Subgroups — #46812790

Order 34432.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₅₃₈	C₂	1	1	yes	maximal, characteristic
2	#31770423	C₂	1	2	yes	maximal
3	#31770380	C₂	1	2	yes	maximal
4	C₄×#16864377	C₂	1	1	yes	maximal, characteristic
5	C₄⋊C₁₆×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₅₃₈	C₄	1	1	yes	characteristic
7	C₂×#16864377	C₂²	1	1	yes	characteristic
8	C₂×#16864377	C₂²	1	1	yes	characteristic
9	C₄⋊#14524130	C₄	1	1	yes	characteristic
10	#22575382	C₄	1	2	yes	—
11	C₈×Dic₂₆₉	C₂²	1	2	yes	—
12	C₂×C₁₆×C₂₆₉	C₂²	1	2	yes	—
13	C₄×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₅₃₈	C₈	1	1	yes	characteristic
15	C₂×#14524130	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₂₆₉	C₈	1	1	yes	characteristic
17	#16864381	C₈	1	2	yes	—
18	C₄×Dic₂₆₉	C₂×C₄	1	2	yes	—
19	#16864377	—	2	1	no	—
20	#16864377	Q₈	1	2	yes	complemented
21	C₂×C₈×C₂₆₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
23	C₄²×C₂₆₉	C₂×C₄	1	1	yes	characteristic
24	C₁₆×C₂₆₉	—	2	2	no	—
25	Dic₅₃₈	—	2	2	no	—
26	C₂×Dic₂₆₉	C₂×C₈	1	2	yes	—
27	C₂×Dic₂₆₉	C₂×C₈	1	2	yes	—
28	#14524130	—	2	2	no	—
29	C₂×C₄×C₂₆₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₂₆₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₂₆₉	—	2	1	no	—
33	C₈×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₂₆₉	#41	1	1	yes	characteristic
35	C₂²×C₂₆₉	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₂₆₉	—	2	1	no	—
37	C₄×C₂₆₉	M₃₂	1	2	yes	—
38	C₄×C₂₆₉	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₂₆₉	#131	1	1	yes	characteristic
40	Dic₂₆₉	—	2	2	no	—
41	Dic₂₆₉	—	2	2	no	—
42	C₂×C₂₆₉	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₂₆₉	C₂×M₃₂	1	1	yes	characteristic
44	C₂×C₂₆₉	#504	1	1	yes	characteristic
45	C₂₆₉	#2183	1	1	yes	characteristic, complemented, Sylow
46	#2183	—	269	1	no	maximal, Sylow
47	C₈×Q₈	—	269	1	no	—
48	C₄⋊C₁₆	—	269	2	no	—
49	C₄⋊C₁₆	D₂₆₉	1	1	yes	characteristic
50	#346	—	269	2	no	—
51	C₄×C₁₆	—	269	1	no	—
52	C₄×Q₈	—	269	1	no	—
53	C₂×C₁₆	—	269	1	no	—
54	C₂×C₁₆	—	269	1	no	—
55	C₂×C₁₆	C₂×D₂₆₉	1	2	yes	—
56	C₄⋊C₈	—	269	2	no	—
57	C₄⋊C₈	—	269	1	no	—
58	C₄×C₈	—	269	2	no	—
59	C₄×C₈	C₂×D₂₆₉	1	1	yes	characteristic
60	C₂×Q₈	—	269	1	no	—
61	C₂×C₈	—	269	2	no	—
62	C₂×C₈	C₄×D₂₆₉	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
64	C₄⋊C₄	—	269	2	no	—
65	C₄⋊C₄	—	269	1	no	—
66	C₄²	—	269	2	no	—
67	C₄²	C₄×D₂₆₉	1	1	yes	characteristic
68	C₁₆	—	2	2	no	—
69	C₁₆	—	269	2	no	—
70	C₁₆	—	538	1	no	—
71	Q₈	—	538	2	no	—
72	C₂×C₄	—	269	2	no	—
73	C₂×C₄	—	269	2	no	—
74	C₂×C₄	C₈×D₂₆₉	1	1	yes	characteristic
75	C₂×C₄	C₈×D₂₆₉	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₂₆₉	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	538	2	no	—
79	C₈	Q₈×D₂₆₉	1	1	yes	characteristic
80	C₈	#16864401	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₂₆₉	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	538	2	no	—
84	C₄	—	538	2	no	—
85	C₄	#22575392	1	2	yes	—
86	C₄	#22575418	1	1	yes	characteristic
87	C₄	#22575461	1	1	yes	characteristic
88	C₂	#31770729	1	1	yes	characteristic
89	C₂	C₂×#22575392	1	1	yes	characteristic
90	C₂	#31771023	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₆₉
commutator, socle, prime radical	C₂²×C₂₆₉	C₂²×C₈
Fitting	C₄⋊C₁₆×C₂₆₉	C₂
cosocle	C₂×C₈×C₂₆₉	C₂³
hypercenter	C₄⋊C₁₆	D₂₆₉
hypocenter	C₂₆₉	#2183
LCS	G ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
20. #16864377	Q₈	semidirect
45. C₂₆₉	#2183	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₅₃₈	2
#31770423	2
#31770380	2
C₄×#16864377	2
C₄⋊C₁₆×C₂₆₉	2
#2183	269