Group #28359065 — subgroups

Subgroups — C₁₀₉×#1811 #28359065

Order 13952.

272 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 120 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	C₂²×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×C₁₀₉	C₂	1	2	yes	maximal, complemented
3	#407×C₁₀₉	C₂	1	2	yes	maximal
4	#341×C₁₀₉	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₁₀₉	—	2	4	no	—
7	C₂×M₁₆×C₁₀₉	C₄	1	4	yes	—
8	C₂×M₁₆×C₁₀₉	C₂²	1	4	yes	—
9	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
11	#108×C₁₀₉	—	2	4	no	—
12	C₂²⋊C₈×C₁₀₉	—	2	4	no	—
13	C₂⁴×C₁₀₉	Q₈	1	1	yes	characteristic
14	C₂²×C₄×C₁₀₉	—	2	2	no	—
15	C₂²×C₄×C₁₀₉	—	2	2	no	—
16	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
18	C₂²×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
20	M₁₆×C₁₀₉	—	2	8	no	—
21	M₁₆×C₁₀₉	—	2	4	no	—
22	M₁₆×C₁₀₉	—	2	4	no	—
23	C₂×C₈×C₁₀₉	—	2	2	no	—
24	C₂×C₈×C₁₀₉	—	2	4	no	—
25	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₁₀₉	D₄	1	4	yes	—
28	C₂³×C₁₀₉	—	2	2	no	—
29	C₂³×C₁₀₉	—	4	2	no	—
30	C₂³×C₁₀₉	C₄⋊C₄	1	2	yes	—
31	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	—	2	2	no	—
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	—	2	2	no	—
35	C₂×C₄×C₁₀₉	—	2	4	no	—
36	C₂×C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	#41	1	2	yes	—
41	C₈×C₁₀₉	—	2	2	no	—
42	C₈×C₁₀₉	—	4	4	no	—
43	C₈×C₁₀₉	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₁₀₉	—	2	2	no	—
45	C₂²×C₁₀₉	—	2	2	no	—
46	C₂²×C₁₀₉	—	2	4	no	—
47	C₂²×C₁₀₉	—	4	4	no	—
48	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₁₀₉	—	2	2	no	—
51	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₁₀₉	#117	1	1	yes	characteristic
53	C₄×C₁₀₉	#121	1	2	yes	—
54	C₂×C₁₀₉	—	2	2	no	—
55	C₂×C₁₀₉	—	4	2	no	—
56	C₂×C₁₀₉	#380	1	1	yes	characteristic
57	C₂×C₁₀₉	#430	1	2	yes	—
58	#1811	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₁₀₉	#1811	1	1	yes	characteristic, direct product factor, Sylow
60	C₂²×M₁₆	C₂×C₁₀₉	1	1	yes	characteristic
61	C₂×#108	C₂×C₁₀₉	1	2	yes	complemented
62	#407	C₂×C₁₀₉	1	2	yes	—
63	#341	C₂×C₁₀₉	1	2	yes	complemented
64	C₂³×C₄	C₂²×C₁₀₉	1	1	yes	characteristic
65	C₂×M₁₆	—	2	4	no	—
66	C₂×M₁₆	C₄×C₁₀₉	1	4	yes	—
67	C₂×M₁₆	C₂²×C₁₀₉	1	4	yes	—
68	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
69	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
70	#108	—	2	4	no	—
71	C₂²⋊C₈	—	2	4	no	—
72	C₂⁴	Q₈×C₁₀₉	1	1	yes	characteristic
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
76	C₂²×C₄	D₄×C₁₀₉	1	2	yes	—
77	C₂²×C₄	Q₈×C₁₀₉	1	1	yes	characteristic
78	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
79	M₁₆	—	2	8	no	—
80	M₁₆	—	2	4	no	—
81	M₁₆	—	2	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
85	C₂×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
86	C₂×C₈	D₄×C₁₀₉	1	4	yes	—
87	C₂³	—	2	2	no	—
88	C₂³	—	4	2	no	—
89	C₂³	C₄⋊C₄×C₁₀₉	1	2	yes	—
90	C₂³	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	C₄⋊C₄×C₁₀₉	1	2	yes	—
96	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
97	C₂×C₄	C₂×D₄×C₁₀₉	1	2	yes	—
98	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
99	C₂×C₄	#41×C₁₀₉	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	4	4	no	—
102	C₈	C₂²⋊C₄×C₁₀₉	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	—	2	4	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
108	C₂²	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
109	C₄	—	2	2	no	—
110	C₄	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
111	C₄	#117×C₁₀₉	1	1	yes	characteristic
112	C₄	#121×C₁₀₉	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	—	4	2	no	—
115	C₂	#380×C₁₀₉	1	1	yes	characteristic
116	C₂	#430×C₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×M₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#108×C₁₀₉

C₂

yes

maximal, complemented

#407×C₁₀₉

C₂

yes

maximal

#341×C₁₀₉

C₂

yes

maximal, complemented

C₂³×C₄×C₁₀₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₀₉

—

C₂×M₁₆×C₁₀₉

C₄

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#108×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

C₂⁴×C₁₀₉

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

Q₈

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₄⋊C₄

yes

—

C₂³×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂²⋊C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#380

yes

characteristic

C₂×C₁₀₉

#430

yes

—

#1811

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#1811

yes

characteristic, direct product factor, Sylow

C₂²×M₁₆

C₂×C₁₀₉

yes

characteristic

C₂×#108

C₂×C₁₀₉

yes

complemented

#407

C₂×C₁₀₉

yes

—

#341

C₂×C₁₀₉

yes

complemented

C₂³×C₄

C₂²×C₁₀₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₄×C₁₀₉

yes

—

C₂×M₁₆

C₂²×C₁₀₉

yes

—

C₂²×C₈

C₂²×C₁₀₉

yes

characteristic

C₂²×C₈

C₂²×C₁₀₉

yes

characteristic

#108

—

C₂²⋊C₈

—

C₂⁴

Q₈×C₁₀₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₁₀₉

yes

—

C₂²×C₄

D₄×C₁₀₉

yes

—

C₂²×C₄

Q₈×C₁₀₉

yes

characteristic

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₁₀₉

yes

—

C₂×C₈

C₂×C₄×C₁₀₉

yes

—

C₂×C₈

D₄×C₁₀₉

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊C₄×C₁₀₉

yes

—

C₂³

C₂×Q₈×C₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₄×C₁₀₉

yes

—

C₂×C₄

C₂²×C₄×C₁₀₉

yes

characteristic

C₂×C₄

C₂×D₄×C₁₀₉

yes

—

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

C₂×C₄

#41×C₁₀₉

yes

—

100

C₈

—

101

C₈

—

102

C₈

C₂²⋊C₄×C₁₀₉

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₄⋊C₄×C₁₀₉

yes

characteristic

108

C₂²

C₂²⋊Q₈×C₁₀₉

yes

—

109

C₄

—

110

C₄

C₂×C₂²⋊C₄×C₁₀₉

yes

characteristic

111

C₄

#117×C₁₀₉

yes

characteristic

112

C₄

#121×C₁₀₉

yes

—

113

C₂

—

114

C₂

—

115

C₂

#380×C₁₀₉

yes

characteristic

116

C₂

#430×C₁₀₉

yes

—

	subgroup	G/H
center	C₂×C₄×C₁₀₉	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₁₀₉
Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
socle	C₂²×C₁₀₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₀₉	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

subgroup

G/H

center

C₂×C₄×C₁₀₉

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₁₀₉

Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₉

socle

C₂²×C₁₀₉

C₂×C₄⋊C₄

prime radical

C₂⁴×C₁₀₉

Q₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₁₀₉ ◁ C₂³×C₄×C₁₀₉ ◁ G

N	H ≅ G/N	type
2. C₂×#108×C₁₀₉	C₂	semidirect
4. #341×C₁₀₉	C₂	semidirect
58. #1811	C₁₀₉	direct (N × H)
59. C₁₀₉	#1811	direct (N × H)
61. C₂×#108	C₂×C₁₀₉	semidirect
63. #341	C₂×C₁₀₉	semidirect

H ≅ G/N

type

2. C₂×#108×C₁₀₉

C₂

semidirect

4. #341×C₁₀₉

C₂

semidirect

58. #1811

C₁₀₉

direct (N × H)

59. C₁₀₉

#1811

direct (N × H)

61. C₂×#108

C₂×C₁₀₉

semidirect

63. #341

C₂×C₁₀₉

semidirect

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

subgroup	index
C₂²×M₁₆×C₁₀₉	2
C₂×#108×C₁₀₉	2
#407×C₁₀₉	2
#341×C₁₀₉	2
#1811	109

subgroup

index

C₂²×M₁₆×C₁₀₉

C₂×#108×C₁₀₉

#407×C₁₀₉

#341×C₁₀₉

#1811

109

Subgroups — C₁₀₉×#1811 #28359065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#1811 #28359065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#1811 #28359065