Subgroups — C₂²×C₂²⋊#11761640 #31771666

Order 17216.

674 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11761640	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊#11761640	C₂	1	12	yes	maximal, direct product factor
3	C₂⁴×D₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11761640	—	2	8	no	—
5	C₂²×#11761640	C₂²	1	12	yes	complemented
6	C₂²⋊#11761640	C₂²	1	16	yes	direct product factor
7	C₂³×D₂₆₉	—	2	8	no	—
8	C₂³×D₂₆₉	C₄	1	1	yes	characteristic, complemented
9	C₂³×D₂₆₉	C₄	1	6	yes	complemented
10	C₂³×D₂₆₉	C₂²	1	1	yes	characteristic
11	C₂³×D₂₆₉	C₂²	1	6	yes	—
12	C₂⁴×C₂₆₉	C₄	1	1	yes	characteristic, complemented
13	C₂×#11761640	—	2	24	no	—
14	C₂×#11761640	C₂³	1	8	yes	complemented
15	C₂²×D₂₆₉	—	2	4	no	—
16	C₂²×D₂₆₉	—	2	24	no	—
17	C₂²×D₂₆₉	—	2	24	no	—
18	C₂²×D₂₆₉	C₂×C₄	1	3	yes	—
19	C₂²×D₂₆₉	C₂×C₄	1	12	yes	complemented
20	C₂²×D₂₆₉	C₂×C₄	1	6	yes	complemented
21	C₂²×D₂₆₉	D₄	1	8	yes	—
22	C₂²×D₂₆₉	C₂³	1	4	yes	—
23	C₂²×D₂₆₉	C₂³	1	3	yes	—
24	C₂³×C₂₆₉	—	2	4	no	—
25	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₂₆₉	C₂×C₄	1	6	yes	complemented
27	C₂²×C₂₆₉	—	2	12	no	—
28	C₂²×C₂₆₉	C₂²⋊C₄	1	4	yes	complemented
29	C₂²×C₂₆₉	C₂²×C₄	1	3	yes	—
30	C₂²×C₂₆₉	C₂²×C₄	1	4	yes	complemented
31	C₂×D₂₆₉	—	2	12	no	—
32	C₂×D₂₆₉	—	2	24	no	—
33	C₂×D₂₆₉	—	2	8	no	—
34	C₂×D₂₆₉	C₂²⋊C₄	1	12	yes	complemented
35	C₂×D₂₆₉	C₂²×C₄	1	3	yes	—
36	C₂×D₂₆₉	C₂²×C₄	1	4	yes	complemented
37	C₂×D₂₆₉	C₂×D₄	1	12	yes	—
38	C₂×D₂₆₉	C₂⁴	1	1	yes	characteristic
39	#11761640	—	2	8	no	—
40	C₂×C₂₆₉	—	2	4	no	—
41	C₂×C₂₆₉	C₂×C₂²⋊C₄	1	6	yes	complemented
42	C₂×C₂₆₉	C₂³×C₄	1	1	yes	characteristic
43	D₂₆₉	—	2	4	no	—
44	D₂₆₉	C₂×C₂²⋊C₄	1	6	yes	complemented
45	D₂₆₉	C₂²×D₄	1	2	yes	—
46	C₂₆₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
47	C₂²×C₂²⋊C₄	—	269	1	no	maximal, Sylow
48	C₂⁵	—	269	1	no	—
49	C₂³×C₄	—	269	2	no	—
50	C₂×C₂²⋊C₄	—	269	12	no	—
51	C₂⁴	—	269	1	no	—
52	C₂⁴	—	269	1	no	—
53	C₂⁴	—	269	6	no	—
54	C₂⁴	—	269	6	no	—
55	C₂⁴	—	538	8	no	—
56	C₂⁴	#11761640	1	1	yes	characteristic, complemented
57	C₂²×C₄	—	269	12	no	—
58	C₂²×C₄	—	538	8	no	—
59	C₂²⋊C₄	—	269	16	no	—
60	C₂³	—	2	4	no	—
61	C₂³	—	269	8	no	—
62	C₂³	—	269	3	no	—
63	C₂³	—	269	3	no	—
64	C₂³	—	269	6	no	—
65	C₂³	—	269	4	no	—
66	C₂³	—	269	12	no	—
67	C₂³	—	538	24	no	—
68	C₂³	—	538	24	no	—
69	C₂³	—	538	4	no	—
70	C₂³	C₂×#11761640	1	1	yes	characteristic
71	C₂³	C₂×#11761640	1	6	yes	complemented
72	C₂×C₄	—	269	8	no	—
73	C₂×C₄	—	538	24	no	—
74	C₂²	—	2	12	no	—
75	C₂²	—	269	12	no	—
76	C₂²	—	269	12	no	—
77	C₂²	—	269	1	no	—
78	C₂²	—	269	3	no	—
79	C₂²	—	269	4	no	—
80	C₂²	—	538	8	no	—
81	C₂²	—	538	24	no	—
82	C₂²	—	538	12	no	—
83	C₂²	C₂²⋊#11761640	1	4	yes	direct product factor
84	C₂²	C₂²×#11761640	1	3	yes	—
85	C₂²	C₂²×#11761640	1	4	yes	complemented
86	C₄	—	538	8	no	—
87	C₂	—	2	4	no	—
88	C₂	—	269	2	no	—
89	C₂	—	269	6	no	—
90	C₂	—	538	4	no	—
91	C₂	C₂×C₂²⋊#11761640	1	6	yes	direct product factor
92	C₂	C₂³×#11761640	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11761640
commutator	C₂×C₂₆₉	C₂³×C₄
Frattini	C₂	C₂³×#11761640
Fitting	C₂⁴×C₂₆₉	C₄
socle	C₂³×C₂₆₉	C₂×C₄
prime radical	C₂⁴×D₂₆₉	C₂
cosocle	C₂×D₂₆₉	C₂⁴
hypercenter	C₂⁴	#11761640
hypocenter	C₂₆₉	C₂²×C₂²⋊C₄
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂³×#11761640	C₂	semidirect
2. C₂×C₂²⋊#11761640	C₂	direct (N × H)
5. C₂²×#11761640	C₂²	semidirect
6. C₂²⋊#11761640	C₂²	direct (N × H)
8. C₂³×D₂₆₉	C₄	semidirect
9. C₂³×D₂₆₉	C₄	semidirect
12. C₂⁴×C₂₆₉	C₄	semidirect
14. C₂×#11761640	C₂³	semidirect
19. C₂²×D₂₆₉	C₂×C₄	semidirect
20. C₂²×D₂₆₉	C₂×C₄	semidirect
26. C₂³×C₂₆₉	C₂×C₄	semidirect
28. C₂²×C₂₆₉	C₂²⋊C₄	semidirect
30. C₂²×C₂₆₉	C₂²×C₄	semidirect
34. C₂×D₂₆₉	C₂²⋊C₄	semidirect
36. C₂×D₂₆₉	C₂²×C₄	semidirect
41. C₂×C₂₆₉	C₂×C₂²⋊C₄	semidirect
44. D₂₆₉	C₂×C₂²⋊C₄	semidirect
46. C₂₆₉	C₂²×C₂²⋊C₄	semidirect
56. C₂⁴	#11761640	semidirect
71. C₂³	C₂×#11761640	semidirect
83. C₂²	C₂²⋊#11761640	direct (N × H)
85. C₂²	C₂²×#11761640	semidirect
91. C₂	C₂×C₂²⋊#11761640	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11761640	2
C₂×C₂²⋊#11761640	2
C₂⁴×D₂₆₉	2
C₂²×C₂²⋊C₄	269

Subgroups — C22×C22⋊#11761640 #31771666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊#11761640 #31771666