Subgroups — #31771549

Order 17216.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11761640	C₂	1	2	yes	maximal
2	(C₂×#11761640)⋊C₄	C₂	1	4	yes	maximal
3	C₄²×D₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11761640	C₂²	1	4	yes	—
5	C₄⋊#11761640	—	2	4	no	—
6	C₂×C₄×D₂₆₉	C₂²	1	2	yes	—
7	C₂×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
8	C₄×Dic₂₆₉	C₄	1	2	yes	complemented
9	C₄×Dic₂₆₉	C₄	1	1	yes	characteristic, complemented
10	C₄²×C₂₆₉	C₄	1	1	yes	characteristic, complemented
11	C₂×#11761640	—	2	4	no	—
12	C₂×#11761640	—	2	8	no	—
13	C₂²×D₂₆₉	C₂³	1	1	yes	characteristic
14	C₂×Dic₂₆₉	C₂×C₄	1	2	yes	—
15	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
16	C₄×D₂₆₉	—	2	4	no	—
17	C₄×D₂₆₉	D₄	1	2	yes	—
18	C₄×D₂₆₉	Q₈	1	2	yes	—
19	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
22	C₂×D₂₆₉	C₂×D₄	1	1	yes	characteristic
23	C₂×D₂₆₉	C₂×Q₈	1	1	yes	characteristic
24	C₂×D₂₆₉	#41	1	4	yes	—
25	#11761640	—	4	4	no	—
26	C₄×C₂₆₉	—	2	2	no	—
27	C₄×C₂₆₉	C₄⋊C₄	1	2	yes	—
28	Dic₂₆₉	—	2	2	no	—
29	Dic₂₆₉	C₄⋊C₄	1	2	yes	—
30	C₂×C₂₆₉	C₂×C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₂₆₉	#117	1	2	yes	—
32	D₂₆₉	#124	1	2	yes	—
33	D₂₆₉	#125	1	2	yes	—
34	C₂₆₉	#382	1	1	yes	characteristic, complemented, Sylow
35	#382	—	269	1	no	maximal, Sylow
36	C₂×C₄⋊C₄	—	269	2	no	—
37	C₂×C₄²	—	269	1	no	—
38	#95	—	269	4	no	—
39	C₂²×C₄	—	269	1	no	—
40	C₂²×C₄	—	269	2	no	—
41	C₂²×C₄	—	269	4	no	—
42	C₄⋊C₄	—	538	4	no	—
43	C₄²	—	269	2	no	—
44	C₄²	—	269	1	no	—
45	C₄²	#11761640	1	1	yes	characteristic, complemented
46	C₂³	—	269	1	no	—
47	C₂×C₄	—	269	1	no	—
48	C₂×C₄	—	269	2	no	—
49	C₂×C₄	—	269	2	no	—
50	C₂×C₄	—	269	2	no	—
51	C₂×C₄	—	538	4	no	—
52	C₂×C₄	—	538	4	no	—
53	C₂×C₄	—	538	8	no	—
54	C₂×C₄	C₂×#11761640	1	2	yes	—
55	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
56	C₂²	—	269	1	no	—
57	C₂²	—	269	1	no	—
58	C₂²	—	269	4	no	—
59	C₂²	C₂²×#11761640	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	269	2	no	—
62	C₄	—	538	2	no	—
63	C₄	—	1076	4	no	—
64	C₄	C₄⋊#11761640	1	2	yes	—
65	C₂	—	269	2	no	—
66	C₂	—	269	2	no	—
67	C₂	C₂×C₄⋊#11761640	1	1	yes	characteristic
68	C₂	#22575532	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator, socle	C₂²×C₂₆₉	C₂²×C₄
Fitting	C₄²×C₂₆₉	C₄
prime radical, cosocle	C₂²×D₂₆₉	C₂³
hypercenter	C₄²	#11761640
hypocenter	C₂₆₉	#382
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
8. C₄×Dic₂₆₉	C₄	semidirect
9. C₄×Dic₂₆₉	C₄	semidirect
10. C₄²×C₂₆₉	C₄	semidirect
34. C₂₆₉	#382	semidirect
45. C₄²	#11761640	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11761640	2
(C₂×#11761640)⋊C₄	2
C₄²×D₂₆₉	2
#382	269