Subgroups — #31770864

Order 17216.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₇₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16864385	C₂	1	1	yes	maximal, characteristic
3	#22575433	C₂	1	2	yes	maximal
4	#22575376	C₂	1	2	yes	maximal, complemented
5	#106×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₃₈	C₂²	1	2	yes	—
7	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
8	Dic₁₀₇₆	—	2	2	no	—
9	#16864385	—	2	2	no	—
10	C₂×#14524130	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₆₉	—	2	2	no	—
12	#16864379	—	2	2	no	—
13	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
15	C₂²×D₂₆₉	D₄	1	1	yes	characteristic
16	Dic₅₃₈	—	2	2	no	—
17	Dic₅₃₈	—	2	2	no	—
18	C₂×Dic₂₆₉	—	2	2	no	—
19	C₂×Dic₂₆₉	D₄	1	1	yes	characteristic
20	C₄×D₂₆₉	—	2	1	no	—
21	C₄×D₂₆₉	—	2	1	no	—
22	#14524130	—	2	1	no	—
23	C₂×C₄×C₂₆₉	—	2	2	no	—
24	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
25	C₈×C₂₆₉	D₄	1	2	yes	—
26	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
27	C₂×D₂₆₉	—	2	1	no	—
28	C₂×D₂₆₉	—	2	2	no	—
29	C₄×C₂₆₉	—	4	2	no	—
30	C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₆₉	#41	1	1	yes	characteristic
32	Dic₂₆₉	—	2	1	no	—
33	Dic₂₆₉	—	4	2	no	—
34	C₂×C₂₆₉	#121	1	1	yes	characteristic
35	C₂×C₂₆₉	#137	1	2	yes	—
36	D₂₆₉	—	4	1	no	—
37	C₂₆₉	#470	1	1	yes	characteristic, complemented, Sylow
38	#470	—	269	1	no	maximal, Sylow
39	C₂×Q₁₆	—	269	1	no	—
40	C₂×M₁₆	—	269	1	no	—
41	C₂²⋊Q₈	—	269	2	no	—
42	#106	D₂₆₉	1	1	yes	characteristic, complemented
43	#103	—	269	2	no	—
44	C₂×Q₈	—	269	2	no	—
45	C₂²×C₄	—	269	1	no	—
46	Q₁₆	—	538	2	no	—
47	M₁₆	—	538	2	no	—
48	C₂×C₈	—	269	1	no	—
49	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
50	C₄⋊C₄	—	538	2	no	—
51	C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
52	C₂²⋊C₄	—	538	2	no	—
53	C₂³	—	269	1	no	—
54	Q₈	—	538	2	no	—
55	Q₈	—	538	2	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	269	1	no	—
58	C₂×C₄	—	538	1	no	—
59	C₂×C₄	—	538	1	no	—
60	C₂×C₄	—	538	2	no	—
61	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
62	C₈	—	538	1	no	—
63	C₈	D₁₀₇₆	1	2	yes	—
64	C₂²	—	538	1	no	—
65	C₂²	—	538	2	no	—
66	C₂²	D₄×D₂₆₉	1	1	yes	characteristic
67	C₄	—	4	2	no	—
68	C₄	—	538	1	no	—
69	C₄	—	1076	2	no	—
70	C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
71	C₄	#16864403	1	1	yes	characteristic
72	C₂	—	1076	1	no	—
73	C₂	#22575428	1	1	yes	characteristic
74	C₂	#22575482	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₂₆₉
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	#106×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×D₄
prime radical	C₂²×D₂₆₉	D₄
hypercenter	#106	D₂₆₉
hypocenter	C₂₆₉	#470
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
1. C₂×Dic₁₀₇₆	C₂	semidirect
4. #22575376	C₂	semidirect
5. #106×C₂₆₉	C₂	semidirect
37. C₂₆₉	#470	semidirect
42. #106	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₀₇₆	2
C₂×#16864385	2
#22575433	2
#22575376	2
#106×C₂₆₉	2
#470	269