Group #41062653 — subgroups

Subgroups — C₂×C₁₀₇×#2845 #41062653

Order 27392.

1742 conjugacy classes of subgroups; only the 974 normal subgroups are shown.

normal subgroups only

This group has 1742 conjugacy classes of subgroups — too many to draw. Only its 974 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	C₁₀₇×#2845	C₂	1	16	yes	maximal, direct product factor
2	C₂×#398×C₁₀₇	C₂	1	12	yes	maximal
3	C₂×#382×C₁₀₇	C₂	1	3	yes	maximal
4	C₂²×C₄⋊C₄×C₁₀₇	C₂²	1	12	yes	—
5	C₂²×C₄⋊C₄×C₁₀₇	C₂²	1	6	yes	—
6	C₂²×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
7	#398×C₁₀₇	C₂²	1	96	yes	—
8	#382×C₁₀₇	C₂²	1	24	yes	—
9	C₂×#95×C₁₀₇	C₂²	1	4	yes	—
10	C₂×#95×C₁₀₇	C₂²	1	12	yes	—
11	C₂³×C₄×C₁₀₇	C₂³	1	12	yes	—
12	C₂³×C₄×C₁₀₇	C₂³	1	3	yes	—
13	C₂×C₄⋊C₄×C₁₀₇	—	4	12	no	—
14	C₂×C₄⋊C₄×C₁₀₇	—	4	24	no	—
15	C₂×C₄⋊C₄×C₁₀₇	C₂³	1	24	yes	—
16	C₂×C₄⋊C₄×C₁₀₇	C₂³	1	48	yes	—
17	C₂×C₄²×C₁₀₇	Q₈	1	8	yes	—
18	C₂×C₄²×C₁₀₇	C₂³	1	4	yes	—
19	#95×C₁₀₇	C₂³	1	48	yes	—
20	#95×C₁₀₇	C₂³	1	16	yes	—
21	C₂⁴×C₁₀₇	C₂⁴	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₇	—	2	6	no	—
23	C₂²×C₄×C₁₀₇	—	2	6	no	—
24	C₂²×C₄×C₁₀₇	—	2	24	no	—
25	C₂²×C₄×C₁₀₇	—	2	24	no	—
26	C₂²×C₄×C₁₀₇	—	2	24	no	—
27	C₂²×C₄×C₁₀₇	C₂×Q₈	1	12	yes	—
28	C₂²×C₄×C₁₀₇	C₂⁴	1	24	yes	—
29	C₂²×C₄×C₁₀₇	C₂⁴	1	6	yes	—
30	C₄⋊C₄×C₁₀₇	—	4	24	no	—
31	C₄⋊C₄×C₁₀₇	—	4	48	no	—
32	C₄²×C₁₀₇	C₂×Q₈	1	16	yes	—
33	C₂³×C₁₀₇	C₂²×Q₈	1	2	yes	—
34	C₂³×C₁₀₇	H(5,𝔽₂)	1	6	yes	—
35	C₂³×C₁₀₇	2₋¹⁺⁴	1	6	yes	—
36	C₂³×C₁₀₇	C₂⁵	1	1	yes	characteristic
37	C₂×C₄×C₁₀₇	—	2	24	no	—
38	C₂×C₄×C₁₀₇	—	2	6	no	—
39	C₂×C₄×C₁₀₇	—	2	6	no	—
40	C₂×C₄×C₁₀₇	—	2	24	no	—
41	C₂×C₄×C₁₀₇	—	2	24	no	—
42	C₂×C₄×C₁₀₇	—	2	24	no	—
43	C₂×C₄×C₁₀₇	—	4	48	no	—
44	C₂×C₄×C₁₀₇	C₂²×Q₈	1	12	yes	—
45	C₂²×C₁₀₇	#544	1	12	yes	—
46	C₂²×C₁₀₇	#562	1	12	yes	—
47	C₂²×C₁₀₇	#564	1	4	yes	—
48	C₂²×C₁₀₇	C₂³×Q₈	1	1	yes	characteristic
49	C₂²×C₁₀₇	C₂×H(5,𝔽₂)	1	3	yes	—
50	C₂²×C₁₀₇	C₂×2₋¹⁺⁴	1	3	yes	—
51	C₄×C₁₀₇	—	2	12	no	—
52	C₄×C₁₀₇	—	4	24	no	—
53	C₂×#2845	C₁₀₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂×C₁₀₇	#2845	1	8	yes	direct product factor
55	C₂×C₁₀₇	C₂×#544	1	3	yes	—
56	C₂×C₁₀₇	C₂×#562	1	3	yes	—
57	C₂×C₁₀₇	C₂×#564	1	1	yes	characteristic
58	#2845	C₂×C₁₀₇	1	16	yes	direct product factor
59	C₂×#398	C₂×C₁₀₇	1	12	yes	—
60	C₂×#382	C₂×C₁₀₇	1	3	yes	—
61	C₁₀₇	C₂×#2845	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²×C₄⋊C₄	C₂²×C₁₀₇	1	12	yes	—
63	C₂²×C₄⋊C₄	C₂²×C₁₀₇	1	6	yes	—
64	C₂²×C₄²	C₂²×C₁₀₇	1	1	yes	characteristic
65	#398	C₂²×C₁₀₇	1	96	yes	—
66	#382	C₂²×C₁₀₇	1	24	yes	—
67	C₂×#95	C₂²×C₁₀₇	1	4	yes	—
68	C₂×#95	C₂²×C₁₀₇	1	12	yes	—
69	C₂³×C₄	C₂³×C₁₀₇	1	12	yes	—
70	C₂³×C₄	C₂³×C₁₀₇	1	3	yes	—
71	C₂×C₄⋊C₄	—	4	12	no	—
72	C₂×C₄⋊C₄	—	4	24	no	—
73	C₂×C₄⋊C₄	C₂³×C₁₀₇	1	24	yes	—
74	C₂×C₄⋊C₄	C₂³×C₁₀₇	1	48	yes	—
75	C₂×C₄²	Q₈×C₁₀₇	1	8	yes	—
76	C₂×C₄²	C₂³×C₁₀₇	1	4	yes	—
77	#95	C₂³×C₁₀₇	1	48	yes	—
78	#95	C₂³×C₁₀₇	1	16	yes	—
79	C₂⁴	C₂⁴×C₁₀₇	1	1	yes	characteristic
80	C₂²×C₄	—	2	6	no	—
81	C₂²×C₄	—	2	6	no	—
82	C₂²×C₄	—	2	24	no	—
83	C₂²×C₄	—	2	24	no	—
84	C₂²×C₄	—	2	24	no	—
85	C₂²×C₄	C₂×Q₈×C₁₀₇	1	12	yes	—
86	C₂²×C₄	C₂⁴×C₁₀₇	1	24	yes	—
87	C₂²×C₄	C₂⁴×C₁₀₇	1	6	yes	—
88	C₄⋊C₄	—	4	24	no	—
89	C₄⋊C₄	—	4	48	no	—
90	C₄²	C₂×Q₈×C₁₀₇	1	16	yes	—
91	C₂³	C₂²×Q₈×C₁₀₇	1	2	yes	—
92	C₂³	H(5,𝔽₂)×C₁₀₇	1	6	yes	—
93	C₂³	2₋¹⁺⁴×C₁₀₇	1	6	yes	—
94	C₂³	C₂⁵×C₁₀₇	1	1	yes	characteristic
95	C₂×C₄	—	2	24	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	6	no	—
98	C₂×C₄	—	2	24	no	—
99	C₂×C₄	—	2	24	no	—
100	C₂×C₄	—	2	24	no	—
101	C₂×C₄	—	4	48	no	—
102	C₂×C₄	C₂²×Q₈×C₁₀₇	1	12	yes	—
103	C₂²	#544×C₁₀₇	1	12	yes	—
104	C₂²	#562×C₁₀₇	1	12	yes	—
105	C₂²	#564×C₁₀₇	1	4	yes	—
106	C₂²	C₂³×Q₈×C₁₀₇	1	1	yes	characteristic
107	C₂²	C₂×H(5,𝔽₂)×C₁₀₇	1	3	yes	—
108	C₂²	C₂×2₋¹⁺⁴×C₁₀₇	1	3	yes	—
109	C₄	—	2	12	no	—
110	C₄	—	4	24	no	—
111	C₂	C₁₀₇×#2845	1	8	yes	direct product factor
112	C₂	C₂×#544×C₁₀₇	1	3	yes	—
113	C₂	C₂×#562×C₁₀₇	1	3	yes	—
114	C₂	C₂×#564×C₁₀₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₀₇×#2845

C₂

yes

maximal, direct product factor

C₂×#398×C₁₀₇

C₂

yes

maximal

C₂×#382×C₁₀₇

C₂

yes

maximal

C₂²×C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₂²×C₄⋊C₄×C₁₀₇

C₂²

yes

—

C₂²×C₄²×C₁₀₇

C₂²

yes

characteristic

#398×C₁₀₇

C₂²

yes

—

#382×C₁₀₇

C₂²

yes

—

C₂×#95×C₁₀₇

C₂²

yes

—

C₂×#95×C₁₀₇

C₂²

yes

—

C₂³×C₄×C₁₀₇

C₂³

yes

—

C₂³×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

—

C₂×C₄⋊C₄×C₁₀₇

C₂³

yes

—

C₂×C₄⋊C₄×C₁₀₇

C₂³

yes

—

C₂×C₄²×C₁₀₇

Q₈

yes

—

C₂×C₄²×C₁₀₇

C₂³

yes

—

#95×C₁₀₇

C₂³

yes

—

#95×C₁₀₇

C₂³

yes

—

C₂⁴×C₁₀₇

C₂⁴

yes

characteristic

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂×Q₈

yes

—

C₂²×C₄×C₁₀₇

C₂⁴

yes

—

C₂²×C₄×C₁₀₇

C₂⁴

yes

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

C₂×Q₈

yes

—

C₂³×C₁₀₇

C₂²×Q₈

yes

—

C₂³×C₁₀₇

H(5,𝔽₂)

yes

—

C₂³×C₁₀₇

2₋¹⁺⁴

yes

—

C₂³×C₁₀₇

C₂⁵

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²×Q₈

yes

—

C₂²×C₁₀₇

#544

yes

—

C₂²×C₁₀₇

#562

yes

—

C₂²×C₁₀₇

#564

yes

—

C₂²×C₁₀₇

C₂³×Q₈

yes

characteristic

C₂²×C₁₀₇

C₂×H(5,𝔽₂)

yes

—

C₂²×C₁₀₇

C₂×2₋¹⁺⁴

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₂×#2845

C₁₀₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₁₀₇

#2845

yes

direct product factor

C₂×C₁₀₇

C₂×#544

yes

—

C₂×C₁₀₇

C₂×#562

yes

—

C₂×C₁₀₇

C₂×#564

yes

characteristic

#2845

C₂×C₁₀₇

yes

direct product factor

C₂×#398

C₂×C₁₀₇

yes

—

C₂×#382

C₂×C₁₀₇

yes

—

C₁₀₇

C₂×#2845

yes

characteristic, direct product factor, Sylow

C₂²×C₄⋊C₄

C₂²×C₁₀₇

yes

—

C₂²×C₄⋊C₄

C₂²×C₁₀₇

yes

—

C₂²×C₄²

C₂²×C₁₀₇

yes

characteristic

#398

C₂²×C₁₀₇

yes

—

#382

C₂²×C₁₀₇

yes

—

C₂×#95

C₂²×C₁₀₇

yes

—

C₂×#95

C₂²×C₁₀₇

yes

—

C₂³×C₄

C₂³×C₁₀₇

yes

—

C₂³×C₄

C₂³×C₁₀₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³×C₁₀₇

yes

—

C₂×C₄⋊C₄

C₂³×C₁₀₇

yes

—

C₂×C₄²

Q₈×C₁₀₇

yes

—

C₂×C₄²

C₂³×C₁₀₇

yes

—

#95

C₂³×C₁₀₇

yes

—

#95

C₂³×C₁₀₇

yes

—

C₂⁴

C₂⁴×C₁₀₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Q₈×C₁₀₇

yes

—

C₂²×C₄

C₂⁴×C₁₀₇

yes

—

C₂²×C₄

C₂⁴×C₁₀₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Q₈×C₁₀₇

yes

—

C₂³

C₂²×Q₈×C₁₀₇

yes

—

C₂³

H(5,𝔽₂)×C₁₀₇

yes

—

C₂³

2₋¹⁺⁴×C₁₀₇

yes

—

C₂³

C₂⁵×C₁₀₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂²×Q₈×C₁₀₇

yes

—

103

C₂²

#544×C₁₀₇

yes

—

104

C₂²

#562×C₁₀₇

yes

—

105

C₂²

#564×C₁₀₇

yes

—

106

C₂²

C₂³×Q₈×C₁₀₇

yes

characteristic

107

C₂²

C₂×H(5,𝔽₂)×C₁₀₇

yes

—

108

C₂²

C₂×2₋¹⁺⁴×C₁₀₇

yes

—

109

C₄

—

110

C₄

—

111

C₂

C₁₀₇×#2845

yes

direct product factor

112

C₂

C₂×#544×C₁₀₇

yes

—

113

C₂

C₂×#562×C₁₀₇

yes

—

114

C₂

C₂×#564×C₁₀₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₀₇	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₀₇
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂⁴×C₁₀₇

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁵×C₁₀₇

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₁₀₇ ◁ G

N	H ≅ G/N	type
1. C₁₀₇×#2845	C₂	direct (N × H)
53. C₂×#2845	C₁₀₇	direct (N × H)
54. C₂×C₁₀₇	#2845	direct (N × H)
58. #2845	C₂×C₁₀₇	direct (N × H)
61. C₁₀₇	C₂×#2845	direct (N × H)
111. C₂	C₁₀₇×#2845	direct (N × H)

H ≅ G/N

type

1. C₁₀₇×#2845

C₂

direct (N × H)

53. C₂×#2845

C₁₀₇

direct (N × H)

54. C₂×C₁₀₇

#2845

direct (N × H)

58. #2845

C₂×C₁₀₇

direct (N × H)

61. C₁₀₇

C₂×#2845

direct (N × H)

111. C₂

C₁₀₇×#2845

direct (N × H)

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

subgroup	index
C₁₀₇×#2845	2
C₂×#398×C₁₀₇	2
C₂×#382×C₁₀₇	2
C₂×#2845	107

subgroup

index

C₁₀₇×#2845

C₂×#398×C₁₀₇

C₂×#382×C₁₀₇

C₂×#2845

107

Subgroups — C₂×C₁₀₇×#2845 #41062653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C107×#2845 #41062653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₀₇×#2845 #41062653