Subgroups — C₁₀₉×#3556 #28360810

Order 13952.

890 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#564×C₁₀₉	C₂	1	4	yes	maximal, complemented
2	#557×C₁₀₉	C₂	1	12	yes	maximal, complemented
3	#541×C₁₀₉	C₂	1	12	yes	maximal
4	#533×C₁₀₉	C₂	1	3	yes	maximal
5	C₄⋊Q₈×C₁₀₉	C₂²	1	6	yes	—
6	C₄⋊Q₈×C₁₀₉	C₂²	1	12	yes	—
7	#126×C₁₀₉	C₂²	1	16	yes	—
8	#125×C₁₀₉	C₂²	1	6	yes	—
9	#125×C₁₀₉	C₂²	1	12	yes	—
10	#125×C₁₀₉	C₂²	1	48	yes	—
11	C₂²⋊Q₈×C₁₀₉	C₂²	1	12	yes	—
12	C₄×Q₈×C₁₀₉	C₂²	1	12	yes	—
13	C₄×Q₈×C₁₀₉	C₂²	1	24	yes	—
14	#117×C₁₀₉	C₂²	1	6	yes	—
15	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₀₉	C₂³	1	12	yes	—
17	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
18	C₄⋊C₄×C₁₀₉	C₂³	1	12	yes	—
19	C₄⋊C₄×C₁₀₉	C₂³	1	12	yes	—
20	C₄⋊C₄×C₁₀₉	C₂³	1	24	yes	—
21	C₄⋊C₄×C₁₀₉	C₂³	1	48	yes	—
22	C₂²⋊C₄×C₁₀₉	C₂³	1	12	yes	—
23	C₄²×C₁₀₉	C₂³	1	3	yes	—
24	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
25	C₄²×C₁₀₉	C₂³	1	12	yes	—
26	C₄²×C₁₀₉	C₂³	1	16	yes	—
27	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
28	Q₈×C₁₀₉	—	2	24	no	—
29	C₂×C₄×C₁₀₉	—	2	6	no	—
30	C₂×C₄×C₁₀₉	C₂⁴	1	24	yes	—
31	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
32	C₂×C₄×C₁₀₉	C₂⁴	1	3	yes	—
33	C₂²×C₁₀₉	—	2	3	no	—
34	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
35	C₄×C₁₀₉	—	2	3	no	—
36	C₄×C₁₀₉	—	2	24	no	—
37	C₄×C₁₀₉	2₋¹⁺⁴	1	6	yes	—
38	C₂×C₁₀₉	—	4	1	no	—
39	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	3	yes	—
40	#3556	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₁₀₉	#3556	1	1	yes	characteristic, direct product factor, Sylow
42	#564	C₂×C₁₀₉	1	4	yes	complemented
43	#557	C₂×C₁₀₉	1	12	yes	complemented
44	#541	C₂×C₁₀₉	1	12	yes	—
45	#533	C₂×C₁₀₉	1	3	yes	—
46	C₄⋊Q₈	C₂²×C₁₀₉	1	6	yes	—
47	C₄⋊Q₈	C₂²×C₁₀₉	1	12	yes	—
48	#126	C₂²×C₁₀₉	1	16	yes	—
49	#125	C₂²×C₁₀₉	1	6	yes	—
50	#125	C₂²×C₁₀₉	1	12	yes	—
51	#125	C₂²×C₁₀₉	1	48	yes	—
52	C₂²⋊Q₈	C₂²×C₁₀₉	1	12	yes	—
53	C₄×Q₈	C₂²×C₁₀₉	1	12	yes	—
54	C₄×Q₈	C₂²×C₁₀₉	1	24	yes	—
55	#117	C₂²×C₁₀₉	1	6	yes	—
56	C₂×C₄²	C₂²×C₁₀₉	1	1	yes	characteristic
57	C₂×Q₈	C₂³×C₁₀₉	1	12	yes	—
58	C₂²×C₄	C₂³×C₁₀₉	1	3	yes	—
59	C₄⋊C₄	C₂³×C₁₀₉	1	12	yes	—
60	C₄⋊C₄	C₂³×C₁₀₉	1	12	yes	—
61	C₄⋊C₄	C₂³×C₁₀₉	1	24	yes	—
62	C₄⋊C₄	C₂³×C₁₀₉	1	48	yes	—
63	C₂²⋊C₄	C₂³×C₁₀₉	1	12	yes	—
64	C₄²	C₂³×C₁₀₉	1	3	yes	—
65	C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
66	C₄²	C₂³×C₁₀₉	1	12	yes	—
67	C₄²	C₂³×C₁₀₉	1	16	yes	—
68	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
69	Q₈	—	2	24	no	—
70	C₂×C₄	—	2	6	no	—
71	C₂×C₄	C₂⁴×C₁₀₉	1	24	yes	—
72	C₂×C₄	C₂⁴×C₁₀₉	1	3	yes	—
73	C₂×C₄	C₂⁴×C₁₀₉	1	3	yes	—
74	C₂²	—	2	3	no	—
75	C₂²	C₂⁵×C₁₀₉	1	1	yes	characteristic
76	C₄	—	2	3	no	—
77	C₄	—	2	24	no	—
78	C₄	2₋¹⁺⁴×C₁₀₉	1	6	yes	—
79	C₂	—	4	1	no	—
80	C₂	C₂×2₋¹⁺⁴×C₁₀₉	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. #564×C₁₀₉	C₂	semidirect
2. #557×C₁₀₉	C₂	semidirect
40. #3556	C₁₀₉	direct (N × H)
41. C₁₀₉	#3556	direct (N × H)
42. #564	C₂×C₁₀₉	semidirect
43. #557	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#564×C₁₀₉	2
#557×C₁₀₉	2
#541×C₁₀₉	2
#533×C₁₀₉	2
#3556	109

Subgroups — C109×#3556 #28360810

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#3556 #28360810