Subgroups — #31770992

Order 17216.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864390	C₂	1	2	yes	maximal, complemented
2	C₂×#16864387	C₂	1	2	yes	maximal, complemented
3	#22575441	C₂	1	1	yes	maximal, characteristic, complemented
4	#22575364	C₂	1	1	yes	maximal, characteristic, complemented
5	#127×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₃₈	C₂²	1	1	yes	characteristic
7	C₂×D₁₀₇₆	C₂²	1	1	yes	characteristic
8	#16864390	—	2	4	no	—
9	#16864387	—	2	4	no	—
10	C₂×#14524130	C₂²	1	2	yes	—
11	#16864379	—	2	2	no	—
12	C₂×D₄×C₂₆₉	—	2	2	no	—
13	C₂×D₄×C₂₆₉	C₂²	1	2	yes	—
14	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
15	C₂²×D₂₆₉	—	2	1	no	—
16	Dic₅₃₈	—	2	2	no	—
17	D₁₀₇₆	—	2	2	no	—
18	C₂×Dic₂₆₉	—	2	1	no	—
19	#14524130	D₄	1	4	yes	complemented
20	C₂³×C₂₆₉	—	2	2	no	—
21	D₄×C₂₆₉	—	2	2	no	—
22	D₄×C₂₆₉	—	2	2	no	—
23	D₄×C₂₆₉	—	4	4	no	—
24	C₂×C₄×C₂₆₉	D₄	1	2	yes	—
25	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
26	C₂²×C₂₆₉	—	4	2	no	—
27	C₂²×C₂₆₉	—	4	4	no	—
28	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
29	C₂×D₂₆₉	—	4	1	no	—
30	C₂×D₂₆₉	—	4	2	no	—
31	C₄×C₂₆₉	—	2	2	no	—
32	C₄×C₂₆₉	C₂×D₄	1	2	yes	—
33	Dic₂₆₉	—	4	1	no	—
34	C₂×C₂₆₉	—	8	2	no	—
35	C₂×C₂₆₉	#127	1	1	yes	characteristic
36	C₂×C₂₆₉	AGL(1,ℤ/8)	1	2	yes	—
37	D₂₆₉	—	8	1	no	—
38	C₂₆₉	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	269	1	no	maximal, Sylow
40	C₂×SD₁₆	—	269	2	no	—
41	C₂×D₈	—	269	2	no	—
42	#127	D₂₆₉	1	1	yes	characteristic, complemented
43	#124	—	269	1	no	—
44	#97	—	269	1	no	—
45	C₂×Q₈	—	269	1	no	—
46	C₂×D₄	—	2	2	no	—
47	C₂×D₄	—	269	1	no	—
48	C₂×D₄	C₂×D₂₆₉	1	2	yes	—
49	SD₁₆	—	538	4	no	—
50	D₈	—	538	4	no	—
51	C₂×C₈	—	269	2	no	—
52	C₂²⋊C₄	—	538	2	no	—
53	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
54	C₂³	—	2	2	no	—
55	C₂³	—	538	1	no	—
56	Q₈	—	538	2	no	—
57	D₄	—	2	2	no	—
58	D₄	—	2	2	no	—
59	D₄	—	4	4	no	—
60	D₄	—	538	2	no	—
61	C₂×C₄	—	538	1	no	—
62	C₂×C₄	C₂²⋊D₂₆₉	1	2	yes	—
63	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
64	C₈	—	269	4	no	—
65	C₂²	—	4	2	no	—
66	C₂²	—	4	4	no	—
67	C₂²	—	1076	1	no	—
68	C₂²	—	1076	2	no	—
69	C₂²	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	1076	1	no	—
72	C₄	D₄×D₂₆₉	1	2	yes	—
73	C₂	—	8	2	no	—
74	C₂	—	2152	1	no	—
75	C₂	#22575449	1	1	yes	characteristic
76	C₂	#22575479	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	#127×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×D₄
hypercenter	#127	D₂₆₉
hypocenter	C₂₆₉	#496
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×#16864390	C₂	semidirect
2. C₂×#16864387	C₂	semidirect
3. #22575441	C₂	semidirect
4. #22575364	C₂	semidirect
5. #127×C₂₆₉	C₂	semidirect
19. #14524130	D₄	semidirect
38. C₂₆₉	#496	semidirect
42. #127	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864390	2
C₂×#16864387	2
#22575441	2
#22575364	2
#127×C₂₆₉	2
#496	269