Subgroups — #31771216

Order 17216.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575448	C₂	1	4	yes	maximal
2	#22575445	C₂	1	1	yes	maximal, characteristic, complemented
3	#22575443	C₂	1	4	yes	maximal, complemented
4	#22575429	C₂	1	2	yes	maximal
5	Q₈×Dic₂₆₉	C₂	1	2	yes	maximal, complemented
6	#22575410	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₆₉	C₂²	1	2	yes	—
10	C₄⋊Dic₂₆₉	C₂²	1	4	yes	—
11	#16864381	C₂²	1	2	yes	—
12	#16864381	C₂²	1	4	yes	—
13	#16864381	C₂²	1	4	yes	—
14	#16864379	C₂²	1	2	yes	—
15	#16864379	C₂²	1	4	yes	—
16	C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
17	C₄×Dic₂₆₉	C₂²	1	4	yes	—
18	C₂×Q₈×C₂₆₉	C₂²	1	2	yes	—
19	C₄⋊C₄×C₂₆₉	C₂²	1	4	yes	—
20	C₄²×C₂₆₉	C₂²	1	1	yes	characteristic
21	C₂²×D₂₆₉	C₂³	1	1	yes	characteristic
22	C₂×Dic₂₆₉	C₂³	1	2	yes	—
23	C₂×Dic₂₆₉	C₂³	1	4	yes	—
24	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
25	C₄×D₂₆₉	—	2	2	no	—
26	Q₈×C₂₆₉	—	2	4	no	—
27	C₂×C₄×C₂₆₉	C₂³	1	2	yes	—
28	C₂×C₄×C₂₆₉	C₂³	1	4	yes	—
29	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
30	C₂²×C₂₆₉	C₂⁴	1	1	yes	characteristic
31	C₂×D₂₆₉	—	2	1	no	—
32	C₂×D₂₆₉	—	2	2	no	—
33	C₄×C₂₆₉	—	2	4	no	—
34	C₄×C₂₆₉	—	2	2	no	—
35	C₄×C₂₆₉	#41	1	2	yes	—
36	Dic₂₆₉	—	2	1	no	—
37	Dic₂₆₉	—	2	2	no	—
38	Dic₂₆₉	—	2	4	no	—
39	C₂×C₂₆₉	C₂×#41	1	1	yes	characteristic
40	C₂×C₂₆₉	2₋¹⁺⁴	1	2	yes	—
41	D₂₆₉	—	4	1	no	—
42	C₂₆₉	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	269	1	no	maximal, Sylow
44	C₄⋊Q₈	D₂₆₉	1	1	yes	characteristic, complemented
45	#126	—	269	4	no	—
46	#125	—	269	4	no	—
47	#125	—	269	1	no	—
48	C₂²⋊Q₈	—	269	2	no	—
49	C₄×Q₈	—	269	2	no	—
50	#117	—	269	1	no	—
51	C₂×Q₈	C₂×D₂₆₉	1	2	yes	—
52	C₂²×C₄	—	269	1	no	—
53	C₄⋊C₄	—	269	2	no	—
54	C₄⋊C₄	—	269	2	no	—
55	C₄⋊C₄	—	269	4	no	—
56	C₄⋊C₄	—	269	4	no	—
57	C₄⋊C₄	—	269	4	no	—
58	C₄⋊C₄	C₂×D₂₆₉	1	4	yes	—
59	C₂²⋊C₄	—	269	2	no	—
60	C₂²⋊C₄	—	269	4	no	—
61	C₄²	—	269	1	no	—
62	C₄²	—	269	4	no	—
63	C₄²	C₂×D₂₆₉	1	1	yes	characteristic
64	C₂³	—	269	1	no	—
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	269	2	no	—
67	C₂×C₄	—	269	4	no	—
68	C₂×C₄	—	269	1	no	—
69	C₂×C₄	—	538	2	no	—
70	C₂×C₄	C₂²×D₂₆₉	1	4	yes	—
71	C₂×C₄	C₂²×D₂₆₉	1	2	yes	—
72	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
73	C₂²	—	538	1	no	—
74	C₂²	—	538	2	no	—
75	C₂²	C₂³×D₂₆₉	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	538	1	no	—
79	C₄	—	538	2	no	—
80	C₄	—	538	4	no	—
81	C₄	#16864401	1	2	yes	—
82	C₂	—	1076	1	no	—
83	C₂	C₂×#16864401	1	1	yes	characteristic
84	C₂	#22575501	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₉
commutator, socle, cosocle	C₂²×C₂₆₉	C₂⁴
Fitting	C₄⋊Q₈×C₂₆₉	C₂
prime radical	C₂²×D₂₆₉	C₂³
hypercenter	C₄⋊Q₈	D₂₆₉
hypocenter	C₂₆₉	#541
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #22575445	C₂	semidirect
3. #22575443	C₂	semidirect
5. Q₈×Dic₂₆₉	C₂	semidirect
7. C₄⋊Q₈×C₂₆₉	C₂	semidirect
42. C₂₆₉	#541	semidirect
44. C₄⋊Q₈	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#22575448	2
#22575445	2
#22575443	2
#22575429	2
Q₈×Dic₂₆₉	2
#22575410	2
C₄⋊Q₈×C₂₆₉	2
#541	269