Subgroups — #31770397

Order 17216.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575449	C₂	1	1	yes	maximal, characteristic
2	#22575368	C₂	1	1	yes	maximal, characteristic, complemented
3	#99×C₂₆₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₀₇₆	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₂₆₉	—	2	1	no	—
6	C₂×C₂²⋊D₂₆₉	—	2	1	no	—
7	C₂²⋊Dic₂₆₉	—	2	1	no	—
8	C₄×Dic₂₆₉	C₄	1	1	yes	characteristic, complemented
9	C₂×D₄×C₂₆₉	C₂²	1	1	yes	characteristic
10	C₂²⋊C₄×C₂₆₉	—	2	1	no	—
11	C₂²×D₂₆₉	—	2	1	no	—
12	D₁₀₇₆	—	4	1	no	—
13	C₂²⋊D₂₆₉	—	4	2	no	—
14	C₂²⋊D₂₆₉	—	4	2	no	—
15	C₂×Dic₂₆₉	—	2	1	no	—
16	C₂×Dic₂₆₉	—	4	1	no	—
17	C₂³×C₂₆₉	D₄	1	1	yes	characteristic
18	C₂³×C₂₆₉	D₄	1	1	yes	characteristic
19	D₄×C₂₆₉	—	4	1	no	—
20	C₂×C₄×C₂₆₉	—	4	1	no	—
21	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₂₆₉	—	2	1	no	—
23	C₂²×C₂₆₉	—	2	1	no	—
24	C₂²×C₂₆₉	—	4	1	no	—
25	C₂²×C₂₆₉	—	4	1	no	—
26	C₂²×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₂₆₉	—	4	1	no	—
28	C₂×D₂₆₉	—	4	1	no	—
29	C₂×D₂₆₉	—	4	1	no	—
30	C₄×C₂₆₉	—	2	1	no	—
31	C₄×C₂₆₉	—	8	1	no	—
32	Dic₂₆₉	—	2	2	no	—
33	Dic₂₆₉	—	8	1	no	—
34	C₂×C₂₆₉	—	2	1	no	—
35	C₂×C₂₆₉	—	4	1	no	—
36	C₂×C₂₆₉	—	4	1	no	—
37	C₂×C₂₆₉	#99	1	1	yes	characteristic
38	D₂₆₉	—	8	1	no	—
39	C₂₆₉	#353	1	1	yes	characteristic, complemented, Sylow
40	#353	—	269	1	no	maximal, Sylow
41	#127	—	269	1	no	—
42	#99	—	269	1	no	—
43	#99	D₂₆₉	1	1	yes	characteristic, complemented
44	C₂×D₄	—	269	1	no	—
45	C₂×D₄	—	538	1	no	—
46	C₂×D₄	—	538	1	no	—
47	C₂×D₄	C₂×D₂₆₉	1	1	yes	characteristic
48	C₂²⋊C₄	—	2	1	no	—
49	C₂²⋊C₄	—	538	1	no	—
50	C₄²	—	269	1	no	—
51	C₂³	—	538	1	no	—
52	C₂³	C₂²⋊D₂₆₉	1	1	yes	characteristic
53	C₂³	D₁₀₇₆	1	1	yes	characteristic
54	D₄	—	4	1	no	—
55	D₄	—	1076	1	no	—
56	D₄	—	1076	2	no	—
57	D₄	—	1076	2	no	—
58	C₂×C₄	—	4	1	no	—
59	C₂×C₄	—	538	1	no	—
60	C₂×C₄	—	1076	1	no	—
61	C₂×C₄	C₄×D₂₆₉	1	1	yes	characteristic
62	C₂²	—	2	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	4	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	1076	1	no	—
67	C₂²	—	1076	1	no	—
68	C₂²	—	1076	1	no	—
69	C₂²	#16864379	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	8	1	no	—
72	C₄	—	538	2	no	—
73	C₄	—	2152	1	no	—
74	C₂	—	2	1	no	—
75	C₂	—	4	1	no	—
76	C₂	—	4	1	no	—
77	C₂	—	2152	1	no	—
78	C₂	#22575367	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22575367
commutator	C₂×C₄×C₂₆₉	C₂×C₄
Frattini	C₂×D₄	C₂×D₂₆₉
Fitting	#99×C₂₆₉	C₂
socle	C₂×C₂₆₉	#99
prime radical	#22575449	C₂
cosocle	C₂×D₄×C₂₆₉	C₂²
hypercenter	#99	D₂₆₉
hypocenter	C₂₆₉	#353
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #99

Complements

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

N	H ≅ G/N	type
2. #22575368	C₂	semidirect
3. #99×C₂₆₉	C₂	semidirect
4. C₂×D₁₀₇₆	C₄	semidirect
8. C₄×Dic₂₆₉	C₄	semidirect
39. C₂₆₉	#353	semidirect
43. #99	D₂₆₉	semidirect

Maximal subgroups

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

subgroup	index
#22575449	2
#22575368	2
#99×C₂₆₉	2
#353	269