Subgroups — #31770544

Order 17216.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₂₆₉	C₂	1	1	yes	maximal, characteristic
2	#22575360	C₂	1	2	yes	maximal
3	#22575360	C₂	1	1	yes	maximal, characteristic
4	#22575360	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₄×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₉	C₂²	1	2	yes	—
7	C₂²×Dic₂₆₉	C₂²	1	2	yes	—
8	C₄⋊Dic₂₆₉	—	2	2	no	—
9	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₆₉	—	2	2	no	—
13	C₂×Dic₂₆₉	—	2	2	no	—
14	C₂×Dic₂₆₉	—	2	2	no	—
15	C₂×Dic₂₆₉	—	2	2	no	—
16	C₂×Dic₂₆₉	—	2	4	no	—
17	C₂×Dic₂₆₉	—	2	2	no	—
18	C₂³×C₂₆₉	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₂₆₉	—	2	1	no	—
20	C₂×C₄×C₂₆₉	—	2	2	no	—
21	C₂×C₄×C₂₆₉	—	2	1	no	—
22	C₂×C₄×C₂₆₉	—	2	1	no	—
23	C₂×C₄×C₂₆₉	—	2	2	no	—
24	C₂×C₄×C₂₆₉	D₄	1	2	yes	—
25	C₂×C₄×C₂₆₉	Q₈	1	2	yes	—
26	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
28	C₂²×C₂₆₉	#41	1	2	yes	—
29	C₂²×C₂₆₉	#41	1	1	yes	characteristic
30	C₂²×C₂₆₉	#41	1	2	yes	—
31	C₄×C₂₆₉	—	2	2	no	—
32	C₄×C₂₆₉	—	2	2	no	—
33	C₄×C₂₆₉	—	4	1	no	—
34	Dic₂₆₉	—	4	2	no	—
35	Dic₂₆₉	—	4	2	no	—
36	C₂×C₂₆₉	C₂²⋊Q₈	1	1	yes	characteristic
37	C₂×C₂₆₉	#123	1	1	yes	characteristic
38	C₂×C₂₆₉	#124	1	1	yes	characteristic
39	C₂×C₂₆₉	#125	1	2	yes	—
40	C₂×C₂₆₉	#126	1	2	yes	—
41	C₂₆₉	#400	1	1	yes	characteristic, complemented, Sylow
42	#400	—	269	1	no	maximal, Sylow
43	C₂×C₄⋊C₄	—	269	1	no	—
44	C₂×C₄⋊C₄	D₂₆₉	1	1	yes	characteristic
45	#95	—	269	1	no	—
46	#95	—	269	2	no	—
47	#95	—	269	2	no	—
48	C₂²×C₄	—	269	2	no	—
49	C₂²×C₄	—	269	2	no	—
50	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
51	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
52	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
53	C₄⋊C₄	—	2	2	no	—
54	C₄⋊C₄	—	538	2	no	—
55	C₂³	C₂²×D₂₆₉	1	1	yes	characteristic
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	2	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	538	2	no	—
62	C₂×C₄	—	538	2	no	—
63	C₂×C₄	—	538	2	no	—
64	C₂×C₄	—	538	4	no	—
65	C₂×C₄	—	538	2	no	—
66	C₂×C₄	C₂²⋊D₂₆₉	1	2	yes	—
67	C₂×C₄	Dic₅₃₈	1	2	yes	—
68	C₂²	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
69	C₂²	C₂×Dic₅₃₈	1	1	yes	characteristic
70	C₂²	#16864401	1	2	yes	—
71	C₂²	#16864402	1	1	yes	characteristic
72	C₂²	#16864403	1	2	yes	—
73	C₄	—	2	2	no	—
74	C₄	—	2	2	no	—
75	C₄	—	4	1	no	—
76	C₄	—	1076	2	no	—
77	C₄	—	1076	2	no	—
78	C₂	#22575432	1	1	yes	characteristic
79	C₂	#22575434	1	1	yes	characteristic
80	C₂	#22575442	1	1	yes	characteristic
81	C₂	#22575444	1	2	yes	—
82	C₂	#22575448	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₂₆₉
commutator, socle, prime radical, cosocle	C₂³×C₂₆₉	C₂³
Fitting	C₂×C₄⋊C₄×C₂₆₉	C₂
hypercenter	C₂×C₄⋊C₄	D₂₆₉
hypocenter	C₂₆₉	#400
LCS	G ▷ C₂³×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
41. C₂₆₉	#400	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₂₆₉	2
#22575360	2
C₂×C₄⋊C₄×C₂₆₉	2
#400	269