Subgroups — #31771512

Order 17216.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575512	C₂	1	2	yes	maximal
2	#22575441	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11761640	—	2	2	no	—
4	C₂×Dic₅₃₈	C₄	1	1	yes	characteristic, complemented
5	C₂×D₁₀₇₆	C₂²	1	1	yes	characteristic
6	#16864379	—	2	2	no	—
7	C₄²×C₂₆₉	C₄	1	1	yes	characteristic, complemented
8	C₂×#11761640	—	4	2	no	—
9	C₂²×D₂₆₉	D₄	1	2	yes	—
10	Dic₅₃₈	—	4	1	no	—
11	D₁₀₇₆	—	4	1	no	—
12	C₂×Dic₂₆₉	—	2	1	no	—
13	C₂×C₄×C₂₆₉	—	2	1	no	—
14	C₂×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
16	C₂×D₂₆₉	—	2	2	no	—
17	C₂×D₂₆₉	—	4	2	no	—
18	#11761640	—	8	2	no	—
19	C₄×C₂₆₉	—	2	1	no	—
20	C₄×C₂₆₉	—	4	1	no	—
21	Dic₂₆₉	—	4	1	no	—
22	C₂×C₂₆₉	—	2	1	no	—
23	C₂×C₂₆₉	#99	1	1	yes	characteristic
24	D₂₆₉	—	4	2	no	—
25	C₂₆₉	#354	1	1	yes	characteristic, complemented, Sylow
26	#354	—	269	1	no	maximal, Sylow
27	#124	—	269	1	no	—
28	#99	—	269	2	no	—
29	C₂×Q₈	—	269	1	no	—
30	C₂×D₄	—	269	1	no	—
31	C₂²⋊C₄	—	538	2	no	—
32	C₂²⋊C₄	—	538	2	no	—
33	C₄²	#11761640	1	1	yes	characteristic, complemented
34	C₂³	—	269	2	no	—
35	Q₈	—	1076	1	no	—
36	D₄	—	1076	1	no	—
37	C₂×C₄	—	2	1	no	—
38	C₂×C₄	—	538	1	no	—
39	C₂×C₄	—	1076	2	no	—
40	C₂×C₄	C₂×#11761640	1	1	yes	characteristic
41	C₂²	—	538	2	no	—
42	C₂²	—	1076	2	no	—
43	C₂²	C₂²⋊#11761640	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	4	1	no	—
46	C₄	—	1076	1	no	—
47	C₄	—	2152	2	no	—
48	C₂	—	2	1	no	—
49	C₂	—	1076	2	no	—
50	C₂	#22575512	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22575512
commutator	C₂×C₄×C₂₆₉	C₂×C₄
Frattini	C₂×C₄	C₂×#11761640
Fitting	C₄²×C₂₆₉	C₄
socle	C₂×C₂₆₉	#99
prime radical, cosocle	C₂×D₁₀₇₆	C₂²
hypercenter	C₄²	#11761640
hypocenter	C₂₆₉	#354
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₅₃₈	C₄	semidirect
7. C₄²×C₂₆₉	C₄	semidirect
25. C₂₆₉	#354	semidirect
33. C₄²	#11761640	semidirect

Maximal subgroups

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

subgroup	index
#22575512	2
#22575441	2
#354	269