Group #28362932 — subgroups

Subgroups — #28362932

Order 13952.

272 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812633	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812609	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₁₀₉	C₂	1	2	yes	maximal
4	#20294788	C₂	1	2	yes	maximal
5	C₂×#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₀₉	—	2	2	no	—
9	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	#16812633	C₄	1	4	yes	complemented
11	#16812609	C₄	1	4	yes	complemented
12	#16812606	—	2	2	no	—
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
15	#106×C₁₀₉	C₄	1	4	yes	complemented
16	C₂²×Dic₁₀₉	—	2	2	no	—
17	C₂²×Dic₁₀₉	D₄	1	2	yes	—
18	C₂×#11724784	—	2	1	no	—
19	C₂×#11724784	—	2	1	no	—
20	C₂×#11724784	C₂×C₄	1	2	yes	—
21	C₄⋊Dic₁₀₉	—	2	2	no	—
22	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
23	C₄×Dic₁₀₉	—	4	4	no	—
24	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₀₉	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₀₉	—	2	2	no	—
27	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
31	C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
33	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₁₀₉	—	2	4	no	—
35	C₂×C₄×C₁₀₉	—	4	2	no	—
36	C₂×C₄×C₁₀₉	—	4	2	no	—
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂²×C₄	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×C₄×C₁₀₉	#41	1	1	yes	characteristic
43	C₂×Dic₁₀₉	—	2	2	no	—
44	C₂×Dic₁₀₉	—	2	2	no	—
45	C₂×Dic₁₀₉	—	2	4	no	—
46	C₂×Dic₁₀₉	—	4	2	no	—
47	C₂×Dic₁₀₉	—	4	2	no	—
48	C₂×Dic₁₀₉	C₄⋊C₄	1	4	yes	—
49	C₈×C₁₀₉	C₄²	1	4	yes	complemented
50	#11724784	—	2	2	no	—
51	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₁₀₉	#137	1	2	yes	—
56	C₄×C₁₀₉	—	4	4	no	—
57	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
58	C₄×C₁₀₉	#117	1	1	yes	characteristic
59	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
61	Dic₁₀₉	—	2	4	no	—
62	Dic₁₀₉	—	4	4	no	—
63	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₁₀₉	#428	1	2	yes	—
65	C₂×C₁₀₉	#440	1	2	yes	—
66	C₂×C₁₀₉	#441	1	1	yes	characteristic
67	C₂×C₁₀₉	#442	1	1	yes	characteristic
68	#1776	—	109	1	no	maximal, Sylow
69	C₁₀₉	#1776	1	1	yes	characteristic, complemented, Sylow
70	C₂×#107	—	109	1	no	—
71	C₂×#106	D₁₀₉	1	1	yes	characteristic
72	C₂×#97	—	109	1	no	—
73	C₄×C₄⋊C₄	—	109	2	no	—
74	#340	—	109	2	no	—
75	C₂²×C₈	—	109	1	no	—
76	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	109	2	no	—
78	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
79	C₂×C₄²	—	109	1	no	—
80	C₂×C₄²	—	218	2	no	—
81	#107	—	109	4	no	—
82	#106	Dic₁₀₉	1	4	yes	complemented
83	#97	—	109	4	no	—
84	#95	—	218	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	109	2	no	—
87	C₂²×C₄	—	218	2	no	—
88	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
89	C₂×C₈	—	109	2	no	—
90	C₂×C₈	—	218	1	no	—
91	C₂×C₈	—	218	1	no	—
92	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
93	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
94	C₂×C₈	C₂×Dic₁₀₉	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	109	4	no	—
97	C₄⋊C₄	—	218	2	no	—
98	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
99	C₄²	—	109	2	no	—
100	C₄²	—	109	2	no	—
101	C₄²	—	436	4	no	—
102	C₂³	D₄×D₁₀₉	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	109	4	no	—
107	C₂×C₄	—	436	2	no	—
108	C₂×C₄	—	436	2	no	—
109	C₂×C₄	—	218	4	no	—
110	C₂×C₄	—	218	2	no	—
111	C₂×C₄	—	218	2	no	—
112	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
114	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
116	C₂×C₄	#13182028	1	1	yes	characteristic
117	C₂×C₄	#13182030	1	1	yes	characteristic
118	C₈	—	218	2	no	—
119	C₈	C₄×Dic₁₀₉	1	4	yes	complemented
120	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
121	C₂²	#16812661	1	1	yes	characteristic
122	C₂²	D₄×Dic₁₀₉	1	1	yes	characteristic
123	C₂²	#16812726	1	2	yes	—
124	C₂²	#16812728	1	2	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	436	4	no	—
127	C₄	—	218	4	no	—
128	C₄	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
129	C₄	#16812657	1	1	yes	characteristic
130	C₄	#16812664	1	1	yes	characteristic
131	C₄	Q₈×Dic₁₀₉	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₁₀₉	1	1	yes	characteristic
133	C₂	#20295065	1	2	yes	—
134	C₂	#20295127	1	2	yes	—
135	C₂	#20295132	1	1	yes	characteristic
136	C₂	#20295137	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812633

C₂

yes

maximal, characteristic

C₂×#16812609

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₁₀₉

C₂

yes

maximal

#20294788

C₂

yes

maximal

C₂×#106×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

#16812633

C₄

yes

complemented

#16812609

C₄

yes

complemented

#16812606

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

#106×C₁₀₉

C₄

yes

complemented

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

D₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

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×Q₈

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₈×C₁₀₉

C₄²

yes

complemented

#11724784

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₂²×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₉

#137

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

#428

yes

—

C₂×C₁₀₉

#440

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1776

yes

characteristic, complemented, Sylow

C₂×#107

—

109

—

C₂×#106

D₁₀₉

yes

characteristic

C₂×#97

—

109

—

C₄×C₄⋊C₄

—

109

—

#340

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

C₂×C₄²

—

218

—

#107

—

109

—

#106

Dic₁₀₉

yes

complemented

#97

—

109

—

#95

—

218

—

C₂²×C₄

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₄×D₁₀₉

yes

—

C₂×C₈

C₂×Dic₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

C₄²

—

109

—

100

C₄²

—

109

—

101

C₄²

—

436

—

102

C₂³

D₄×D₁₀₉

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

109

—

107

C₂×C₄

—

436

—

108

C₂×C₄

—

436

—

109

C₂×C₄

—

218

—

110

C₂×C₄

—

218

—

111

C₂×C₄

—

218

—

112

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

113

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

114

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

115

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

116

C₂×C₄

#13182028

yes

characteristic

117

C₂×C₄

#13182030

yes

characteristic

118

C₈

—

218

—

119

C₈

C₄×Dic₁₀₉

yes

complemented

120

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

121

C₂²

#16812661

yes

characteristic

122

C₂²

D₄×Dic₁₀₉

yes

characteristic

123

C₂²

#16812726

yes

—

124

C₂²

#16812728

yes

—

125

C₄

—

126

C₄

—

436

—

127

C₄

—

218

—

128

C₄

C₂×C₄×Dic₁₀₉

yes

characteristic

129

C₄

#16812657

yes

characteristic

130

C₄

#16812664

yes

characteristic

131

C₄

Q₈×Dic₁₀₉

yes

characteristic

132

C₂

C₄⋊C₄×Dic₁₀₉

yes

characteristic

133

C₂

#20295065

yes

—

134

C₂

#20295127

yes

—

135

C₂

#20295132

yes

characteristic

136

C₂

#20295137

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×#106×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×#106	D₁₀₉
hypocenter	C₁₀₉	#1776
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×#106×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
10. #16812633	C₄	semidirect
11. #16812609	C₄	semidirect
15. #106×C₁₀₉	C₄	semidirect
49. C₈×C₁₀₉	C₄²	semidirect
69. C₁₀₉	#1776	semidirect
82. #106	Dic₁₀₉	semidirect
119. C₈	C₄×Dic₁₀₉	semidirect

H ≅ G/N

type

10. #16812633

C₄

semidirect

11. #16812609

C₄

semidirect

15. #106×C₁₀₉

C₄

semidirect

49. C₈×C₁₀₉

C₄²

semidirect

69. C₁₀₉

#1776

semidirect

82. #106

Dic₁₀₉

semidirect

119. C₈

C₄×Dic₁₀₉

semidirect

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

subgroup	index
C₂×#16812633	2
C₂×#16812609	2
C₄⋊C₄×Dic₁₀₉	2
#20294788	2
C₂×#106×C₁₀₉	2
#1776	109

subgroup

index

C₄⋊C₄×Dic₁₀₉

C₂×#106×C₁₀₉

109

Subgroups — #28362932

Conjugacy classes of subgroups

Complements

Maximal subgroups