Group #42128354 — subgroups

Subgroups — C₁₀₉×#10360 #42128354

Order 27904.

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

normal subgroups only

This group has 476 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×#2845	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₀₉×#1961	C₂	1	3	yes	maximal
3	C₁₀₉×#1295	C₂	1	3	yes	maximal, complemented
4	C₂×#127×C₁₀₉	C₂²	1	1	yes	characteristic
5	C₂×#102×C₁₀₉	—	2	6	no	—
6	C₂×#97×C₁₀₉	C₂²	1	3	yes	—
7	#398×C₁₀₉	—	2	6	no	—
8	#382×C₁₀₉	C₂²	1	3	yes	—
9	C₂²×D₄×C₁₀₉	—	2	3	no	—
10	C₂²×C₈×C₁₀₉	—	2	3	no	—
11	#127×C₁₀₉	—	2	3	no	—
12	#127×C₁₀₉	Q₈	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₀₉	—	2	6	no	—
14	C₂×C₄⋊C₄×C₁₀₉	D₄	1	6	yes	—
15	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
16	#102×C₁₀₉	—	4	12	no	—
17	#97×C₁₀₉	—	2	6	no	—
18	#95×C₁₀₉	—	2	2	no	—
19	#95×C₁₀₉	—	2	6	no	—
20	C₂⁴×C₁₀₉	—	4	1	no	—
21	C₂×D₄×C₁₀₉	—	2	6	no	—
22	C₂×D₄×C₁₀₉	—	4	6	no	—
23	C₂×D₄×C₁₀₉	—	4	6	no	—
24	C₂×D₄×C₁₀₉	—	4	3	no	—
25	C₂²×C₄×C₁₀₉	—	2	6	no	—
26	C₂²×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
27	C₂×C₈×C₁₀₉	—	2	6	no	—
28	C₂×C₈×C₁₀₉	—	4	3	no	—
29	C₂×C₈×C₁₀₉	—	4	3	no	—
30	C₄⋊C₄×C₁₀₉	—	4	6	no	—
31	C₄⋊C₄×C₁₀₉	—	8	6	no	—
32	C₄²×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₁₀₉	#41	1	3	yes	—
34	C₂³×C₁₀₉	—	4	2	no	—
35	C₂³×C₁₀₉	—	8	3	no	—
36	C₂³×C₁₀₉	—	8	3	no	—
37	C₂³×C₁₀₉	#120	1	1	yes	characteristic
38	D₄×C₁₀₉	—	4	12	no	—
39	D₄×C₁₀₉	—	8	6	no	—
40	C₂×C₄×C₁₀₉	—	2	3	no	—
41	C₂×C₄×C₁₀₉	—	2	3	no	—
42	C₂×C₄×C₁₀₉	—	4	6	no	—
43	C₂×C₄×C₁₀₉	—	4	6	no	—
44	C₂×C₄×C₁₀₉	—	4	6	no	—
45	C₂×C₄×C₁₀₉	C₂²⋊Q₈	1	3	yes	—
46	C₂×C₄×C₁₀₉	#124	1	3	yes	—
47	C₈×C₁₀₉	—	4	6	no	—
48	C₂²×C₁₀₉	—	8	6	no	—
49	C₂²×C₁₀₉	—	16	3	no	—
50	C₂²×C₁₀₉	—	16	1	no	—
51	C₂²×C₁₀₉	#393	1	1	yes	characteristic
52	C₂²×C₁₀₉	#453	1	3	yes	—
53	C₂²×C₁₀₉	#454	1	3	yes	—
54	C₄×C₁₀₉	—	2	3	no	—
55	C₄×C₁₀₉	—	2	3	no	—
56	C₄×C₁₀₉	—	8	6	no	—
57	#10360	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂×C₁₀₉	—	16	2	no	—
59	C₂×C₁₀₉	#1656	1	3	yes	—
60	C₂×C₁₀₉	#1658	1	1	yes	characteristic
61	C₂×C₁₀₉	#2027	1	3	yes	—
62	#2845	C₂×C₁₀₉	1	1	yes	characteristic, complemented
63	#1961	C₂×C₁₀₉	1	3	yes	—
64	#1295	C₂×C₁₀₉	1	3	yes	complemented
65	C₁₀₉	#10360	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×#127	C₂²×C₁₀₉	1	1	yes	characteristic
67	C₂×#102	—	2	6	no	—
68	C₂×#97	C₂²×C₁₀₉	1	3	yes	—
69	#398	—	2	6	no	—
70	#382	C₂²×C₁₀₉	1	3	yes	—
71	C₂²×D₄	—	2	3	no	—
72	C₂²×C₈	—	2	3	no	—
73	#127	—	2	3	no	—
74	#127	Q₈×C₁₀₉	1	2	yes	—
75	C₂×C₄⋊C₄	—	2	6	no	—
76	C₂×C₄⋊C₄	D₄×C₁₀₉	1	6	yes	—
77	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
78	#102	—	4	12	no	—
79	#97	—	2	6	no	—
80	#95	—	2	2	no	—
81	#95	—	2	6	no	—
82	C₂⁴	—	4	1	no	—
83	C₂×D₄	—	2	6	no	—
84	C₂×D₄	—	4	6	no	—
85	C₂×D₄	—	4	6	no	—
86	C₂×D₄	—	4	3	no	—
87	C₂²×C₄	—	2	6	no	—
88	C₂²×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
89	C₂×C₈	—	2	6	no	—
90	C₂×C₈	—	4	3	no	—
91	C₂×C₈	—	4	3	no	—
92	C₄⋊C₄	—	4	6	no	—
93	C₄⋊C₄	—	8	6	no	—
94	C₄²	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
95	C₄²	#41×C₁₀₉	1	3	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	—	8	3	no	—
98	C₂³	—	8	3	no	—
99	C₂³	#120×C₁₀₉	1	1	yes	characteristic
100	D₄	—	4	12	no	—
101	D₄	—	8	6	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	C₂²⋊Q₈×C₁₀₉	1	3	yes	—
108	C₂×C₄	#124×C₁₀₉	1	3	yes	—
109	C₈	—	4	6	no	—
110	C₂²	—	8	6	no	—
111	C₂²	—	16	1	no	—
112	C₂²	—	16	3	no	—
113	C₂²	#393×C₁₀₉	1	1	yes	characteristic
114	C₂²	#453×C₁₀₉	1	3	yes	—
115	C₂²	#454×C₁₀₉	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	16	2	no	—
120	C₂	C₁₀₉×#1656	1	3	yes	—
121	C₂	C₁₀₉×#1658	1	1	yes	characteristic
122	C₂	C₁₀₉×#2027	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₉×#2845

C₂

yes

maximal, characteristic, complemented

C₁₀₉×#1961

C₂

yes

maximal

C₁₀₉×#1295

C₂

yes

maximal, complemented

C₂×#127×C₁₀₉

C₂²

yes

characteristic

C₂×#102×C₁₀₉

—

C₂×#97×C₁₀₉

C₂²

yes

—

#398×C₁₀₉

—

#382×C₁₀₉

C₂²

yes

—

C₂²×D₄×C₁₀₉

—

C₂²×C₈×C₁₀₉

—

#127×C₁₀₉

—

#127×C₁₀₉

Q₈

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂×C₄²×C₁₀₉

C₂³

yes

characteristic

#102×C₁₀₉

—

#97×C₁₀₉

—

#95×C₁₀₉

—

#95×C₁₀₉

—

C₂⁴×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×Q₈

yes

characteristic

C₄²×C₁₀₉

#41

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

#120

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₀₉

#124

yes

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#393

yes

characteristic

C₂²×C₁₀₉

#453

yes

—

C₂²×C₁₀₉

#454

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

#10360

C₁₀₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₉

—

C₂×C₁₀₉

#1656

yes

—

C₂×C₁₀₉

#1658

yes

characteristic

C₂×C₁₀₉

#2027

yes

—

#2845

C₂×C₁₀₉

yes

characteristic, complemented

#1961

C₂×C₁₀₉

yes

—

#1295

C₂×C₁₀₉

yes

complemented

C₁₀₉

#10360

yes

characteristic, direct product factor, Sylow

C₂×#127

C₂²×C₁₀₉

yes

characteristic

C₂×#102

—

C₂×#97

C₂²×C₁₀₉

yes

—

#398

—

#382

C₂²×C₁₀₉

yes

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

Q₈×C₁₀₉

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₁₀₉

yes

—

C₂×C₄²

C₂³×C₁₀₉

yes

—

—

—

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₁₀₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Q₈×C₁₀₉

yes

characteristic

C₄²

#41×C₁₀₉

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#120×C₁₀₉

yes

characteristic

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂²⋊Q₈×C₁₀₉

yes

—

108

C₂×C₄

#124×C₁₀₉

yes

—

109

C₈

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

#393×C₁₀₉

yes

characteristic

114

C₂²

#453×C₁₀₉

yes

—

115

C₂²

#454×C₁₀₉

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

120

C₂

C₁₀₉×#1656

yes

—

121

C₂

C₁₀₉×#1658

yes

characteristic

122

C₂

C₁₀₉×#2027

yes

—

	subgroup	G/H
center, socle	C₂³×C₁₀₉	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₀₉
prime radical	C₂×#127×C₁₀₉	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

subgroup

G/H

center, socle

C₂³×C₁₀₉

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₁₀₉

prime radical

C₂×#127×C₁₀₉

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₁₀₉ ◁ C₂×C₄²×C₁₀₉ ◁ G

N	H ≅ G/N	type
1. C₁₀₉×#2845	C₂	semidirect
3. C₁₀₉×#1295	C₂	semidirect
57. #10360	C₁₀₉	direct (N × H)
62. #2845	C₂×C₁₀₉	semidirect
64. #1295	C₂×C₁₀₉	semidirect
65. C₁₀₉	#10360	direct (N × H)

H ≅ G/N

type

1. C₁₀₉×#2845

C₂

semidirect

3. C₁₀₉×#1295

C₂

semidirect

57. #10360

C₁₀₉

direct (N × H)

62. #2845

C₂×C₁₀₉

semidirect

64. #1295

C₂×C₁₀₉

semidirect

65. C₁₀₉

#10360

direct (N × H)

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

subgroup	index
C₁₀₉×#2845	2
C₁₀₉×#1961	2
C₁₀₉×#1295	2
#10360	109

subgroup

index

C₁₀₉×#2845

C₁₀₉×#1961

C₁₀₉×#1295

#10360

109

Subgroups — C₁₀₉×#10360 #42128354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C109×#10360 #42128354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#10360 #42128354