Group #42130512 — subgroups

Subgroups — C₁₀₉×#12518 #42130512

Order 27904.

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

normal subgroups only

This group has 330 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×#2248	C₂	1	2	yes	maximal, complemented
2	C₁₀₉×#2247	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₀₉×#2246	C₂	1	1	yes	maximal, characteristic
4	C₁₀₉×#1709	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₃₂×C₁₀₉	—	2	2	no	—
7	C₂×SD₃₂×C₁₀₉	—	2	4	no	—
8	C₂×D₁₆×C₁₀₉	—	2	2	no	—
9	#494×C₁₀₉	C₂²	1	2	yes	—
10	#493×C₁₀₉	C₂²	1	2	yes	—
11	#486×C₁₀₉	—	2	4	no	—
12	C₄×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
16	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
17	C₂×D₈×C₁₀₉	—	2	2	no	—
18	C₂×D₈×C₁₀₉	—	2	2	no	—
19	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
20	#127×C₁₀₉	—	2	2	no	—
21	Q₃₂×C₁₀₉	—	4	4	no	—
22	SD₃₂×C₁₀₉	—	4	8	no	—
23	D₁₆×C₁₀₉	—	4	4	no	—
24	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
25	C₂×C₁₆×C₁₀₉	D₄	1	2	yes	—
26	#102×C₁₀₉	—	4	8	no	—
27	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
28	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₁₀₉	—	4	2	no	—
30	C₂×D₄×C₁₀₉	—	4	2	no	—
31	C₂×D₄×C₁₀₉	—	4	4	no	—
32	Q₁₆×C₁₀₉	—	4	4	no	—
33	Q₁₆×C₁₀₉	—	4	4	no	—
34	D₈×C₁₀₉	—	4	4	no	—
35	D₈×C₁₀₉	—	4	4	no	—
36	C₂×C₈×C₁₀₉	D₈	1	4	yes	—
37	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₁₀₉	—	4	4	no	—
40	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₁₀₉	D₈	1	4	yes	—
42	C₁₆×C₁₀₉	D₈	1	4	yes	—
43	C₂³×C₁₀₉	—	8	2	no	—
44	Q₈×C₁₀₉	—	8	2	no	—
45	Q₈×C₁₀₉	—	8	2	no	—
46	D₄×C₁₀₉	—	8	2	no	—
47	D₄×C₁₀₉	—	8	2	no	—
48	D₄×C₁₀₉	—	8	8	no	—
49	C₂×C₄×C₁₀₉	—	8	2	no	—
50	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂×D₈	1	2	yes	—
52	C₈×C₁₀₉	—	2	4	no	—
53	C₈×C₁₀₉	C₂×D₈	1	2	yes	—
54	C₈×C₁₀₉	C₂×D₈	1	2	yes	—
55	C₂²×C₁₀₉	—	16	2	no	—
56	C₂²×C₁₀₉	—	16	4	no	—
57	C₂²×C₁₀₉	#493	1	1	yes	characteristic
58	C₄×C₁₀₉	—	16	2	no	—
59	C₄×C₁₀₉	#493	1	1	yes	characteristic
60	C₄×C₁₀₉	#493	1	1	yes	characteristic
61	C₄×C₁₀₉	#508	1	4	yes	—
62	#12518	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×C₁₀₉	—	32	2	no	—
64	C₂×C₁₀₉	#1712	1	1	yes	characteristic
65	C₂×C₁₀₉	#2249	1	2	yes	—
66	#2248	C₂×C₁₀₉	1	2	yes	complemented
67	#2247	C₂×C₁₀₉	1	1	yes	characteristic, complemented
68	#2246	C₂×C₁₀₉	1	1	yes	characteristic
69	#1709	C₂×C₁₀₉	1	2	yes	complemented
70	C₈×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
71	C₁₀₉	#12518	1	1	yes	characteristic, direct product factor, Sylow
72	C₂×Q₃₂	—	2	2	no	—
73	C₂×SD₃₂	—	2	4	no	—
74	C₂×D₁₆	—	2	2	no	—
75	#494	C₂²×C₁₀₉	1	2	yes	—
76	#493	C₂²×C₁₀₉	1	2	yes	—
77	#486	—	2	4	no	—
78	C₄×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
79	C₄×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
80	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×D₈	—	2	2	no	—
84	C₂×D₈	—	2	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	#127	—	2	2	no	—
87	Q₃₂	—	4	4	no	—
88	SD₃₂	—	4	8	no	—
89	D₁₆	—	4	4	no	—
90	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
91	C₂×C₁₆	D₄×C₁₀₉	1	2	yes	—
92	#102	—	4	8	no	—
93	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
94	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	4	2	no	—
97	C₂×D₄	—	4	4	no	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	4	4	no	—
100	D₈	—	4	4	no	—
101	D₈	—	4	4	no	—
102	C₂×C₈	D₈×C₁₀₉	1	4	yes	—
103	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
104	C₂×C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
105	C₄⋊C₄	—	4	4	no	—
106	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
107	C₁₆	D₈×C₁₀₉	1	4	yes	—
108	C₁₆	D₈×C₁₀₉	1	4	yes	—
109	C₂³	—	8	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	8	8	no	—
113	D₄	—	8	2	no	—
114	D₄	—	8	2	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
117	C₂×C₄	C₂×D₈×C₁₀₉	1	2	yes	—
118	C₈	—	2	4	no	—
119	C₈	C₂×D₈×C₁₀₉	1	2	yes	—
120	C₈	C₂×D₈×C₁₀₉	1	2	yes	—
121	C₂²	—	16	2	no	—
122	C₂²	—	16	4	no	—
123	C₂²	#493×C₁₀₉	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	#493×C₁₀₉	1	1	yes	characteristic
126	C₄	#493×C₁₀₉	1	1	yes	characteristic
127	C₄	#508×C₁₀₉	1	4	yes	—
128	C₂	—	32	2	no	—
129	C₂	C₁₀₉×#1712	1	1	yes	characteristic
130	C₂	C₁₀₉×#2249	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2248

C₂

yes

maximal, complemented

C₁₀₉×#2247

C₂

yes

maximal, characteristic, complemented

C₁₀₉×#2246

C₂

yes

maximal, characteristic

C₁₀₉×#1709

C₂

yes

maximal, complemented

C₈×C₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₃₂×C₁₀₉

—

C₂×SD₃₂×C₁₀₉

—

C₂×D₁₆×C₁₀₉

—

#494×C₁₀₉

C₂²

yes

—

#493×C₁₀₉

C₂²

yes

—

#486×C₁₀₉

—

C₄×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₄×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₈²×C₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₂×D₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

#127×C₁₀₉

—

Q₃₂×C₁₀₉

—

SD₃₂×C₁₀₉

—

D₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

D₄

yes

—

C₂×C₁₆×C₁₀₉

D₄

yes

—

#102×C₁₀₉

—

C₄×C₈×C₁₀₉

D₄

yes

—

C₄×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

D₈×C₁₀₉

—

D₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₈

yes

—

C₂×C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

characteristic

C₁₆×C₁₀₉

D₈

yes

—

C₁₆×C₁₀₉

D₈

yes

—

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#127

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₈

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×D₈

yes

—

C₈×C₁₀₉

C₂×D₈

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#493

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

#12518

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#1712

yes

characteristic

C₂×C₁₀₉

#2249

yes

—

#2248

C₂×C₁₀₉

yes

complemented

#2247

C₂×C₁₀₉

yes

characteristic, complemented

#2246

C₂×C₁₀₉

yes

characteristic

#1709

C₂×C₁₀₉

yes

complemented

C₈×C₁₆

C₂×C₁₀₉

yes

characteristic, complemented

C₁₀₉

#12518

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

—

C₂×SD₃₂

—

C₂×D₁₆

—

#494

C₂²×C₁₀₉

yes

—

#493

C₂²×C₁₀₉

yes

—

#486

—

C₄×C₁₆

C₂²×C₁₀₉

yes

characteristic

C₄×C₁₆

C₂²×C₁₀₉

yes

characteristic

C₈²

C₂²×C₁₀₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

D₄×C₁₀₉

yes

—

C₂×C₁₆

D₄×C₁₀₉

yes

—

#102

—

C₄×C₈

D₄×C₁₀₉

yes

—

C₄×C₈

C₂³×C₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

100

D₈

—

101

D₈

—

102

C₂×C₈

D₈×C₁₀₉

yes

—

103

C₂×C₈

C₂×D₄×C₁₀₉

yes

characteristic

104

C₂×C₈

C₂×D₄×C₁₀₉

yes

characteristic

105

C₄⋊C₄

—

106

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

107

C₁₆

D₈×C₁₀₉

yes

—

108

C₁₆

D₈×C₁₀₉

yes

—

109

C₂³

—

110

Q₈

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

#127×C₁₀₉

yes

characteristic

117

C₂×C₄

C₂×D₈×C₁₀₉

yes

—

118

C₈

—

119

C₈

C₂×D₈×C₁₀₉

yes

—

120

C₈

C₂×D₈×C₁₀₉

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

#493×C₁₀₉

yes

characteristic

124

C₄

—

125

C₄

#493×C₁₀₉

yes

characteristic

126

C₄

#493×C₁₀₉

yes

characteristic

127

C₄

#508×C₁₀₉

yes

—

128

C₂

—

129

C₂

C₁₀₉×#1712

yes

characteristic

130

C₂

C₁₀₉×#2249

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₀₉
prime radical	C₁₀₉×#2246	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×#2246

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×#2248	C₂	semidirect
2. C₁₀₉×#2247	C₂	semidirect
4. C₁₀₉×#1709	C₂	semidirect
5. C₈×C₁₆×C₁₀₉	C₂	semidirect
62. #12518	C₁₀₉	direct (N × H)
66. #2248	C₂×C₁₀₉	semidirect
67. #2247	C₂×C₁₀₉	semidirect
69. #1709	C₂×C₁₀₉	semidirect
70. C₈×C₁₆	C₂×C₁₀₉	semidirect
71. C₁₀₉	#12518	direct (N × H)

H ≅ G/N

type

1. C₁₀₉×#2248

C₂

semidirect

2. C₁₀₉×#2247

C₂

semidirect

4. C₁₀₉×#1709

C₂

semidirect

5. C₈×C₁₆×C₁₀₉

C₂

semidirect

62. #12518

C₁₀₉

direct (N × H)

66. #2248

C₂×C₁₀₉

semidirect

67. #2247

C₂×C₁₀₉

semidirect

69. #1709

C₂×C₁₀₉

semidirect

70. C₈×C₁₆

C₂×C₁₀₉

semidirect

71. C₁₀₉

#12518

direct (N × H)

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

subgroup	index
C₁₀₉×#2248	2
C₁₀₉×#2247	2
C₁₀₉×#2246	2
C₁₀₉×#1709	2
C₈×C₁₆×C₁₀₉	2
#12518	109

subgroup

index

C₁₀₉×#2248

C₁₀₉×#2247

C₁₀₉×#2246

C₁₀₉×#1709

C₈×C₁₆×C₁₀₉

#12518

109

Subgroups — C₁₀₉×#12518 #42130512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12518 #42130512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12518 #42130512