Subgroups — #31770613

Order 17216.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#16864386	C₂	1	1	yes	maximal, characteristic
3	#22575376	C₂	1	4	yes	maximal, complemented
4	#117×C₂₆₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₃₈	—	2	1	no	—
6	C₂×Dic₅₃₈	—	2	2	no	—
7	C₂×Dic₅₃₈	C₄	1	4	yes	complemented
8	C₂×Dic₅₃₈	C₂²	1	2	yes	—
9	C₂²×Dic₂₆₉	—	2	1	no	—
10	#16864386	—	2	2	no	—
11	C₂×#14524130	C₂²	1	2	yes	—
12	C₂²×C₄×C₂₆₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₆₉	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
15	C₄²×C₂₆₉	—	2	1	no	—
16	Dic₅₃₈	—	2	4	no	—
17	Dic₅₃₈	—	2	2	no	—
18	Dic₅₃₈	C₂×C₄	1	4	yes	—
19	C₂×Dic₂₆₉	—	2	4	no	—
20	C₂×Dic₂₆₉	—	2	2	no	—
21	#14524130	—	2	2	no	—
22	C₂³×C₂₆₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₆₉	—	2	2	no	—
24	C₂×C₄×C₂₆₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₂₆₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₆₉	D₄	1	2	yes	—
27	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
28	C₂²×C₂₆₉	—	2	2	no	—
29	C₂²×C₂₆₉	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₂₆₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₂₆₉	—	4	2	no	—
32	C₄×C₂₆₉	C₂²⋊C₄	1	2	yes	—
33	C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
34	C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
35	Dic₂₆₉	—	2	4	no	—
36	C₂×C₂₆₉	—	2	2	no	—
37	C₂×C₂₆₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₂₆₉	#137	1	2	yes	—
39	C₂₆₉	#419	1	1	yes	characteristic, complemented, Sylow
40	#419	—	269	1	no	maximal, Sylow
41	C₂²×Q₈	—	269	1	no	—
42	C₂×M₁₆	—	269	1	no	—
43	#117	D₂₆₉	1	1	yes	characteristic
44	#103	—	269	4	no	—
45	C₂×Q₈	—	269	4	no	—
46	C₂×Q₈	—	269	2	no	—
47	C₂×Q₈	—	538	1	no	—
48	C₂×Q₈	—	538	2	no	—
49	C₂²×C₄	—	538	1	no	—
50	C₂²×C₄	C₂×D₂₆₉	1	1	yes	characteristic
51	M₁₆	—	538	2	no	—
52	C₂×C₈	—	269	2	no	—
53	C₄⋊C₄	C₂×D₂₆₉	1	2	yes	—
54	C₂²⋊C₄	—	2	1	no	—
55	C₄²	—	2	1	no	—
56	C₂³	C₂²⋊D₂₆₉	1	1	yes	characteristic
57	Q₈	—	269	4	no	—
58	Q₈	—	538	4	no	—
59	Q₈	—	538	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	538	4	no	—
62	C₂×C₄	—	538	2	no	—
63	C₂×C₄	C₄×D₂₆₉	1	2	yes	—
64	C₂×C₄	C₂²⋊D₂₆₉	1	1	yes	characteristic
65	C₂×C₄	D₁₀₇₆	1	2	yes	—
66	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
67	C₈	—	538	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	#16864379	1	2	yes	—
70	C₂²	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
71	C₄	—	4	2	no	—
72	C₄	—	538	4	no	—
73	C₄	#16864379	1	2	yes	—
74	C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
75	C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
76	C₂	—	2	2	no	—
77	C₂	C₂×#16864379	1	1	yes	characteristic
78	C₂	#22575483	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₂₆₉
commutator	C₄×C₂₆₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	#117×C₂₆₉	C₂
socle	C₂²×C₂₆₉	C₂×D₄
prime radical	C₂³×C₂₆₉	D₄
cosocle	C₂×C₄×C₂₆₉	C₂³
hypercenter	#117	D₂₆₉
hypocenter	C₂₆₉	#419
LCS	G ▷ C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #22575376	C₂	semidirect
7. C₂×Dic₅₃₈	C₄	semidirect
39. C₂₆₉	#419	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₃₈	2
C₂×#16864386	2
#22575376	2
#117×C₂₆₉	2
#419	269