Group #42128606 — subgroups

Subgroups — C₁₀₉×#10612 #42128606

Order 27904.

328 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 98 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×#1574	C₂	1	3	yes	maximal, complemented
2	C₁₀₉×#1545	C₂	1	3	yes	maximal
3	C₁₀₉×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#127×C₁₀₉	C₂²	1	1	yes	characteristic
5	#431×C₁₀₉	C₂²	1	3	yes	—
6	#423×C₁₀₉	C₂²	1	3	yes	—
7	#335×C₁₀₉	—	2	3	no	—
8	#334×C₁₀₉	—	2	3	no	—
9	#331×C₁₀₉	—	2	3	no	—
10	#331×C₁₀₉	—	2	3	no	—
11	C₂²×D₄×C₁₀₉	—	2	3	no	—
12	C₂×M₁₆×C₁₀₉	—	2	3	no	—
13	#127×C₁₀₉	—	2	3	no	—
14	#127×C₁₀₉	Q₈	1	2	yes	—
15	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₉	—	2	3	no	—
17	C₄⋊C₈×C₁₀₉	—	2	3	no	—
18	C₄⋊C₈×C₁₀₉	D₄	1	3	yes	—
19	C₄⋊C₈×C₁₀₉	D₄	1	3	yes	—
20	C₂²⋊C₈×C₁₀₉	—	2	3	no	—
21	C₄×C₈×C₁₀₉	—	2	3	no	—
22	C₂⁴×C₁₀₉	—	4	1	no	—
23	C₂×D₄×C₁₀₉	—	2	6	no	—
24	C₂×D₄×C₁₀₉	—	4	3	no	—
25	C₂×D₄×C₁₀₉	—	8	3	no	—
26	C₂×D₄×C₁₀₉	—	8	3	no	—
27	C₂²×C₄×C₁₀₉	#41	1	3	yes	—
28	M₁₆×C₁₀₉	—	8	3	no	—
29	C₂×C₈×C₁₀₉	—	2	3	no	—
30	C₂×C₈×C₁₀₉	—	2	3	no	—
31	C₂×C₈×C₁₀₉	—	4	3	no	—
32	C₄²×C₁₀₉	C₂×D₄	1	3	yes	—
33	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
34	C₂³×C₁₀₉	—	4	2	no	—
35	C₂³×C₁₀₉	—	8	6	no	—
36	C₂³×C₁₀₉	#126	1	1	yes	characteristic
37	D₄×C₁₀₉	—	8	6	no	—
38	D₄×C₁₀₉	—	8	3	no	—
39	D₄×C₁₀₉	—	8	3	no	—
40	C₂×C₄×C₁₀₉	—	2	3	no	—
41	C₂×C₄×C₁₀₉	—	2	3	no	—
42	C₂×C₄×C₁₀₉	#121	1	3	yes	—
43	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	3	yes	—
44	C₈×C₁₀₉	—	4	3	no	—
45	C₈×C₁₀₉	—	4	3	no	—
46	C₈×C₁₀₉	—	8	3	no	—
47	C₂²×C₁₀₉	—	2	3	no	—
48	C₂²×C₁₀₉	—	8	6	no	—
49	C₂²×C₁₀₉	—	16	4	no	—
50	C₂²×C₁₀₉	#396	1	1	yes	characteristic
51	C₄×C₁₀₉	—	2	3	no	—
52	C₄×C₁₀₉	#471	1	3	yes	—
53	C₄×C₁₀₉	#472	1	3	yes	—
54	#10612	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₁₀₉	—	4	1	no	—
56	C₂×C₁₀₉	—	16	2	no	—
57	C₂×C₁₀₉	#2063	1	3	yes	—
58	#1574	C₂×C₁₀₉	1	3	yes	complemented
59	#1545	C₂×C₁₀₉	1	3	yes	—
60	#1309	C₂×C₁₀₉	1	1	yes	characteristic, complemented
61	C₁₀₉	#10612	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×#127	C₂²×C₁₀₉	1	1	yes	characteristic
63	#431	C₂²×C₁₀₉	1	3	yes	—
64	#423	C₂²×C₁₀₉	1	3	yes	—
65	#335	—	2	3	no	—
66	#334	—	2	3	no	—
67	#331	—	2	3	no	—
68	#331	—	2	3	no	—
69	C₂²×D₄	—	2	3	no	—
70	C₂×M₁₆	—	2	3	no	—
71	#127	—	2	3	no	—
72	#127	Q₈×C₁₀₉	1	2	yes	—
73	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	—	2	3	no	—
76	C₄⋊C₈	D₄×C₁₀₉	1	3	yes	—
77	C₄⋊C₈	D₄×C₁₀₉	1	3	yes	—
78	C₂²⋊C₈	—	2	3	no	—
79	C₄×C₈	—	2	3	no	—
80	C₂⁴	—	4	1	no	—
81	C₂×D₄	—	2	6	no	—
82	C₂×D₄	—	4	3	no	—
83	C₂×D₄	—	8	3	no	—
84	C₂×D₄	—	8	3	no	—
85	C₂²×C₄	#41×C₁₀₉	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	2	3	no	—
88	C₂×C₈	—	2	3	no	—
89	C₂×C₈	—	4	3	no	—
90	C₄²	C₂×D₄×C₁₀₉	1	3	yes	—
91	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
92	C₂³	—	4	2	no	—
93	C₂³	—	8	6	no	—
94	C₂³	#126×C₁₀₉	1	1	yes	characteristic
95	D₄	—	8	6	no	—
96	D₄	—	8	3	no	—
97	D₄	—	8	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#121×C₁₀₉	1	3	yes	—
101	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	3	yes	—
102	C₈	—	4	3	no	—
103	C₈	—	4	3	no	—
104	C₈	—	8	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	8	6	no	—
107	C₂²	—	16	4	no	—
108	C₂²	#396×C₁₀₉	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#471×C₁₀₉	1	3	yes	—
111	C₄	#472×C₁₀₉	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	16	2	no	—
114	C₂	C₁₀₉×#2063	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#1574

C₂

yes

maximal, complemented

C₁₀₉×#1545

C₂

yes

maximal

C₁₀₉×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₀₉

C₂²

yes

characteristic

#431×C₁₀₉

C₂²

yes

—

#423×C₁₀₉

C₂²

yes

—

#335×C₁₀₉

—

#334×C₁₀₉

—

#331×C₁₀₉

—

#331×C₁₀₉

—

C₂²×D₄×C₁₀₉

—

C₂×M₁₆×C₁₀₉

—

#127×C₁₀₉

—

#127×C₁₀₉

Q₈

yes

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

D₄

yes

—

C₄⋊C₈×C₁₀₉

D₄

yes

—

C₂²⋊C₈×C₁₀₉

—

C₄×C₈×C₁₀₉

—

C₂⁴×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

#41

yes

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

—

C₄²×C₁₀₉

C₂×Q₈

yes

characteristic

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

#126

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#121

yes

—

C₂×C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#396

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

#471

yes

—

C₄×C₁₀₉

#472

yes

—

#10612

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#2063

yes

—

#1574

C₂×C₁₀₉

yes

complemented

#1545

C₂×C₁₀₉

yes

—

#1309

C₂×C₁₀₉

yes

characteristic, complemented

C₁₀₉

#10612

yes

characteristic, direct product factor, Sylow

C₂×#127

C₂²×C₁₀₉

yes

characteristic

#431

C₂²×C₁₀₉

yes

—

#423

C₂²×C₁₀₉

yes

—

—

—

—

—

C₂²×D₄

—

C₂×M₁₆

—

#127

—

#127

Q₈×C₁₀₉

yes

—

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₄×C₁₀₉

yes

—

C₄⋊C₈

D₄×C₁₀₉

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

#41×C₁₀₉

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₂×D₄×C₁₀₉

yes

—

C₄²

C₂×Q₈×C₁₀₉

yes

characteristic

C₂³

—

C₂³

—

C₂³

#126×C₁₀₉

yes

characteristic

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#121×C₁₀₉

yes

—

101

C₂×C₄

C₂²⋊Q₈×C₁₀₉

yes

—

102

C₈

—

103

C₈

—

104

C₈

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

#396×C₁₀₉

yes

characteristic

109

C₄

—

110

C₄

#471×C₁₀₉

yes

—

111

C₄

#472×C₁₀₉

yes

—

112

C₂

—

113

C₂

—

114

C₂

C₁₀₉×#2063

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂×#127×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#396

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

prime radical

C₂×#127×C₁₀₉

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
1. C₁₀₉×#1574	C₂	semidirect
3. C₁₀₉×#1309	C₂	semidirect
54. #10612	C₁₀₉	direct (N × H)
58. #1574	C₂×C₁₀₉	semidirect
60. #1309	C₂×C₁₀₉	semidirect
61. C₁₀₉	#10612	direct (N × H)

H ≅ G/N

type

1. C₁₀₉×#1574

C₂

semidirect

3. C₁₀₉×#1309

C₂

semidirect

54. #10612

C₁₀₉

direct (N × H)

58. #1574

C₂×C₁₀₉

semidirect

60. #1309

C₂×C₁₀₉

semidirect

61. C₁₀₉

#10612

direct (N × H)

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

subgroup	index
C₁₀₉×#1574	2
C₁₀₉×#1545	2
C₁₀₉×#1309	2
#10612	109

subgroup

index

C₁₀₉×#1574

C₁₀₉×#1545

C₁₀₉×#1309

#10612

109

Subgroups — C₁₀₉×#10612 #42128606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10612 #42128606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10612 #42128606