Group #28359252 — subgroups

Subgroups — C₁₀₉×#1998 #28359252

Order 13952.

228 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 112 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×#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#343×C₁₀₉	C₂	1	4	yes	maximal
5	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
6	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
7	#125×C₁₀₉	—	2	1	no	—
8	#125×C₁₀₉	—	2	1	no	—
9	#125×C₁₀₉	—	2	2	no	—
10	C₂×C₄⋊C₄×C₁₀₉	C₄	1	4	yes	—
11	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₀₉	—	2	2	no	—
14	#97×C₁₀₉	—	2	1	no	—
15	C₄×C₈×C₁₀₉	—	2	1	no	—
16	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
19	M₁₆×C₁₀₉	—	2	2	no	—
20	M₁₆×C₁₀₉	—	2	4	no	—
21	M₁₆×C₁₀₉	Q₈	1	4	yes	—
22	C₂×C₈×C₁₀₉	—	2	2	no	—
23	C₂×C₈×C₁₀₉	—	2	2	no	—
24	C₄⋊C₄×C₁₀₉	—	2	4	no	—
25	C₄⋊C₄×C₁₀₉	—	2	4	no	—
26	C₄⋊C₄×C₁₀₉	—	2	2	no	—
27	C₄⋊C₄×C₁₀₉	—	2	2	no	—
28	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
29	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
30	C₄²×C₁₀₉	D₄	1	2	yes	—
31	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	—	2	4	no	—
33	C₂×C₄×C₁₀₉	—	4	2	no	—
34	C₂×C₄×C₁₀₉	—	4	4	no	—
35	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₀₉	#41	1	2	yes	—
41	C₈×C₁₀₉	—	2	4	no	—
42	C₈×C₁₀₉	—	4	2	no	—
43	C₂²×C₁₀₉	—	2	2	no	—
44	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₀₉	C₄×Q₈	1	2	yes	—
46	C₄×C₁₀₉	—	2	2	no	—
47	C₄×C₁₀₉	—	2	2	no	—
48	C₄×C₁₀₉	—	4	4	no	—
49	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₁₀₉	#124	1	1	yes	characteristic
51	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₁₀₉	—	2	2	no	—
53	C₂×C₁₀₉	#391	1	1	yes	characteristic
54	C₂×C₁₀₉	#413	1	2	yes	—
55	#1998	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₁₀₉	#1998	1	1	yes	characteristic, direct product factor, Sylow
57	C₂×#125	C₂×C₁₀₉	1	1	yes	characteristic
58	#423	C₂×C₁₀₉	1	1	yes	characteristic
59	C₄×M₁₆	C₂×C₁₀₉	1	1	yes	characteristic
60	#343	C₂×C₁₀₉	1	4	yes	—
61	C₂×M₁₆	C₂²×C₁₀₉	1	2	yes	—
62	C₂×M₁₆	C₂²×C₁₀₉	1	2	yes	—
63	#125	—	2	1	no	—
64	#125	—	2	1	no	—
65	#125	—	2	2	no	—
66	C₂×C₄⋊C₄	C₄×C₁₀₉	1	4	yes	—
67	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	2	yes	—
68	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
69	C₄⋊C₈	—	2	2	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	C₂×C₄×C₁₀₉	1	4	yes	—
73	C₂²×C₄	C₂×C₄×C₁₀₉	1	2	yes	—
74	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	2	4	no	—
77	M₁₆	Q₈×C₁₀₉	1	4	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
85	C₄²	D₄×C₁₀₉	1	1	yes	characteristic
86	C₄²	D₄×C₁₀₉	1	2	yes	—
87	C₂³	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
92	C₂×C₄	C₂²⋊C₄×C₁₀₉	1	2	yes	—
93	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
94	C₂×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
95	C₂×C₄	C₂×Q₈×C₁₀₉	1	2	yes	—
96	C₂×C₄	#41×C₁₀₉	1	2	yes	—
97	C₈	—	2	4	no	—
98	C₈	—	4	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
101	C₂²	C₄×Q₈×C₁₀₉	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	C₂²⋊Q₈×C₁₀₉	1	2	yes	—
106	C₄	#124×C₁₀₉	1	1	yes	characteristic
107	C₄	C₄⋊Q₈×C₁₀₉	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#391×C₁₀₉	1	1	yes	characteristic
110	C₂	#413×C₁₀₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125×C₁₀₉

C₂

yes

maximal, characteristic

#423×C₁₀₉

C₂

yes

maximal, characteristic

C₄×M₁₆×C₁₀₉

C₂

yes

maximal, characteristic

#343×C₁₀₉

C₂

yes

maximal

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

#125×C₁₀₉

—

#125×C₁₀₉

—

#125×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

#124

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

#391

yes

characteristic

C₂×C₁₀₉

#413

yes

—

#1998

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₁₀₉

#1998

yes

characteristic, direct product factor, Sylow

C₂×#125

C₂×C₁₀₉

yes

characteristic

#423

C₂×C₁₀₉

yes

characteristic

C₄×M₁₆

C₂×C₁₀₉

yes

characteristic

#343

C₂×C₁₀₉

yes

—

C₂×M₁₆

C₂²×C₁₀₉

yes

—

C₂×M₁₆

C₂²×C₁₀₉

yes

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

C₄×C₁₀₉

yes

—

C₂×C₄⋊C₄

C₂²×C₁₀₉

yes

—

C₂×C₄²

C₂²×C₁₀₉

yes

characteristic

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂²×C₄

C₂×C₄×C₁₀₉

yes

—

C₂²×C₄

C₂×C₄×C₁₀₉

yes

—

C₂²×C₄

C₂³×C₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

Q₈×C₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₄×C₁₀₉

yes

characteristic

C₄²

D₄×C₁₀₉

yes

characteristic

C₄²

D₄×C₁₀₉

yes

—

C₂³

C₂²×C₄×C₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₁₀₉

yes

—

C₂×C₄

C₂²⋊C₄×C₁₀₉

yes

—

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

C₂×C₄

C₂×D₄×C₁₀₉

yes

characteristic

C₂×C₄

C₂×Q₈×C₁₀₉

yes

—

C₂×C₄

#41×C₁₀₉

yes

—

C₈

—

C₈

—

C₂²

—

100

C₂²

C₂×C₂²⋊C₄×C₁₀₉

yes

characteristic

101

C₂²

C₄×Q₈×C₁₀₉

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₂²⋊Q₈×C₁₀₉

yes

—

106

C₄

#124×C₁₀₉

yes

characteristic

107

C₄

C₄⋊Q₈×C₁₀₉

yes

characteristic

108

C₂

—

109

C₂

#391×C₁₀₉

yes

characteristic

110

C₂

#413×C₁₀₉

yes

—

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

subgroup

G/H

center, socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₁₀₉

Frattini, cosocle

C₂²×C₄

C₂³×C₁₀₉

prime radical

C₂³×C₁₀₉

C₂²×C₄

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
55. #1998	C₁₀₉	direct (N × H)
56. C₁₀₉	#1998	direct (N × H)

H ≅ G/N

type

55. #1998

C₁₀₉

direct (N × H)

56. C₁₀₉

#1998

direct (N × H)

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

subgroup	index
C₂×#125×C₁₀₉	2
#423×C₁₀₉	2
C₄×M₁₆×C₁₀₉	2
#343×C₁₀₉	2
#1998	109

subgroup

index

C₂×#125×C₁₀₉

#423×C₁₀₉

C₄×M₁₆×C₁₀₉

#343×C₁₀₉

#1998

109

Subgroups — C₁₀₉×#1998 #28359252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#1998 #28359252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#1998 #28359252