Group #42130522 — subgroups

Subgroups — C₁₀₉×#12528 #42130522

Order 27904.

274 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 102 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	C₁₀₉×#2247	C₂	1	2	yes	maximal
2	C₁₀₉×#2241	C₂	1	2	yes	maximal
3	C₁₀₉×#1713	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1707	C₂	1	1	yes	maximal, characteristic
5	C₁₀₉×#1312	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₁₀₉	—	2	4	no	—
7	#500×C₁₀₉	—	2	1	no	—
8	#500×C₁₀₉	—	2	1	no	—
9	#500×C₁₀₉	C₂²	1	1	yes	characteristic
10	#498×C₁₀₉	—	2	1	no	—
11	#494×C₁₀₉	C₂²	1	1	yes	characteristic
12	#494×C₁₀₉	C₂²	1	2	yes	—
13	#487×C₁₀₉	—	2	1	no	—
14	#358×C₁₀₉	—	2	4	no	—
15	C₄×C₁₆×C₁₀₉	C₂²	1	2	yes	—
16	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
18	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
19	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
20	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
21	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
22	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
23	Q₃₂×C₁₀₉	—	4	8	no	—
24	C₂×C₁₆×C₁₀₉	D₄	1	4	yes	—
25	#107×C₁₀₉	—	2	2	no	—
26	#107×C₁₀₉	—	4	2	no	—
27	#107×C₁₀₉	—	4	2	no	—
28	#106×C₁₀₉	—	4	2	no	—
29	#103×C₁₀₉	—	4	2	no	—
30	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
32	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₁₀₉	—	4	1	no	—
34	C₂×Q₈×C₁₀₉	—	4	2	no	—
35	C₂×Q₈×C₁₀₉	—	4	1	no	—
36	Q₁₆×C₁₀₉	—	4	4	no	—
37	Q₁₆×C₁₀₉	—	4	4	no	—
38	Q₁₆×C₁₀₉	—	4	4	no	—
39	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
40	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
41	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₁₀₉	—	4	1	no	—
43	C₄⋊C₄×C₁₀₉	—	4	1	no	—
44	C₄⋊C₄×C₁₀₉	—	4	1	no	—
45	C₄⋊C₄×C₁₀₉	—	4	1	no	—
46	C₄⋊C₄×C₁₀₉	—	4	2	no	—
47	C₄⋊C₄×C₁₀₉	—	4	2	no	—
48	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₁₀₉	—	2	4	no	—
50	Q₈×C₁₀₉	—	8	2	no	—
51	Q₈×C₁₀₉	—	8	2	no	—
52	Q₈×C₁₀₉	—	8	2	no	—
53	Q₈×C₁₀₉	—	8	2	no	—
54	C₂×C₄×C₁₀₉	—	8	1	no	—
55	C₂×C₄×C₁₀₉	—	8	2	no	—
56	C₂×C₄×C₁₀₉	—	8	1	no	—
57	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
58	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
60	C₈×C₁₀₉	—	2	2	no	—
61	C₈×C₁₀₉	Q₃₂	1	4	yes	complemented
62	C₈×C₁₀₉	AGL(1,ℤ/8)	1	4	yes	—
63	C₂²×C₁₀₉	#493	1	1	yes	characteristic
64	C₄×C₁₀₉	—	16	1	no	—
65	C₄×C₁₀₉	—	16	2	no	—
66	C₄×C₁₀₉	—	16	1	no	—
67	C₄×C₁₀₉	#496	1	2	yes	—
68	C₄×C₁₀₉	C₂×Q₃₂	1	2	yes	—
69	C₄×C₁₀₉	#510	1	2	yes	—
70	#12528	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂×C₁₀₉	#1721	1	1	yes	characteristic
72	C₂×C₁₀₉	#2247	1	1	yes	characteristic
73	C₂×C₁₀₉	#2251	1	1	yes	characteristic
74	#2247	C₂×C₁₀₉	1	2	yes	—
75	#2241	C₂×C₁₀₉	1	2	yes	—
76	#1713	C₂×C₁₀₉	1	1	yes	characteristic
77	#1707	C₂×C₁₀₉	1	1	yes	characteristic
78	#1312	C₂×C₁₀₉	1	1	yes	characteristic
79	C₁₀₉	#12528	1	1	yes	characteristic, direct product factor, Sylow
80	C₂×Q₃₂	—	2	4	no	—
81	#500	—	2	1	no	—
82	#500	—	2	1	no	—
83	#500	C₂²×C₁₀₉	1	1	yes	characteristic
84	#498	—	2	1	no	—
85	#494	C₂²×C₁₀₉	1	1	yes	characteristic
86	#494	C₂²×C₁₀₉	1	2	yes	—
87	#487	—	2	1	no	—
88	#358	—	2	4	no	—
89	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
90	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	2	2	no	—
93	C₂×Q₁₆	—	2	2	no	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄⋊Q₈	—	2	1	no	—
96	C₄⋊Q₈	—	2	2	no	—
97	Q₃₂	—	4	8	no	—
98	C₂×C₁₆	D₄×C₁₀₉	1	4	yes	—
99	#107	—	2	2	no	—
100	#107	—	4	2	no	—
101	#107	—	4	2	no	—
102	#106	—	4	2	no	—
103	#103	—	4	2	no	—
104	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
105	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
106	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
107	C₂×Q₈	—	4	1	no	—
108	C₂×Q₈	—	4	2	no	—
109	C₂×Q₈	—	4	1	no	—
110	Q₁₆	—	4	4	no	—
111	Q₁₆	—	4	4	no	—
112	Q₁₆	—	4	4	no	—
113	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
114	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
115	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	4	1	no	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
123	C₁₆	—	2	4	no	—
124	Q₈	—	8	2	no	—
125	Q₈	—	8	2	no	—
126	Q₈	—	8	2	no	—
127	Q₈	—	8	2	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
132	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
133	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	Q₃₂×C₁₀₉	1	4	yes	complemented
136	C₈	AGL(1,ℤ/8)×C₁₀₉	1	4	yes	—
137	C₂²	#493×C₁₀₉	1	1	yes	characteristic
138	C₄	—	16	1	no	—
139	C₄	—	16	2	no	—
140	C₄	—	16	1	no	—
141	C₄	#496×C₁₀₉	1	2	yes	—
142	C₄	C₂×Q₃₂×C₁₀₉	1	2	yes	—
143	C₄	#510×C₁₀₉	1	2	yes	—
144	C₂	C₁₀₉×#1721	1	1	yes	characteristic
145	C₂	C₁₀₉×#2247	1	1	yes	characteristic
146	C₂	C₁₀₉×#2251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2247

C₂

yes

maximal

C₁₀₉×#2241

C₂

yes

maximal

C₁₀₉×#1713

C₂

yes

maximal, characteristic

C₁₀₉×#1707

C₂

yes

maximal, characteristic

C₁₀₉×#1312

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₁₀₉

—

#500×C₁₀₉

—

#500×C₁₀₉

—

#500×C₁₀₉

C₂²

yes

characteristic

#498×C₁₀₉

—

#494×C₁₀₉

C₂²

yes

characteristic

#494×C₁₀₉

C₂²

yes

—

#487×C₁₀₉

—

#358×C₁₀₉

—

C₄×C₁₆×C₁₀₉

C₂²

yes

—

C₈²×C₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

Q₃₂×C₁₀₉

—

C₂×C₁₆×C₁₀₉

D₄

yes

—

#107×C₁₀₉

—

#107×C₁₀₉

—

#107×C₁₀₉

—

#106×C₁₀₉

—

#103×C₁₀₉

—

C₄×C₈×C₁₀₉

D₄

yes

characteristic

C₄×C₈×C₁₀₉

D₄

yes

characteristic

C₄×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

D₈

yes

—

C₂×C₈×C₁₀₉

D₈

yes

—

C₂×C₈×C₁₀₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

characteristic

C₁₆×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#127

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₈

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

Q₃₂

yes

complemented

C₈×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₀₉

#493

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#496

yes

—

C₄×C₁₀₉

C₂×Q₃₂

yes

—

C₄×C₁₀₉

#510

yes

—

#12528

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

#2241

C₂×C₁₀₉

yes

—

#1713

C₂×C₁₀₉

yes

characteristic

#1707

C₂×C₁₀₉

yes

characteristic

#1312

C₂×C₁₀₉

yes

characteristic

C₁₀₉

#12528

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

—

#500

—

#500

—

#500

C₂²×C₁₀₉

yes

characteristic

#498

—

#494

C₂²×C₁₀₉

yes

characteristic

#494

C₂²×C₁₀₉

yes

—

#487

—

#358

—

C₄×C₁₆

C₂²×C₁₀₉

yes

—

C₈²

C₂²×C₁₀₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

D₄×C₁₀₉

yes

—

#107

—

100

#107

—

101

#107

—

102

#106

—

103

#103

—

104

C₄×C₈

D₄×C₁₀₉

yes

characteristic

105

C₄×C₈

D₄×C₁₀₉

yes

characteristic

106

C₄×C₈

C₂³×C₁₀₉

yes

characteristic

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

Q₁₆

—

111

Q₁₆

—

112

Q₁₆

—

113

C₂×C₈

D₈×C₁₀₉

yes

—

114

C₂×C₈

D₈×C₁₀₉

yes

—

115

C₂×C₈

C₂×D₄×C₁₀₉

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

123

C₁₆

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#127×C₁₀₉

yes

characteristic

132

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

133

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

134

C₈

—

135

C₈

Q₃₂×C₁₀₉

yes

complemented

136

C₈

AGL(1,ℤ/8)×C₁₀₉

yes

—

137

C₂²

#493×C₁₀₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#496×C₁₀₉

yes

—

142

C₄

C₂×Q₃₂×C₁₀₉

yes

—

143

C₄

#510×C₁₀₉

yes

—

144

C₂

C₁₀₉×#1721

yes

characteristic

145

C₂

C₁₀₉×#2247

yes

characteristic

146

C₂

C₁₀₉×#2251

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₀₉	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈²×C₁₀₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₀₉

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₀₉

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈²×C₁₀₉ ◁ G

N	H ≅ G/N	type
61. C₈×C₁₀₉	Q₃₂	semidirect
70. #12528	C₁₀₉	direct (N × H)
79. C₁₀₉	#12528	direct (N × H)
135. C₈	Q₃₂×C₁₀₉	semidirect

H ≅ G/N

type

61. C₈×C₁₀₉

Q₃₂

semidirect

70. #12528

C₁₀₉

direct (N × H)

79. C₁₀₉

#12528

direct (N × H)

135. C₈

Q₃₂×C₁₀₉

semidirect

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

subgroup	index
C₁₀₉×#2247	2
C₁₀₉×#2241	2
C₁₀₉×#1713	2
C₁₀₉×#1707	2
C₁₀₉×#1312	2
#12528	109

subgroup

index

C₁₀₉×#2247

C₁₀₉×#2241

C₁₀₉×#1713

C₁₀₉×#1707

C₁₀₉×#1312

#12528

109

Subgroups — C₁₀₉×#12528 #42130522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12528 #42130522

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12528 #42130522