Subgroups — C₁₀₉×#2842 #28360096

Order 13952.

548 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#399×C₁₀₉	C₂	1	4	yes	maximal, complemented
2	#396×C₁₀₉	C₂	1	8	yes	maximal, complemented
3	#379×C₁₀₉	C₂	1	3	yes	maximal
4	C₂⁵×C₁₀₉	C₂²	1	1	yes	characteristic
5	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	12	yes	complemented
6	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	12	yes	—
7	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	6	yes	—
8	#95×C₁₀₉	C₂²	1	4	yes	complemented
9	C₂⁴×C₁₀₉	—	4	6	no	—
10	C₂⁴×C₁₀₉	Q₈	1	4	yes	—
11	C₂⁴×C₁₀₉	C₂³	1	2	yes	—
12	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
13	C₂²×C₄×C₁₀₉	C₂³	1	12	yes	—
14	C₄⋊C₄×C₁₀₉	—	4	12	no	—
15	C₂²⋊C₄×C₁₀₉	—	4	12	no	—
16	C₂²⋊C₄×C₁₀₉	—	4	6	no	—
17	C₂³×C₁₀₉	—	2	12	no	—
18	C₂³×C₁₀₉	—	2	6	no	—
19	C₂³×C₁₀₉	—	4	12	no	—
20	C₂³×C₁₀₉	—	4	16	no	—
21	C₂³×C₁₀₉	C₂×Q₈	1	4	yes	—
22	C₂³×C₁₀₉	C₂×Q₈	1	2	yes	—
23	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	—	2	24	no	—
25	C₂×C₄×C₁₀₉	—	2	12	no	—
26	C₂²×C₁₀₉	—	2	12	no	—
27	C₂²×C₁₀₉	—	2	6	no	—
28	C₂²×C₁₀₉	—	4	16	no	—
29	C₂²×C₁₀₉	—	4	8	no	—
30	C₂²×C₁₀₉	—	4	4	no	—
31	C₂²×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
32	C₂²×C₁₀₉	H(5,𝔽₂)	1	6	yes	—
33	C₄×C₁₀₉	—	4	12	no	—
34	C₂×C₁₀₉	—	4	4	no	—
35	C₂×C₁₀₉	—	4	2	no	—
36	C₂×C₁₀₉	#543	1	3	yes	—
37	C₂×C₁₀₉	#560	1	4	yes	—
38	#2842	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₁₀₉	#2842	1	1	yes	characteristic, direct product factor, Sylow
40	#399	C₂×C₁₀₉	1	4	yes	complemented
41	#396	C₂×C₁₀₉	1	8	yes	complemented
42	#379	C₂×C₁₀₉	1	3	yes	—
43	C₂⁵	C₂²×C₁₀₉	1	1	yes	characteristic
44	C₂×C₄⋊C₄	C₂²×C₁₀₉	1	12	yes	complemented
45	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	12	yes	—
46	C₂×C₂²⋊C₄	C₂²×C₁₀₉	1	6	yes	—
47	#95	C₂²×C₁₀₉	1	4	yes	complemented
48	C₂⁴	—	4	6	no	—
49	C₂⁴	Q₈×C₁₀₉	1	4	yes	—
50	C₂⁴	C₂³×C₁₀₉	1	2	yes	—
51	C₂⁴	C₂³×C₁₀₉	1	1	yes	characteristic
52	C₂²×C₄	C₂³×C₁₀₉	1	12	yes	—
53	C₄⋊C₄	—	4	12	no	—
54	C₂²⋊C₄	—	4	12	no	—
55	C₂²⋊C₄	—	4	6	no	—
56	C₂³	—	2	12	no	—
57	C₂³	—	2	6	no	—
58	C₂³	—	4	12	no	—
59	C₂³	—	4	16	no	—
60	C₂³	C₂×Q₈×C₁₀₉	1	4	yes	—
61	C₂³	C₂×Q₈×C₁₀₉	1	2	yes	—
62	C₂³	C₂⁴×C₁₀₉	1	1	yes	characteristic
63	C₂×C₄	—	2	24	no	—
64	C₂×C₄	—	2	12	no	—
65	C₂²	—	2	12	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	4	16	no	—
68	C₂²	—	4	8	no	—
69	C₂²	—	4	4	no	—
70	C₂²	C₂²×Q₈×C₁₀₉	1	1	yes	characteristic
71	C₂²	H(5,𝔽₂)×C₁₀₉	1	6	yes	—
72	C₄	—	4	12	no	—
73	C₂	—	4	4	no	—
74	C₂	—	4	2	no	—
75	C₂	#543×C₁₀₉	1	3	yes	—
76	C₂	#560×C₁₀₉	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. #399×C₁₀₉	C₂	semidirect
2. #396×C₁₀₉	C₂	semidirect
5. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
8. #95×C₁₀₉	C₂²	semidirect
38. #2842	C₁₀₉	direct (N × H)
39. C₁₀₉	#2842	direct (N × H)
40. #399	C₂×C₁₀₉	semidirect
41. #396	C₂×C₁₀₉	semidirect
44. C₂×C₄⋊C₄	C₂²×C₁₀₉	semidirect
47. #95	C₂²×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#399×C₁₀₉	2
#396×C₁₀₉	2
#379×C₁₀₉	2
#2842	109

Subgroups — C109×#2842 #28360096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2842 #28360096