Subgroups — #27197453

Order 13184.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20268223	C₂	1	2	yes	maximal, complemented
2	Dic₁₆₄₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#20267630	C₂	1	2	yes	maximal, complemented
4	D₃₂₉₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₂×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#15693043	—	2	2	no	—
7	Dic₈₂₄	C₂²	1	2	yes	—
8	#15692966	—	2	2	no	—
9	D₁₆₄₈	C₂²	1	2	yes	—
10	C₂×C₁₆×C₁₀₃	C₂²	1	1	yes	characteristic
11	C₃₂×C₁₀₃	C₂²	1	1	yes	characteristic
12	C₃₂×C₁₀₃	C₂²	1	1	yes	characteristic
13	#13110105	—	4	2	no	—
14	Dic₄₁₂	—	2	2	no	—
15	#13110094	—	4	2	no	—
16	D₈₂₄	—	2	2	no	—
17	C₂×C₈×C₁₀₃	D₄	1	1	yes	characteristic
18	C₁₆×C₁₀₃	D₄	1	1	yes	characteristic
19	C₁₆×C₁₀₃	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₁₀₃	D₈	1	1	yes	characteristic
21	C₂²⋊D₁₀₃	—	8	2	no	—
22	D₄₁₂	—	4	2	no	—
23	C₄×D₁₀₃	—	8	2	no	—
24	Dic₂₀₆	—	4	2	no	—
25	C₈×C₁₀₃	D₈	1	1	yes	characteristic
26	C₈×C₁₀₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₁₀₃	D₁₆	1	1	yes	characteristic
28	C₂×D₁₀₃	—	8	2	no	—
29	C₄×C₁₀₃	D₁₆	1	1	yes	characteristic
30	C₄×C₁₀₃	C₂×D₈	1	1	yes	characteristic
31	Dic₁₀₃	—	8	2	no	—
32	C₂×C₁₀₃	—	2	1	no	—
33	C₂×C₁₀₃	C₂×D₁₆	1	1	yes	characteristic
34	D₁₀₃	—	16	2	no	—
35	#2262	—	103	1	no	maximal, Sylow
36	C₁₀₃	#2262	1	1	yes	characteristic, complemented, Sylow
37	#508	—	103	2	no	—
38	Q₆₄	—	103	1	no	—
39	SD₆₄	—	103	2	no	—
40	D₃₂	—	103	1	no	—
41	C₂×C₃₂	D₁₀₃	1	1	yes	characteristic, complemented
42	#135	—	206	2	no	—
43	Q₃₂	—	103	2	no	—
44	SD₃₂	—	206	2	no	—
45	D₁₆	—	103	2	no	—
46	C₂×C₁₆	C₂×D₁₀₃	1	1	yes	characteristic
47	C₃₂	C₂×D₁₀₃	1	1	yes	characteristic
48	C₃₂	C₂×D₁₀₃	1	1	yes	characteristic
49	#41	—	412	2	no	—
50	Q₁₆	—	206	2	no	—
51	SD₁₆	—	412	2	no	—
52	D₈	—	206	2	no	—
53	C₂×C₈	D₄₁₂	1	1	yes	characteristic
54	C₁₆	D₄₁₂	1	1	yes	characteristic
55	C₁₆	C₂²×D₁₀₃	1	1	yes	characteristic
56	Q₈	—	412	2	no	—
57	D₄	—	824	2	no	—
58	D₄	—	412	2	no	—
59	C₂×C₄	—	824	2	no	—
60	C₂×C₄	D₈₂₄	1	1	yes	characteristic
61	C₈	D₈₂₄	1	1	yes	characteristic
62	C₈	C₂×D₄₁₂	1	1	yes	characteristic
63	C₂²	—	824	2	no	—
64	C₂²	D₁₆₄₈	1	1	yes	characteristic
65	C₄	—	824	2	no	—
66	C₄	D₁₆₄₈	1	1	yes	characteristic
67	C₄	C₂×D₈₂₄	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1648	2	no	—
70	C₂	C₂×D₁₆₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₆₄₈
commutator, cosocle	C₁₆×C₁₀₃	C₂³
Frattini	C₁₆	C₂²×D₁₀₃
Fitting	C₂×C₃₂×C₁₀₃	C₂
socle	C₂×C₁₀₃	C₂×D₁₆
hypercenter	C₂×C₃₂	D₁₀₃
hypocenter	C₁₀₃	#2262
LCS	G ▷ C₁₆×C₁₀₃ ▷ C₈×C₁₀₃ ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #20268223	C₂	semidirect
2. Dic₁₆₄₈	C₂	semidirect
3. #20267630	C₂	semidirect
4. D₃₂₉₆	C₂	semidirect
5. C₂×C₃₂×C₁₀₃	C₂	semidirect
36. C₁₀₃	#2262	semidirect
41. C₂×C₃₂	D₁₀₃	semidirect

Maximal subgroups

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

subgroup	index
#20268223	2
Dic₁₆₄₈	2
#20267630	2
D₃₂₉₆	2
C₂×C₃₂×C₁₀₃	2
#2262	103