Subgroups — #46825891

Order 34432.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31771497	C₂	1	2	yes	maximal
2	#31770975	C₂	1	1	yes	maximal, characteristic
3	#22575527	—	2	2	no	—
4	#22575453	C₄	1	1	yes	characteristic
5	#22575376	—	2	2	no	—
6	#22575364	C₂²	1	1	yes	characteristic
7	#125×C₂₆₉	C₄	1	1	yes	characteristic
8	#16864405	—	2	4	no	—
9	C₂×Dic₅₃₈	—	2	1	no	—
10	C₂×#14524130	D₄	1	2	yes	—
11	C₄⋊Dic₂₆₉	—	4	1	no	—
12	C₄⋊C₄×C₂₆₉	—	2	1	no	—
13	C₄⋊C₄×C₂₆₉	—	4	1	no	—
14	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
15	Dic₅₃₈	—	4	2	no	—
16	C₂×Dic₂₆₉	—	4	1	no	—
17	#14524130	—	2	2	no	—
18	C₂×C₄×C₂₆₉	—	2	1	no	—
19	C₂×C₄×C₂₆₉	—	4	1	no	—
20	C₂×C₄×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
21	C₂²×C₂₆₉	#100	1	1	yes	characteristic
22	C₄×C₂₆₉	—	4	1	no	—
23	C₄×C₂₆₉	—	8	1	no	—
24	C₄×C₂₆₉	#104	1	2	yes	—
25	Dic₂₆₉	—	8	1	no	—
26	C₂×C₂₆₉	—	2	1	no	—
27	C₂×C₂₆₉	#329	1	1	yes	characteristic
28	C₂₆₉	#1356	1	1	yes	characteristic, complemented, Sylow
29	#1356	—	269	1	no	maximal, Sylow
30	#491	—	269	1	no	—
31	#347	—	269	2	no	—
32	C₄⋊Q₈	—	269	1	no	—
33	#125	#11761640	1	1	yes	characteristic
34	M₃₂	—	538	2	no	—
35	#103	—	538	2	no	—
36	#97	—	269	1	no	—
37	C₂×Q₈	—	538	1	no	—
38	C₂×C₈	—	269	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	4	1	no	—
41	C₄⋊C₄	—	1076	1	no	—
42	C₄²	C₂×#11761640	1	1	yes	characteristic
43	C₁₆	—	538	4	no	—
44	Q₈	—	1076	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	4	1	no	—
47	C₂×C₄	—	1076	1	no	—
48	C₂×C₄	C₂²⋊#11761640	1	1	yes	characteristic
49	C₈	—	538	2	no	—
50	C₂²	#22575513	1	1	yes	characteristic
51	C₄	—	4	1	no	—
52	C₄	—	8	1	no	—
53	C₄	—	2152	1	no	—
54	C₄	#22575518	1	2	yes	—
55	C₂	—	2	1	no	—
56	C₂	#31771461	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31771461
commutator	C₄²×C₂₆₉	C₂×C₄
Frattini	C₄²	C₂×#11761640
Fitting	#125×C₂₆₉	C₄
socle	C₂×C₂₆₉	#329
prime radical	C₂²×C₂₆₉	#100
cosocle	#22575364	C₂²
hypercenter	#125	#11761640
hypocenter	C₂₆₉	#1356
LCS	G ▷ C₄²×C₂₆₉ ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
28. C₂₆₉	#1356	semidirect

Maximal subgroups

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

subgroup	index
#31771497	2
#31770975	2
#1356	269