Subgroups — #31771654

Order 17216.

102 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	C₄×#14524137	C₂	1	1	yes	maximal, characteristic
4	C₄×#14524137	C₂	1	2	yes	maximal
5	#22575418	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524137	C₂²	1	2	yes	—
7	C₂×#14524137	C₂²	1	1	yes	characteristic
8	C₂×#14524137	C₂²	1	1	yes	characteristic
9	C₂×Dic₅₃₈	C₄	1	1	yes	characteristic
10	#16864381	C₄	1	2	yes	—
11	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
12	C₄×Dic₂₆₉	C₂²	1	2	yes	—
13	C₄⋊C₄×C₂₆₉	C₄	1	1	yes	characteristic
14	#14524137	—	2	2	no	—
15	#14524137	—	2	1	no	—
16	#14524137	Q₈	1	2	yes	complemented
17	Dic₅₃₈	C₈	1	4	yes	complemented
18	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₂₆₉	C₂×C₄	1	2	yes	—
20	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
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	C₂×C₈	1	2	yes	—
26	Dic₂₆₉	—	2	1	no	—
27	Dic₂₆₉	C₂×C₈	1	4	yes	—
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	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₂₆₉	#131	1	1	yes	characteristic
33	C₂₆₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₈×Q₈	—	269	1	no	maximal, Sylow
35	C₄×Q₈	—	269	1	no	—
36	C₄⋊C₈	—	269	1	no	—
37	C₄⋊C₈	—	269	2	no	—
38	C₄×C₈	—	269	2	no	—
39	C₄×C₈	—	269	1	no	—
40	C₂×Q₈	—	269	1	no	—
41	C₂×C₈	—	269	1	no	—
42	C₂×C₈	—	269	2	no	—
43	C₂×C₈	—	269	1	no	—
44	C₄⋊C₄	—	269	2	no	—
45	C₄⋊C₄	#11761640	1	1	yes	characteristic
46	C₄²	—	269	2	no	—
47	C₄²	—	269	1	no	—
48	Q₈	—	269	4	no	—
49	C₂×C₄	—	269	1	no	—
50	C₂×C₄	—	269	2	no	—
51	C₂×C₄	—	269	1	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	2	no	—
56	C₈	—	538	1	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	—	269	4	no	—
62	C₄	—	538	1	no	—
63	C₄	D₂₆₉⋊C₈	1	2	yes	—
64	C₂	Q₈×#11761640	1	1	yes	characteristic
65	C₂	C₂×D₂₆₉⋊C₈	1	1	yes	characteristic
66	C₂	#22575540	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator	C₂×C₂₆₉	C₂²×C₈
Fitting	C₄⋊C₄×C₂₆₉	C₄
socle, prime radical	C₂²×C₂₆₉	C₂²×C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₄⋊C₄	#11761640
hypocenter	C₂₆₉	C₈×Q₈
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
16. #14524137	Q₈	semidirect
17. Dic₅₃₈	C₈	semidirect
33. C₂₆₉	C₈×Q₈	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
C₄×#14524137	2
#22575418	2
C₈×Q₈	269