Group #42130674 — subgroups

Subgroups — C₁₀₉×#12680 #42130674

Order 27904.

958 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 230 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	C₂²×#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊C₁₆×C₁₀₉	C₂	1	6	yes	maximal, complemented
3	C₂²×#41×C₁₀₉	C₄	1	1	yes	characteristic
4	C₂×#131×C₁₀₉	—	2	12	no	—
5	C₂²×M₁₆×C₁₀₉	C₄	1	3	yes	—
6	C₂³×C₈×C₁₀₉	C₂²	1	3	yes	—
7	C₂²×C₁₆×C₁₀₉	C₂²	1	4	yes	—
8	C₂²⋊C₁₆×C₁₀₉	—	2	12	no	—
9	C₂×#41×C₁₀₉	—	2	6	no	—
10	C₂×#41×C₁₀₉	—	2	6	no	—
11	C₂²×Q₈×C₁₀₉	C₈	1	1	yes	characteristic
12	C₂²×D₄×C₁₀₉	C₈	1	3	yes	—
13	C₂³×C₄×C₁₀₉	C₂×C₄	1	3	yes	—
14	#131×C₁₀₉	—	2	32	no	—
15	C₂×M₁₆×C₁₀₉	—	2	6	no	—
16	C₂×M₁₆×C₁₀₉	—	2	12	no	—
17	C₂²×C₈×C₁₀₉	—	2	12	no	—
18	C₂²×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
19	C₂²×C₈×C₁₀₉	D₄	1	12	yes	—
20	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×C₁₆×C₁₀₉	—	2	12	no	—
22	C₂⁴×C₁₀₉	C₂×C₈	1	3	yes	—
23	#41×C₁₀₉	—	2	8	no	—
24	#41×C₁₀₉	—	2	24	no	—
25	C₂×Q₈×C₁₀₉	—	2	6	no	—
26	C₂×D₄×C₁₀₉	—	2	12	no	—
27	C₂×D₄×C₁₀₉	—	2	6	no	—
28	C₂²×C₄×C₁₀₉	—	2	6	no	—
29	C₂²×C₄×C₁₀₉	—	2	6	no	—
30	C₂²×C₄×C₁₀₉	C₂²⋊C₄	1	6	yes	—
31	C₂²×C₄×C₁₀₉	C₂×C₈	1	3	yes	—
32	C₂²×C₄×C₁₀₉	M₁₆	1	6	yes	—
33	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	M₁₆×C₁₀₉	—	2	24	no	—
35	C₂×C₈×C₁₀₉	—	2	6	no	—
36	C₂×C₈×C₁₀₉	—	2	24	no	—
37	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	6	yes	—
38	C₂×C₈×C₁₀₉	C₂×D₄	1	6	yes	—
39	C₁₆×C₁₀₉	—	4	4	no	—
40	C₂³×C₁₀₉	—	2	6	no	—
41	C₂³×C₁₀₉	—	2	12	no	—
42	C₂³×C₁₀₉	C₂²⋊C₈	1	6	yes	—
43	C₂³×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
44	Q₈×C₁₀₉	—	2	8	no	—
45	D₄×C₁₀₉	—	2	24	no	—
46	C₂×C₄×C₁₀₉	—	2	6	no	—
47	C₂×C₄×C₁₀₉	—	2	6	no	—
48	C₂×C₄×C₁₀₉	—	2	6	no	—
49	C₂×C₄×C₁₀₉	—	2	12	no	—
50	C₂×C₄×C₁₀₉	C₂²⋊C₈	1	6	yes	—
51	C₂×C₄×C₁₀₉	C₂×C₂²⋊C₄	1	3	yes	—
52	C₂×C₄×C₁₀₉	C₂×M₁₆	1	3	yes	—
53	C₈×C₁₀₉	—	2	6	no	—
54	C₈×C₁₀₉	#120	1	4	yes	—
55	C₂²×C₁₀₉	—	2	12	no	—
56	C₂²×C₁₀₉	—	2	6	no	—
57	C₂²×C₁₀₉	—	4	12	no	—
58	C₂²×C₁₀₉	C₂×C₂²⋊C₈	1	3	yes	—
59	C₂²×C₁₀₉	#504	1	4	yes	—
60	C₄×C₁₀₉	—	2	6	no	—
61	C₄×C₁₀₉	#379	1	1	yes	characteristic
62	C₄×C₁₀₉	#407	1	3	yes	—
63	#12680	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₁₀₉	—	4	6	no	—
65	C₂×C₁₀₉	#1779	1	1	yes	characteristic
66	C₂×C₁₀₉	#2113	1	6	yes	—
67	C₂²×#131	C₂×C₁₀₉	1	1	yes	characteristic
68	C₂×C₂²⋊C₁₆	C₂×C₁₀₉	1	6	yes	complemented
69	C₁₀₉	#12680	1	1	yes	characteristic, direct product factor, Sylow
70	C₂²×#41	C₄×C₁₀₉	1	1	yes	characteristic
71	C₂×#131	—	2	12	no	—
72	C₂²×M₁₆	C₄×C₁₀₉	1	3	yes	—
73	C₂³×C₈	C₂²×C₁₀₉	1	3	yes	—
74	C₂²×C₁₆	C₂²×C₁₀₉	1	4	yes	—
75	C₂²⋊C₁₆	—	2	12	no	—
76	C₂×#41	—	2	6	no	—
77	C₂×#41	—	2	6	no	—
78	C₂²×Q₈	C₈×C₁₀₉	1	1	yes	characteristic
79	C₂²×D₄	C₈×C₁₀₉	1	3	yes	—
80	C₂³×C₄	C₂×C₄×C₁₀₉	1	3	yes	—
81	#131	—	2	32	no	—
82	C₂×M₁₆	—	2	6	no	—
83	C₂×M₁₆	—	2	12	no	—
84	C₂²×C₈	—	2	12	no	—
85	C₂²×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
86	C₂²×C₈	D₄×C₁₀₉	1	12	yes	—
87	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
88	C₂×C₁₆	—	2	12	no	—
89	C₂⁴	C₂×C₈×C₁₀₉	1	3	yes	—
90	#41	—	2	8	no	—
91	#41	—	2	24	no	—
92	C₂×Q₈	—	2	6	no	—
93	C₂×D₄	—	2	12	no	—
94	C₂×D₄	—	2	6	no	—
95	C₂²×C₄	—	2	6	no	—
96	C₂²×C₄	—	2	6	no	—
97	C₂²×C₄	C₂²⋊C₄×C₁₀₉	1	6	yes	—
98	C₂²×C₄	C₂×C₈×C₁₀₉	1	3	yes	—
99	C₂²×C₄	M₁₆×C₁₀₉	1	6	yes	—
100	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
101	M₁₆	—	2	24	no	—
102	C₂×C₈	—	2	6	no	—
103	C₂×C₈	—	2	24	no	—
104	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	6	yes	—
105	C₂×C₈	C₂×D₄×C₁₀₉	1	6	yes	—
106	C₁₆	—	4	4	no	—
107	C₂³	—	2	12	no	—
108	C₂³	—	2	6	no	—
109	C₂³	C₂²⋊C₈×C₁₀₉	1	6	yes	—
110	C₂³	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
111	Q₈	—	2	8	no	—
112	D₄	—	2	24	no	—
113	C₂×C₄	—	2	6	no	—
114	C₂×C₄	—	2	6	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	2	12	no	—
117	C₂×C₄	C₂²⋊C₈×C₁₀₉	1	6	yes	—
118	C₂×C₄	C₂×C₂²⋊C₄×C₁₀₉	1	3	yes	—
119	C₂×C₄	C₂×M₁₆×C₁₀₉	1	3	yes	—
120	C₈	—	2	6	no	—
121	C₈	#120×C₁₀₉	1	4	yes	—
122	C₂²	—	2	6	no	—
123	C₂²	—	2	12	no	—
124	C₂²	—	4	12	no	—
125	C₂²	C₂×C₂²⋊C₈×C₁₀₉	1	3	yes	—
126	C₂²	#504×C₁₀₉	1	4	yes	—
127	C₄	—	2	6	no	—
128	C₄	#379×C₁₀₉	1	1	yes	characteristic
129	C₄	#407×C₁₀₉	1	3	yes	—
130	C₂	—	4	6	no	—
131	C₂	C₁₀₉×#1779	1	1	yes	characteristic
132	C₂	C₁₀₉×#2113	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#131×C₁₀₉

C₂

yes

maximal, characteristic

C₂×C₂²⋊C₁₆×C₁₀₉

C₂

yes

maximal, complemented

C₂²×#41×C₁₀₉

C₄

yes

characteristic

C₂×#131×C₁₀₉

—

C₂²×M₁₆×C₁₀₉

C₄

yes

—

C₂³×C₈×C₁₀₉

C₂²

yes

—

C₂²×C₁₆×C₁₀₉

C₂²

yes

—

C₂²⋊C₁₆×C₁₀₉

—

C₂×#41×C₁₀₉

—

C₂×#41×C₁₀₉

—

C₂²×Q₈×C₁₀₉

C₈

yes

characteristic

C₂²×D₄×C₁₀₉

C₈

yes

—

C₂³×C₄×C₁₀₉

C₂×C₄

yes

—

#131×C₁₀₉

—

C₂×M₁₆×C₁₀₉

—

C₂×M₁₆×C₁₀₉

—

C₂²×C₈×C₁₀₉

—

C₂²×C₈×C₁₀₉

C₂×C₄

yes

—

C₂²×C₈×C₁₀₉

D₄

yes

—

C₂²×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₁₆×C₁₀₉

—

C₂⁴×C₁₀₉

C₂×C₈

yes

—

#41×C₁₀₉

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₈

yes

—

C₂²×C₄×C₁₀₉

M₁₆

yes

—

C₂²×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₈×C₁₀₉

C₂×D₄

yes

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₂²⋊C₈

yes

—

C₂³×C₁₀₉

C₂²×C₈

yes

characteristic

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₈

yes

—

C₂×C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×M₁₆

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

#120

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₈

yes

—

C₂²×C₁₀₉

#504

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#379

yes

characteristic

C₄×C₁₀₉

#407

yes

—

#12680

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#1779

yes

characteristic

C₂×C₁₀₉

#2113

yes

—

C₂²×#131

C₂×C₁₀₉

yes

characteristic

C₂×C₂²⋊C₁₆

C₂×C₁₀₉

yes

complemented

C₁₀₉

#12680

yes

characteristic, direct product factor, Sylow

C₂²×#41

C₄×C₁₀₉

yes

characteristic

C₂×#131

—

C₂²×M₁₆

C₄×C₁₀₉

yes

—

C₂³×C₈

C₂²×C₁₀₉

yes

—

C₂²×C₁₆

C₂²×C₁₀₉

yes

—

C₂²⋊C₁₆

—

C₂×#41

—

C₂×#41

—

C₂²×Q₈

C₈×C₁₀₉

yes

characteristic

C₂²×D₄

C₈×C₁₀₉

yes

—

C₂³×C₄

C₂×C₄×C₁₀₉

yes

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₁₀₉

yes

—

C₂²×C₈

D₄×C₁₀₉

yes

—

C₂²×C₈

C₂³×C₁₀₉

yes

characteristic

C₂×C₁₆

—

C₂⁴

C₂×C₈×C₁₀₉

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₁₀₉

yes

—

C₂²×C₄

C₂×C₈×C₁₀₉

yes

—

C₂²×C₄

M₁₆×C₁₀₉

yes

—

100

C₂²×C₄

C₂²×C₄×C₁₀₉

yes

characteristic

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂²⋊C₄×C₁₀₉

yes

—

105

C₂×C₈

C₂×D₄×C₁₀₉

yes

—

106

C₁₆

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²⋊C₈×C₁₀₉

yes

—

110

C₂³

C₂²×C₈×C₁₀₉

yes

characteristic

111

Q₈

—

112

D₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²⋊C₈×C₁₀₉

yes

—

118

C₂×C₄

C₂×C₂²⋊C₄×C₁₀₉

yes

—

119

C₂×C₄

C₂×M₁₆×C₁₀₉

yes

—

120

C₈

—

121

C₈

#120×C₁₀₉

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×C₂²⋊C₈×C₁₀₉

yes

—

126

C₂²

#504×C₁₀₉

yes

—

127

C₄

—

128

C₄

#379×C₁₀₉

yes

characteristic

129

C₄

#407×C₁₀₉

yes

—

130

C₂

—

131

C₂

C₁₀₉×#1779

yes

characteristic

132

C₂

C₁₀₉×#2113

yes

—

	subgroup	G/H
center	C₂²×C₈×C₁₀₉	C₂³
commutator	C₂³	C₂²×C₈×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂³×C₁₀₉	C₂²×C₈
prime radical	C₂²×#41×C₁₀₉	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₀₉ ◁ G

subgroup

G/H

center

C₂²×C₈×C₁₀₉

C₂³

commutator

C₂³

C₂²×C₈×C₁₀₉

Frattini, cosocle

C₂²×C₈

C₂³×C₁₀₉

socle

C₂³×C₁₀₉

C₂²×C₈

prime radical

C₂²×#41×C₁₀₉

C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
2. C₂×C₂²⋊C₁₆×C₁₀₉	C₂	semidirect
63. #12680	C₁₀₉	direct (N × H)
68. C₂×C₂²⋊C₁₆	C₂×C₁₀₉	semidirect
69. C₁₀₉	#12680	direct (N × H)

H ≅ G/N

type

2. C₂×C₂²⋊C₁₆×C₁₀₉

C₂

semidirect

63. #12680

C₁₀₉

direct (N × H)

68. C₂×C₂²⋊C₁₆

C₂×C₁₀₉

semidirect

69. C₁₀₉

#12680

direct (N × H)

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

subgroup	index
C₂²×#131×C₁₀₉	2
C₂×C₂²⋊C₁₆×C₁₀₉	2
#12680	109

subgroup

index

C₂²×#131×C₁₀₉

C₂×C₂²⋊C₁₆×C₁₀₉

#12680

109

Subgroups — C₁₀₉×#12680 #42130674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12680 #42130674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12680 #42130674