 52 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×#16812769	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812767	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812638	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182039	—	2	1	no	—
5	C₂×C₄⋊#89137	—	2	1	no	—
6	#16812769	—	2	1	no	—
7	#16812769	C₄	1	2	yes	complemented
8	#16812767	—	2	1	no	—
9	#16812767	C₄	1	2	yes	—
10	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
11	#16812638	—	2	1	no	—
12	#16812638	—	2	1	no	—
13	C₂×#13182004	C₄	1	1	yes	characteristic, complemented
14	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
15	C₂²×#89137	—	4	1	no	—
16	#13182039	—	2	2	no	—
17	#13182039	—	4	1	no	—
18	C₂×#11724786	—	4	1	no	—
19	C₄⋊#89137	—	2	2	no	—
20	C₄⋊#89137	—	4	1	no	—
21	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
22	C₂×#11724784	Q₈	1	1	yes	characteristic
23	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₈×D₁₀₉	D₄	1	1	yes	characteristic
26	C₈×D₁₀₉	D₄	1	1	yes	characteristic
27	#13182004	—	2	1	no	—
28	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₁₆×C₁₀₉	—	2	1	no	—
30	C₂²×D₁₀₉	SD₁₆	1	1	yes	characteristic
31	C₂×#89137	—	4	2	no	—
32	C₂×#89137	—	8	1	no	—
33	C₂×#89137	—	8	1	no	—
34	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	SD₁₆	1	1	yes	characteristic
36	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
37	C₄×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×D₁₀₉	D₈	1	1	yes	characteristic
39	C₄×D₁₀₉	Q₁₆	1	1	yes	characteristic
40	#11724786	—	4	2	no	—
41	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
42	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
43	#11724784	C₄⋊C₄	1	1	yes	characteristic
44	#11724784	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₀₉	#106	1	1	yes	characteristic
46	C₂×D₁₀₉	—	2	1	no	—
47	C₂×D₁₀₉	—	2	1	no	—
48	C₂×D₁₀₉	#102	1	1	yes	characteristic
49	C₂×D₁₀₉	#103	1	1	yes	characteristic
50	#89137	—	8	2	no	—
51	C₄×C₁₀₉	#95	1	1	yes	characteristic
52	C₄×C₁₀₉	#107	1	1	yes	characteristic
53	Dic₁₀₉	#102	1	1	yes	characteristic
54	Dic₁₀₉	#103	1	1	yes	characteristic
55	C₂×C₁₀₉	#340	1	1	yes	characteristic
56	C₂×C₁₀₉	#365	1	1	yes	characteristic
57	C₂×C₁₀₉	#365	1	1	yes	characteristic
58	D₁₀₉	—	2	1	no	—
59	D₁₀₉	—	2	1	no	—
60	#1383	—	109	1	no	maximal, Sylow
61	C₁₀₉	#1383	1	1	yes	characteristic, complemented, Sylow
62	C₂×M₃₂	—	109	1	no	—
63	C₂×#108	—	109	1	no	—
64	C₂×#106	—	109	1	no	—
65	C₂×M₁₆	—	218	1	no	—
66	C₂²×C₈	—	109	1	no	—
67	C₂×C₄⋊C₄	—	218	1	no	—
68	M₃₂	—	218	1	no	—
69	M₃₂	—	218	1	no	—
70	C₂×C₁₆	—	109	1	no	—
71	C₂×C₁₆	#89137	1	1	yes	characteristic, complemented
72	#108	—	109	2	no	—
73	#108	—	218	1	no	—
74	#106	—	109	2	no	—
75	#106	—	218	1	no	—
76	C₂²×C₄	—	109	1	no	—
77	C₂²×C₄	—	436	1	no	—
78	M₁₆	—	436	1	no	—
79	M₁₆	—	218	2	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	—	109	1	no	—
82	C₂×C₈	—	109	1	no	—
83	C₂×C₈	—	109	1	no	—
84	C₂×C₈	—	109	1	no	—
85	C₂×C₈	—	436	1	no	—
86	C₂×C₈	C₂×#89137	1	1	yes	characteristic
87	C₄⋊C₄	—	436	1	no	—
88	C₄⋊C₄	—	218	2	no	—
89	C₁₆	—	2	1	no	—
90	C₁₆	—	218	1	no	—
91	C₂³	—	109	1	no	—
92	C₂×C₄	—	872	1	no	—
93	C₂×C₄	—	872	1	no	—
94	C₂×C₄	—	109	1	no	—
95	C₂×C₄	—	109	1	no	—
96	C₂×C₄	—	109	1	no	—
97	C₂×C₄	—	109	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	436	2	no	—
100	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
101	C₈	—	109	1	no	—
102	C₈	—	109	1	no	—
103	C₈	—	436	2	no	—
104	C₈	C₄×#89137	1	1	yes	characteristic
105	C₈	C₂²⋊#89137	1	1	yes	characteristic
106	C₂²	—	109	1	no	—
107	C₂²	—	109	1	no	—
108	C₂²	—	218	1	no	—
109	C₂²	—	218	1	no	—
110	C₂²	#16812767	1	1	yes	characteristic
111	C₄	—	872	2	no	—
112	C₄	—	109	1	no	—
113	C₄	—	109	1	no	—
114	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
115	C₄	#16812768	1	1	yes	characteristic
116	C₂	—	218	1	no	—
117	C₂	—	218	1	no	—
118	C₂	#20295900	1	1	yes	characteristic
119	C₂	#20295950	1	1	yes	characteristic
120	C₂	#20295951	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812769

C₂

yes

maximal, characteristic

C₂×#16812767

C₂

yes

maximal, characteristic

C₂×#16812638

C₂

yes

maximal, characteristic

C₂×#13182039

—

C₂×C₄⋊#89137

—

#16812769

—

#16812769

C₄

yes

complemented

#16812767

—

#16812767

C₄

yes

—

C₂×C₈×D₁₀₉

C₂²

yes

—

—

C₄

yes

characteristic, complemented

C₂×C₁₆×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

—

—

—

—

—

C₂×C₄×D₁₀₉

D₄

yes

characteristic

C₂×#11724784

Q₈

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

#13182004

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

—

C₂²×D₁₀₉

SD₁₆

yes

—

—

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂×Dic₁₀₉

SD₁₆

yes

characteristic

C₄×D₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×D₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×D₁₀₉

D₈

yes

characteristic

C₄×D₁₀₉

Q₁₆

yes

characteristic

#11724786

—

C₈×C₁₀₉

C₄²

yes

characteristic

C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#106

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

yes

Dic₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

D₁₀₉

—

#1383

—

109

maximal, Sylow

C₁₀₉

#1383

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂×#108

—

109

—

C₂×#106

—

109

—

C₂×M₁₆

—

218

—

C₂²×C₈

—

109

—

C₂×C₄⋊C₄

—

218

—

M₃₂

—

218

—

M₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

#89137

yes

characteristic, complemented

#108

—

109

—

#108

—

218

—

#106

—

109

—

#106

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

436

—

M₁₆

—

436

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

436

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

—

436

—

C₄⋊C₄

—

218

—

C₁₆

—

C₁₆

—

218

—

C₂³

—

109

—

C₂×C₄

—

872

—

C₂×C₄

—

872

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

100

C₂×C₄

C₄⋊#89137

yes

characteristic

101

C₈

—

109

—

102

C₈

—

109

—

103

C₈

—

436

—

104

C₈

C₄×#89137

yes

characteristic

105

C₈

C₂²⋊#89137

yes

characteristic

106

C₂²

—

109

—

107

C₂²

—

109

—

108

C₂²

—

218

—

109

C₂²

—

218

—

110

C₂²

#16812767

yes

characteristic

111

C₄

—

872

—

112

C₄

—

109

—

113

C₄

—

109

—

114

C₄

(C₂×#89137)⋊C₄

yes

characteristic

115

C₄

#16812768

yes

characteristic

116

C₂

—

218

—

117

C₂

—

218

—

118

C₂

#20295900

yes

characteristic

119

C₂

#20295950

yes

characteristic

120

C₂

#20295951

yes

characteristic

	subgroup	G/H
center	C₂²	#16812767
commutator	C₈×C₁₀₉	C₄²
Frattini	C₂×C₈	C₂×#89137
Fitting	C₂×C₁₆×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#106
prime radical	C₂²×D₁₀₉	SD₁₆
cosocle	C₂×C₈×D₁₀₉	C₂²
hypercenter	C₂×C₁₆	#89137
hypocenter	C₁₀₉	#1383
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₂²

#16812767

commutator

C₈×C₁₀₉

C₄²

Frattini

C₂×C₈

C₂×#89137

Fitting

C₂×C₁₆×C₁₀₉

C₄

socle

C₂²×C₁₀₉

#106

prime radical

C₂²×D₁₀₉

SD₁₆

cosocle

C₂×C₈×D₁₀₉

C₂²

C₂×C₁₆

C₁₀₉

G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
7. #16812769	C₄	semidirect
13. C₂×#13182004	C₄	semidirect
14. C₂×C₁₆×C₁₀₉	C₄	semidirect
61. C₁₀₉	#1383	semidirect
71. C₂×C₁₆	#89137	semidirect

H ≅ G/N

type

7. #16812769

C₄

semidirect

13. C₂×#13182004

C₄

semidirect

14. C₂×C₁₆×C₁₀₉

C₄

semidirect

61. C₁₀₉

#1383

semidirect

71. C₂×C₁₆

#89137

semidirect

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

subgroup	index
C₂×#16812769	2
C₂×#16812767	2
C₂×#16812638	2
#1383	109

subgroup

index

109

Subgroups — #28378097

Conjugacy classes of subgroups

Complements

Maximal subgroups