Subgroups — #31771577

Order 17216.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16864411	C₂	1	2	yes	maximal, complemented
2	#22575510	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×D₂₆₉	C₂	1	1	yes	maximal, characteristic
4	#16864411	—	2	4	no	—
5	C₂×#14524137	C₂²	1	4	yes	complemented
6	C₂³×D₂₆₉	C₄	1	1	yes	characteristic
7	C₂²×Dic₂₆₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₂₆₉	—	2	4	no	—
9	C₂×C₄×D₂₆₉	C₄	1	2	yes	—
10	C₂×C₄×D₂₆₉	C₂²	1	2	yes	—
11	C₂²×C₄×C₂₆₉	C₄	1	1	yes	characteristic
12	#14524137	—	2	4	no	—
13	C₂²×D₂₆₉	—	2	2	no	—
14	C₂²×D₂₆₉	—	2	4	no	—
15	C₂²×D₂₆₉	C₂×C₄	1	2	yes	—
16	C₂×Dic₂₆₉	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₂₆₉	D₄	1	4	yes	—
18	C₂×Dic₂₆₉	C₂³	1	1	yes	characteristic
19	C₄×D₂₆₉	—	2	4	no	—
20	C₄×D₂₆₉	—	2	4	no	—
21	C₂³×C₂₆₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₆₉	—	2	2	no	—
23	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
24	C₂²×C₂₆₉	—	2	2	no	—
25	C₂²×C₂₆₉	C₂²⋊C₄	1	2	yes	—
26	C₂²×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₂₆₉	—	2	8	no	—
28	C₂×D₂₆₉	—	2	2	no	—
29	C₂×D₂₆₉	—	2	2	no	—
30	C₂×D₂₆₉	M₁₆	1	4	yes	complemented
31	C₄×C₂₆₉	—	2	2	no	—
32	Dic₂₆₉	C₂²⋊C₄	1	2	yes	—
33	Dic₂₆₉	C₂×D₄	1	2	yes	—
34	C₂×C₂₆₉	—	2	2	no	—
35	C₂×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₂₆₉	C₂×M₁₆	1	2	yes	—
37	D₂₆₉	—	2	4	no	—
38	C₂₆₉	#407	1	1	yes	characteristic, complemented, Sylow
39	#407	—	269	1	no	maximal, Sylow
40	C₂³×C₄	—	269	1	no	—
41	C₂×M₁₆	—	269	2	no	—
42	C₂²⋊C₈	—	269	4	no	—
43	C₂⁴	—	269	1	no	—
44	C₂²×C₄	—	269	1	no	—
45	C₂²×C₄	—	269	2	no	—
46	C₂²×C₄	—	269	2	no	—
47	C₂²×C₄	—	538	4	no	—
48	C₂²×C₄	#11761640	1	1	yes	characteristic
49	M₁₆	—	538	4	no	—
50	C₂×C₈	—	269	4	no	—
51	C₂³	—	269	2	no	—
52	C₂³	—	538	4	no	—
53	C₂³	—	538	2	no	—
54	C₂³	C₂×#11761640	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	269	4	no	—
57	C₂×C₄	—	269	1	no	—
58	C₂×C₄	—	269	1	no	—
59	C₂×C₄	—	538	4	no	—
60	C₂×C₄	—	538	4	no	—
61	C₂×C₄	C₂×#11761640	1	2	yes	—
62	C₈	—	538	4	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	269	4	no	—
65	C₂²	—	538	8	no	—
66	C₂²	—	538	2	no	—
67	C₂²	—	538	2	no	—
68	C₂²	C₂²⋊#11761640	1	2	yes	—
69	C₂²	C₂²×#11761640	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	269	2	no	—
72	C₄	—	269	2	no	—
73	C₂	—	2	2	no	—
74	C₂	—	538	4	no	—
75	C₂	C₂×C₂²⋊#11761640	1	1	yes	characteristic
76	C₂	D₂₆₉⋊M₁₆	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11761640
commutator, socle	C₂²×C₂₆₉	C₂²×C₄
Fitting	C₂²×C₄×C₂₆₉	C₄
prime radical	C₂³×D₂₆₉	C₄
cosocle	C₂×Dic₂₆₉	C₂³
hypercenter	C₂²×C₄	#11761640
hypocenter	C₂₆₉	#407
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16864411	C₂	semidirect
2. #22575510	C₂	semidirect
5. C₂×#14524137	C₂²	semidirect
30. C₂×D₂₆₉	M₁₆	semidirect
38. C₂₆₉	#407	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16864411	2
#22575510	2
C₂²×C₄×D₂₆₉	2
#407	269