Subgroups — #31771646

Order 17216.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×#11761640	C₂	1	2	yes	maximal, complemented
2	#22575521	C₂	1	1	yes	maximal, characteristic, complemented
3	#22575515	C₂	1	2	yes	maximal, complemented
4	#22575507	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×D₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11761640	—	2	2	no	—
7	D₂₆₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊#11761640	C₂²	1	2	yes	—
9	C₂²⋊#11761640	—	2	2	no	—
10	C₄×#11761640	C₂²	1	1	yes	characteristic
11	D₄×D₂₆₉	C₂²	1	2	yes	—
12	D₂₁₅₂	C₄	1	1	yes	characteristic, complemented
13	#16864387	C₄	1	2	yes	complemented
14	C₈×D₂₆₉	C₂²	1	1	yes	characteristic
15	D₈×C₂₆₉	C₄	1	1	yes	characteristic, complemented
16	C₂×#11761640	—	2	2	no	—
17	C₂×#11761640	—	4	4	no	—
18	C₂×#11761640	D₄	1	2	yes	—
19	#14524137	—	2	1	no	—
20	C₂²×D₂₆₉	—	2	2	no	—
21	D₁₀₇₆	C₂×C₄	1	2	yes	complemented
22	C₂²⋊D₂₆₉	—	2	2	no	—
23	C₄×D₂₆₉	C₂³	1	1	yes	characteristic
24	#14524130	C₂×C₄	1	1	yes	characteristic, complemented
25	D₄×C₂₆₉	C₂×C₄	1	2	yes	complemented
26	C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic, complemented
27	C₂²×C₂₆₉	—	2	2	no	—
28	C₂×D₂₆₉	—	2	2	no	—
29	C₂×D₂₆₉	—	4	2	no	—
30	C₂×D₂₆₉	—	4	2	no	—
31	C₂×D₂₆₉	C₂×D₄	1	1	yes	characteristic
32	#11761640	—	2	2	no	—
33	#11761640	—	4	2	no	—
34	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
35	Dic₂₆₉	#41	1	1	yes	characteristic
36	C₂×C₂₆₉	—	4	2	no	—
37	C₂×C₂₆₉	C₄×D₄	1	1	yes	characteristic
38	D₂₆₉	—	4	2	no	—
39	D₂₆₉	AGL(1,ℤ/8)	1	2	yes	—
40	C₂₆₉	#442	1	1	yes	characteristic, complemented, Sylow
41	#442	—	269	1	no	maximal, Sylow
42	C₂×D₈	—	269	1	no	—
43	C₄×D₄	—	269	2	no	—
44	#106	—	269	1	no	—
45	#102	—	269	2	no	—
46	#97	—	269	1	no	—
47	C₂×D₄	—	269	2	no	—
48	C₂²×C₄	—	538	2	no	—
49	D₈	—	269	2	no	—
50	D₈	—	269	1	no	—
51	D₈	#11761640	1	1	yes	characteristic, complemented
52	C₂×C₈	—	269	1	no	—
53	C₂×C₈	—	269	1	no	—
54	C₄⋊C₄	—	269	2	no	—
55	C₂²⋊C₄	—	538	2	no	—
56	C₄²	—	269	1	no	—
57	C₂³	—	538	2	no	—
58	D₄	—	269	2	no	—
59	D₄	—	538	2	no	—
60	D₄	C₂×#11761640	1	2	yes	complemented
61	C₂×C₄	—	269	1	no	—
62	C₂×C₄	—	269	2	no	—
63	C₂×C₄	—	538	2	no	—
64	C₂×C₄	—	1076	4	no	—
65	C₈	—	269	1	no	—
66	C₈	—	538	1	no	—
67	C₈	C₂×#11761640	1	1	yes	characteristic, complemented
68	C₂²	—	2	2	no	—
69	C₂²	—	269	1	no	—
70	C₂²	—	538	2	no	—
71	C₂²	—	1076	2	no	—
72	C₂²	—	1076	2	no	—
73	C₄	—	269	1	no	—
74	C₄	—	538	2	no	—
75	C₄	—	1076	2	no	—
76	C₄	C₂²×#11761640	1	1	yes	characteristic
77	C₂	—	4	2	no	—
78	C₂	—	269	2	no	—
79	C₂	—	1076	2	no	—
80	C₂	D₄×#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#11761640
commutator	C₄×C₂₆₉	C₂²×C₄
Frattini	C₄	C₂²×#11761640
Fitting	D₈×C₂₆₉	C₄
socle	C₂×C₂₆₉	C₄×D₄
prime radical	D₈×D₂₆₉	C₂
cosocle	C₄×D₂₆₉	C₂³
hypercenter	D₈	#11761640
hypocenter	C₂₆₉	#442
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. D₄×#11761640	C₂	semidirect
2. #22575521	C₂	semidirect
3. #22575515	C₂	semidirect
4. #22575507	C₂	semidirect
12. D₂₁₅₂	C₄	semidirect
13. #16864387	C₄	semidirect
15. D₈×C₂₆₉	C₄	semidirect
21. D₁₀₇₆	C₂×C₄	semidirect
24. #14524130	C₂×C₄	semidirect
25. D₄×C₂₆₉	C₂×C₄	semidirect
26. C₈×C₂₆₉	C₂×C₄	semidirect
40. C₂₆₉	#442	semidirect
51. D₈	#11761640	semidirect
60. D₄	C₂×#11761640	semidirect
67. C₈	C₂×#11761640	semidirect

Maximal subgroups

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

subgroup	index
D₄×#11761640	2
#22575521	2
#22575515	2
#22575507	2
D₈×D₂₆₉	2
#442	269