Subgroups — #20268539

Order 6592.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15693009	C₂	1	2	yes	maximal, complemented
2	#15693006	C₂	1	4	yes	maximal, complemented
3	#15692999	C₂	1	4	yes	maximal, complemented
4	#15692996	C₂	1	4	yes	maximal, complemented
5	#125×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₀₆	C₂²	1	2	yes	—
7	C₂×D₄₁₂	C₂²	1	4	yes	—
8	C₂×D₄₁₂	C₂²	1	2	yes	—
9	C₂×C₄×D₁₀₃	C₂²	1	4	yes	—
10	Dic₁₀₃⋊C₄	C₂²	1	4	yes	—
11	(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	—
12	(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	—
13	(C₂×D₁₀₃)⋊C₄	C₂²	1	4	yes	—
14	C₄⋊C₄×C₁₀₃	C₂²	1	4	yes	—
15	C₄⋊C₄×C₁₀₃	C₂²	1	2	yes	—
16	C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
17	C₂²×D₁₀₃	C₂³	1	4	yes	—
18	C₂×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₁₀₃	C₂³	1	2	yes	—
20	C₂×C₄×C₁₀₃	C₂³	1	4	yes	—
21	C₂×Dic₁₀₃	C₂³	1	4	yes	—
22	D₄₁₂	—	4	2	no	—
23	D₄₁₂	—	4	4	no	—
24	C₄×D₁₀₃	—	4	4	no	—
25	Dic₂₀₆	—	4	2	no	—
26	C₂²×C₁₀₃	C₂⁴	1	1	yes	characteristic
27	C₂×D₁₀₃	—	2	4	no	—
28	C₂×D₁₀₃	—	2	4	no	—
29	C₂×D₁₀₃	—	2	4	no	—
30	C₄×C₁₀₃	—	2	2	no	—
31	C₄×C₁₀₃	—	2	4	no	—
32	C₄×C₁₀₃	—	2	1	no	—
33	Dic₁₀₃	—	2	4	no	—
34	C₂×C₁₀₃	H(5,𝔽₂)	1	2	yes	—
35	C₂×C₁₀₃	2₋¹⁺⁴	1	1	yes	characteristic
36	D₁₀₃	—	4	4	no	—
37	C₁₀₃	#562	1	1	yes	characteristic, complemented, Sylow
38	#562	—	103	1	no	maximal, Sylow
39	#125	D₁₀₃	1	1	yes	characteristic, complemented
40	#124	—	103	2	no	—
41	#123	—	103	4	no	—
42	C₂²⋊Q₈	—	103	4	no	—
43	#121	—	103	4	no	—
44	C₂×Q₈	—	103	2	no	—
45	C₂×D₄	—	103	4	no	—
46	C₂×D₄	—	103	2	no	—
47	C₂²×C₄	—	103	4	no	—
48	C₄⋊C₄	—	103	4	no	—
49	C₄⋊C₄	C₂×D₁₀₃	1	4	yes	—
50	C₄⋊C₄	C₂×D₁₀₃	1	2	yes	—
51	C₂²⋊C₄	—	103	4	no	—
52	C₂²⋊C₄	—	103	4	no	—
53	C₂²⋊C₄	—	103	4	no	—
54	C₄²	C₂×D₁₀₃	1	1	yes	characteristic
55	C₂³	—	103	4	no	—
56	Q₈	—	412	2	no	—
57	D₄	—	412	4	no	—
58	D₄	—	412	2	no	—
59	C₂×C₄	—	103	4	no	—
60	C₂×C₄	—	412	4	no	—
61	C₂×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₀₃	1	2	yes	—
63	C₂×C₄	C₂²×D₁₀₃	1	4	yes	—
64	C₂²	—	206	4	no	—
65	C₂²	—	206	4	no	—
66	C₂²	—	206	4	no	—
67	C₂²	C₂³×D₁₀₃	1	1	yes	characteristic
68	C₄	—	2	4	no	—
69	C₄	—	2	2	no	—
70	C₄	—	2	1	no	—
71	C₄	—	206	4	no	—
72	C₂	—	412	4	no	—
73	C₂	#15693067	1	2	yes	—
74	C₂	#15693069	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₃
commutator, socle	C₂²×C₁₀₃	C₂⁴
Fitting	#125×C₁₀₃	C₂
hypercenter	#125	D₁₀₃
hypocenter	C₁₀₃	#562
LCS	G ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. #15693009	C₂	semidirect
2. #15693006	C₂	semidirect
3. #15692999	C₂	semidirect
4. #15692996	C₂	semidirect
5. #125×C₁₀₃	C₂	semidirect
37. C₁₀₃	#562	semidirect
39. #125	D₁₀₃	semidirect