Group #42130529 — subgroups

Subgroups — C₁₀₉×#12535 #42130529

Order 27904.

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

normal subgroups only

This group has 250 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×#2254	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	#500×C₁₀₉	—	2	1	no	—
7	#500×C₁₀₉	C₂²	1	1	yes	characteristic
8	#500×C₁₀₉	C₂²	1	2	yes	—
9	#498×C₁₀₉	—	2	1	no	—
10	#494×C₁₀₉	—	2	1	no	—
11	#494×C₁₀₉	C₂²	1	1	yes	characteristic
12	#487×C₁₀₉	—	2	1	no	—
13	#367×C₁₀₉	—	2	4	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	—	4	2	no	—
19	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
20	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
21	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
22	C₂×C₁₆×C₁₀₉	D₄	1	4	yes	—
23	#107×C₁₀₉	—	2	2	no	—
24	#107×C₁₀₉	—	2	2	no	—
25	#107×C₁₀₉	—	2	2	no	—
26	#107×C₁₀₉	—	4	2	no	—
27	#106×C₁₀₉	—	4	2	no	—
28	#103×C₁₀₉	—	4	2	no	—
29	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
30	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
31	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂×Q₈×C₁₀₉	—	4	1	no	—
33	C₂×Q₈×C₁₀₉	—	4	1	no	—
34	C₂×Q₈×C₁₀₉	—	4	2	no	—
35	Q₁₆×C₁₀₉	—	4	4	no	—
36	Q₁₆×C₁₀₉	—	8	4	no	—
37	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
38	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
39	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
40	C₄⋊C₄×C₁₀₉	—	4	2	no	—
41	C₄⋊C₄×C₁₀₉	—	4	1	no	—
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	2	no	—
46	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₁₆×C₁₀₉	—	2	4	no	—
48	Q₈×C₁₀₉	—	8	2	no	—
49	Q₈×C₁₀₉	—	8	4	no	—
50	Q₈×C₁₀₉	—	8	2	no	—
51	C₂×C₄×C₁₀₉	—	8	1	no	—
52	C₂×C₄×C₁₀₉	—	8	2	no	—
53	C₂×C₄×C₁₀₉	—	8	1	no	—
54	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
55	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
57	C₈×C₁₀₉	—	2	2	no	—
58	C₈×C₁₀₉	SD₃₂	1	4	yes	—
59	C₈×C₁₀₉	#137	1	4	yes	—
60	C₂²×C₁₀₉	#493	1	1	yes	characteristic
61	C₄×C₁₀₉	—	16	1	no	—
62	C₄×C₁₀₉	—	16	2	no	—
63	C₄×C₁₀₉	—	16	1	no	—
64	C₄×C₁₀₉	#497	1	2	yes	—
65	C₄×C₁₀₉	C₂×SD₃₂	1	2	yes	—
66	C₄×C₁₀₉	#510	1	2	yes	—
67	#12535	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₁₀₉	#1722	1	1	yes	characteristic
69	C₂×C₁₀₉	#2248	1	1	yes	characteristic
70	C₂×C₁₀₉	#2251	1	1	yes	characteristic
71	#2254	C₂×C₁₀₉	1	2	yes	—
72	#2241	C₂×C₁₀₉	1	2	yes	—
73	#1713	C₂×C₁₀₉	1	1	yes	characteristic
74	#1707	C₂×C₁₀₉	1	1	yes	characteristic
75	#1312	C₂×C₁₀₉	1	1	yes	characteristic
76	C₁₀₉	#12535	1	1	yes	characteristic, direct product factor, Sylow
77	#500	—	2	1	no	—
78	#500	C₂²×C₁₀₉	1	1	yes	characteristic
79	#500	C₂²×C₁₀₉	1	2	yes	—
80	#498	—	2	1	no	—
81	#494	—	2	1	no	—
82	#494	C₂²×C₁₀₉	1	1	yes	characteristic
83	#487	—	2	1	no	—
84	#367	—	2	4	no	—
85	#358	—	2	4	no	—
86	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
87	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
88	C₂×Q₁₆	—	2	2	no	—
89	C₂×Q₁₆	—	4	2	no	—
90	C₄⋊Q₈	—	2	1	no	—
91	C₄⋊Q₈	—	2	1	no	—
92	C₄⋊Q₈	—	2	2	no	—
93	C₂×C₁₆	D₄×C₁₀₉	1	4	yes	—
94	#107	—	2	2	no	—
95	#107	—	2	2	no	—
96	#107	—	2	2	no	—
97	#107	—	4	2	no	—
98	#106	—	4	2	no	—
99	#103	—	4	2	no	—
100	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
101	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
102	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
103	C₂×Q₈	—	4	1	no	—
104	C₂×Q₈	—	4	1	no	—
105	C₂×Q₈	—	4	2	no	—
106	Q₁₆	—	4	4	no	—
107	Q₁₆	—	8	4	no	—
108	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
109	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
110	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	1	no	—
113	C₄⋊C₄	—	4	1	no	—
114	C₄⋊C₄	—	4	1	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
118	C₁₆	—	2	4	no	—
119	Q₈	—	8	4	no	—
120	Q₈	—	8	2	no	—
121	Q₈	—	8	2	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	—	8	2	no	—
124	C₂×C₄	—	8	1	no	—
125	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
126	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
127	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	SD₃₂×C₁₀₉	1	4	yes	—
130	C₈	#137×C₁₀₉	1	4	yes	—
131	C₂²	#493×C₁₀₉	1	1	yes	characteristic
132	C₄	—	16	1	no	—
133	C₄	—	16	2	no	—
134	C₄	—	16	1	no	—
135	C₄	#497×C₁₀₉	1	2	yes	—
136	C₄	C₂×SD₃₂×C₁₀₉	1	2	yes	—
137	C₄	#510×C₁₀₉	1	2	yes	—
138	C₂	C₁₀₉×#1722	1	1	yes	characteristic
139	C₂	C₁₀₉×#2248	1	1	yes	characteristic
140	C₂	C₁₀₉×#2251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2254

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

#500×C₁₀₉

—

#500×C₁₀₉

C₂²

yes

characteristic

#500×C₁₀₉

C₂²

yes

—

#498×C₁₀₉

—

#494×C₁₀₉

—

#494×C₁₀₉

C₂²

yes

characteristic

#487×C₁₀₉

—

#367×C₁₀₉

—

#358×C₁₀₉

—

C₄×C₁₆×C₁₀₉

C₂²

yes

—

C₈²×C₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₂×C₁₆×C₁₀₉

D₄

yes

—

#107×C₁₀₉

—

#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

—

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

—

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

SD₃₂

yes

—

C₈×C₁₀₉

#137

yes

—

C₂²×C₁₀₉

#493

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#497

yes

—

C₄×C₁₀₉

C₂×SD₃₂

yes

—

C₄×C₁₀₉

#510

yes

—

#12535

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

#12535

yes

characteristic, direct product factor, Sylow

#500

—

#500

C₂²×C₁₀₉

yes

characteristic

#500

C₂²×C₁₀₉

yes

—

#498

—

#494

—

#494

C₂²×C₁₀₉

yes

—

—

—

C₄×C₁₆

C₂²×C₁₀₉

yes

—

C₈²

C₂²×C₁₀₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

D₄×C₁₀₉

yes

—

—

—

—

—

—

—

100

C₄×C₈

D₄×C₁₀₉

yes

characteristic

101

C₄×C₈

D₄×C₁₀₉

yes

characteristic

102

C₄×C₈

C₂³×C₁₀₉

yes

characteristic

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

Q₁₆

—

107

Q₁₆

—

108

C₂×C₈

D₈×C₁₀₉

yes

—

109

C₂×C₈

D₈×C₁₀₉

yes

—

110

C₂×C₈

C₂×D₄×C₁₀₉

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

118

C₁₆

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#127×C₁₀₉

yes

characteristic

126

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

127

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

128

C₈

—

129

C₈

SD₃₂×C₁₀₉

yes

—

130

C₈

#137×C₁₀₉

yes

—

131

C₂²

#493×C₁₀₉

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

#497×C₁₀₉

yes

—

136

C₄

C₂×SD₃₂×C₁₀₉

yes

—

137

C₄

#510×C₁₀₉

yes

—

138

C₂

C₁₀₉×#1722

yes

characteristic

139

C₂

C₁₀₉×#2248

yes

characteristic

140

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
67. #12535	C₁₀₉	direct (N × H)
76. C₁₀₉	#12535	direct (N × H)

H ≅ G/N

type

67. #12535

C₁₀₉

direct (N × H)

76. C₁₀₉

#12535

direct (N × H)

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

subgroup	index
C₁₀₉×#2254	2
C₁₀₉×#2241	2
C₁₀₉×#1713	2
C₁₀₉×#1707	2
C₁₀₉×#1312	2
#12535	109

subgroup

index

C₁₀₉×#2254

C₁₀₉×#2241

C₁₀₉×#1713

C₁₀₉×#1707

C₁₀₉×#1312

#12535

109

Subgroups — C₁₀₉×#12535 #42130529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12535 #42130529

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12535 #42130529