Subgroups — #28378090

Order 13952.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812768	C₂	1	2	yes	maximal
2	C₂×C₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#89137	—	2	2	no	—
4	#16812768	—	2	2	no	—
5	#16812768	C₄	1	4	yes	complemented
6	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×#13182004	C₄	1	1	yes	characteristic, complemented
8	C₁₆×D₁₀₉	—	2	2	no	—
9	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
10	C₂²×#89137	—	4	2	no	—
11	C₄⋊#89137	—	2	4	no	—
12	C₄⋊#89137	—	4	2	no	—
13	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
14	C₂×#11724784	Q₈	1	1	yes	characteristic
15	C₈×D₁₀₉	C₂×C₄	1	2	yes	—
16	C₈×D₁₀₉	D₄	1	2	yes	—
17	#13182004	—	2	1	no	—
18	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₀₉	—	2	1	no	—
20	C₂²×D₁₀₉	D₈	1	1	yes	characteristic
21	C₂×#89137	—	4	4	no	—
22	C₂×#89137	—	8	2	no	—
23	C₂×#89137	—	8	2	no	—
24	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
25	C₂×Dic₁₀₉	Q₁₆	1	1	yes	characteristic
26	C₄×D₁₀₉	C₂²⋊C₄	1	2	yes	—
27	C₄×D₁₀₉	SD₁₆	1	2	yes	—
28	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
29	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
30	#11724784	C₄⋊C₄	1	2	yes	—
31	C₂²×C₁₀₉	#107	1	1	yes	characteristic
32	C₂×D₁₀₉	#102	1	2	yes	—
33	C₂×D₁₀₉	D₁₆	1	1	yes	characteristic
34	C₂×D₁₀₉	SD₃₂	1	2	yes	—
35	C₂×D₁₀₉	Q₃₂	1	1	yes	characteristic
36	#89137	—	8	4	no	—
37	C₄×C₁₀₉	#95	1	1	yes	characteristic
38	C₄×C₁₀₉	#106	1	1	yes	characteristic
39	Dic₁₀₉	#103	1	2	yes	—
40	C₂×C₁₀₉	#340	1	1	yes	characteristic
41	C₂×C₁₀₉	#366	1	1	yes	characteristic
42	C₂×C₁₀₉	#367	1	1	yes	characteristic
43	D₁₀₉	#357	1	2	yes	—
44	D₁₀₉	#358	1	2	yes	—
45	#1380	—	109	1	no	maximal, Sylow
46	C₁₀₉	#1380	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₁₆	—	109	1	no	—
48	C₂×#107	—	109	2	no	—
49	C₂²×C₈	—	109	1	no	—
50	C₂×C₄⋊C₄	—	218	2	no	—
51	C₂×C₁₆	—	109	1	no	—
52	C₂×C₁₆	—	218	2	no	—
53	C₂×C₁₆	#89137	1	1	yes	characteristic, complemented
54	#107	—	109	4	no	—
55	#107	—	218	2	no	—
56	C₂²×C₄	—	109	1	no	—
57	C₂²×C₄	—	436	2	no	—
58	C₂×C₈	—	109	1	no	—
59	C₂×C₈	—	109	2	no	—
60	C₂×C₈	—	109	2	no	—
61	C₂×C₈	C₂×#89137	1	1	yes	characteristic
62	C₄⋊C₄	—	436	2	no	—
63	C₄⋊C₄	—	218	4	no	—
64	C₁₆	—	2	1	no	—
65	C₁₆	—	218	1	no	—
66	C₂³	—	109	1	no	—
67	C₂×C₄	—	872	2	no	—
68	C₂×C₄	—	872	2	no	—
69	C₂×C₄	—	109	1	no	—
70	C₂×C₄	—	109	2	no	—
71	C₂×C₄	—	109	2	no	—
72	C₂×C₄	—	436	4	no	—
73	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
74	C₈	—	109	2	no	—
75	C₈	C₄×#89137	1	1	yes	characteristic
76	C₈	C₂²⋊#89137	1	1	yes	characteristic
77	C₂²	—	109	1	no	—
78	C₂²	—	109	1	no	—
79	C₂²	—	109	2	no	—
80	C₂²	—	109	2	no	—
81	C₂²	#16812768	1	1	yes	characteristic
82	C₄	—	872	4	no	—
83	C₄	—	109	2	no	—
84	C₄	(C₂×#89137)⋊C₄	1	1	yes	characteristic
85	C₄	#16812767	1	1	yes	characteristic
86	C₂	—	109	2	no	—
87	C₂	—	109	2	no	—
88	C₂	#20295900	1	1	yes	characteristic
89	C₂	#20295952	1	1	yes	characteristic
90	C₂	#20295953	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812768
commutator	C₈×C₁₀₉	C₄²
Frattini	C₂×C₈	C₂×#89137
Fitting	C₂×C₁₆×C₁₀₉	C₄
socle	C₂²×C₁₀₉	#107
prime radical	C₂²×D₁₀₉	D₈
cosocle	C₂×C₈×D₁₀₉	C₂²
hypercenter	C₂×C₁₆	#89137
hypocenter	C₁₀₉	#1380
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
5. #16812768	C₄	semidirect
7. C₂×#13182004	C₄	semidirect
9. C₂×C₁₆×C₁₀₉	C₄	semidirect
46. C₁₀₉	#1380	semidirect
53. C₂×C₁₆	#89137	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812768	2
C₂×C₁₆×D₁₀₉	2
#1380	109