Group #42125236 — subgroups

Subgroups — C₁₀₉×#7242 #42125236

Order 27904.

190 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 94 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×#1568	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₁₀₉	C₄	1	2	yes	—
4	#432×C₁₀₉	—	2	2	no	—
5	C₂×C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₄⋊C₁₆×C₁₀₉	—	2	2	no	—
7	#335×C₁₀₉	—	2	2	no	—
8	#334×C₁₀₉	C₄	1	4	yes	—
9	C₂²×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₀₉	Q₈	1	1	yes	characteristic
11	C₂×C₄²×C₁₀₉	D₄	1	1	yes	characteristic
12	C₂×C₁₆×C₁₀₉	—	2	2	no	—
13	C₂×C₁₆×C₁₀₉	—	4	2	no	—
14	C₄⋊C₈×C₁₀₉	—	2	4	no	—
15	C₂²⋊C₈×C₁₀₉	—	4	2	no	—
16	#97×C₁₀₉	—	4	2	no	—
17	C₄×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₁₀₉	D₄	1	2	yes	—
19	C₂²×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₀₉	D₈	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₉	Q₁₆	1	1	yes	characteristic
22	C₂×C₈×C₁₀₉	—	2	1	no	—
23	C₂×C₈×C₁₀₉	—	2	1	no	—
24	C₂×C₈×C₁₀₉	—	4	1	no	—
25	C₂×C₈×C₁₀₉	—	4	4	no	—
26	C₂×C₈×C₁₀₉	C₄²	1	1	yes	characteristic
27	C₂×C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	C₄⋊C₄	1	2	yes	—
29	C₄²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₁₀₉	SD₁₆	1	2	yes	—
31	C₁₆×C₁₀₉	—	4	2	no	—
32	C₂³×C₁₀₉	#107	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	—	2	2	no	—
34	C₂×C₄×C₁₀₉	—	2	1	no	—
35	C₂×C₄×C₁₀₉	—	2	1	no	—
36	C₂×C₄×C₁₀₉	—	4	1	no	—
37	C₂×C₄×C₁₀₉	#95	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	#102	1	2	yes	—
39	C₂×C₄×C₁₀₉	#103	1	2	yes	—
40	C₂×C₄×C₁₀₉	#106	1	1	yes	characteristic
41	C₈×C₁₀₉	—	2	2	no	—
42	C₈×C₁₀₉	—	2	1	no	—
43	C₈×C₁₀₉	—	8	4	no	—
44	C₈×C₁₀₉	#100	1	1	yes	characteristic
45	C₈×C₁₀₉	#101	1	1	yes	characteristic
46	C₂²×C₁₀₉	—	2	1	no	—
47	C₂²×C₁₀₉	—	2	1	no	—
48	C₂²×C₁₀₉	#340	1	1	yes	characteristic
49	C₂²×C₁₀₉	#368	1	2	yes	—
50	C₄×C₁₀₉	—	2	1	no	—
51	C₄×C₁₀₉	—	2	2	no	—
52	C₄×C₁₀₉	—	2	2	no	—
53	C₄×C₁₀₉	#338	1	1	yes	characteristic
54	C₄×C₁₀₉	#343	1	1	yes	characteristic
55	#7242	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×C₁₀₉	—	2	2	no	—
57	C₂×C₁₀₉	#1292	1	1	yes	characteristic
58	C₂×C₁₀₉	#1381	1	1	yes	characteristic
59	C₂×C₁₀₉	#1384	1	1	yes	characteristic
60	C₂×C₄⋊C₁₆	C₂×C₁₀₉	1	1	yes	characteristic
61	#1568	C₂×C₁₀₉	1	2	yes	—
62	C₁₀₉	#7242	1	1	yes	characteristic, direct product factor, Sylow
63	C₂²×C₁₆	C₄×C₁₀₉	1	2	yes	—
64	#432	—	2	2	no	—
65	C₂×C₄×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
66	C₄⋊C₁₆	—	2	2	no	—
67	#335	—	2	2	no	—
68	#334	C₄×C₁₀₉	1	4	yes	—
69	C₂²×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
70	C₂²×C₈	Q₈×C₁₀₉	1	1	yes	characteristic
71	C₂×C₄²	D₄×C₁₀₉	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	4	2	no	—
74	C₄⋊C₈	—	2	4	no	—
75	C₂²⋊C₈	—	4	2	no	—
76	#97	—	4	2	no	—
77	C₄×C₈	C₂×C₄×C₁₀₉	1	2	yes	—
78	C₄×C₈	D₄×C₁₀₉	1	2	yes	—
79	C₂²×C₄	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
80	C₂²×C₄	D₈×C₁₀₉	1	1	yes	characteristic
81	C₂²×C₄	Q₁₆×C₁₀₉	1	1	yes	characteristic
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	4	1	no	—
85	C₂×C₈	—	4	4	no	—
86	C₂×C₈	C₄²×C₁₀₉	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊C₄×C₁₀₉	1	1	yes	characteristic
88	C₂×C₈	C₄⋊C₄×C₁₀₉	1	2	yes	—
89	C₄²	C₂²⋊C₄×C₁₀₉	1	2	yes	—
90	C₄²	SD₁₆×C₁₀₉	1	2	yes	—
91	C₁₆	—	4	2	no	—
92	C₂³	#107×C₁₀₉	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	#95×C₁₀₉	1	1	yes	characteristic
98	C₂×C₄	#102×C₁₀₉	1	2	yes	—
99	C₂×C₄	#103×C₁₀₉	1	2	yes	—
100	C₂×C₄	#106×C₁₀₉	1	1	yes	characteristic
101	C₈	—	2	2	no	—
102	C₈	—	2	1	no	—
103	C₈	—	8	4	no	—
104	C₈	#100×C₁₀₉	1	1	yes	characteristic
105	C₈	#101×C₁₀₉	1	1	yes	characteristic
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#340×C₁₀₉	1	1	yes	characteristic
109	C₂²	#368×C₁₀₉	1	2	yes	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	2	no	—
113	C₄	#338×C₁₀₉	1	1	yes	characteristic
114	C₄	#343×C₁₀₉	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	C₁₀₉×#1292	1	1	yes	characteristic
117	C₂	C₁₀₉×#1381	1	1	yes	characteristic
118	C₂	C₁₀₉×#1384	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊C₁₆×C₁₀₉

C₂

yes

maximal, characteristic

C₁₀₉×#1568

C₂

yes

maximal

C₂²×C₁₆×C₁₀₉

C₄

yes

—

#432×C₁₀₉

—

C₂×C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₁₀₉

—

#335×C₁₀₉

—

#334×C₁₀₉

C₄

yes

—

C₂²×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₁₀₉

Q₈

yes

characteristic

C₂×C₄²×C₁₀₉

D₄

yes

characteristic

C₂×C₁₆×C₁₀₉

—

C₂×C₁₆×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

—

#97×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂×C₄

yes

—

C₄×C₈×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₈

yes

characteristic

C₂²×C₄×C₁₀₉

Q₁₆

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₄²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₄⋊C₄

yes

—

C₄²×C₁₀₉

C₂²⋊C₄

yes

—

C₄²×C₁₀₉

SD₁₆

yes

—

C₁₆×C₁₀₉

—

C₂³×C₁₀₉

#107

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#95

yes

characteristic

C₂×C₄×C₁₀₉

#102

yes

—

C₂×C₄×C₁₀₉

#103

yes

—

C₂×C₄×C₁₀₉

#106

yes

characteristic

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

yes

C₈×C₁₀₉

yes

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#340

yes

characteristic

C₂²×C₁₀₉

#368

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₄⋊C₁₆

C₂×C₁₀₉

yes

characteristic

#1568

C₂×C₁₀₉

yes

—

C₁₀₉

#7242

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

C₄×C₁₀₉

yes

—

#432

—

C₂×C₄×C₈

C₂²×C₁₀₉

yes

characteristic

C₄⋊C₁₆

—

#335

—

#334

C₄×C₁₀₉

yes

—

C₂²×C₈

C₂×C₄×C₁₀₉

yes

characteristic

C₂²×C₈

Q₈×C₁₀₉

yes

characteristic

C₂×C₄²

D₄×C₁₀₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂×C₄×C₁₀₉

yes

—

C₄×C₈

D₄×C₁₀₉

yes

—

C₂²×C₄

C₄⋊C₄×C₁₀₉

yes

characteristic

C₂²×C₄

D₈×C₁₀₉

yes

characteristic

C₂²×C₄

Q₁₆×C₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₁₀₉

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₁₀₉

yes

characteristic

C₂×C₈

C₄⋊C₄×C₁₀₉

yes

—

C₄²

C₂²⋊C₄×C₁₀₉

yes

—

C₄²

SD₁₆×C₁₀₉

yes

—

C₁₆

—

C₂³

#107×C₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#95×C₁₀₉

yes

characteristic

C₂×C₄

#102×C₁₀₉

yes

—

C₂×C₄

#103×C₁₀₉

yes

—

100

C₂×C₄

#106×C₁₀₉

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

—

104

C₈

#100×C₁₀₉

yes

characteristic

105

C₈

#101×C₁₀₉

yes

characteristic

106

C₂²

—

107

C₂²

—

108

C₂²

#340×C₁₀₉

yes

characteristic

109

C₂²

#368×C₁₀₉

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#338×C₁₀₉

yes

characteristic

114

C₄

#343×C₁₀₉

yes

characteristic

115

C₂

—

116

C₂

C₁₀₉×#1292

yes

characteristic

117

C₂

C₁₀₉×#1381

yes

characteristic

118

C₂

C₁₀₉×#1384

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#340
commutator	C₂×C₈	C₄²×C₁₀₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₀₉
prime radical	C₂³×C₁₀₉	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#340

commutator

C₂×C₈

C₄²×C₁₀₉

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₁₀₉

prime radical

C₂³×C₁₀₉

#107

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ C₂×C₄×C₈×C₁₀₉ ◁ G

N	H ≅ G/N	type
55. #7242	C₁₀₉	direct (N × H)
62. C₁₀₉	#7242	direct (N × H)

H ≅ G/N

type

55. #7242

C₁₀₉

direct (N × H)

62. C₁₀₉

#7242

direct (N × H)

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×#1568	2
#7242	109

subgroup

index

C₂×C₄⋊C₁₆×C₁₀₉

C₁₀₉×#1568

#7242

109

Subgroups — C₁₀₉×#7242 #42125236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#7242 #42125236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#7242 #42125236