Subgroups — C₁₀₉×#2017 #28359271

Order 13952.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#575×C₁₀₉	C₂	1	3	yes	maximal, complemented
2	#408×C₁₀₉	C₂	1	3	yes	maximal, complemented
3	#338×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#41×C₁₀₉	C₂²	1	3	yes	—
5	#137×C₁₀₉	—	2	6	no	—
6	AGL(1,ℤ/8)×C₁₀₉	—	2	6	no	—
7	#135×C₁₀₉	—	2	6	no	—
8	C₂×M₁₆×C₁₀₉	C₂²	1	3	yes	—
9	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₂²⋊C₈×C₁₀₉	—	2	6	no	—
11	#41×C₁₀₉	—	2	6	no	—
12	#41×C₁₀₉	—	2	6	no	—
13	C₂×Q₈×C₁₀₉	D₄	1	3	yes	—
14	C₂×D₄×C₁₀₉	—	2	3	no	—
15	C₂×D₄×C₁₀₉	D₄	1	3	yes	—
16	C₂²×C₄×C₁₀₉	—	2	3	no	—
17	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
18	Q₁₆×C₁₀₉	—	2	6	no	—
19	SD₁₆×C₁₀₉	—	2	12	no	—
20	D₈×C₁₀₉	—	2	6	no	—
21	M₁₆×C₁₀₉	—	2	6	no	—
22	C₂×C₈×C₁₀₉	—	2	3	no	—
23	C₂×C₈×C₁₀₉	D₄	1	6	yes	—
24	C₂³×C₁₀₉	—	2	3	no	—
25	C₂³×C₁₀₉	#41	1	1	yes	characteristic
26	Q₈×C₁₀₉	—	2	6	no	—
27	D₄×C₁₀₉	—	2	6	no	—
28	D₄×C₁₀₉	—	4	6	no	—
29	C₂×C₄×C₁₀₉	—	2	3	no	—
30	C₂×C₄×C₁₀₉	—	4	3	no	—
31	C₂×C₄×C₁₀₉	—	4	3	no	—
32	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
34	C₈×C₁₀₉	—	2	6	no	—
35	C₈×C₁₀₉	—	4	1	no	—
36	C₂²×C₁₀₉	—	4	1	no	—
37	C₂²×C₁₀₉	—	4	6	no	—
38	C₂²×C₁₀₉	—	4	3	no	—
39	C₂²×C₁₀₉	#121	1	3	yes	—
40	C₄×C₁₀₉	—	4	3	no	—
41	C₄×C₁₀₉	#120	1	3	yes	—
42	C₄×C₁₀₉	#127	1	1	yes	characteristic
43	C₂×C₁₀₉	—	2	3	no	—
44	C₂×C₁₀₉	—	8	3	no	—
45	C₂×C₁₀₉	#392	1	1	yes	characteristic
46	#2017	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₁₀₉	#2017	1	1	yes	characteristic, direct product factor, Sylow
48	#575	C₂×C₁₀₉	1	3	yes	complemented
49	#408	C₂×C₁₀₉	1	3	yes	complemented
50	#338	C₂×C₁₀₉	1	1	yes	characteristic, complemented
51	C₂×#41	C₂²×C₁₀₉	1	3	yes	—
52	#137	—	2	6	no	—
53	AGL(1,ℤ/8)	—	2	6	no	—
54	#135	—	2	6	no	—
55	C₂×M₁₆	C₂²×C₁₀₉	1	3	yes	—
56	C₂²×C₈	C₂²×C₁₀₉	1	1	yes	characteristic
57	C₂²⋊C₈	—	2	6	no	—
58	#41	—	2	6	no	—
59	#41	—	2	6	no	—
60	C₂×Q₈	D₄×C₁₀₉	1	3	yes	—
61	C₂×D₄	—	2	3	no	—
62	C₂×D₄	D₄×C₁₀₉	1	3	yes	—
63	C₂²×C₄	—	2	3	no	—
64	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
65	Q₁₆	—	2	6	no	—
66	SD₁₆	—	2	12	no	—
67	D₈	—	2	6	no	—
68	M₁₆	—	2	6	no	—
69	C₂×C₈	—	2	3	no	—
70	C₂×C₈	D₄×C₁₀₉	1	6	yes	—
71	C₂³	—	2	3	no	—
72	C₂³	#41×C₁₀₉	1	1	yes	characteristic
73	Q₈	—	2	6	no	—
74	D₄	—	2	6	no	—
75	D₄	—	4	6	no	—
76	C₂×C₄	—	2	3	no	—
77	C₂×C₄	—	4	3	no	—
78	C₂×C₄	—	4	3	no	—
79	C₂×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
80	C₂×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
81	C₈	—	2	6	no	—
82	C₈	—	4	1	no	—
83	C₂²	—	4	1	no	—
84	C₂²	—	4	6	no	—
85	C₂²	—	4	3	no	—
86	C₂²	#121×C₁₀₉	1	3	yes	—
87	C₄	—	4	3	no	—
88	C₄	#120×C₁₀₉	1	3	yes	—
89	C₄	#127×C₁₀₉	1	1	yes	characteristic
90	C₂	—	2	3	no	—
91	C₂	—	8	3	no	—
92	C₂	#392×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₀₉	#127
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#392
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. #575×C₁₀₉	C₂	semidirect
2. #408×C₁₀₉	C₂	semidirect
3. #338×C₁₀₉	C₂	semidirect
46. #2017	C₁₀₉	direct (N × H)
47. C₁₀₉	#2017	direct (N × H)
48. #575	C₂×C₁₀₉	semidirect
49. #408	C₂×C₁₀₉	semidirect
50. #338	C₂×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#575×C₁₀₉	2
#408×C₁₀₉	2
#338×C₁₀₉	2
#2017	109

Subgroups — C109×#2017 #28359271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2017 #28359271