Group #42131641 — subgroups

Subgroups — C₁₀₉×#13647 #42131641

Order 27904.

186 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2258	C₂	1	2	yes	maximal, complemented
2	C₁₀₉×#2170	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1422	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₀₉×#1401	C₂	1	2	yes	maximal, complemented
5	C₄×C₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#504×C₁₀₉	C₂²	1	2	yes	—
7	#443×C₁₀₉	C₄	1	1	yes	characteristic
8	M₆₄×C₁₀₉	—	2	2	no	—
9	M₆₄×C₁₀₉	C₂²	1	2	yes	—
10	C₂×C₃₂×C₁₀₉	—	2	2	no	—
11	C₂×C₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₁₀₉	C₂²	1	1	yes	characteristic
13	#364×C₁₀₉	C₄	1	1	yes	characteristic
14	#350×C₁₀₉	C₄	1	2	yes	—
15	C₄×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
16	#135×C₁₀₉	C₈	1	1	yes	characteristic
17	#131×C₁₀₉	C₂×C₄	1	2	yes	—
18	M₃₂×C₁₀₉	—	2	2	no	—
19	M₃₂×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₁₀₉	—	2	2	no	—
21	C₂×C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₁₀₉	C₂³	1	1	yes	characteristic
23	#108×C₁₀₉	C₈	1	1	yes	characteristic
24	#104×C₁₀₉	C₈	1	2	yes	—
25	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₃₂×C₁₀₉	—	2	2	no	—
27	C₃₂×C₁₀₉	—	2	1	no	—
28	C₃₂×C₁₀₉	D₄	1	1	yes	characteristic
29	C₃₂×C₁₀₉	D₄	1	1	yes	characteristic
30	#41×C₁₀₉	C₂×C₈	1	2	yes	—
31	Q₁₆×C₁₀₉	C₁₆	1	1	yes	characteristic
32	SD₁₆×C₁₀₉	C₁₆	1	2	yes	—
33	D₈×C₁₀₉	C₁₆	1	1	yes	characteristic
34	M₁₆×C₁₀₉	—	2	2	no	—
35	M₁₆×C₁₀₉	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₁₀₉	—	2	2	no	—
37	C₂×C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₁₀₉	C₂×C₈	1	1	yes	characteristic
40	C₁₆×C₁₀₉	—	2	2	no	—
41	C₁₆×C₁₀₉	—	2	1	no	—
42	C₁₆×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₁₀₉	#41	1	1	yes	characteristic
44	Q₈×C₁₀₉	C₂×C₁₆	1	2	yes	—
45	D₄×C₁₀₉	—	2	2	no	—
46	D₄×C₁₀₉	C₂×C₁₆	1	2	yes	—
47	C₂×C₄×C₁₀₉	—	2	1	no	—
48	C₂×C₄×C₁₀₉	—	2	2	no	—
49	C₂×C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₁₀₉	—	2	2	no	—
51	C₈×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
52	C₈×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
53	C₈×C₁₀₉	C₄×D₄	1	1	yes	characteristic
54	C₈×C₁₀₉	#131	1	1	yes	characteristic
55	C₂²×C₁₀₉	—	2	2	no	—
56	C₂²×C₁₀₉	#504	1	1	yes	characteristic
57	C₄×C₁₀₉	—	2	2	no	—
58	C₄×C₁₀₉	—	2	2	no	—
59	C₄×C₁₀₉	C₈×D₄	1	1	yes	characteristic
60	C₄×C₁₀₉	C₂²×C₁₆	1	1	yes	characteristic
61	#13647	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₁₀₉	—	2	1	no	—
63	C₂×C₁₀₉	—	4	2	no	—
64	C₂×C₁₀₉	D₄×C₁₆	1	1	yes	characteristic
65	#2258	C₂×C₁₀₉	1	2	yes	complemented
66	#2170	C₂×C₁₀₉	1	1	yes	characteristic
67	#1422	C₂×C₁₀₉	1	1	yes	characteristic, complemented
68	#1401	C₂×C₁₀₉	1	2	yes	complemented
69	C₄×C₃₂	C₂×C₁₀₉	1	1	yes	characteristic, complemented
70	C₁₀₉	#13647	1	1	yes	characteristic, direct product factor, Sylow
71	#504	C₂²×C₁₀₉	1	2	yes	—
72	#443	C₄×C₁₀₉	1	1	yes	characteristic
73	M₆₄	—	2	2	no	—
74	M₆₄	C₂²×C₁₀₉	1	2	yes	—
75	C₂×C₃₂	—	2	2	no	—
76	C₂×C₃₂	C₂²×C₁₀₉	1	1	yes	characteristic
77	C₂×C₃₂	C₂²×C₁₀₉	1	1	yes	characteristic
78	#364	C₄×C₁₀₉	1	1	yes	characteristic
79	#350	C₄×C₁₀₉	1	2	yes	—
80	C₄×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
81	#135	C₈×C₁₀₉	1	1	yes	characteristic
82	#131	C₂×C₄×C₁₀₉	1	2	yes	—
83	M₃₂	—	2	2	no	—
84	M₃₂	C₂×C₄×C₁₀₉	1	2	yes	—
85	C₂×C₁₆	—	2	2	no	—
86	C₂×C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
87	C₂×C₁₆	C₂³×C₁₀₉	1	1	yes	characteristic
88	#108	C₈×C₁₀₉	1	1	yes	characteristic
89	#104	C₈×C₁₀₉	1	2	yes	—
90	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
91	C₃₂	—	2	2	no	—
92	C₃₂	—	2	1	no	—
93	C₃₂	D₄×C₁₀₉	1	1	yes	characteristic
94	C₃₂	D₄×C₁₀₉	1	1	yes	characteristic
95	#41	C₂×C₈×C₁₀₉	1	2	yes	—
96	Q₁₆	C₁₆×C₁₀₉	1	1	yes	characteristic
97	SD₁₆	C₁₆×C₁₀₉	1	2	yes	—
98	D₈	C₁₆×C₁₀₉	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	C₂×C₈×C₁₀₉	1	2	yes	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
103	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
104	C₄²	C₂×C₈×C₁₀₉	1	1	yes	characteristic
105	C₁₆	—	2	2	no	—
106	C₁₆	—	2	1	no	—
107	C₁₆	C₂×D₄×C₁₀₉	1	1	yes	characteristic
108	C₁₆	#41×C₁₀₉	1	1	yes	characteristic
109	Q₈	C₂×C₁₆×C₁₀₉	1	2	yes	—
110	D₄	—	2	2	no	—
111	D₄	C₂×C₁₆×C₁₀₉	1	2	yes	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
115	C₈	—	2	2	no	—
116	C₈	C₂×C₁₆×C₁₀₉	1	1	yes	characteristic
117	C₈	C₂×C₁₆×C₁₀₉	1	1	yes	characteristic
118	C₈	C₄×D₄×C₁₀₉	1	1	yes	characteristic
119	C₈	#131×C₁₀₉	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	#504×C₁₀₉	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	C₈×D₄×C₁₀₉	1	1	yes	characteristic
125	C₄	C₂²×C₁₆×C₁₀₉	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	4	2	no	—
128	C₂	D₄×C₁₆×C₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2258

C₂

yes

maximal, complemented

C₁₀₉×#2170

C₂

yes

maximal, characteristic

C₁₀₉×#1422

C₂

yes

maximal, characteristic, complemented

C₁₀₉×#1401

C₂

yes

maximal, complemented

C₄×C₃₂×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#504×C₁₀₉

C₂²

yes

—

#443×C₁₀₉

C₄

yes

characteristic

M₆₄×C₁₀₉

—

M₆₄×C₁₀₉

C₂²

yes

—

C₂×C₃₂×C₁₀₉

—

C₂×C₃₂×C₁₀₉

C₂²

yes

characteristic

C₂×C₃₂×C₁₀₉

C₂²

yes

characteristic

#364×C₁₀₉

C₄

yes

characteristic

#350×C₁₀₉

C₄

yes

—

C₄×C₁₆×C₁₀₉

C₂²

yes

characteristic

#135×C₁₀₉

C₈

yes

characteristic

#131×C₁₀₉

C₂×C₄

yes

—

M₃₂×C₁₀₉

—

M₃₂×C₁₀₉

C₂×C₄

yes

—

C₂×C₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂³

yes

characteristic

#108×C₁₀₉

C₈

yes

characteristic

#104×C₁₀₉

C₈

yes

—

C₄×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₃₂×C₁₀₉

—

C₃₂×C₁₀₉

—

C₃₂×C₁₀₉

D₄

yes

characteristic

C₃₂×C₁₀₉

D₄

yes

characteristic

#41×C₁₀₉

C₂×C₈

yes

—

Q₁₆×C₁₀₉

C₁₆

yes

characteristic

SD₁₆×C₁₀₉

C₁₆

yes

—

D₈×C₁₀₉

C₁₆

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

C₂×C₈

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₈×C₁₀₉

C₂²×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₈

yes

characteristic

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

C₂×D₄

yes

characteristic

C₁₆×C₁₀₉

#41

yes

characteristic

Q₈×C₁₀₉

C₂×C₁₆

yes

—

D₄×C₁₀₉

—

D₄×C₁₀₉

C₂×C₁₆

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×C₁₆

yes

characteristic

C₈×C₁₀₉

C₂×C₁₆

yes

characteristic

C₈×C₁₀₉

C₄×D₄

yes

characteristic

C₈×C₁₀₉

#131

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#504

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₈×D₄

yes

characteristic

C₄×C₁₀₉

C₂²×C₁₆

yes

characteristic

#13647

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

D₄×C₁₆

yes

characteristic

#2258

C₂×C₁₀₉

yes

complemented

#2170

C₂×C₁₀₉

yes

characteristic

#1422

C₂×C₁₀₉

yes

characteristic, complemented

#1401

C₂×C₁₀₉

yes

complemented

C₄×C₃₂

C₂×C₁₀₉

yes

characteristic, complemented

C₁₀₉

#13647

yes

characteristic, direct product factor, Sylow

#504

C₂²×C₁₀₉

yes

—

#443

C₄×C₁₀₉

yes

characteristic

M₆₄

—

M₆₄

C₂²×C₁₀₉

yes

—

C₂×C₃₂

—

C₂×C₃₂

C₂²×C₁₀₉

yes

characteristic

C₂×C₃₂

C₂²×C₁₀₉

yes

characteristic

#364

C₄×C₁₀₉

yes

characteristic

#350

C₄×C₁₀₉

yes

—

C₄×C₁₆

C₂²×C₁₀₉

yes

characteristic

#135

C₈×C₁₀₉

yes

characteristic

#131

C₂×C₄×C₁₀₉

yes

—

M₃₂

—

M₃₂

C₂×C₄×C₁₀₉

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₁₀₉

yes

characteristic

C₂×C₁₆

C₂³×C₁₀₉

yes

characteristic

#108

C₈×C₁₀₉

yes

characteristic

#104

C₈×C₁₀₉

yes

—

C₄×C₈

C₂×C₄×C₁₀₉

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

D₄×C₁₀₉

yes

characteristic

C₃₂

D₄×C₁₀₉

yes

characteristic

#41

C₂×C₈×C₁₀₉

yes

—

Q₁₆

C₁₆×C₁₀₉

yes

characteristic

SD₁₆

C₁₆×C₁₀₉

yes

—

D₈

C₁₆×C₁₀₉

yes

characteristic

M₁₆

—

100

M₁₆

C₂×C₈×C₁₀₉

yes

—

101

C₂×C₈

—

102

C₂×C₈

C₂×C₈×C₁₀₉

yes

characteristic

103

C₂×C₈

C₂²×C₄×C₁₀₉

yes

characteristic

104

C₄²

C₂×C₈×C₁₀₉

yes

characteristic

105

C₁₆

—

106

C₁₆

—

107

C₁₆

C₂×D₄×C₁₀₉

yes

characteristic

108

C₁₆

#41×C₁₀₉

yes

characteristic

109

Q₈

C₂×C₁₆×C₁₀₉

yes

—

110

D₄

—

111

D₄

C₂×C₁₆×C₁₀₉

yes

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²×C₈×C₁₀₉

yes

characteristic

115

C₈

—

116

C₈

C₂×C₁₆×C₁₀₉

yes

characteristic

117

C₈

C₂×C₁₆×C₁₀₉

yes

characteristic

118

C₈

C₄×D₄×C₁₀₉

yes

characteristic

119

C₈

#131×C₁₀₉

yes

characteristic

120

C₂²

—

121

C₂²

#504×C₁₀₉

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

C₈×D₄×C₁₀₉

yes

characteristic

125

C₄

C₂²×C₁₆×C₁₀₉

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

D₄×C₁₆×C₁₀₉

yes

characteristic

	subgroup	G/H
center	C₃₂×C₁₀₉	D₄
commutator	C₄	C₂²×C₁₆×C₁₀₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₀₉
socle	C₂×C₁₀₉	D₄×C₁₆
prime radical	#135×C₁₀₉	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂×C₁₀₉ ◁ C₂×C₃₂×C₁₀₉ ◁ G

subgroup

G/H

center

C₃₂×C₁₀₉

D₄

commutator

C₄

C₂²×C₁₆×C₁₀₉

Frattini, cosocle

C₂×C₁₆

C₂³×C₁₀₉

socle

C₂×C₁₀₉

D₄×C₁₆

prime radical

#135×C₁₀₉

C₈

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₃₂×C₁₀₉ ◁ C₂×C₃₂×C₁₀₉ ◁ G

N	H ≅ G/N	type
1. C₁₀₉×#2258	C₂	semidirect
3. C₁₀₉×#1422	C₂	semidirect
4. C₁₀₉×#1401	C₂	semidirect
5. C₄×C₃₂×C₁₀₉	C₂	semidirect
61. #13647	C₁₀₉	direct (N × H)
65. #2258	C₂×C₁₀₉	semidirect
67. #1422	C₂×C₁₀₉	semidirect
68. #1401	C₂×C₁₀₉	semidirect
69. C₄×C₃₂	C₂×C₁₀₉	semidirect
70. C₁₀₉	#13647	direct (N × H)

H ≅ G/N

type

1. C₁₀₉×#2258

C₂

semidirect

3. C₁₀₉×#1422

C₂

semidirect

4. C₁₀₉×#1401

C₂

semidirect

5. C₄×C₃₂×C₁₀₉

C₂

semidirect

61. #13647

C₁₀₉

direct (N × H)

65. #2258

C₂×C₁₀₉

semidirect

67. #1422

C₂×C₁₀₉

semidirect

68. #1401

C₂×C₁₀₉

semidirect

69. C₄×C₃₂

C₂×C₁₀₉

semidirect

70. C₁₀₉

#13647

direct (N × H)

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

subgroup	index
C₁₀₉×#2258	2
C₁₀₉×#2170	2
C₁₀₉×#1422	2
C₁₀₉×#1401	2
C₄×C₃₂×C₁₀₉	2
#13647	109

subgroup

index

C₁₀₉×#2258

C₁₀₉×#2170

C₁₀₉×#1422

C₁₀₉×#1401

C₄×C₃₂×C₁₀₉

#13647

109

Subgroups — C₁₀₉×#13647 #42131641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#13647 #42131641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#13647 #42131641