Subgroups — #31771656

Order 17216.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575520	C₂	1	2	yes	maximal
2	C₄⋊#14524137	C₂	1	1	yes	maximal, characteristic
3	#22575508	C₂	1	2	yes	maximal
4	C₄×#14524137	C₂	1	1	yes	maximal, characteristic
5	Q₈×Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524137	C₂²	1	1	yes	characteristic
7	C₂×#14524137	C₂²	1	2	yes	—
8	C₂×#14524137	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₆₉	C₄	1	2	yes	—
10	C₄⋊Dic₂₆₉	C₄	1	1	yes	characteristic
11	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
12	C₄×Dic₂₆₉	C₂²	1	2	yes	—
13	C₂×Q₈×C₂₆₉	C₄	1	1	yes	characteristic
14	#14524137	—	2	1	no	—
15	#14524137	—	2	2	no	—
16	#14524137	Q₈	1	2	yes	complemented
17	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₂₆₉	C₂×C₄	1	2	yes	—
19	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
20	Q₈×C₂₆₉	—	2	2	no	—
21	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₂₆₉	—	2	2	no	—
25	C₄×C₂₆₉	M₁₆	1	2	yes	—
26	Dic₂₆₉	—	2	1	no	—
27	Dic₂₆₉	—	2	2	no	—
28	Dic₂₆₉	C₂×Q₈	1	1	yes	characteristic
29	Dic₂₆₉	#41	1	1	yes	characteristic
30	C₂×C₂₆₉	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₂₆₉	C₂×M₁₆	1	1	yes	characteristic
32	C₂×C₂₆₉	#131	1	1	yes	characteristic
33	C₂₆₉	#446	1	1	yes	characteristic, complemented, Sylow
34	#446	—	269	1	no	maximal, Sylow
35	C₄×Q₈	—	269	1	no	—
36	C₄⋊C₈	—	269	1	no	—
37	C₄⋊C₈	—	269	2	no	—
38	#97	—	269	2	no	—
39	C₄×C₈	—	269	1	no	—
40	C₂×Q₈	#11761640	1	1	yes	characteristic
41	C₂×C₈	—	269	1	no	—
42	C₂×C₈	—	269	1	no	—
43	C₂×C₈	—	269	2	no	—
44	C₄⋊C₄	—	269	2	no	—
45	C₄⋊C₄	—	269	1	no	—
46	C₄²	—	269	1	no	—
47	C₄²	—	269	2	no	—
48	Q₈	—	2	2	no	—
49	C₂×C₄	—	269	1	no	—
50	C₂×C₄	—	269	1	no	—
51	C₂×C₄	—	269	2	no	—
52	C₂×C₄	C₂×#11761640	1	2	yes	—
53	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
54	C₈	—	269	2	no	—
55	C₈	—	538	1	no	—
56	C₈	—	538	2	no	—
57	C₂²	C₂²×#11761640	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	269	1	no	—
60	C₄	—	269	1	no	—
61	C₄	—	538	1	no	—
62	C₄	—	538	2	no	—
63	C₄	#16864411	1	2	yes	—
64	C₂	Q₈×#11761640	1	1	yes	characteristic
65	C₂	C₂×#16864411	1	1	yes	characteristic
66	C₂	#22575541	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator, socle, prime radical	C₂²×C₂₆₉	C₂²×C₄
Fitting	C₂×Q₈×C₂₆₉	C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₂×Q₈	#11761640
hypocenter	C₂₆₉	#446
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
16. #14524137	Q₈	semidirect
33. C₂₆₉	#446	semidirect

Maximal subgroups

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

subgroup	index
#22575520	2
C₄⋊#14524137	2
#22575508	2
C₄×#14524137	2
Q₈×Dic₂₆₉	2
#446	269