Subgroups — #31771341

Order 17216.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575440	C₂	1	4	yes	maximal, complemented
2	#22575434	C₂	1	4	yes	maximal, complemented
3	C₄×Dic₅₃₈	C₂	1	2	yes	maximal, complemented
4	D₄×Dic₂₆₉	C₂	1	4	yes	maximal, complemented
5	#127×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₈	C₂²	1	2	yes	—
7	C₂²×Dic₂₆₉	C₂²	1	4	yes	—
8	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
9	#16864381	C₂²	1	4	yes	—
10	C₂²⋊Dic₂₆₉	C₂²	1	8	yes	—
11	C₂²⋊Dic₂₆₉	C₂²	1	4	yes	—
12	C₄×Dic₂₆₉	C₂²	1	4	yes	—
13	C₂×D₄×C₂₆₉	C₂²	1	4	yes	—
14	C₂×D₄×C₂₆₉	C₂²	1	2	yes	—
15	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
16	Dic₅₃₈	—	2	4	no	—
17	C₂×Dic₂₆₉	—	4	4	no	—
18	C₂×Dic₂₆₉	C₂³	1	4	yes	—
19	C₂×Dic₂₆₉	C₂³	1	4	yes	—
20	C₂³×C₂₆₉	C₂³	1	4	yes	—
21	D₄×C₂₆₉	—	2	8	no	—
22	D₄×C₂₆₉	—	4	2	no	—
23	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
25	C₂²×C₂₆₉	—	2	8	no	—
26	C₂²×C₂₆₉	—	2	4	no	—
27	C₂²×C₂₆₉	C₂⁴	1	1	yes	characteristic
28	C₄×C₂₆₉	—	2	1	no	—
29	C₄×C₂₆₉	#41	1	4	yes	—
30	Dic₂₆₉	—	2	4	no	—
31	Dic₂₆₉	—	2	4	no	—
32	C₂×C₂₆₉	—	4	4	no	—
33	C₂×C₂₆₉	C₂×#41	1	2	yes	—
34	C₂×C₂₆₉	H(5,𝔽₂)	1	1	yes	characteristic
35	C₂₆₉	#559	1	1	yes	characteristic, complemented, Sylow
36	#559	—	269	1	no	maximal, Sylow
37	#127	D₂₆₉	1	1	yes	characteristic
38	#124	—	269	4	no	—
39	#123	—	269	4	no	—
40	C₄×Q₈	—	269	2	no	—
41	C₄×D₄	—	269	4	no	—
42	C₂×Q₈	—	269	2	no	—
43	C₂×D₄	C₂×D₂₆₉	1	4	yes	—
44	C₂×D₄	C₂×D₂₆₉	1	2	yes	—
45	C₂²×C₄	—	269	4	no	—
46	C₄⋊C₄	—	269	2	no	—
47	C₄⋊C₄	—	269	4	no	—
48	C₂²⋊C₄	—	269	8	no	—
49	C₂²⋊C₄	—	269	4	no	—
50	C₄²	—	269	4	no	—
51	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
52	C₂³	C₂²×D₂₆₉	1	4	yes	—
53	Q₈	—	538	4	no	—
54	D₄	—	2	8	no	—
55	D₄	—	4	2	no	—
56	C₂×C₄	—	269	4	no	—
57	C₂×C₄	—	269	4	no	—
58	C₂×C₄	—	1076	4	no	—
59	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
60	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
61	C₂²	—	2	8	no	—
62	C₂²	—	2	4	no	—
63	C₂²	C₂³×D₂₆₉	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	—	538	4	no	—
66	C₄	—	538	4	no	—
67	C₄	#16864401	1	4	yes	—
68	C₂	—	4	4	no	—
69	C₂	C₂×#16864401	1	2	yes	—
70	C₂	#22575498	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁴
Fitting, prime radical	#127×C₂₆₉	C₂
hypercenter	#127	D₂₆₉
hypocenter	C₂₆₉	#559
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #22575440	C₂	semidirect
2. #22575434	C₂	semidirect
3. C₄×Dic₅₃₈	C₂	semidirect
4. D₄×Dic₂₆₉	C₂	semidirect
35. C₂₆₉	#559	semidirect

Maximal subgroups

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

subgroup	index
#22575440	2
#22575434	2
C₄×Dic₅₃₈	2
D₄×Dic₂₆₉	2
#127×C₂₆₉	2
#559	269