Subgroups — #31771652

Order 17216.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575540	C₂	1	2	yes	maximal, complemented
2	#22575523	C₂	1	1	yes	maximal, characteristic, complemented
3	#22575517	C₂	1	2	yes	maximal, complemented
4	#22575507	C₂	1	1	yes	maximal, characteristic, complemented
5	#22575472	C₂	1	1	yes	maximal, characteristic
6	#16864412	C₂²	1	2	yes	—
7	#16864411	—	2	2	no	—
8	D₂₆₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#14524137	—	2	2	no	—
10	C₄×#11761640	C₂²	1	1	yes	characteristic
11	#16864401	C₂²	1	2	yes	—
12	Dic₁₀₇₆	C₄	1	1	yes	characteristic, complemented
13	#16864390	C₄	1	2	yes	complemented
14	C₈×D₂₆₉	C₂²	1	1	yes	characteristic
15	D₈×C₂₆₉	C₄	1	1	yes	characteristic, complemented
16	C₂×#11761640	—	2	1	no	—
17	#14524137	—	2	2	no	—
18	#14524137	D₄	1	2	yes	—
19	Dic₅₃₈	C₂×C₄	1	2	yes	—
20	C₂²⋊D₂₆₉	—	2	2	no	—
21	C₂×Dic₂₆₉	—	2	2	no	—
22	C₄×D₂₆₉	C₂³	1	1	yes	characteristic
23	#14524130	C₂×C₄	1	1	yes	characteristic
24	D₄×C₂₆₉	C₂×C₄	1	2	yes	—
25	C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₂₆₉	—	2	2	no	—
27	C₂×D₂₆₉	#41	1	1	yes	characteristic
28	#11761640	—	4	1	no	—
29	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
30	Dic₂₆₉	—	2	2	no	—
31	Dic₂₆₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₂₆₉	—	4	2	no	—
33	C₂×C₂₆₉	C₄×D₄	1	1	yes	characteristic
34	D₂₆₉	—	2	1	no	—
35	C₂₆₉	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	269	1	no	maximal, Sylow
37	#135	—	269	1	no	—
38	#131	—	269	2	no	—
39	#108	—	269	1	no	—
40	#104	—	269	2	no	—
41	#97	—	269	1	no	—
42	#41	—	269	2	no	—
43	Q₁₆	—	269	1	no	—
44	SD₁₆	—	269	2	no	—
45	D₈	#11761640	1	1	yes	characteristic, complemented
46	M₁₆	—	269	2	no	—
47	M₁₆	—	538	2	no	—
48	C₂×C₈	—	269	1	no	—
49	C₂×C₈	—	269	1	no	—
50	C₂×C₈	—	538	2	no	—
51	C₄²	—	269	1	no	—
52	Q₈	—	269	2	no	—
53	D₄	—	538	2	no	—
54	D₄	C₂×#11761640	1	2	yes	—
55	C₂×C₄	—	269	1	no	—
56	C₂×C₄	—	538	2	no	—
57	C₂×C₄	—	538	1	no	—
58	C₈	—	269	2	no	—
59	C₈	—	269	1	no	—
60	C₈	—	538	2	no	—
61	C₈	C₂×#11761640	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	—	269	1	no	—
64	C₄	—	269	1	no	—
65	C₄	—	538	2	no	—
66	C₄	—	1076	1	no	—
67	C₄	C₂²×#11761640	1	1	yes	characteristic
68	C₂	—	4	2	no	—
69	C₂	—	538	1	no	—
70	C₂	D₄×#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11761640
commutator	C₄×C₂₆₉	C₂²×C₄
Frattini	C₄	C₂²×#11761640
Fitting	D₈×C₂₆₉	C₄
socle	C₂×C₂₆₉	C₄×D₄
prime radical	#22575472	C₂
cosocle	C₄×D₂₆₉	C₂³
hypercenter	D₈	#11761640
hypocenter	C₂₆₉	#444
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #22575540	C₂	semidirect
2. #22575523	C₂	semidirect
3. #22575517	C₂	semidirect
4. #22575507	C₂	semidirect
12. Dic₁₀₇₆	C₄	semidirect
13. #16864390	C₄	semidirect
15. D₈×C₂₆₉	C₄	semidirect
35. C₂₆₉	#444	semidirect
45. D₈	#11761640	semidirect

Maximal subgroups

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

subgroup	index
#22575540	2
#22575523	2
#22575517	2
#22575507	2
#22575472	2
#444	269