Subgroups — #31770880

Order 17216.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22575483	C₂	1	2	yes	maximal
2	C₂×Dic₁₀₇₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#22575389	C₂	1	1	yes	maximal, characteristic, complemented
4	#22575371	C₂	1	2	yes	maximal, complemented
5	#131×C₂₆₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₈	C₂²	1	2	yes	—
7	Dic₁₀₇₆	—	2	1	no	—
8	Dic₁₀₇₆	—	2	1	no	—
9	#16864392	—	2	2	no	—
10	#16864390	—	2	2	no	—
11	#16864386	C₂²	1	2	yes	—
12	#41×C₂₆₉	C₂²	1	1	yes	characteristic
13	M₁₆×C₂₆₉	—	2	1	no	—
14	M₁₆×C₂₆₉	C₂²	1	1	yes	characteristic
15	C₂×C₈×C₂₆₉	—	2	1	no	—
16	C₂×C₈×C₂₆₉	C₂²	1	1	yes	characteristic
17	Dic₅₃₈	—	2	2	no	—
18	Dic₅₃₈	—	2	2	no	—
19	C₂×Dic₂₆₉	—	2	2	no	—
20	#14524130	—	2	2	no	—
21	Q₈×C₂₆₉	D₄	1	1	yes	characteristic
22	D₄×C₂₆₉	—	2	1	no	—
23	D₄×C₂₆₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₂₆₉	—	2	1	no	—
25	C₂×C₄×C₂₆₉	C₂³	1	1	yes	characteristic
26	C₈×C₂₆₉	—	2	1	no	—
27	C₈×C₂₆₉	D₄	1	1	yes	characteristic
28	C₈×C₂₆₉	D₄	1	1	yes	characteristic
29	C₂²×C₂₆₉	—	2	1	no	—
30	C₂²×C₂₆₉	#41	1	1	yes	characteristic
31	C₄×C₂₆₉	—	2	1	no	—
32	C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₆₉	C₂×D₄	1	1	yes	characteristic
34	Dic₂₆₉	—	4	2	no	—
35	C₂×C₂₆₉	—	2	1	no	—
36	C₂×C₂₆₉	—	4	1	no	—
37	C₂×C₂₆₉	#121	1	1	yes	characteristic
38	C₂₆₉	#473	1	1	yes	characteristic, complemented, Sylow
39	#473	—	269	1	no	maximal, Sylow
40	#137	—	269	2	no	—
41	C₂×Q₁₆	—	269	1	no	—
42	#131	D₂₆₉	1	1	yes	characteristic
43	#108	—	269	1	no	—
44	#101	—	269	2	no	—
45	#41	C₂×D₂₆₉	1	1	yes	characteristic
46	C₂×Q₈	—	269	2	no	—
47	Q₁₆	—	538	2	no	—
48	Q₁₆	—	538	1	no	—
49	Q₁₆	—	538	1	no	—
50	SD₁₆	—	538	2	no	—
51	M₁₆	—	2	1	no	—
52	M₁₆	—	269	2	no	—
53	M₁₆	C₂×D₂₆₉	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	C₂×D₂₆₉	1	1	yes	characteristic
56	Q₈	—	538	2	no	—
57	Q₈	—	538	2	no	—
58	Q₈	D₁₀₇₆	1	1	yes	characteristic
59	D₄	—	2	1	no	—
60	D₄	D₁₀₇₆	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	538	2	no	—
63	C₂×C₄	C₂²×D₂₆₉	1	1	yes	characteristic
64	C₈	—	2	1	no	—
65	C₈	—	538	2	no	—
66	C₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
67	C₈	C₂²⋊D₂₆₉	1	1	yes	characteristic
68	C₂²	—	2	1	no	—
69	C₂²	#16864402	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	1076	2	no	—
72	C₄	C₂×C₂²⋊D₂₆₉	1	1	yes	characteristic
73	C₄	C₂×D₁₀₇₆	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	4	1	no	—
76	C₂	#22575427	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22575427
commutator, cosocle	C₂×C₄×C₂₆₉	C₂³
Frattini	C₂×C₄	C₂²×D₂₆₉
Fitting	#131×C₂₆₉	C₂
socle	C₂×C₂₆₉	#121
prime radical	#41×C₂₆₉	C₂²
hypercenter	#131	D₂₆₉
hypocenter	C₂₆₉	#473
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₀₇₆	C₂	semidirect
3. #22575389	C₂	semidirect
4. #22575371	C₂	semidirect
38. C₂₆₉	#473	semidirect

Maximal subgroups

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

subgroup	index
#22575483	2
C₂×Dic₁₀₇₆	2
#22575389	2
#22575371	2
#131×C₂₆₉	2
#473	269