Group #42128297 — subgroups

Subgroups — C₁₀₉×#10303 #42128297

Order 27904.

478 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₁₀₉×#3255	C₂	1	3	yes	maximal
2	C₁₀₉×#1994	C₂	1	3	yes	maximal
3	C₁₀₉×#1309	C₂	1	1	yes	maximal, characteristic
4	#533×C₁₀₉	C₂²	1	3	yes	—
5	#475×C₁₀₉	—	2	6	no	—
6	#470×C₁₀₉	—	2	6	no	—
7	#462×C₁₀₉	—	2	6	no	—
8	#423×C₁₀₉	C₂²	1	3	yes	—
9	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
10	#339×C₁₀₉	—	2	6	no	—
11	C₂×Q₁₆×C₁₀₉	—	2	6	no	—
12	C₂×M₁₆×C₁₀₉	D₄	1	6	yes	—
13	C₄⋊Q₈×C₁₀₉	D₄	1	6	yes	—
14	#125×C₁₀₉	—	2	3	no	—
15	C₂²⋊Q₈×C₁₀₉	—	2	6	no	—
16	C₄×Q₈×C₁₀₉	—	2	6	no	—
17	#117×C₁₀₉	—	2	3	no	—
18	C₂×C₄²×C₁₀₉	—	2	3	no	—
19	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
20	#106×C₁₀₉	—	2	6	no	—
21	C₄⋊C₈×C₁₀₉	—	2	6	no	—
22	#103×C₁₀₉	—	2	12	no	—
23	C₂×Q₈×C₁₀₉	—	2	6	no	—
24	C₂²×C₄×C₁₀₉	—	4	1	no	—
25	C₂²×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
26	Q₁₆×C₁₀₉	—	8	6	no	—
27	M₁₆×C₁₀₉	—	4	6	no	—
28	C₂×C₈×C₁₀₉	—	2	6	no	—
29	C₄⋊C₄×C₁₀₉	—	2	6	no	—
30	C₄⋊C₄×C₁₀₉	—	2	6	no	—
31	C₄⋊C₄×C₁₀₉	—	4	6	no	—
32	C₂²⋊C₄×C₁₀₉	—	4	3	no	—
33	C₄²×C₁₀₉	—	2	6	no	—
34	C₄²×C₁₀₉	—	4	3	no	—
35	C₄²×C₁₀₉	—	4	3	no	—
36	C₄²×C₁₀₉	C₂×D₄	1	3	yes	—
37	C₄²×C₁₀₉	#41	1	1	yes	characteristic
38	C₂³×C₁₀₉	#127	1	1	yes	characteristic
39	Q₈×C₁₀₉	—	4	6	no	—
40	Q₈×C₁₀₉	—	4	6	no	—
41	C₂×C₄×C₁₀₉	—	2	3	no	—
42	C₂×C₄×C₁₀₉	—	2	3	no	—
43	C₂×C₄×C₁₀₉	—	4	6	no	—
44	C₂×C₄×C₁₀₉	—	4	6	no	—
45	C₂×C₄×C₁₀₉	#120	1	3	yes	—
46	C₂×C₄×C₁₀₉	#121	1	3	yes	—
47	C₈×C₁₀₉	—	4	6	no	—
48	C₂²×C₁₀₉	—	2	3	no	—
49	C₂²×C₁₀₉	#392	1	1	yes	characteristic
50	C₄×C₁₀₉	—	2	3	no	—
51	C₄×C₁₀₉	—	4	4	no	—
52	C₄×C₁₀₉	—	8	6	no	—
53	C₄×C₁₀₉	#455	1	3	yes	—
54	C₄×C₁₀₉	#456	1	3	yes	—
55	#10303	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×C₁₀₉	—	4	1	no	—
57	C₂×C₁₀₉	#2007	1	3	yes	—
58	#3255	C₂×C₁₀₉	1	3	yes	—
59	#1994	C₂×C₁₀₉	1	3	yes	—
60	#1309	C₂×C₁₀₉	1	1	yes	characteristic
61	C₁₀₉	#10303	1	1	yes	characteristic, direct product factor, Sylow
62	#533	C₂²×C₁₀₉	1	3	yes	—
63	#475	—	2	6	no	—
64	#470	—	2	6	no	—
65	#462	—	2	6	no	—
66	#423	C₂²×C₁₀₉	1	3	yes	—
67	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
68	#339	—	2	6	no	—
69	C₂×Q₁₆	—	2	6	no	—
70	C₂×M₁₆	D₄×C₁₀₉	1	6	yes	—
71	C₄⋊Q₈	D₄×C₁₀₉	1	6	yes	—
72	#125	—	2	3	no	—
73	C₂²⋊Q₈	—	2	6	no	—
74	C₄×Q₈	—	2	6	no	—
75	#117	—	2	3	no	—
76	C₂×C₄²	—	2	3	no	—
77	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
78	#106	—	2	6	no	—
79	C₄⋊C₈	—	2	6	no	—
80	#103	—	2	12	no	—
81	C₂×Q₈	—	2	6	no	—
82	C₂²×C₄	—	4	1	no	—
83	C₂²×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
84	Q₁₆	—	8	6	no	—
85	M₁₆	—	4	6	no	—
86	C₂×C₈	—	2	6	no	—
87	C₄⋊C₄	—	2	6	no	—
88	C₄⋊C₄	—	2	6	no	—
89	C₄⋊C₄	—	4	6	no	—
90	C₂²⋊C₄	—	4	3	no	—
91	C₄²	—	2	6	no	—
92	C₄²	—	4	3	no	—
93	C₄²	—	4	3	no	—
94	C₄²	C₂×D₄×C₁₀₉	1	3	yes	—
95	C₄²	#41×C₁₀₉	1	1	yes	characteristic
96	C₂³	#127×C₁₀₉	1	1	yes	characteristic
97	Q₈	—	4	6	no	—
98	Q₈	—	4	6	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	—	2	3	no	—
101	C₂×C₄	—	4	6	no	—
102	C₂×C₄	—	4	6	no	—
103	C₂×C₄	#120×C₁₀₉	1	3	yes	—
104	C₂×C₄	#121×C₁₀₉	1	3	yes	—
105	C₈	—	4	6	no	—
106	C₂²	—	2	3	no	—
107	C₂²	#392×C₁₀₉	1	1	yes	characteristic
108	C₄	—	2	3	no	—
109	C₄	—	4	4	no	—
110	C₄	—	8	6	no	—
111	C₄	#455×C₁₀₉	1	3	yes	—
112	C₄	#456×C₁₀₉	1	3	yes	—
113	C₂	—	4	1	no	—
114	C₂	C₁₀₉×#2007	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#3255

C₂

yes

maximal

C₁₀₉×#1994

C₂

yes

maximal

C₁₀₉×#1309

C₂

yes

maximal, characteristic

#533×C₁₀₉

C₂²

yes

—

#475×C₁₀₉

—

#470×C₁₀₉

—

#462×C₁₀₉

—

#423×C₁₀₉

C₂²

yes

—

C₄³×C₁₀₉

C₂²

yes

characteristic

#339×C₁₀₉

—

C₂×Q₁₆×C₁₀₉

—

C₂×M₁₆×C₁₀₉

D₄

yes

—

C₄⋊Q₈×C₁₀₉

D₄

yes

—

#125×C₁₀₉

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

#117×C₁₀₉

—

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

#106×C₁₀₉

—

C₄⋊C₈×C₁₀₉

—

#103×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

Q₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×D₄

yes

—

C₄²×C₁₀₉

yes

C₂³×C₁₀₉

yes

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

#120

yes

—

C₂×C₄×C₁₀₉

#121

yes

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#392

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#455

yes

—

C₄×C₁₀₉

#456

yes

—

#10303

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#2007

yes

—

#3255

C₂×C₁₀₉

yes

—

#1994

C₂×C₁₀₉

yes

—

#1309

C₂×C₁₀₉

yes

characteristic

C₁₀₉

#10303

yes

characteristic, direct product factor, Sylow

#533

C₂²×C₁₀₉

yes

—

—

—

—

C₂²×C₁₀₉

yes

—

C₄³

C₂²×C₁₀₉

yes

characteristic

#339

—

C₂×Q₁₆

—

C₂×M₁₆

D₄×C₁₀₉

yes

—

C₄⋊Q₈

D₄×C₁₀₉

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂³×C₁₀₉

yes

characteristic

#106

—

C₄⋊C₈

—

#103

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₄×C₁₀₉

yes

—

C₄²

#41×C₁₀₉

yes

characteristic

C₂³

#127×C₁₀₉

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#120×C₁₀₉

yes

—

104

C₂×C₄

#121×C₁₀₉

yes

—

105

C₈

—

106

C₂²

—

107

C₂²

#392×C₁₀₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#455×C₁₀₉

yes

—

112

C₄

#456×C₁₀₉

yes

—

113

C₂

—

114

C₂

C₁₀₉×#2007

yes

—

	subgroup	G/H
center, socle	C₂²×C₁₀₉	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₀₉

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

prime radical

C₂³×C₁₀₉

#127

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
55. #10303	C₁₀₉	direct (N × H)
61. C₁₀₉	#10303	direct (N × H)

H ≅ G/N

type

55. #10303

C₁₀₉

direct (N × H)

61. C₁₀₉

#10303

direct (N × H)

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

subgroup	index
C₁₀₉×#3255	2
C₁₀₉×#1994	2
C₁₀₉×#1309	2
#10303	109

subgroup

index

C₁₀₉×#3255

C₁₀₉×#1994

C₁₀₉×#1309

#10303

109

Subgroups — C₁₀₉×#10303 #42128297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10303 #42128297

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10303 #42128297