Group #42130886 — subgroups

Subgroups — C₁₀₉×#12892 #42130886

Order 27904.

274 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 158 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	C₂×#346×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1848	C₂	1	2	yes	maximal
3	C₁₀₉×#1381	C₂	1	4	yes	maximal
4	C₂²×C₁₆×C₁₀₉	C₂²	1	2	yes	—
5	C₂×#108×C₁₀₉	—	2	2	no	—
6	C₂×#108×C₁₀₉	C₂²	1	4	yes	—
7	#423×C₁₀₉	—	2	2	no	—
8	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
9	#346×C₁₀₉	C₄	1	4	yes	—
10	C₂×M₁₆×C₁₀₉	—	2	4	no	—
11	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
13	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₁₀₉	—	2	4	no	—
15	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
16	#108×C₁₀₉	—	2	8	no	—
17	#108×C₁₀₉	—	4	4	no	—
18	C₄⋊C₈×C₁₀₉	—	4	4	no	—
19	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₁₀₉	Q₈	1	2	yes	—
21	C₂²×C₄×C₁₀₉	Q₁₆	1	2	yes	—
22	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
23	M₁₆×C₁₀₉	—	4	4	no	—
24	M₁₆×C₁₀₉	—	4	4	no	—
25	C₂×C₈×C₁₀₉	—	4	4	no	—
26	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₁₀₉	SD₁₆	1	4	yes	—
28	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	#41	1	4	yes	—
31	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
32	C₄²×C₁₀₉	D₈	1	2	yes	—
33	C₁₆×C₁₀₉	—	2	4	no	—
34	C₂³×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	—	4	2	no	—
36	C₂×C₄×C₁₀₉	#107	1	4	yes	—
37	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	#124	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
41	C₈×C₁₀₉	—	8	4	no	—
42	C₈×C₁₀₉	#106	1	4	yes	—
43	C₈×C₁₀₉	#117	1	2	yes	—
44	C₈×C₁₀₉	#125	1	2	yes	—
45	C₂²×C₁₀₉	—	2	2	no	—
46	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
47	C₂²×C₁₀₉	#487	1	2	yes	—
48	C₄×C₁₀₉	—	2	4	no	—
49	C₄×C₁₀₉	#382	1	1	yes	characteristic
50	C₄×C₁₀₉	C₂×#106	1	1	yes	characteristic
51	C₄×C₁₀₉	#486	1	2	yes	—
52	#12892	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×C₁₀₉	—	2	2	no	—
54	C₂×C₁₀₉	#1834	1	1	yes	characteristic
55	C₂×C₁₀₉	#2161	1	2	yes	—
56	C₂×#346	C₂×C₁₀₉	1	1	yes	characteristic
57	#1848	C₂×C₁₀₉	1	2	yes	—
58	#1381	C₂×C₁₀₉	1	4	yes	—
59	C₁₀₉	#12892	1	1	yes	characteristic, direct product factor, Sylow
60	C₂²×C₁₆	C₂²×C₁₀₉	1	2	yes	—
61	C₂×#108	—	2	2	no	—
62	C₂×#108	C₂²×C₁₀₉	1	4	yes	—
63	#423	—	2	2	no	—
64	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
65	#346	C₄×C₁₀₉	1	4	yes	—
66	C₂×M₁₆	—	2	4	no	—
67	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
68	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
69	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
70	C₂×C₁₆	—	2	4	no	—
71	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
72	#108	—	2	8	no	—
73	#108	—	4	4	no	—
74	C₄⋊C₈	—	4	4	no	—
75	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
76	C₄×C₈	Q₈×C₁₀₉	1	2	yes	—
77	C₂²×C₄	Q₁₆×C₁₀₉	1	2	yes	—
78	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
79	M₁₆	—	4	4	no	—
80	M₁₆	—	4	4	no	—
81	C₂×C₈	—	4	4	no	—
82	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
83	C₂×C₈	SD₁₆×C₁₀₉	1	4	yes	—
84	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
85	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
86	C₂×C₈	#41×C₁₀₉	1	4	yes	—
87	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
88	C₄²	D₈×C₁₀₉	1	2	yes	—
89	C₁₆	—	2	4	no	—
90	C₂³	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	#107×C₁₀₉	1	4	yes	—
93	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
94	C₂×C₄	#124×C₁₀₉	1	2	yes	—
95	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
96	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
97	C₈	—	8	4	no	—
98	C₈	#106×C₁₀₉	1	4	yes	—
99	C₈	#117×C₁₀₉	1	2	yes	—
100	C₈	#125×C₁₀₉	1	2	yes	—
101	C₂²	—	2	2	no	—
102	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
103	C₂²	#487×C₁₀₉	1	2	yes	—
104	C₄	—	2	4	no	—
105	C₄	#382×C₁₀₉	1	1	yes	characteristic
106	C₄	C₂×#106×C₁₀₉	1	1	yes	characteristic
107	C₄	#486×C₁₀₉	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₁₀₉×#1834	1	1	yes	characteristic
110	C₂	C₁₀₉×#2161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#346×C₁₀₉

C₂

yes

maximal, characteristic

C₁₀₉×#1848

C₂

yes

maximal

C₁₀₉×#1381

C₂

yes

maximal

C₂²×C₁₆×C₁₀₉

C₂²

yes

—

C₂×#108×C₁₀₉

—

C₂×#108×C₁₀₉

C₂²

yes

—

#423×C₁₀₉

—

C₂×C₄×C₈×C₁₀₉

C₂²

yes

characteristic

#346×C₁₀₉

C₄

yes

—

C₂×M₁₆×C₁₀₉

—

C₂²×C₈×C₁₀₉

D₄

yes

characteristic

C₂²×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₄²×C₁₀₉

D₄

yes

characteristic

C₂×C₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

C₂×C₄

yes

—

#108×C₁₀₉

—

#108×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂×C₄

yes

—

C₄×C₈×C₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

Q₁₆

yes

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₈×C₁₀₉

SD₁₆

yes

—

C₂×C₈×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₀₉

#41

yes

—

C₄²×C₁₀₉

C₄⋊C₄

yes

—

C₄²×C₁₀₉

D₈

yes

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#107

yes

—

C₂×C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

#124

yes

—

C₂×C₄×C₁₀₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

#106

yes

—

C₈×C₁₀₉

#117

yes

—

C₈×C₁₀₉

#125

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×#107

yes

characteristic

C₂²×C₁₀₉

#487

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

#12892

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#1834

yes

characteristic

C₂×C₁₀₉

#2161

yes

—

C₂×#346

C₂×C₁₀₉

yes

characteristic

#1848

C₂×C₁₀₉

yes

—

#1381

C₂×C₁₀₉

yes

—

C₁₀₉

#12892

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

C₂²×C₁₀₉

yes

—

C₂×#108

—

C₂×#108

C₂²×C₁₀₉

yes

—

#423

—

C₂×C₄×C₈

C₂²×C₁₀₉

yes

characteristic

#346

C₄×C₁₀₉

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₁₀₉

yes

characteristic

C₂²×C₈

C₂³×C₁₀₉

yes

characteristic

C₂×C₄²

D₄×C₁₀₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₁₀₉

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₁₀₉

yes

—

C₄×C₈

Q₈×C₁₀₉

yes

—

C₂²×C₄

Q₁₆×C₁₀₉

yes

—

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₁₀₉

yes

—

C₂×C₈

SD₁₆×C₁₀₉

yes

—

C₂×C₈

C₂²×C₄×C₁₀₉

yes

characteristic

C₂×C₈

C₂×Q₈×C₁₀₉

yes

characteristic

C₂×C₈

#41×C₁₀₉

yes

—

C₄²

C₄⋊C₄×C₁₀₉

yes

—

C₄²

D₈×C₁₀₉

yes

—

C₁₆

—

C₂³

C₂×Q₁₆×C₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

#107×C₁₀₉

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₁₀₉

yes

characteristic

C₂×C₄

#124×C₁₀₉

yes

—

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

C₂×C₄

C₂×SD₁₆×C₁₀₉

yes

—

C₈

—

C₈

#106×C₁₀₉

yes

—

C₈

#117×C₁₀₉

yes

—

100

C₈

#125×C₁₀₉

yes

—

101

C₂²

—

102

C₂²

C₂×#107×C₁₀₉

yes

characteristic

103

C₂²

#487×C₁₀₉

yes

—

104

C₄

—

105

C₄

#382×C₁₀₉

yes

characteristic

106

C₄

C₂×#106×C₁₀₉

yes

characteristic

107

C₄

#486×C₁₀₉

yes

—

108

C₂

—

109

C₂

C₁₀₉×#1834

yes

characteristic

110

C₂

C₁₀₉×#2161

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×#107
prime radical	C₂³×C₁₀₉	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₀₉

C₂×D₈

commutator

C₂×C₈

C₂²×C₄×C₁₀₉

Frattini, cosocle

C₂²×C₈

C₂³×C₁₀₉

socle

C₂²×C₁₀₉

C₂×#107

prime radical

C₂³×C₁₀₉

C₂×Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
52. #12892	C₁₀₉	direct (N × H)
59. C₁₀₉	#12892	direct (N × H)

H ≅ G/N

type

52. #12892

C₁₀₉

direct (N × H)

59. C₁₀₉

#12892

direct (N × H)

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

subgroup	index
C₂×#346×C₁₀₉	2
C₁₀₉×#1848	2
C₁₀₉×#1381	2
#12892	109

subgroup

index

C₂×#346×C₁₀₉

C₁₀₉×#1848

C₁₀₉×#1381

#12892

109

Subgroups — C₁₀₉×#12892 #42130886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12892 #42130886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12892 #42130886