Group #28359341 — subgroups

Subgroups — C₁₀₉×#2087 #28359341

Order 13952.

224 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 104 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×#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#389×C₁₀₉	C₂	1	2	yes	maximal
4	#340×C₁₀₉	C₂	1	2	yes	maximal
5	#340×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
12	#107×C₁₀₉	—	2	2	no	—
13	C₄⋊C₈×C₁₀₉	—	2	2	no	—
14	#95×C₁₀₉	—	2	2	no	—
15	C₂²×C₄×C₁₀₉	—	2	2	no	—
16	C₂²×C₄×C₁₀₉	—	2	2	no	—
17	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	—	2	2	no	—
21	C₂×C₈×C₁₀₉	—	2	1	no	—
22	C₂×C₈×C₁₀₉	—	2	2	no	—
23	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
24	C₄⋊C₄×C₁₀₉	—	2	2	no	—
25	C₄⋊C₄×C₁₀₉	—	2	2	no	—
26	C₄⋊C₄×C₁₀₉	—	2	2	no	—
27	C₄⋊C₄×C₁₀₉	—	4	4	no	—
28	C₄⋊C₄×C₁₀₉	Q₈	1	4	yes	—
29	C₄²×C₁₀₉	—	2	1	no	—
30	C₄²×C₁₀₉	—	2	1	no	—
31	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	—	2	1	no	—
33	C₂×C₄×C₁₀₉	—	2	1	no	—
34	C₂×C₄×C₁₀₉	—	2	1	no	—
35	C₂×C₄×C₁₀₉	—	2	4	no	—
36	C₂×C₄×C₁₀₉	—	2	2	no	—
37	C₂×C₄×C₁₀₉	—	4	2	no	—
38	C₂×C₄×C₁₀₉	—	4	2	no	—
39	C₂×C₄×C₁₀₉	—	4	2	no	—
40	C₂×C₄×C₁₀₉	—	4	2	no	—
41	C₂×C₄×C₁₀₉	—	4	2	no	—
42	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	—
43	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₀₉	#41	1	2	yes	—
46	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
47	C₈×C₁₀₉	—	2	2	no	—
48	C₈×C₁₀₉	—	4	1	no	—
49	C₂²×C₁₀₉	#121	1	1	yes	characteristic
50	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
52	C₂²×C₁₀₉	#135	1	1	yes	characteristic
53	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
54	C₄×C₁₀₉	—	2	2	no	—
55	C₄×C₁₀₉	—	4	1	no	—
56	C₄×C₁₀₉	—	4	4	no	—
57	C₄×C₁₀₉	—	8	2	no	—
58	C₄×C₁₀₉	#123	1	1	yes	characteristic
59	C₄×C₁₀₉	#125	1	2	yes	—
60	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₁₀₉	#398	1	1	yes	characteristic
62	C₂×C₁₀₉	#467	1	1	yes	characteristic
63	C₂×C₁₀₉	#469	1	1	yes	characteristic
64	C₂×C₁₀₉	#477	1	2	yes	—
65	C₂×C₁₀₉	#478	1	2	yes	—
66	#2087	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₁₀₉	#2087	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×#107	C₂×C₁₀₉	1	1	yes	characteristic
69	C₂×C₄⋊C₈	C₂×C₁₀₉	1	1	yes	characteristic
70	#389	C₂×C₁₀₉	1	2	yes	—
71	#340	C₂×C₁₀₉	1	2	yes	—
72	#340	C₂×C₁₀₉	1	1	yes	characteristic
73	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
74	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	2	2	no	—
76	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
77	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
78	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
79	#107	—	2	2	no	—
80	C₄⋊C₈	—	2	2	no	—
81	#95	—	2	2	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
85	C₂²×C₄	D₄×C₁₀₉	1	1	yes	characteristic
86	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	D₄×C₁₀₉	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	Q₈×C₁₀₉	1	4	yes	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	C₂×D₄×C₁₀₉	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	Q₁₆×C₁₀₉	1	2	yes	—
110	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
111	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
112	C₂×C₄	#41×C₁₀₉	1	2	yes	—
113	C₂×C₄	#41×C₁₀₉	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	4	1	no	—
116	C₂²	#121×C₁₀₉	1	1	yes	characteristic
117	C₂²	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
118	C₂²	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
119	C₂²	#135×C₁₀₉	1	1	yes	characteristic
120	C₂²	AGL(1,ℤ/8)×C₁₀₉	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	4	1	no	—
123	C₄	—	4	4	no	—
124	C₄	—	8	2	no	—
125	C₄	#123×C₁₀₉	1	1	yes	characteristic
126	C₄	#125×C₁₀₉	1	2	yes	—
127	C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
128	C₂	#398×C₁₀₉	1	1	yes	characteristic
129	C₂	#467×C₁₀₉	1	1	yes	characteristic
130	C₂	#469×C₁₀₉	1	1	yes	characteristic
131	C₂	#477×C₁₀₉	1	2	yes	—
132	C₂	#478×C₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#107×C₁₀₉

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₀₉

C₂

yes

maximal, characteristic

#389×C₁₀₉

C₂

yes

maximal

#340×C₁₀₉

C₂

yes

maximal

#340×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

#95×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

Q₈

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

Q₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

#121

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₂²×C₁₀₉

C₂×Q₁₆

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#123

yes

characteristic

C₄×C₁₀₉

#125

yes

—

C₄×C₁₀₉

C₄⋊Q₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

C₂×C₁₀₉

#478

yes

—

#2087

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#2087

yes

characteristic, direct product factor, Sylow

C₂×#107

C₂×C₁₀₉

yes

characteristic

C₂×C₄⋊C₈

C₂×C₁₀₉

yes

characteristic

#389

C₂×C₁₀₉

yes

—

#340

C₂×C₁₀₉

yes

—

#340

C₂×C₁₀₉

yes

characteristic

C₂²×C₈

C₂²×C₁₀₉

yes

characteristic

C₂²×C₈

C₂²×C₁₀₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄²

C₂²×C₁₀₉

yes

characteristic

#107

—

C₄⋊C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₀₉

yes

characteristic

C₂²×C₄

D₄×C₁₀₉

yes

characteristic

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Q₈×C₁₀₉

yes

—

C₄²

—

C₄²

—

C₂³

C₂×D₄×C₁₀₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Q₁₆×C₁₀₉

yes

—

110

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

111

C₂×C₄

C₂×Q₈×C₁₀₉

yes

—

112

C₂×C₄

#41×C₁₀₉

yes

—

113

C₂×C₄

#41×C₁₀₉

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

#121×C₁₀₉

yes

characteristic

117

C₂²

C₂²⋊Q₈×C₁₀₉

yes

—

118

C₂²

C₂×Q₁₆×C₁₀₉

yes

characteristic

119

C₂²

#135×C₁₀₉

yes

characteristic

120

C₂²

AGL(1,ℤ/8)×C₁₀₉

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#123×C₁₀₉

yes

characteristic

126

C₄

#125×C₁₀₉

yes

—

127

C₄

C₄⋊Q₈×C₁₀₉

yes

characteristic

128

C₂

#398×C₁₀₉

yes

characteristic

129

C₂

#467×C₁₀₉

yes

characteristic

130

C₂

#469×C₁₀₉

yes

characteristic

131

C₂

#477×C₁₀₉

yes

—

132

C₂

#478×C₁₀₉

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₀₉	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₀₉

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₉

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
66. #2087	C₁₀₉	direct (N × H)
67. C₁₀₉	#2087	direct (N × H)

H ≅ G/N

type

66. #2087

C₁₀₉

direct (N × H)

67. C₁₀₉

#2087

direct (N × H)

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

subgroup	index
C₂×#107×C₁₀₉	2
C₂×C₄⋊C₈×C₁₀₉	2
#389×C₁₀₉	2
#340×C₁₀₉	2
#2087	109

subgroup

index

C₂×#107×C₁₀₉

C₂×C₄⋊C₈×C₁₀₉

#389×C₁₀₉

#340×C₁₀₉

#2087

109

Subgroups — C₁₀₉×#2087 #28359341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#2087 #28359341

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2087 #28359341