Subgroups — C₂²×#126×C₁₀₉ #28360692

Order 13952.

1320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#126×C₁₀₉	C₂	1	24	yes	maximal, direct product factor
2	C₂²×C₄⋊C₄×C₁₀₉	C₂	1	3	yes	maximal, complemented
3	C₂²×C₂²⋊C₄×C₁₀₉	C₂	1	3	yes	maximal
4	C₂²×C₄²×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
6	C₂³×C₄×C₁₀₉	C₂²	1	3	yes	—
7	C₂³×C₄×C₁₀₉	C₂²	1	3	yes	—
8	#126×C₁₀₉	C₂²	1	64	yes	direct product factor
9	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	36	yes	complemented
10	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	36	yes	—
11	C₂×C₄²×C₁₀₉	C₂²	1	12	yes	complemented
12	C₂⁴×C₁₀₉	—	2	12	no	—
13	C₂⁴×C₁₀₉	C₂³	1	6	yes	—
14	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
15	C₂²×C₄×C₁₀₉	—	2	12	no	—
16	C₂²×C₄×C₁₀₉	—	2	12	no	—
17	C₂²×C₄×C₁₀₉	C₂³	1	18	yes	—
18	C₂²×C₄×C₁₀₉	C₂³	1	18	yes	—
19	C₄⋊C₄×C₁₀₉	C₂³	1	48	yes	complemented
20	C₂²⋊C₄×C₁₀₉	C₂³	1	48	yes	—
21	C₄²×C₁₀₉	C₂³	1	16	yes	complemented
22	C₂³×C₁₀₉	—	2	36	no	—
23	C₂³×C₁₀₉	—	4	16	no	—
24	C₂³×C₁₀₉	#41	1	12	yes	—
25	C₂³×C₁₀₉	C₂⁴	1	4	yes	—
26	C₂³×C₁₀₉	C₂⁴	1	3	yes	—
27	C₂×C₄×C₁₀₉	—	2	36	no	—
28	C₂×C₄×C₁₀₉	—	2	36	no	—
29	C₂×C₄×C₁₀₉	C₂⁴	1	12	yes	—
30	C₂×C₄×C₁₀₉	C₂⁴	1	12	yes	—
31	C₂²×C₁₀₉	—	2	12	no	—
32	C₂²×C₁₀₉	—	4	24	no	—
33	C₂²×C₁₀₉	#126	1	16	yes	direct product factor
34	C₂²×C₁₀₉	C₂×#41	1	18	yes	—
35	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
36	C₄×C₁₀₉	—	2	12	no	—
37	C₄×C₁₀₉	—	2	12	no	—
38	C₂×C₁₀₉	—	4	4	no	—
39	C₂×C₁₀₉	C₂×#126	1	12	yes	direct product factor
40	C₂×C₁₀₉	C₂²×#41	1	3	yes	—
41	C₂²×#126	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₁₀₉	C₂²×#126	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×#126	C₂×C₁₀₉	1	24	yes	direct product factor
44	C₂²×C₄⋊C₄	C₂×C₁₀₉	1	3	yes	complemented
45	C₂²×C₂²⋊C₄	C₂×C₁₀₉	1	3	yes	—
46	C₂²×C₄²	C₂×C₁₀₉	1	1	yes	characteristic, complemented
47	C₂⁵	C₂²×C₁₀₉	1	1	yes	characteristic
48	C₂³×C₄	C₂²×C₁₀₉	1	3	yes	—
49	C₂³×C₄	C₂²×C₁₀₉	1	3	yes	—
50	#126	C₂²×C₁₀₉	1	64	yes	direct product factor
51	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	36	yes	complemented
52	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	36	yes	—
53	C₂×C₄²	C₂²×C₁₀₉	1	12	yes	complemented
54	C₂⁴	—	2	12	no	—
55	C₂⁴	C₂³×C₁₀₉	1	6	yes	—
56	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
57	C₂²×C₄	—	2	12	no	—
58	C₂²×C₄	—	2	12	no	—
59	C₂²×C₄	C₂³×C₁₀₉	1	18	yes	—
60	C₂²×C₄	C₂³×C₁₀₉	1	18	yes	—
61	C₄⋊C₄	C₂³×C₁₀₉	1	48	yes	complemented
62	C₂²⋊C₄	C₂³×C₁₀₉	1	48	yes	—
63	C₄²	C₂³×C₁₀₉	1	16	yes	complemented
64	C₂³	—	2	36	no	—
65	C₂³	—	4	16	no	—
66	C₂³	#41×C₁₀₉	1	12	yes	—
67	C₂³	C₂⁴×C₁₀₉	1	4	yes	—
68	C₂³	C₂⁴×C₁₀₉	1	3	yes	—
69	C₂×C₄	—	2	36	no	—
70	C₂×C₄	—	2	36	no	—
71	C₂×C₄	C₂⁴×C₁₀₉	1	12	yes	—
72	C₂×C₄	C₂⁴×C₁₀₉	1	12	yes	—
73	C₂²	—	2	12	no	—
74	C₂²	—	4	24	no	—
75	C₂²	#126×C₁₀₉	1	16	yes	direct product factor
76	C₂²	C₂×#41×C₁₀₉	1	18	yes	—
77	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
78	C₄	—	2	12	no	—
79	C₄	—	2	12	no	—
80	C₂	—	4	4	no	—
81	C₂	C₂×#126×C₁₀₉	1	12	yes	direct product factor
82	C₂	C₂²×#41×C₁₀₉	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₀₉	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₀₉
prime radical	C₂⁵×C₁₀₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₀₉ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#126×C₁₀₉	C₂	direct (N × H)
2. C₂²×C₄⋊C₄×C₁₀₉	C₂	semidirect
4. C₂²×C₄²×C₁₀₉	C₂	semidirect
8. #126×C₁₀₉	C₂²	direct (N × H)
9. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
11. C₂×C₄²×C₁₀₉	C₂²	semidirect
19. C₄⋊C₄×C₁₀₉	C₂³	semidirect
21. C₄²×C₁₀₉	C₂³	semidirect
33. C₂²×C₁₀₉	#126	direct (N × H)
39. C₂×C₁₀₉	C₂×#126	direct (N × H)
41. C₂²×#126	C₁₀₉	direct (N × H)
42. C₁₀₉	C₂²×#126	direct (N × H)
43. C₂×#126	C₂×C₁₀₉	direct (N × H)
44. C₂²×C₄⋊C₄	C₂×C₁₀₉	semidirect
46. C₂²×C₄²	C₂×C₁₀₉	semidirect
50. #126	C₂²×C₁₀₉	direct (N × H)
51. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
53. C₂×C₄²	C₂²×C₁₀₉	semidirect
61. C₄⋊C₄	C₂³×C₁₀₉	semidirect
63. C₄²	C₂³×C₁₀₉	semidirect
75. C₂²	#126×C₁₀₉	direct (N × H)
81. C₂	C₂×#126×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#126×C₁₀₉	2
C₂²×C₄⋊C₄×C₁₀₉	2
C₂²×C₂²⋊C₄×C₁₀₉	2
C₂²×C₄²×C₁₀₉	2
C₂²×#126	109

Subgroups — C22×#126×C109 #28360692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#126×C₁₀₉ #28360692