Group #42130504 — subgroups

Subgroups — C₁₀₉×#12510 #42130504

Order 27904.

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

normal subgroups only

This group has 262 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×#2241	C₂	1	4	yes	maximal
2	C₁₀₉×#1713	C₂	1	1	yes	maximal, characteristic
3	C₁₀₉×#1713	C₂	1	1	yes	maximal, characteristic
4	C₈×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#500×C₁₀₉	—	2	1	no	—
6	#500×C₁₀₉	—	2	1	no	—
7	#500×C₁₀₉	—	2	1	no	—
8	#500×C₁₀₉	C₂²	1	2	yes	—
9	#494×C₁₀₉	—	2	1	no	—
10	#494×C₁₀₉	C₂²	1	2	yes	—
11	#358×C₁₀₉	—	2	8	no	—
12	C₄×C₁₆×C₁₀₉	C₂²	1	2	yes	—
13	C₈²×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆×C₁₀₉	—	2	4	no	—
15	C₂×Q₁₆×C₁₀₉	—	4	2	no	—
16	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
17	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
18	C₂×C₁₆×C₁₀₉	D₄	1	4	yes	—
19	#107×C₁₀₉	—	2	4	no	—
20	#107×C₁₀₉	—	4	2	no	—
21	#107×C₁₀₉	—	4	2	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×Q₈×C₁₀₉	—	4	2	no	—
28	Q₁₆×C₁₀₉	—	4	8	no	—
29	Q₁₆×C₁₀₉	—	8	4	no	—
30	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
31	C₂×C₈×C₁₀₉	D₈	1	2	yes	—
32	C₂×C₈×C₁₀₉	C₂×D₄	1	2	yes	—
33	C₄⋊C₄×C₁₀₉	—	4	2	no	—
34	C₄⋊C₄×C₁₀₉	—	4	2	no	—
35	C₄⋊C₄×C₁₀₉	—	4	2	no	—
36	C₄⋊C₄×C₁₀₉	—	4	2	no	—
37	C₄²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₁₆×C₁₀₉	—	2	4	no	—
39	Q₈×C₁₀₉	—	8	4	no	—
40	Q₈×C₁₀₉	—	8	4	no	—
41	C₂×C₄×C₁₀₉	—	8	2	no	—
42	C₂×C₄×C₁₀₉	—	8	2	no	—
43	C₂×C₄×C₁₀₉	#127	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂×D₈	1	1	yes	characteristic
46	C₈×C₁₀₉	SD₃₂	1	4	yes	—
47	C₈×C₁₀₉	Q₃₂	1	4	yes	—
48	C₈×C₁₀₉	#135	1	4	yes	—
49	C₂²×C₁₀₉	#493	1	1	yes	characteristic
50	C₄×C₁₀₉	—	16	2	no	—
51	C₄×C₁₀₉	—	16	2	no	—
52	C₄×C₁₀₉	#495	1	2	yes	—
53	C₄×C₁₀₉	C₂×SD₃₂	1	2	yes	—
54	C₄×C₁₀₉	C₂×Q₃₂	1	2	yes	—
55	#12510	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×C₁₀₉	#1709	1	1	yes	characteristic
57	C₂×C₁₀₉	#2247	1	1	yes	characteristic
58	C₂×C₁₀₉	#2248	1	1	yes	characteristic
59	#2241	C₂×C₁₀₉	1	4	yes	—
60	#1713	C₂×C₁₀₉	1	1	yes	characteristic
61	#1713	C₂×C₁₀₉	1	1	yes	characteristic
62	C₈×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
63	C₁₀₉	#12510	1	1	yes	characteristic, direct product factor, Sylow
64	#500	—	2	1	no	—
65	#500	—	2	1	no	—
66	#500	—	2	1	no	—
67	#500	C₂²×C₁₀₉	1	2	yes	—
68	#494	—	2	1	no	—
69	#494	C₂²×C₁₀₉	1	2	yes	—
70	#358	—	2	8	no	—
71	C₄×C₁₆	C₂²×C₁₀₉	1	2	yes	—
72	C₈²	C₂²×C₁₀₉	1	1	yes	characteristic
73	C₂×Q₁₆	—	2	4	no	—
74	C₂×Q₁₆	—	4	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	C₂×C₁₆	D₄×C₁₀₉	1	4	yes	—
78	#107	—	2	4	no	—
79	#107	—	4	2	no	—
80	#107	—	4	2	no	—
81	#107	—	4	2	no	—
82	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
83	C₄×C₈	D₄×C₁₀₉	1	1	yes	characteristic
84	C₄×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	4	2	no	—
87	Q₁₆	—	4	8	no	—
88	Q₁₆	—	8	4	no	—
89	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
90	C₂×C₈	D₈×C₁₀₉	1	2	yes	—
91	C₂×C₈	C₂×D₄×C₁₀₉	1	2	yes	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	4	2	no	—
96	C₄²	C₂×D₄×C₁₀₉	1	1	yes	characteristic
97	C₁₆	—	2	4	no	—
98	Q₈	—	8	4	no	—
99	Q₈	—	8	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	#127×C₁₀₉	1	1	yes	characteristic
103	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
104	C₂×C₄	C₂×D₈×C₁₀₉	1	1	yes	characteristic
105	C₈	SD₃₂×C₁₀₉	1	4	yes	—
106	C₈	Q₃₂×C₁₀₉	1	4	yes	—
107	C₈	#135×C₁₀₉	1	4	yes	—
108	C₂²	#493×C₁₀₉	1	1	yes	characteristic
109	C₄	—	16	2	no	—
110	C₄	—	16	2	no	—
111	C₄	#495×C₁₀₉	1	2	yes	—
112	C₄	C₂×SD₃₂×C₁₀₉	1	2	yes	—
113	C₄	C₂×Q₃₂×C₁₀₉	1	2	yes	—
114	C₂	C₁₀₉×#1709	1	1	yes	characteristic
115	C₂	C₁₀₉×#2247	1	1	yes	characteristic
116	C₂	C₁₀₉×#2248	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2241

C₂

yes

maximal

C₁₀₉×#1713

C₂

yes

maximal, characteristic

C₁₀₉×#1713

C₂

yes

maximal, characteristic

C₈×C₁₆×C₁₀₉

C₂

yes

maximal, characteristic

#500×C₁₀₉

—

#500×C₁₀₉

—

#500×C₁₀₉

—

#500×C₁₀₉

C₂²

yes

—

#494×C₁₀₉

—

#494×C₁₀₉

C₂²

yes

—

#358×C₁₀₉

—

C₄×C₁₆×C₁₀₉

C₂²

yes

—

C₈²×C₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

—

C₂×C₁₆×C₁₀₉

D₄

yes

—

#107×C₁₀₉

—

#107×C₁₀₉

—

#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×Q₈×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

D₈

yes

—

C₂×C₈×C₁₀₉

D₈

yes

—

C₂×C₈×C₁₀₉

C₂×D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

characteristic

C₁₆×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×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

SD₃₂

yes

—

C₈×C₁₀₉

Q₃₂

yes

—

C₈×C₁₀₉

#135

yes

—

C₂²×C₁₀₉

#493

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#495

yes

—

C₄×C₁₀₉

C₂×SD₃₂

yes

—

C₄×C₁₀₉

C₂×Q₃₂

yes

—

#12510

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

#1713

C₂×C₁₀₉

yes

characteristic

#1713

C₂×C₁₀₉

yes

characteristic

C₈×C₁₆

C₂×C₁₀₉

yes

characteristic

C₁₀₉

#12510

yes

characteristic, direct product factor, Sylow

—

—

—

C₂²×C₁₀₉

yes

—

#494

—

#494

C₂²×C₁₀₉

yes

—

#358

—

C₄×C₁₆

C₂²×C₁₀₉

yes

—

C₈²

C₂²×C₁₀₉

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

D₄×C₁₀₉

yes

—

—

—

—

—

C₄×C₈

D₄×C₁₀₉

yes

characteristic

C₄×C₈

D₄×C₁₀₉

yes

characteristic

C₄×C₈

C₂³×C₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₈×C₁₀₉

yes

—

C₂×C₈

D₈×C₁₀₉

yes

—

C₂×C₈

C₂×D₄×C₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄×C₁₀₉

yes

characteristic

C₁₆

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#127×C₁₀₉

yes

characteristic

103

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

104

C₂×C₄

C₂×D₈×C₁₀₉

yes

characteristic

105

C₈

SD₃₂×C₁₀₉

yes

—

106

C₈

Q₃₂×C₁₀₉

yes

—

107

C₈

#135×C₁₀₉

yes

—

108

C₂²

#493×C₁₀₉

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

#495×C₁₀₉

yes

—

112

C₄

C₂×SD₃₂×C₁₀₉

yes

—

113

C₄

C₂×Q₃₂×C₁₀₉

yes

—

114

C₂

C₁₀₉×#1709

yes

characteristic

115

C₂

C₁₀₉×#2247

yes

characteristic

116

C₂

C₁₀₉×#2248

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂²×C₁₀₉

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₀₉

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

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

N	H ≅ G/N	type
55. #12510	C₁₀₉	direct (N × H)
63. C₁₀₉	#12510	direct (N × H)

H ≅ G/N

type

55. #12510

C₁₀₉

direct (N × H)

63. C₁₀₉

#12510

direct (N × H)

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

subgroup	index
C₁₀₉×#2241	2
C₁₀₉×#1713	2
C₈×C₁₆×C₁₀₉	2
#12510	109

subgroup

index

C₁₀₉×#2241

C₁₀₉×#1713

C₈×C₁₆×C₁₀₉

#12510

109

Subgroups — C₁₀₉×#12510 #42130504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12510 #42130504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12510 #42130504