Subgroups — #46825847

Order 34432.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771527	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×#14524137	C₂	1	1	yes	maximal, characteristic, complemented
3	#31770722	C₂	1	1	yes	maximal, characteristic
4	#22575527	—	2	1	no	—
5	C₄×#14524137	—	2	1	no	—
6	#22575460	—	2	1	no	—
7	#22575389	C₄	1	1	yes	characteristic
8	#22575380	—	2	1	no	—
9	C₈×Dic₂₆₉	C₂²	1	1	yes	characteristic
10	#135×C₂₆₉	C₄	1	1	yes	characteristic
11	C₂×#14524137	—	2	2	no	—
12	#16864405	—	4	1	no	—
13	#16864386	—	2	1	no	—
14	#16864386	—	4	1	no	—
15	C₂×#14524130	—	4	1	no	—
16	C₂×#14524130	D₄	1	1	yes	characteristic
17	C₄×Dic₂₆₉	D₄	1	1	yes	characteristic
18	#41×C₂₆₉	—	2	1	no	—
19	Q₁₆×C₂₆₉	C₈	1	1	yes	characteristic, complemented
20	SD₁₆×C₂₆₉	—	2	1	no	—
21	D₈×C₂₆₉	C₈	1	1	yes	characteristic, complemented
22	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
23	#14524137	—	2	4	no	—
24	C₂×Dic₂₆₉	—	2	1	no	—
25	#14524130	—	4	1	no	—
26	#14524130	D₈	1	1	yes	characteristic
27	#14524130	SD₁₆	1	1	yes	characteristic
28	Q₈×C₂₆₉	—	2	1	no	—
29	D₄×C₂₆₉	—	2	1	no	—
30	D₄×C₂₆₉	—	4	1	no	—
31	C₂×C₄×C₂₆₉	—	4	1	no	—
32	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₈×C₂₆₉	C₂×C₈	1	1	yes	characteristic
34	C₈×C₂₆₉	M₁₆	1	1	yes	characteristic
35	C₂²×C₂₆₉	—	4	1	no	—
36	C₂²×C₂₆₉	#104	1	1	yes	characteristic
37	C₄×C₂₆₉	—	4	1	no	—
38	C₄×C₂₆₉	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₂₆₉	#102	1	1	yes	characteristic
40	Dic₂₆₉	—	2	2	no	—
41	C₂×C₂₆₉	—	2	1	no	—
42	C₂×C₂₆₉	—	8	1	no	—
43	C₂×C₂₆₉	D₄⋊C₈	1	1	yes	characteristic
44	C₂₆₉	#1335	1	1	yes	characteristic, complemented, Sylow
45	#1335	—	269	1	no	maximal, Sylow
46	#443	—	269	1	no	—
47	#364	—	269	1	no	—
48	C₈²	—	269	1	no	—
49	#135	#11761640	1	1	yes	characteristic
50	#131	—	538	1	no	—
51	M₃₂	—	538	1	no	—
52	#108	—	269	1	no	—
53	#104	—	538	1	no	—
54	C₄×C₈	—	269	1	no	—
55	C₄×C₈	—	538	1	no	—
56	#41	—	2	1	no	—
57	Q₁₆	#14524137	1	1	yes	characteristic, complemented
58	SD₁₆	—	2	1	no	—
59	D₈	#14524137	1	1	yes	characteristic, complemented
60	M₁₆	—	538	1	no	—
61	M₁₆	—	1076	1	no	—
62	C₂×C₈	—	269	1	no	—
63	C₂×C₈	—	538	2	no	—
64	C₂×C₈	—	1076	1	no	—
65	C₂×C₈	C₂×#11761640	1	1	yes	characteristic
66	C₄²	—	269	1	no	—
67	C₁₆	—	1076	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	4	1	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	538	1	no	—
73	C₂×C₄	C₂²⋊#11761640	1	1	yes	characteristic
74	C₈	—	269	1	no	—
75	C₈	—	269	1	no	—
76	C₈	—	538	4	no	—
77	C₈	—	1076	1	no	—
78	C₈	C₂×#14524137	1	1	yes	characteristic
79	C₈	#16864411	1	1	yes	characteristic
80	C₂²	—	4	1	no	—
81	C₂²	#22575517	1	1	yes	characteristic
82	C₄	—	4	1	no	—
83	C₄	—	538	2	no	—
84	C₄	C₂²⋊#14524137	1	1	yes	characteristic
85	C₄	#22575515	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	D₄⋊#14524137	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#22575515
commutator	C₈×C₂₆₉	C₂×C₈
Frattini	C₂×C₈	C₂×#11761640
Fitting, prime radical	#135×C₂₆₉	C₄
socle	C₂×C₂₆₉	D₄⋊C₈
cosocle	C₈×Dic₂₆₉	C₂²
hypercenter	#135	#11761640
hypocenter	C₂₆₉	#1335
LCS	G ▷ C₈×C₂₆₉ ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #31771527	C₂	semidirect
2. C₈×#14524137	C₂	semidirect
19. Q₁₆×C₂₆₉	C₈	semidirect
21. D₈×C₂₆₉	C₈	semidirect
44. C₂₆₉	#1335	semidirect
57. Q₁₆	#14524137	semidirect
59. D₈	#14524137	semidirect

Maximal subgroups

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

subgroup	index
#31771527	2
C₈×#14524137	2
#31770722	2
#1335	269