Group #42130883 — subgroups

Subgroups — C₁₀₉×#12889 #42130883

Order 27904.

310 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 174 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×C₄×C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₁₀₉×#1846	C₂	1	2	yes	maximal
3	C₁₀₉×#1380	C₂	1	4	yes	maximal
4	C₂²×C₁₆×C₁₀₉	C₂²	1	2	yes	—
5	C₂×#107×C₁₀₉	C₂²	1	4	yes	—
6	C₂×#106×C₁₀₉	—	2	2	no	—
7	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
8	#382×C₁₀₉	—	2	2	no	—
9	C₄×C₁₆×C₁₀₉	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₀₉	D₄	1	1	yes	characteristic
11	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₀₉	—	2	4	no	—
13	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₁₀₉	—	2	4	no	—
15	C₂×C₁₆×C₁₀₉	C₂×C₄	1	4	yes	—
16	#107×C₁₀₉	—	2	8	no	—
17	#106×C₁₀₉	—	4	4	no	—
18	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₁₀₉	Q₈	1	2	yes	—
20	#95×C₁₀₉	—	4	4	no	—
21	C₂²×C₄×C₁₀₉	—	4	4	no	—
22	C₂²×C₄×C₁₀₉	D₈	1	2	yes	—
23	C₂²×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
25	C₂×C₈×C₁₀₉	SD₁₆	1	4	yes	—
26	C₂×C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	#41	1	4	yes	—
29	C₄⋊C₄×C₁₀₉	—	4	4	no	—
30	C₄⋊C₄×C₁₀₉	—	4	4	no	—
31	C₄²×C₁₀₉	C₄⋊C₄	1	2	yes	—
32	C₄²×C₁₀₉	Q₁₆	1	2	yes	—
33	C₁₆×C₁₀₉	—	2	4	no	—
34	C₂³×C₁₀₉	C₂×D₈	1	1	yes	characteristic
35	C₂×C₄×C₁₀₉	—	2	4	no	—
36	C₂×C₄×C₁₀₉	—	8	4	no	—
37	C₂×C₄×C₁₀₉	—	8	8	no	—
38	C₂×C₄×C₁₀₉	#107	1	4	yes	—
39	C₂×C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	#124	1	2	yes	—
41	C₂×C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
43	C₈×C₁₀₉	#106	1	4	yes	—
44	C₈×C₁₀₉	#117	1	2	yes	—
45	C₈×C₁₀₉	#125	1	2	yes	—
46	C₂²×C₁₀₉	C₂×#107	1	1	yes	characteristic
47	C₂²×C₁₀₉	#486	1	2	yes	—
48	C₂²×C₁₀₉	#508	1	4	yes	—
49	C₄×C₁₀₉	—	2	4	no	—
50	C₄×C₁₀₉	—	16	4	no	—
51	C₄×C₁₀₉	#382	1	1	yes	characteristic
52	C₄×C₁₀₉	C₂×#106	1	1	yes	characteristic
53	C₄×C₁₀₉	#487	1	2	yes	—
54	#12889	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₁₀₉	#1834	1	1	yes	characteristic
56	C₂×C₁₀₉	#2158	1	2	yes	—
57	C₂×C₁₀₉	#2242	1	4	yes	—
58	C₂×C₄×C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
59	#1846	C₂×C₁₀₉	1	2	yes	—
60	#1380	C₂×C₁₀₉	1	4	yes	—
61	C₁₀₉	#12889	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²×C₁₆	C₂²×C₁₀₉	1	2	yes	—
63	C₂×#107	C₂²×C₁₀₉	1	4	yes	—
64	C₂×#106	—	2	2	no	—
65	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
66	#382	—	2	2	no	—
67	C₄×C₁₆	C₄×C₁₀₉	1	4	yes	complemented
68	C₂²×C₈	D₄×C₁₀₉	1	1	yes	characteristic
69	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	2	4	no	—
71	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
72	C₂×C₁₆	—	2	4	no	—
73	C₂×C₁₆	C₂×C₄×C₁₀₉	1	4	yes	—
74	#107	—	2	8	no	—
75	#106	—	4	4	no	—
76	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
77	C₄×C₈	Q₈×C₁₀₉	1	2	yes	—
78	#95	—	4	4	no	—
79	C₂²×C₄	—	4	4	no	—
80	C₂²×C₄	D₈×C₁₀₉	1	2	yes	—
81	C₂²×C₄	C₂×D₄×C₁₀₉	1	1	yes	characteristic
82	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
83	C₂×C₈	SD₁₆×C₁₀₉	1	4	yes	—
84	C₂×C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
85	C₂×C₈	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
86	C₂×C₈	#41×C₁₀₉	1	4	yes	—
87	C₄⋊C₄	—	4	4	no	—
88	C₄⋊C₄	—	4	4	no	—
89	C₄²	C₄⋊C₄×C₁₀₉	1	2	yes	—
90	C₄²	Q₁₆×C₁₀₉	1	2	yes	—
91	C₁₆	—	2	4	no	—
92	C₂³	C₂×D₈×C₁₀₉	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	8	8	no	—
95	C₂×C₄	—	8	4	no	—
96	C₂×C₄	#107×C₁₀₉	1	4	yes	—
97	C₂×C₄	C₂×C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
98	C₂×C₄	#124×C₁₀₉	1	2	yes	—
99	C₂×C₄	C₂×SD₁₆×C₁₀₉	1	2	yes	—
100	C₂×C₄	C₂×Q₁₆×C₁₀₉	1	1	yes	characteristic
101	C₈	#106×C₁₀₉	1	4	yes	—
102	C₈	#117×C₁₀₉	1	2	yes	—
103	C₈	#125×C₁₀₉	1	2	yes	—
104	C₂²	C₂×#107×C₁₀₉	1	1	yes	characteristic
105	C₂²	#486×C₁₀₉	1	2	yes	—
106	C₂²	#508×C₁₀₉	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	16	4	no	—
109	C₄	#382×C₁₀₉	1	1	yes	characteristic
110	C₄	C₂×#106×C₁₀₉	1	1	yes	characteristic
111	C₄	#487×C₁₀₉	1	2	yes	—
112	C₂	C₁₀₉×#1834	1	1	yes	characteristic
113	C₂	C₁₀₉×#2158	1	2	yes	—
114	C₂	C₁₀₉×#2242	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₁₀₉×#1846

C₂

yes

maximal

C₁₀₉×#1380

C₂

yes

maximal

C₂²×C₁₆×C₁₀₉

C₂²

yes

—

C₂×#107×C₁₀₉

C₂²

yes

—

C₂×#106×C₁₀₉

—

C₂×C₄×C₈×C₁₀₉

C₂²

yes

characteristic

#382×C₁₀₉

—

C₄×C₁₆×C₁₀₉

C₄

yes

complemented

C₂²×C₈×C₁₀₉

D₄

yes

characteristic

C₂²×C₈×C₁₀₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

D₄

yes

characteristic

C₂×C₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

C₂×C₄

yes

—

#107×C₁₀₉

—

#106×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂×C₄

yes

—

C₄×C₈×C₁₀₉

Q₈

yes

—

#95×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₈

yes

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₈×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₈×C₁₀₉

SD₁₆

yes

—

C₂×C₈×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₀₉

#41

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₄⋊C₄

yes

—

C₄²×C₁₀₉

Q₁₆

yes

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

C₂×D₈

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#107

yes

—

C₂×C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

#124

yes

—

C₂×C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₂×C₄×C₁₀₉

C₂×Q₁₆

yes

characteristic

C₈×C₁₀₉

#106

yes

—

C₈×C₁₀₉

#117

yes

—

C₈×C₁₀₉

#125

yes

—

C₂²×C₁₀₉

C₂×#107

yes

characteristic

C₂²×C₁₀₉

#486

yes

—

C₂²×C₁₀₉

#508

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

#12889

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

#1834

yes

characteristic

C₂×C₁₀₉

#2158

yes

—

C₂×C₁₀₉

#2242

yes

—

C₂×C₄×C₁₆

C₂×C₁₀₉

yes

characteristic

#1846

C₂×C₁₀₉

yes

—

#1380

C₂×C₁₀₉

yes

—

C₁₀₉

#12889

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

C₂²×C₁₀₉

yes

—

C₂×#107

C₂²×C₁₀₉

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂²×C₁₀₉

yes

characteristic

#382

—

C₄×C₁₆

C₄×C₁₀₉

yes

complemented

C₂²×C₈

D₄×C₁₀₉

yes

characteristic

C₂²×C₈

C₂³×C₁₀₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

D₄×C₁₀₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₁₀₉

yes

—

#107

—

#106

—

C₄×C₈

C₂×C₄×C₁₀₉

yes

—

C₄×C₈

Q₈×C₁₀₉

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

D₈×C₁₀₉

yes

—

C₂²×C₄

C₂×D₄×C₁₀₉

yes

characteristic

C₂×C₈

C₄⋊C₄×C₁₀₉

yes

—

C₂×C₈

SD₁₆×C₁₀₉

yes

—

C₂×C₈

C₂²×C₄×C₁₀₉

yes

characteristic

C₂×C₈

C₂×Q₈×C₁₀₉

yes

characteristic

C₂×C₈

#41×C₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄⋊C₄×C₁₀₉

yes

—

C₄²

Q₁₆×C₁₀₉

yes

—

C₁₆

—

C₂³

C₂×D₈×C₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#107×C₁₀₉

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₁₀₉

yes

characteristic

C₂×C₄

#124×C₁₀₉

yes

—

C₂×C₄

C₂×SD₁₆×C₁₀₉

yes

—

100

C₂×C₄

C₂×Q₁₆×C₁₀₉

yes

characteristic

101

C₈

#106×C₁₀₉

yes

—

102

C₈

#117×C₁₀₉

yes

—

103

C₈

#125×C₁₀₉

yes

—

104

C₂²

C₂×#107×C₁₀₉

yes

characteristic

105

C₂²

#486×C₁₀₉

yes

—

106

C₂²

#508×C₁₀₉

yes

—

107

C₄

—

108

C₄

—

109

C₄

#382×C₁₀₉

yes

characteristic

110

C₄

C₂×#106×C₁₀₉

yes

characteristic

111

C₄

#487×C₁₀₉

yes

—

112

C₂

C₁₀₉×#1834

yes

characteristic

113

C₂

C₁₀₉×#2158

yes

—

114

C₂

C₁₀₉×#2242

yes

—

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

subgroup

G/H

center, socle, prime radical

C₂³×C₁₀₉

C₂×D₈

commutator

C₂×C₈

C₂²×C₄×C₁₀₉

Frattini, cosocle

C₂²×C₈

C₂³×C₁₀₉

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
9. C₄×C₁₆×C₁₀₉	C₄	semidirect
54. #12889	C₁₀₉	direct (N × H)
61. C₁₀₉	#12889	direct (N × H)
67. C₄×C₁₆	C₄×C₁₀₉	semidirect

H ≅ G/N

type

9. C₄×C₁₆×C₁₀₉

C₄

semidirect

54. #12889

C₁₀₉

direct (N × H)

61. C₁₀₉

#12889

direct (N × H)

67. C₄×C₁₆

C₄×C₁₀₉

semidirect

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

subgroup	index
C₂×C₄×C₁₆×C₁₀₉	2
C₁₀₉×#1846	2
C₁₀₉×#1380	2
#12889	109

subgroup

index

C₂×C₄×C₁₆×C₁₀₉

C₁₀₉×#1846

C₁₀₉×#1380

#12889

109

Subgroups — C₁₀₉×#12889 #42130883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#12889 #42130883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12889 #42130883