Group #28359463 — subgroups

Subgroups — C₁₀₉×#2209 #28359463

Order 13952.

154 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 68 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×Q₃₂×C₁₀₉	C₂	1	2	yes	maximal
2	#494×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#358×C₁₀₉	C₂	1	2	yes	maximal
6	C₂×Q₁₆×C₁₀₉	—	2	1	no	—
7	C₂×Q₁₆×C₁₀₉	C₂²	1	2	yes	—
8	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈×C₁₀₉	—	2	1	no	—
10	C₄×Q₈×C₁₀₉	—	2	1	no	—
11	Q₃₂×C₁₀₉	—	2	4	no	—
12	C₂×C₁₆×C₁₀₉	C₂²	1	2	yes	—
13	#107×C₁₀₉	C₂²	1	1	yes	characteristic
14	#103×C₁₀₉	—	2	1	no	—
15	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₀₉	—	2	1	no	—
17	C₂×Q₈×C₁₀₉	—	2	2	no	—
18	Q₁₆×C₁₀₉	—	2	2	no	—
19	Q₁₆×C₁₀₉	—	2	1	no	—
20	Q₁₆×C₁₀₉	—	2	2	no	—
21	Q₁₆×C₁₀₉	—	4	2	no	—
22	Q₁₆×C₁₀₉	D₄	1	2	yes	—
23	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₀₉	—	2	1	no	—
26	C₄⋊C₄×C₁₀₉	—	4	1	no	—
27	C₄⋊C₄×C₁₀₉	—	4	1	no	—
28	C₄⋊C₄×C₁₀₉	—	4	1	no	—
29	C₄²×C₁₀₉	—	4	1	no	—
30	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
31	C₁₆×C₁₀₉	—	2	2	no	—
32	Q₈×C₁₀₉	—	2	2	no	—
33	Q₈×C₁₀₉	—	4	1	no	—
34	Q₈×C₁₀₉	—	4	2	no	—
35	Q₈×C₁₀₉	—	4	2	no	—
36	C₂×C₄×C₁₀₉	—	4	1	no	—
37	C₂×C₄×C₁₀₉	—	4	1	no	—
38	C₂×C₄×C₁₀₉	—	4	2	no	—
39	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₈×C₁₀₉	—	2	1	no	—
43	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₀₉	#41	1	1	yes	characteristic
45	C₂²×C₁₀₉	C₂×D₈	1	1	yes	characteristic
46	C₄×C₁₀₉	—	2	1	no	—
47	C₄×C₁₀₉	—	4	2	no	—
48	C₄×C₁₀₉	—	8	2	no	—
49	C₄×C₁₀₉	—	8	1	no	—
50	C₄×C₁₀₉	Q₃₂	1	2	yes	complemented
51	C₄×C₁₀₉	#121	1	1	yes	characteristic
52	C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
53	C₂×C₁₀₉	#459	1	1	yes	characteristic
54	C₂×C₁₀₉	C₂×Q₃₂	1	1	yes	characteristic
55	C₂×C₁₀₉	#510	1	1	yes	characteristic
56	#2209	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₁₀₉	#2209	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×Q₃₂	C₂×C₁₀₉	1	2	yes	—
59	#494	C₂×C₁₀₉	1	1	yes	characteristic
60	C₄×Q₁₆	C₂×C₁₀₉	1	1	yes	characteristic
61	C₄⋊C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
62	#358	C₂×C₁₀₉	1	2	yes	—
63	C₂×Q₁₆	—	2	1	no	—
64	C₂×Q₁₆	C₂²×C₁₀₉	1	2	yes	—
65	C₂×Q₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
66	C₄⋊Q₈	—	2	1	no	—
67	C₄×Q₈	—	2	1	no	—
68	Q₃₂	—	2	4	no	—
69	C₂×C₁₆	C₂²×C₁₀₉	1	2	yes	—
70	#107	C₂²×C₁₀₉	1	1	yes	characteristic
71	#103	—	2	1	no	—
72	C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	2	2	no	—
75	Q₁₆	—	2	2	no	—
76	Q₁₆	—	2	1	no	—
77	Q₁₆	—	2	2	no	—
78	Q₁₆	—	4	2	no	—
79	Q₁₆	D₄×C₁₀₉	1	2	yes	—
80	C₂×C₈	D₄×C₁₀₉	1	1	yes	characteristic
81	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
82	C₄⋊C₄	—	2	1	no	—
83	C₄⋊C₄	—	4	1	no	—
84	C₄⋊C₄	—	4	1	no	—
85	C₄⋊C₄	—	4	1	no	—
86	C₄²	—	4	1	no	—
87	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	2	2	no	—
90	Q₈	—	4	1	no	—
91	Q₈	—	4	2	no	—
92	Q₈	—	4	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	D₈×C₁₀₉	1	1	yes	characteristic
97	C₂×C₄	D₈×C₁₀₉	1	1	yes	characteristic
98	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	C₂×D₄×C₁₀₉	1	1	yes	characteristic
101	C₈	#41×C₁₀₉	1	1	yes	characteristic
102	C₂²	C₂×D₈×C₁₀₉	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	8	1	no	—
106	C₄	—	8	2	no	—
107	C₄	Q₃₂×C₁₀₉	1	2	yes	complemented
108	C₄	#121×C₁₀₉	1	1	yes	characteristic
109	C₄	AGL(1,ℤ/8)×C₁₀₉	1	1	yes	characteristic
110	C₂	#459×C₁₀₉	1	1	yes	characteristic
111	C₂	C₂×Q₃₂×C₁₀₉	1	1	yes	characteristic
112	C₂	#510×C₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₃₂×C₁₀₉

C₂

yes

maximal

#494×C₁₀₉

C₂

yes

maximal, characteristic

C₄×Q₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₄⋊C₁₆×C₁₀₉

C₂

yes

maximal, characteristic

#358×C₁₀₉

C₂

yes

maximal

C₂×Q₁₆×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

C₂²

yes

—

C₂×Q₁₆×C₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

Q₃₂×C₁₀₉

—

C₂×C₁₆×C₁₀₉

C₂²

yes

—

#107×C₁₀₉

C₂²

yes

characteristic

#103×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

—

Q₁₆×C₁₀₉

D₄

yes

—

C₂×C₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₁₆×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

D₈

yes

characteristic

C₂×C₄×C₁₀₉

D₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

#41

yes

characteristic

C₂²×C₁₀₉

C₂×D₈

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

Q₃₂

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

C₂×Q₃₂

yes

C₂×C₁₀₉

yes

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#2209

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

C₂×C₁₀₉

yes

—

#494

C₂×C₁₀₉

yes

characteristic

C₄×Q₁₆

C₂×C₁₀₉

yes

characteristic

C₄⋊C₁₆

C₂×C₁₀₉

yes

characteristic

#358

C₂×C₁₀₉

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂²×C₁₀₉

yes

—

C₂×Q₁₆

C₂²×C₁₀₉

yes

characteristic

C₄⋊Q₈

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

C₂²×C₁₀₉

yes

—

#107

C₂²×C₁₀₉

yes

characteristic

#103

—

C₄×C₈

C₂²×C₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

D₄×C₁₀₉

yes

—

C₂×C₈

D₄×C₁₀₉

yes

characteristic

C₂×C₈

C₂³×C₁₀₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₄×C₁₀₉

yes

characteristic

C₁₆

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₈×C₁₀₉

yes

characteristic

C₂×C₄

D₈×C₁₀₉

yes

characteristic

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

C₈

—

100

C₈

C₂×D₄×C₁₀₉

yes

characteristic

101

C₈

#41×C₁₀₉

yes

characteristic

102

C₂²

C₂×D₈×C₁₀₉

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

Q₃₂×C₁₀₉

yes

complemented

108

C₄

#121×C₁₀₉

yes

characteristic

109

C₄

AGL(1,ℤ/8)×C₁₀₉

yes

characteristic

110

C₂

#459×C₁₀₉

yes

characteristic

111

C₂

C₂×Q₃₂×C₁₀₉

yes

characteristic

112

C₂

#510×C₁₀₉

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂²×C₁₀₉

C₂×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₁₀₉

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₄²×C₁₀₉ ◁ C₄×C₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
50. C₄×C₁₀₉	Q₃₂	semidirect
56. #2209	C₁₀₉	direct (N × H)
57. C₁₀₉	#2209	direct (N × H)
107. C₄	Q₃₂×C₁₀₉	semidirect

H ≅ G/N

type

50. C₄×C₁₀₉

Q₃₂

semidirect

56. #2209

C₁₀₉

direct (N × H)

57. C₁₀₉

#2209

direct (N × H)

107. C₄

Q₃₂×C₁₀₉

semidirect

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

subgroup	index
C₂×Q₃₂×C₁₀₉	2
#494×C₁₀₉	2
C₄×Q₁₆×C₁₀₉	2
C₄⋊C₁₆×C₁₀₉	2
#358×C₁₀₉	2
#2209	109

subgroup

index

C₂×Q₃₂×C₁₀₉

#494×C₁₀₉

C₄×Q₁₆×C₁₀₉

C₄⋊C₁₆×C₁₀₉

#358×C₁₀₉

#2209

109

Subgroups — C₁₀₉×#2209 #28359463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#2209 #28359463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2209 #28359463