Group #42129406 — subgroups

Subgroups — C₁₀₉×#11412 #42129406

Order 27904.

260 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 190 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×#2162	C₂	1	2	yes	maximal
2	C₁₀₉×#1450	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1312	C₂	1	2	yes	maximal, complemented
4	C₁₀₉×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#431×C₁₀₉	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₄⋊C₁₆×C₁₀₉	C₄	1	4	yes	—
9	C₂²⋊C₁₆×C₁₀₉	C₄	1	4	yes	—
10	C₄×C₁₆×C₁₀₉	C₂²	1	2	yes	—
11	C₄×C₁₆×C₁₀₉	C₂²	1	2	yes	—
12	#322×C₁₀₉	C₄	1	2	yes	—
13	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
18	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
19	C₄⋊C₈×C₁₀₉	C₈	1	4	yes	—
20	C₂²⋊C₈×C₁₀₉	C₈	1	4	yes	—
21	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₉	C₄²	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₁₀₉	—	2	4	no	—
29	C₂×C₈×C₁₀₉	C₄²	1	2	yes	—
30	C₂×C₈×C₁₀₉	C₂×C₈	1	4	yes	—
31	C₂×C₈×C₁₀₉	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
33	C₄²×C₁₀₉	C₄²	1	1	yes	characteristic
34	C₄²×C₁₀₉	C₂×C₈	1	2	yes	—
35	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₁₆×C₁₀₉	—	2	4	no	—
37	C₁₆×C₁₀₉	—	2	4	no	—
38	C₂³×C₁₀₉	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄×C₁₀₉	—	2	4	no	—
41	C₂×C₄×C₁₀₉	C₄×C₈	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂²×C₈	1	2	yes	—
45	C₈×C₁₀₉	—	2	2	no	—
46	C₈×C₁₀₉	—	2	4	no	—
47	C₈×C₁₀₉	—	2	4	no	—
48	C₈×C₁₀₉	#131	1	4	yes	—
49	C₂²×C₁₀₉	—	2	2	no	—
50	C₂²×C₁₀₉	—	2	1	no	—
51	C₂²×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
52	C₄×C₁₀₉	—	2	2	no	—
53	C₄×C₁₀₉	—	2	1	no	—
54	C₄×C₁₀₉	#405	1	2	yes	—
55	C₄×C₁₀₉	#504	1	4	yes	—
56	#11412	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×C₁₀₉	—	4	1	no	—
58	C₂×C₁₀₉	#1450	1	1	yes	characteristic
59	C₂×C₁₀₉	#2108	1	2	yes	—
60	#2162	C₂×C₁₀₉	1	2	yes	—
61	#1450	C₂×C₁₀₉	1	1	yes	characteristic
62	#1312	C₂×C₁₀₉	1	2	yes	complemented
63	#1311	C₂×C₁₀₉	1	1	yes	characteristic, complemented
64	C₈×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic, complemented
65	C₁₀₉	#11412	1	1	yes	characteristic, direct product factor, Sylow
66	#431	C₄×C₁₀₉	1	2	yes	—
67	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
68	C₄⋊C₁₆	C₄×C₁₀₉	1	4	yes	—
69	C₂²⋊C₁₆	C₄×C₁₀₉	1	4	yes	—
70	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
71	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
72	#322	C₄×C₁₀₉	1	2	yes	—
73	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
74	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
75	C₂²×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
76	C₂×C₄²	C₂×C₄×C₁₀₉	1	1	yes	characteristic
77	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
78	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
79	C₄⋊C₈	C₈×C₁₀₉	1	4	yes	—
80	C₂²⋊C₈	C₈×C₁₀₉	1	4	yes	—
81	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
82	C₄×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
83	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
84	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
85	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
86	C₂²×C₄	C₄²×C₁₀₉	1	1	yes	characteristic
87	C₂²×C₄	C₂×C₈×C₁₀₉	1	2	yes	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	C₄²×C₁₀₉	1	2	yes	—
90	C₂×C₈	C₂×C₈×C₁₀₉	1	4	yes	—
91	C₂×C₈	C₂×C₈×C₁₀₉	1	4	yes	—
92	C₂×C₈	C₂²×C₄×C₁₀₉	1	2	yes	—
93	C₄²	C₄²×C₁₀₉	1	1	yes	characteristic
94	C₄²	C₂×C₈×C₁₀₉	1	2	yes	—
95	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
96	C₁₆	—	2	4	no	—
97	C₁₆	—	2	4	no	—
98	C₂³	C₄×C₈×C₁₀₉	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄×C₈×C₁₀₉	1	2	yes	—
102	C₂×C₄	C₄×C₈×C₁₀₉	1	1	yes	characteristic
103	C₂×C₄	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
104	C₂×C₄	C₂²×C₈×C₁₀₉	1	2	yes	—
105	C₈	—	2	4	no	—
106	C₈	—	2	4	no	—
107	C₈	—	2	2	no	—
108	C₈	#131×C₁₀₉	1	4	yes	—
109	C₂²	—	2	1	no	—
110	C₂²	—	2	2	no	—
111	C₂²	C₂×C₄×C₈×C₁₀₉	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	2	no	—
114	C₄	#405×C₁₀₉	1	2	yes	—
115	C₄	#504×C₁₀₉	1	4	yes	—
116	C₂	—	4	1	no	—
117	C₂	C₁₀₉×#1450	1	1	yes	characteristic
118	C₂	C₁₀₉×#2108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2162

C₂

yes

maximal

C₁₀₉×#1450

C₂

yes

maximal, characteristic

C₁₀₉×#1312

C₂

yes

maximal, complemented

C₁₀₉×#1311

C₂

yes

maximal, characteristic, complemented

C₈×C₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#431×C₁₀₉

C₄

yes

—

C₂×C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₀₉

C₄

yes

—

C₂²⋊C₁₆×C₁₀₉

C₄

yes

—

C₄×C₁₆×C₁₀₉

C₂²

yes

—

C₄×C₁₆×C₁₀₉

C₂²

yes

—

#322×C₁₀₉

C₄

yes

—

C₈²×C₁₀₉

C₂²

yes

characteristic

C₈²×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂×C₄

yes

—

C₂×C₁₆×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₈×C₁₀₉

C₈

yes

—

C₂²⋊C₈×C₁₀₉

C₈

yes

—

C₄×C₈×C₁₀₉

C₂×C₄

yes

—

C₄×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄×C₈×C₁₀₉

C₂×C₄

yes

—

C₄×C₈×C₁₀₉

C₂×C₄

yes

—

C₄×C₈×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₄²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₄²

yes

—

C₂×C₈×C₁₀₉

C₂×C₈

yes

—

C₂×C₈×C₁₀₉

C₂×C₈

yes

—

C₂×C₈×C₁₀₉

C₂²×C₄

yes

—

C₄²×C₁₀₉

C₄²

yes

characteristic

C₄²×C₁₀₉

C₂×C₈

yes

—

C₄²×C₁₀₉

C₂²×C₄

yes

characteristic

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

C₄×C₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄×C₈

yes

—

C₂×C₄×C₁₀₉

C₄×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₈

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

#131

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄×C₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#405

yes

—

C₄×C₁₀₉

#504

yes

—

#11412

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#1450

yes

characteristic

C₂×C₁₀₉

#2108

yes

—

#2162

C₂×C₁₀₉

yes

—

#1450

C₂×C₁₀₉

yes

characteristic

#1312

C₂×C₁₀₉

yes

complemented

#1311

C₂×C₁₀₉

yes

characteristic, complemented

C₈×C₁₆

C₂×C₁₀₉

yes

characteristic, complemented

C₁₀₉

#11412

yes

characteristic, direct product factor, Sylow

#431

C₄×C₁₀₉

yes

—

C₂×C₄×C₈

C₂²×C₁₀₉

yes

characteristic

C₄⋊C₁₆

C₄×C₁₀₉

yes

—

C₂²⋊C₁₆

C₄×C₁₀₉

yes

—

C₄×C₁₆

C₂²×C₁₀₉

yes

—

C₄×C₁₆

C₂²×C₁₀₉

yes

—

#322

C₄×C₁₀₉

yes

—

C₈²

C₂²×C₁₀₉

yes

characteristic

C₈²

C₂²×C₁₀₉

yes

characteristic

C₂²×C₈

C₂×C₄×C₁₀₉

yes

—

C₂×C₄²

C₂×C₄×C₁₀₉

yes

characteristic

C₂×C₁₆

C₂×C₄×C₁₀₉

yes

—

C₂×C₁₆

C₂×C₄×C₁₀₉

yes

—

C₄⋊C₈

C₈×C₁₀₉

yes

—

C₂²⋊C₈

C₈×C₁₀₉

yes

—

C₄×C₈

C₂×C₄×C₁₀₉

yes

—

C₄×C₈

C₂×C₄×C₁₀₉

yes

characteristic

C₄×C₈

C₂×C₄×C₁₀₉

yes

—

C₄×C₈

C₂×C₄×C₁₀₉

yes

—

C₄×C₈

C₂³×C₁₀₉

yes

characteristic

C₂²×C₄

C₄²×C₁₀₉

yes

characteristic

C₂²×C₄

C₂×C₈×C₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

C₄²×C₁₀₉

yes

—

C₂×C₈

C₂×C₈×C₁₀₉

yes

—

C₂×C₈

C₂×C₈×C₁₀₉

yes

—

C₂×C₈

C₂²×C₄×C₁₀₉

yes

—

C₄²

C₄²×C₁₀₉

yes

characteristic

C₄²

C₂×C₈×C₁₀₉

yes

—

C₄²

C₂²×C₄×C₁₀₉

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

C₄×C₈×C₁₀₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄×C₈×C₁₀₉

yes

—

102

C₂×C₄

C₄×C₈×C₁₀₉

yes

characteristic

103

C₂×C₄

C₂×C₄²×C₁₀₉

yes

characteristic

104

C₂×C₄

C₂²×C₈×C₁₀₉

yes

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

#131×C₁₀₉

yes

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂×C₄×C₈×C₁₀₉

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#405×C₁₀₉

yes

—

115

C₄

#504×C₁₀₉

yes

—

116

C₂

—

117

C₂

C₁₀₉×#1450

yes

characteristic

118

C₂

C₁₀₉×#2108

yes

—

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

subgroup

G/H

center

C₄×C₈×C₁₀₉

C₂³

commutator

C₂²

C₂×C₄×C₈×C₁₀₉

Frattini, cosocle

C₄×C₈

C₂³×C₁₀₉

socle

C₂²×C₁₀₉

C₂×C₄×C₈

prime radical

C₂³×C₁₀₉

C₄×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₄×C₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
3. C₁₀₉×#1312	C₂	semidirect
4. C₁₀₉×#1311	C₂	semidirect
5. C₈×C₁₆×C₁₀₉	C₂	semidirect
56. #11412	C₁₀₉	direct (N × H)
62. #1312	C₂×C₁₀₉	semidirect
63. #1311	C₂×C₁₀₉	semidirect
64. C₈×C₁₆	C₂×C₁₀₉	semidirect
65. C₁₀₉	#11412	direct (N × H)

H ≅ G/N

type

3. C₁₀₉×#1312

C₂

semidirect

4. C₁₀₉×#1311

C₂

semidirect

5. C₈×C₁₆×C₁₀₉

C₂

semidirect

56. #11412

C₁₀₉

direct (N × H)

62. #1312

C₂×C₁₀₉

semidirect

63. #1311

C₂×C₁₀₉

semidirect

64. C₈×C₁₆

C₂×C₁₀₉

semidirect

65. C₁₀₉

#11412

direct (N × H)

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

subgroup	index
C₁₀₉×#2162	2
C₁₀₉×#1450	2
C₁₀₉×#1312	2
C₁₀₉×#1311	2
C₈×C₁₆×C₁₀₉	2
#11412	109

subgroup

index

C₁₀₉×#2162

C₁₀₉×#1450

C₁₀₉×#1312

C₁₀₉×#1311

C₈×C₁₆×C₁₀₉

#11412

109

Subgroups — C₁₀₉×#11412 #42129406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#11412 #42129406

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#11412 #42129406