Subgroups — #46826035

Order 34432.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₂₆₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#31771531	C₂	1	1	yes	maximal, characteristic
3	Q₃₂×D₂₆₉	C₂	1	1	yes	maximal, characteristic
4	#22575522	—	2	1	no	—
5	#22575503	—	2	1	no	—
6	Q₁₆×D₂₆₉	—	2	1	no	—
7	Dic₂₁₅₂	C₄	1	1	yes	characteristic, complemented
8	#22575399	—	2	1	no	—
9	C₁₆×D₂₆₉	C₂²	1	1	yes	characteristic
10	Q₃₂×C₂₆₉	C₄	1	1	yes	characteristic, complemented
11	C₄⋊#11761640	—	4	1	no	—
12	Q₈×D₂₆₉	—	4	1	no	—
13	Dic₁₀₇₆	—	2	1	no	—
14	#16864392	—	4	1	no	—
15	C₈×D₂₆₉	D₄	1	1	yes	characteristic
16	#16864377	D₄	1	1	yes	characteristic
17	Q₁₆×C₂₆₉	—	2	1	no	—
18	C₁₆×C₂₆₉	C₂×C₄	1	1	yes	characteristic
19	C₂×#11761640	—	8	1	no	—
20	Dic₅₃₈	—	4	1	no	—
21	Dic₅₃₈	—	8	1	no	—
22	C₄×D₂₆₉	—	8	1	no	—
23	C₄×D₂₆₉	D₈	1	1	yes	characteristic
24	#14524130	SD₁₆	1	1	yes	characteristic
25	Q₈×C₂₆₉	—	4	1	no	—
26	C₈×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₂₆₉	D₁₆	1	1	yes	characteristic
28	#11761640	—	16	1	no	—
29	C₄×C₂₆₉	—	8	1	no	—
30	C₄×C₂₆₉	#102	1	1	yes	characteristic
31	Dic₂₆₉	—	8	1	no	—
32	Dic₂₆₉	SD₃₂	1	1	yes	characteristic
33	C₂×C₂₆₉	#357	1	1	yes	characteristic
34	D₂₆₉	SD₆₄	1	1	yes	characteristic
35	D₂₆₉	Q₆₄	1	1	yes	characteristic
36	C₂₆₉	#1416	1	1	yes	characteristic, complemented, Sylow
37	#1416	—	269	1	no	maximal, Sylow
38	C₂×Q₃₂	—	269	1	no	—
39	C₂×C₃₂	—	269	1	no	—
40	#366	—	269	1	no	—
41	C₂×Q₁₆	—	538	1	no	—
42	Q₃₂	—	269	1	no	—
43	Q₃₂	—	538	1	no	—
44	Q₃₂	#11761640	1	1	yes	characteristic, complemented
45	C₂×C₁₆	—	269	1	no	—
46	#107	—	538	1	no	—
47	C₃₂	—	538	1	no	—
48	C₂×Q₈	—	1076	1	no	—
49	Q₁₆	—	2	1	no	—
50	Q₁₆	—	538	1	no	—
51	Q₁₆	—	1076	1	no	—
52	C₂×C₈	—	269	1	no	—
53	C₄⋊C₄	—	1076	1	no	—
54	C₁₆	—	269	1	no	—
55	C₁₆	C₂×#11761640	1	1	yes	characteristic
56	Q₈	—	4	1	no	—
57	Q₈	—	1076	1	no	—
58	Q₈	—	2152	1	no	—
59	C₂×C₄	—	269	1	no	—
60	C₂×C₄	—	2152	1	no	—
61	C₂×C₄	—	2152	1	no	—
62	C₈	—	269	1	no	—
63	C₈	C₂²⋊#11761640	1	1	yes	characteristic
64	C₂²	—	269	1	no	—
65	C₄	—	8	1	no	—
66	C₄	—	269	1	no	—
67	C₄	—	2152	1	no	—
68	C₄	—	4304	1	no	—
69	C₄	#22575515	1	1	yes	characteristic
70	C₂	—	269	1	no	—
71	C₂	—	269	1	no	—
72	C₂	#31771517	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31771517
commutator	C₁₆×C₂₆₉	C₂×C₄
Frattini	C₁₆	C₂×#11761640
Fitting	Q₃₂×C₂₆₉	C₄
socle	C₂×C₂₆₉	#357
prime radical	C₂×D₂₆₉	D₁₆
cosocle	C₁₆×D₂₆₉	C₂²
hypercenter	Q₃₂	#11761640
hypocenter	C₂₆₉	#1416
LCS	G ▷ C₁₆×C₂₆₉ ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
7. Dic₂₁₅₂	C₄	semidirect
10. Q₃₂×C₂₆₉	C₄	semidirect
36. C₂₆₉	#1416	semidirect
44. Q₃₂	#11761640	semidirect

Maximal subgroups

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

subgroup	index
D₂₆₉⋊C₃₂	2
#31771531	2
Q₃₂×D₂₆₉	2
#1416	269