Subgroups — #31771683

Order 17216.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575540	C₂	1	8	yes	maximal, complemented
2	D₂₆₉⋊M₁₆	C₂	1	2	yes	maximal, complemented
3	C₂×#16864411	C₂	1	4	yes	maximal, complemented
4	C₂×#16864401	C₂	1	1	yes	maximal, characteristic
5	#16864412	C₂²	1	4	yes	complemented
6	#16864411	C₂²	1	4	yes	—
7	#16864411	C₂²	1	8	yes	complemented
8	D₂₆₉⋊C₈	C₂²	1	4	yes	complemented
9	C₂×#14524137	C₂²	1	8	yes	complemented
10	#16864401	C₄	1	4	yes	—
11	#16864401	C₂²	1	4	yes	—
12	C₂×Dic₅₃₈	C₄	1	1	yes	characteristic
13	C₂×C₂²⋊D₂₆₉	C₄	1	2	yes	—
14	C₂²×Dic₂₆₉	C₂²	1	2	yes	—
15	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₂₆₉	C₄	1	1	yes	characteristic
17	#14524137	C₂³	1	8	yes	—
18	C₂²×D₂₆₉	C₂×C₄	1	1	yes	characteristic
19	Dic₅₃₈	C₂×C₄	1	4	yes	—
20	C₂²⋊D₂₆₉	C₂×C₄	1	8	yes	—
21	C₂×Dic₂₆₉	C₂×C₄	1	2	yes	—
22	C₂×Dic₂₆₉	C₂×C₄	1	4	yes	—
23	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₆₉	C₂³	1	4	yes	—
25	C₄×D₂₆₉	C₂×C₄	1	2	yes	—
26	C₄×D₂₆₉	C₂³	1	2	yes	—
27	C₂³×C₂₆₉	C₂×C₄	1	2	yes	—
28	D₄×C₂₆₉	C₂×C₄	1	4	yes	—
29	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₂₆₉	—	4	2	no	—
31	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₂₆₉	C₂²×C₄	1	4	yes	—
33	C₂×D₂₆₉	—	4	1	no	—
34	C₂×D₂₆₉	C₂²×C₄	1	2	yes	—
35	C₄×C₂₆₉	C₂²×C₄	1	2	yes	—
36	Dic₂₆₉	C₂²×C₄	1	1	yes	characteristic
37	Dic₂₆₉	C₂²×C₄	1	4	yes	—
38	Dic₂₆₉	C₂⁴	1	1	yes	characteristic
39	C₂×C₂₆₉	—	2	1	no	—
40	C₂×C₂₆₉	—	2	4	no	—
41	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
42	D₂₆₉	—	2	2	no	—
43	C₂₆₉	#568	1	1	yes	characteristic, complemented, Sylow
44	#568	—	269	1	no	maximal, Sylow
45	C₂×#41	—	269	1	no	—
46	#131	—	269	8	no	—
47	C₂×M₁₆	—	269	4	no	—
48	C₂×M₁₆	—	269	2	no	—
49	#41	—	269	4	no	—
50	#41	—	269	4	no	—
51	C₂×Q₈	—	269	1	no	—
52	C₂×D₄	—	269	2	no	—
53	C₂×D₄	#11761640	1	1	yes	characteristic
54	C₂²×C₄	—	269	2	no	—
55	C₂²×C₄	—	269	1	no	—
56	M₁₆	—	269	4	no	—
57	M₁₆	—	269	8	no	—
58	M₁₆	—	269	4	no	—
59	C₂×C₈	—	269	8	no	—
60	C₂×C₈	—	269	4	no	—
61	C₂³	—	269	1	no	—
62	C₂³	C₂×#11761640	1	2	yes	—
63	Q₈	—	269	4	no	—
64	D₄	—	269	8	no	—
65	D₄	C₂×#11761640	1	4	yes	—
66	C₂×C₄	—	269	4	no	—
67	C₂×C₄	—	269	1	no	—
68	C₂×C₄	—	269	2	no	—
69	C₂×C₄	—	269	4	no	—
70	C₂×C₄	—	269	2	no	—
71	C₂×C₄	—	269	2	no	—
72	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
73	C₈	—	269	8	no	—
74	C₂²	—	4	2	no	—
75	C₂²	—	269	2	no	—
76	C₂²	—	1076	1	no	—
77	C₂²	C₂²×#11761640	1	1	yes	characteristic
78	C₂²	C₂²×#11761640	1	4	yes	—
79	C₄	—	269	1	no	—
80	C₄	—	269	1	no	—
81	C₄	—	269	4	no	—
82	C₄	C₂²×#11761640	1	2	yes	—
83	C₂	—	2	1	no	—
84	C₂	—	2	4	no	—
85	C₂	—	538	2	no	—
86	C₂	C₂³×#11761640	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#11761640
commutator, socle	C₂×C₂₆₉	C₂³×C₄
Fitting	C₂×D₄×C₂₆₉	C₄
prime radical	C₂×#16864401	C₂
cosocle	Dic₂₆₉	C₂⁴
hypercenter	C₂×D₄	#11761640
hypocenter	C₂₆₉	#568
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #22575540	C₂	semidirect
2. D₂₆₉⋊M₁₆	C₂	semidirect
3. C₂×#16864411	C₂	semidirect
5. #16864412	C₂²	semidirect
7. #16864411	C₂²	semidirect
8. D₂₆₉⋊C₈	C₂²	semidirect
9. C₂×#14524137	C₂²	semidirect
43. C₂₆₉	#568	semidirect

Maximal subgroups

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

subgroup	index
#22575540	2
D₂₆₉⋊M₁₆	2
C₂×#16864411	2
C₂×#16864401	2
#568	269