Subgroups — #46812402

Order 34432.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31770654	C₂	1	3	yes	maximal, complemented
2	#31770568	C₂	1	3	yes	maximal
3	#338×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₂₆₉	C₂²	1	1	yes	characteristic
5	C₂×#16864386	C₂²	1	3	yes	—
6	#22575389	—	2	6	no	—
7	#22575365	—	2	6	no	—
8	C₂×M₁₆×C₂₆₉	C₂²	1	3	yes	—
9	C₂³×D₂₆₉	Q₈	1	1	yes	characteristic
10	C₂²×Dic₂₆₉	Q₈	1	1	yes	characteristic
11	C₂×C₄×D₂₆₉	—	2	3	no	—
12	C₂×C₄×D₂₆₉	—	2	3	no	—
13	#16864386	—	2	6	no	—
14	C₂×#14524130	—	2	3	no	—
15	C₂²×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
16	M₁₆×C₂₆₉	—	2	6	no	—
17	C₂×C₈×C₂₆₉	D₄	1	6	yes	—
18	C₂²×D₂₆₉	—	2	3	no	—
19	C₂²×D₂₆₉	—	4	2	no	—
20	C₂×Dic₂₆₉	—	2	3	no	—
21	C₄×D₂₆₉	—	4	3	no	—
22	C₄×D₂₆₉	—	4	1	no	—
23	#14524130	—	4	3	no	—
24	C₂³×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
25	C₂×C₄×C₂₆₉	C₂×D₄	1	3	yes	—
26	C₂×C₄×C₂₆₉	#41	1	3	yes	—
27	C₈×C₂₆₉	—	2	6	no	—
28	C₂²×C₂₆₉	—	4	1	no	—
29	C₂²×C₂₆₉	C₂²⋊Q₈	1	3	yes	—
30	C₂×D₂₆₉	—	2	6	no	—
31	C₂×D₂₆₉	—	4	3	no	—
32	C₂×D₂₆₉	—	4	1	no	—
33	C₄×C₂₆₉	#123	1	3	yes	—
34	C₄×C₂₆₉	#127	1	1	yes	characteristic
35	Dic₂₆₉	—	4	1	no	—
36	C₂×C₂₆₉	—	2	3	no	—
37	C₂×C₂₆₉	#399	1	1	yes	characteristic
38	D₂₆₉	—	4	2	no	—
39	C₂₆₉	#2091	1	1	yes	characteristic, complemented, Sylow
40	#2091	—	269	1	no	maximal, Sylow
41	#430	—	269	3	no	—
42	#407	—	269	3	no	—
43	#338	D₂₆₉	1	1	yes	characteristic, complemented
44	C₂³×C₄	—	269	1	no	—
45	C₂×M₁₆	—	269	3	no	—
46	C₂×M₁₆	C₂×D₂₆₉	1	3	yes	—
47	#108	—	538	6	no	—
48	C₂²⋊C₈	—	538	6	no	—
49	C₂⁴	—	269	1	no	—
50	C₂²×C₄	—	269	1	no	—
51	C₂²×C₄	—	538	3	no	—
52	C₂²×C₄	—	538	3	no	—
53	C₂²×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
54	M₁₆	—	2	6	no	—
55	M₁₆	—	538	6	no	—
56	C₂×C₈	—	538	3	no	—
57	C₂×C₈	D₁₀₇₆	1	6	yes	—
58	C₂³	—	538	3	no	—
59	C₂³	—	1076	2	no	—
60	C₂³	Q₈×D₂₆₉	1	1	yes	characteristic
61	C₂×C₄	—	538	3	no	—
62	C₂×C₄	—	1076	3	no	—
63	C₂×C₄	—	1076	1	no	—
64	C₂×C₄	C₂×D₁₀₇₆	1	3	yes	—
65	C₂×C₄	#16864401	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	—	1076	3	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	538	6	no	—
70	C₂²	—	1076	3	no	—
71	C₂²	—	1076	1	no	—
72	C₂²	#22575433	1	3	yes	—
73	C₄	—	1076	1	no	—
74	C₄	#22575437	1	3	yes	—
75	C₄	#22575450	1	1	yes	characteristic
76	C₂	—	2	3	no	—
77	C₂	—	1076	2	no	—
78	C₂	#31770542	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#22575450
commutator, cosocle	C₂²×C₄×C₂₆₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₆₉
Fitting	#338×C₂₆₉	C₂
socle	C₂×C₂₆₉	#399
prime radical	C₂³×D₂₆₉	Q₈
hypercenter	#338	D₂₆₉
hypocenter	C₂₆₉	#2091
LCS	G ▷ C₂²×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #31770654	C₂	semidirect
3. #338×C₂₆₉	C₂	semidirect
39. C₂₆₉	#2091	semidirect
43. #338	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#31770654	2
#31770568	2
#338×C₂₆₉	2
#2091	269