Group #42130503 — subgroups

Subgroups — C₁₀₉×#12509 #42130503

Order 27904.

306 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 110 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×#2240	C₂	1	4	yes	maximal, complemented
2	C₁₀₉×#1713	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#1712	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#500×C₁₀₉	—	2	1	no	—
6	#500×C₁₀₉	C₂²	1	2	yes	—
7	#494×C₁₀₉	—	2	1	no	—
8	#493×C₁₀₉	—	2	1	no	—
9	#493×C₁₀₉	—	2	1	no	—
10	#493×C₁₀₉	C₂²	1	2	yes	—
11	#357×C₁₀₉	—	2	8	no	—
12	C₄×C₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₁₀₉	—	4	2	no	—
15	C₂×D₈×C₁₀₉	—	2	4	no	—
16	C₂×D₈×C₁₀₉	—	4	2	no	—
17	C₂×D₈×C₁₀₉	—	4	2	no	—
18	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
19	#127×C₁₀₉	—	2	2	no	—
20	C₂×C₁₆×C₁₀₉	D₄	1	4	yes	—
21	#107×C₁₀₉	—	2	4	no	—
22	#107×C₁₀₉	—	4	2	no	—
23	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₀₉	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₁₀₉	—	4	2	no	—
27	C₂×D₄×C₁₀₉	—	4	2	no	—
28	C₂×D₄×C₁₀₉	—	4	2	no	—
29	C₂×D₄×C₁₀₉	—	4	2	no	—
30	Q₁₆×C₁₀₉	—	8	4	no	—
31	D₈×C₁₀₉	—	4	8	no	—
32	D₈×C₁₀₉	—	8	4	no	—
33	D₈×C₁₀₉	—	8	4	no	—
34	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
35	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
36	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
37	C₄⋊C₄×C₁₀₉	—	4	2	no	—
38	C₄⋊C₄×C₁₀₉	—	4	2	no	—
39	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₁₀₉	—	2	4	no	—
41	C₂³×C₁₀₉	—	8	2	no	—
42	Q₈×C₁₀₉	—	8	4	no	—
43	D₄×C₁₀₉	—	8	4	no	—
44	D₄×C₁₀₉	—	8	4	no	—
45	D₄×C₁₀₉	—	8	4	no	—
46	C₂×C₄×C₁₀₉	—	8	2	no	—
47	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
50	C₈×C₁₀₉	D₁₆	1	4	yes	—
51	C₈×C₁₀₉	SD₃₂	1	4	yes	—
52	C₈×C₁₀₉	#135	1	4	yes	—
53	C₂²×C₁₀₉	—	16	2	no	—
54	C₂²×C₁₀₉	—	16	2	no	—
55	C₂²×C₁₀₉	—	16	2	no	—
56	C₂²×C₁₀₉	#493	1	1	yes	characteristic
57	C₄×C₁₀₉	—	16	2	no	—
58	C₄×C₁₀₉	#495	1	2	yes	—
59	C₄×C₁₀₉	C₂×D₁₆	1	2	yes	—
60	C₄×C₁₀₉	C₂×SD₃₂	1	2	yes	—
61	#12509	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₁₀₉	—	32	2	no	—
63	C₂×C₁₀₉	#1709	1	1	yes	characteristic
64	C₂×C₁₀₉	#2246	1	1	yes	characteristic
65	C₂×C₁₀₉	#2248	1	1	yes	characteristic
66	#2240	C₂×C₁₀₉	1	4	yes	complemented
67	#1713	C₂×C₁₀₉	1	1	yes	characteristic, complemented
68	#1712	C₂×C₁₀₉	1	1	yes	characteristic
69	C₈×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
70	C₁₀₉	#12509	1	1	yes	characteristic, direct product factor, Sylow
71	#500	—	2	1	no	—
72	#500	C₂²×C₁₀₉	1	2	yes	—
73	#494	—	2	1	no	—
74	#493	—	2	1	no	—
75	#493	—	2	1	no	—
76	#493	C₂²×C₁₀₉	1	2	yes	—
77	#357	—	2	8	no	—
78	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
79	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
80	C₂×Q₁₆	—	4	2	no	—
81	C₂×D₈	—	2	4	no	—
82	C₂×D₈	—	4	2	no	—
83	C₂×D₈	—	4	2	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	#127	—	2	2	no	—
86	C₂×C₁₆	D₄×C₁₀₉	1	4	yes	—
87	#107	—	2	4	no	—
88	#107	—	4	2	no	—
89	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
90	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
91	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
92	C₂×Q₈	—	4	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	—	4	2	no	—
95	C₂×D₄	—	4	2	no	—
96	Q₁₆	—	8	4	no	—
97	D₈	—	4	8	no	—
98	D₈	—	8	4	no	—
99	D₈	—	8	4	no	—
100	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
101	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
102	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
106	C₁₆	—	2	4	no	—
107	C₂³	—	8	2	no	—
108	Q₈	—	8	4	no	—
109	D₄	—	8	4	no	—
110	D₄	—	8	4	no	—
111	D₄	—	8	4	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
114	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
115	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
116	C₈	D₁₆×C₁₀₉	1	4	yes	—
117	C₈	SD₃₂×C₁₀₉	1	4	yes	—
118	C₈	#135×C₁₀₉	1	4	yes	—
119	C₂²	—	16	2	no	—
120	C₂²	—	16	2	no	—
121	C₂²	—	16	2	no	—
122	C₂²	#493×C₁₀₉	1	1	yes	characteristic
123	C₄	—	16	2	no	—
124	C₄	#495×C₁₀₉	1	2	yes	—
125	C₄	C₂×D₁₆×C₁₀₉	1	2	yes	—
126	C₄	C₂×SD₃₂×C₁₀₉	1	2	yes	—
127	C₂	—	32	2	no	—
128	C₂	C₁₀₉×#1709	1	1	yes	characteristic
129	C₂	C₁₀₉×#2246	1	1	yes	characteristic
130	C₂	C₁₀₉×#2248	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2240

C₂

yes

maximal, complemented

C₁₀₉×#1713

C₂

yes

maximal, characteristic, complemented

C₁₀₉×#1712

C₂

yes

maximal, characteristic

C₈×C₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#500×C₁₀₉

—

#500×C₁₀₉

C₂²

yes

—

#494×C₁₀₉

—

#493×C₁₀₉

—

#493×C₁₀₉

—

#493×C₁₀₉

C₂²

yes

—

#357×C₁₀₉

—

C₄×C₁₆×C₁₀₉

C₂²

yes

—

C₈²×C₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

#127×C₁₀₉

—

C₂×C₁₆×C₁₀₉

D₄

yes

—

#107×C₁₀₉

—

#107×C₁₀₉

—

C₄×C₈×C₁₀₉

D₄

yes

characteristic

C₄×C₈×C₁₀₉

D₄

yes

characteristic

C₄×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

Q₁₆×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₈

yes

—

C₂×C₈×C₁₀₉

D₈

yes

—

C₂×C₈×C₁₀₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

characteristic

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×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

D₁₆

yes

—

C₈×C₁₀₉

SD₃₂

yes

—

C₈×C₁₀₉

#135

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#493

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

#495

yes

—

C₄×C₁₀₉

C₂×D₁₆

yes

—

C₄×C₁₀₉

C₂×SD₃₂

yes

—

#12509

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

complemented

#1713

C₂×C₁₀₉

yes

characteristic, complemented

#1712

C₂×C₁₀₉

yes

characteristic

C₈×C₁₆

C₂×C₁₀₉

yes

characteristic, complemented

C₁₀₉

#12509

yes

characteristic, direct product factor, Sylow

#500

—

#500

C₂²×C₁₀₉

yes

—

—

—

—

C₂²×C₁₀₉

yes

—

#357

—

C₄×C₁₆

C₂²×C₁₀₉

yes

—

C₈²

C₂²×C₁₀₉

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₄×C₁₀₉

yes

—

#107

—

#107

—

C₄×C₈

D₄×C₁₀₉

yes

characteristic

C₄×C₈

D₄×C₁₀₉

yes

characteristic

C₄×C₈

C₂³×C₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

D₈

—

D₈

—

100

C₂×C₈

D₈×C₁₀₉

yes

—

101

C₂×C₈

D₈×C₁₀₉

yes

—

102

C₂×C₈

C₂×D₄×C₁₀₉

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

106

C₁₆

—

107

C₂³

—

108

Q₈

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

#127×C₁₀₉

yes

characteristic

114

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

115

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

116

C₈

D₁₆×C₁₀₉

yes

—

117

C₈

SD₃₂×C₁₀₉

yes

—

118

C₈

#135×C₁₀₉

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#493×C₁₀₉

yes

characteristic

123

C₄

—

124

C₄

#495×C₁₀₉

yes

—

125

C₄

C₂×D₁₆×C₁₀₉

yes

—

126

C₄

C₂×SD₃₂×C₁₀₉

yes

—

127

C₂

—

128

C₂

C₁₀₉×#1709

yes

characteristic

129

C₂

C₁₀₉×#2246

yes

characteristic

130

C₂

C₁₀₉×#2248

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
prime radical	C₁₀₉×#1712	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₈²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₀₉

prime radical

C₁₀₉×#1712

C₂

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
1. C₁₀₉×#2240	C₂	semidirect
2. C₁₀₉×#1713	C₂	semidirect
4. C₈×C₁₆×C₁₀₉	C₂	semidirect
61. #12509	C₁₀₉	direct (N × H)
66. #2240	C₂×C₁₀₉	semidirect
67. #1713	C₂×C₁₀₉	semidirect
69. C₈×C₁₆	C₂×C₁₀₉	semidirect
70. C₁₀₉	#12509	direct (N × H)

H ≅ G/N

type

1. C₁₀₉×#2240

C₂

semidirect

2. C₁₀₉×#1713

C₂

semidirect

4. C₈×C₁₆×C₁₀₉

C₂

semidirect

61. #12509

C₁₀₉

direct (N × H)

66. #2240

C₂×C₁₀₉

semidirect

67. #1713

C₂×C₁₀₉

semidirect

69. C₈×C₁₆

C₂×C₁₀₉

semidirect

70. C₁₀₉

#12509

direct (N × H)

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

subgroup	index
C₁₀₉×#2240	2
C₁₀₉×#1713	2
C₁₀₉×#1712	2
C₈×C₁₆×C₁₀₉	2
#12509	109

subgroup

index

C₁₀₉×#2240

C₁₀₉×#1713

C₁₀₉×#1712

C₈×C₁₆×C₁₀₉

#12509

109

Subgroups — C₁₀₉×#12509 #42130503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12509 #42130503

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12509 #42130503