Group #28378840 — subgroups

Subgroups — #28378840

Order 13952.

224 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 90 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×#16812770	C₂	1	1	yes	maximal, characteristic
2	#20295991	C₂	1	2	yes	maximal
3	C₂×#16812753	C₂	1	1	yes	maximal, characteristic
4	#20295896	C₂	1	2	yes	maximal
5	#20295061	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	C₂²	1	2	yes	—
7	C₂×D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×#89137	C₂²	1	1	yes	characteristic
9	#16812770	C₄	1	4	yes	complemented
10	#16812754	—	2	2	no	—
11	#16812753	C₄	1	4	yes	complemented
12	C₄×#11724786	—	2	2	no	—
13	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
14	#16812657	C₂²	1	2	yes	—
15	#16812632	C₄	1	2	yes	complemented
16	#16812631	C₄	1	1	yes	characteristic, complemented
17	#106×C₁₀₉	C₄	1	1	yes	characteristic, complemented
18	C₂²×#89137	—	2	1	no	—
19	#13182039	—	2	2	no	—
20	#13182039	C₂×C₄	1	4	yes	—
21	D₁₀₉⋊C₈	C₂×C₄	1	2	yes	—
22	D₁₀₉⋊C₈	Q₈	1	2	yes	—
23	C₂×#11724786	—	2	2	no	—
24	C₂×#11724786	D₄	1	2	yes	—
25	C₄×#89137	—	2	1	no	—
26	C₄×#89137	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
29	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
30	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
32	#13182006	—	2	2	no	—
33	C₄×Dic₁₀₉	—	2	2	no	—
34	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₂²×D₁₀₉	#41	1	1	yes	characteristic
37	C₂×#89137	—	2	2	no	—
38	C₂×#89137	—	4	2	no	—
39	C₂×C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₁₀₉	—	2	2	no	—
42	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
45	C₄×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
46	C₄×D₁₀₉	#41	1	1	yes	characteristic
47	#11724786	—	2	4	no	—
48	#11724786	C₄⋊C₄	1	4	yes	—
49	C₈×C₁₀₉	C₄²	1	2	yes	—
50	#11724784	C₄²	1	2	yes	—
51	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
52	C₂×D₁₀₉	—	2	2	no	—
53	C₂×D₁₀₉	#117	1	1	yes	characteristic
54	C₂×D₁₀₉	C₄×Q₈	1	1	yes	characteristic
55	#89137	—	4	2	no	—
56	C₄×C₁₀₉	—	4	2	no	—
57	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
58	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
59	Dic₁₀₉	—	4	2	no	—
60	Dic₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
61	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
62	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₁₀₉	#444	1	2	yes	—
64	D₁₀₉	—	2	1	no	—
65	D₁₀₉	—	2	1	no	—
66	#1773	—	109	1	no	maximal, Sylow
67	C₁₀₉	#1773	1	1	yes	characteristic, complemented, Sylow
68	C₂×#108	—	109	1	no	—
69	#427	—	109	1	no	—
70	#405	—	109	2	no	—
71	C₂×#97	—	109	1	no	—
72	#339	—	109	2	no	—
73	C₂×M₁₆	—	109	2	no	—
74	C₂²×C₈	—	109	1	no	—
75	C₂²×C₈	—	109	1	no	—
76	#117	—	109	2	no	—
77	C₂×C₄²	—	109	1	no	—
78	#108	—	109	4	no	—
79	#107	—	109	2	no	—
80	#106	—	109	1	no	—
81	#106	#89137	1	1	yes	characteristic, complemented
82	#97	—	109	4	no	—
83	#97	—	218	2	no	—
84	C₄×C₈	—	218	2	no	—
85	C₂²×C₄	—	109	1	no	—
86	C₂²×C₄	—	218	1	no	—
87	M₁₆	—	109	4	no	—
88	M₁₆	—	218	2	no	—
89	C₂×C₈	—	109	2	no	—
90	C₂×C₈	—	109	2	no	—
91	C₂×C₈	—	109	2	no	—
92	C₂×C₈	—	109	1	no	—
93	C₂×C₈	—	109	2	no	—
94	C₂×C₈	—	109	2	no	—
95	C₂×C₈	—	218	2	no	—
96	C₂×C₈	C₂×#89137	1	1	yes	characteristic
97	C₄⋊C₄	—	109	2	no	—
98	C₄⋊C₄	C₂×#89137	1	2	yes	—
99	C₂²⋊C₄	—	218	2	no	—
100	C₄²	—	109	2	no	—
101	C₄²	—	218	2	no	—
102	C₄²	—	218	1	no	—
103	C₂³	—	109	1	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	109	1	no	—
106	C₂×C₄	—	109	1	no	—
107	C₂×C₄	—	109	1	no	—
108	C₂×C₄	—	109	1	no	—
109	C₂×C₄	—	109	1	no	—
110	C₂×C₄	—	436	2	no	—
111	C₂×C₄	—	218	2	no	—
112	C₂×C₄	—	218	2	no	—
113	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
114	C₈	—	109	4	no	—
115	C₈	—	109	2	no	—
116	C₈	—	218	4	no	—
117	C₈	C₄×#89137	1	2	yes	—
118	C₂²	—	109	1	no	—
119	C₂²	—	109	1	no	—
120	C₂²	—	218	2	no	—
121	C₂²	D₄×#89137	1	1	yes	characteristic
122	C₄	—	4	2	no	—
123	C₄	—	109	1	no	—
124	C₄	—	109	1	no	—
125	C₄	—	436	2	no	—
126	C₄	—	436	2	no	—
127	C₄	C₂×C₄×#89137	1	1	yes	characteristic
128	C₄	Q₈×#89137	1	1	yes	characteristic
129	C₂	—	218	1	no	—
130	C₂	—	218	1	no	—
131	C₂	C₄⋊C₄×#89137	1	1	yes	characteristic
132	C₂	#20296071	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812770

C₂

yes

maximal, characteristic

#20295991

C₂

yes

maximal

C₂×#16812753

C₂

yes

maximal, characteristic

#20295896

C₂

yes

maximal

#20295061

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

C₂×D₁₀₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×#89137

C₂²

yes

characteristic

#16812770

C₄

yes

complemented

#16812754

—

#16812753

C₄

yes

complemented

C₄×#11724786

—

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812657

C₂²

yes

—

#16812632

C₄

yes

complemented

#16812631

C₄

yes

characteristic, complemented

#106×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

#13182039

—

#13182039

C₂×C₄

yes

—

D₁₀₉⋊C₈

C₂×C₄

yes

—

D₁₀₉⋊C₈

Q₈

yes

—

C₂×#11724786

—

C₂×#11724786

D₄

yes

—

C₄×#89137

—

C₄×#89137

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₂×C₄

yes

—

C₈×D₁₀₉

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182006

—

C₄×Dic₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

yes

—

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

C₂²×C₄

yes

characteristic

C₄×D₁₀₉

C₂²×C₄

yes

characteristic

C₄×D₁₀₉

C₂×Q₈

yes

C₄×D₁₀₉

yes

—

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₄²

yes

—

#11724784

C₄²

yes

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

#117

yes

characteristic

C₂×D₁₀₉

C₄×Q₈

yes

characteristic

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂×C₄⋊C₄

yes

characteristic

Dic₁₀₉

C₄×D₄

yes

characteristic

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

#444

yes

—

D₁₀₉

—

D₁₀₉

—

#1773

—

109

maximal, Sylow

C₁₀₉

#1773

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

#427

—

109

—

#405

—

109

—

C₂×#97

—

109

—

#339

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

#117

—

109

—

C₂×C₄²

—

109

—

#108

—

109

—

#107

—

109

—

#106

—

109

—

#106

#89137

yes

characteristic, complemented

#97

—

109

—

#97

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×#89137

yes

—

C₂²⋊C₄

—

218

—

100

C₄²

—

109

—

101

C₄²

—

218

—

102

C₄²

—

218

—

103

C₂³

—

109

—

104

C₂×C₄

—

105

C₂×C₄

—

109

—

106

C₂×C₄

—

109

—

107

C₂×C₄

—

109

—

108

C₂×C₄

—

109

—

109

C₂×C₄

—

109

—

110

C₂×C₄

—

436

—

111

C₂×C₄

—

218

—

112

C₂×C₄

—

218

—

113

C₂×C₄

C₂²×#89137

yes

characteristic

114

C₈

—

109

—

115

C₈

—

109

—

116

C₈

—

218

—

117

C₈

C₄×#89137

yes

—

118

C₂²

—

109

—

119

C₂²

—

109

—

120

C₂²

—

218

—

121

C₂²

D₄×#89137

yes

characteristic

122

C₄

—

123

C₄

—

109

—

124

C₄

—

109

—

125

C₄

—

436

—

126

C₄

—

436

—

127

C₄

C₂×C₄×#89137

yes

characteristic

128

C₄

Q₈×#89137

yes

characteristic

129

C₂

—

218

—

130

C₂

—

218

—

131

C₂

C₄⋊C₄×#89137

yes

characteristic

132

C₂

#20296071

yes

—

	subgroup	G/H
center	C₂²	D₄×#89137
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×#89137
Fitting	#106×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₄×D₄
prime radical	C₂²×D₁₀₉	#41
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	#106	#89137
hypocenter	C₁₀₉	#1773
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

subgroup

G/H

center

C₂²

D₄×#89137

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂×C₄

C₂²×#89137

Fitting

#106×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₄×D₄

prime radical

C₂²×D₁₀₉

#41

cosocle

C₂×C₄×D₁₀₉

C₂³

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

N	H ≅ G/N	type
9. #16812770	C₄	semidirect
11. #16812753	C₄	semidirect
15. #16812632	C₄	semidirect
16. #16812631	C₄	semidirect
17. #106×C₁₀₉	C₄	semidirect
67. C₁₀₉	#1773	semidirect
81. #106	#89137	semidirect

H ≅ G/N

type

9. #16812770

C₄

semidirect

11. #16812753

C₄

semidirect

15. #16812632

C₄

semidirect

16. #16812631

C₄

semidirect

17. #106×C₁₀₉

C₄

semidirect

67. C₁₀₉

#1773

semidirect

81. #106

#89137

semidirect

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

subgroup	index
C₂×#16812770	2
#20295991	2
C₂×#16812753	2
#20295896	2
#20295061	2
#1773	109

subgroup

index

109

Subgroups — #28378840

Conjugacy classes of subgroups

Complements

Maximal subgroups