Subgroups — #36348828

Order 22112.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26010865	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₆₉₁	C₂	1	1	yes	maximal, characteristic, complemented
3	Dic₂₇₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	#26010856	C₂	1	1	yes	maximal, characteristic, complemented
5	#26010854	C₂	1	1	yes	maximal, characteristic, complemented
6	#26010849	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₁₆×C₆₉₁	C₂	1	1	yes	maximal, characteristic, complemented
8	Dic₁₃₈₂	—	2	1	no	—
9	Dic₁₃₈₂	C₂²	1	1	yes	characteristic
10	Dic₁₃₈₂	C₂²	1	1	yes	characteristic
11	C₂²⋊D₆₉₁	—	2	1	no	—
12	C₂×Dic₆₉₁	—	2	1	no	—
13	C₄×D₆₉₁	—	2	1	no	—
14	C₄×D₆₉₁	C₂²	1	1	yes	characteristic
15	C₆₉₁⋊C₈	C₂²	1	1	yes	characteristic
16	Q₈×C₆₉₁	C₂²	1	1	yes	characteristic
17	D₄×C₆₉₁	C₂²	1	1	yes	characteristic
18	C₈×C₆₉₁	C₂²	1	1	yes	characteristic
19	C₂²×C₆₉₁	—	2	1	no	—
20	C₂×D₆₉₁	D₄	1	1	yes	characteristic
21	C₄×C₆₉₁	—	2	1	no	—
22	C₄×C₆₉₁	C₂³	1	1	yes	characteristic
23	Dic₆₉₁	—	2	1	no	—
24	Dic₆₉₁	—	2	1	no	—
25	Dic₆₉₁	D₄	1	1	yes	characteristic
26	C₂×C₆₉₁	—	4	1	no	—
27	C₂×C₆₉₁	C₂×D₄	1	1	yes	characteristic
28	D₆₉₁	—	2	1	no	—
29	C₆₉₁	#137	1	1	yes	characteristic, complemented, Sylow
30	#137	—	691	1	no	maximal, Sylow
31	#41	—	691	1	no	—
32	C₂×Q₈	—	691	1	no	—
33	Q₁₆	—	691	1	no	—
34	Q₁₆	—	691	1	no	—
35	SD₁₆	—	691	1	no	—
36	SD₁₆	D₆₉₁	1	1	yes	characteristic, complemented
37	M₁₆	—	691	1	no	—
38	Q₈	—	1382	1	no	—
39	Q₈	—	691	1	no	—
40	Q₈	—	691	1	no	—
41	Q₈	C₂×D₆₉₁	1	1	yes	characteristic
42	D₄	—	1382	1	no	—
43	D₄	C₂×D₆₉₁	1	1	yes	characteristic
44	C₂×C₄	—	1382	1	no	—
45	C₂×C₄	—	1382	1	no	—
46	C₂×C₄	—	691	1	no	—
47	C₈	—	691	1	no	—
48	C₈	C₂×D₆₉₁	1	1	yes	characteristic
49	C₂²	—	2	1	no	—
50	C₂²	—	691	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	1382	1	no	—
53	C₄	—	1382	1	no	—
54	C₄	—	691	1	no	—
55	C₄	C₂²×D₆₉₁	1	1	yes	characteristic
56	C₂	—	4	1	no	—
57	C₂	—	1382	1	no	—
58	C₂	D₄×D₆₉₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₆₉₁
commutator, cosocle	C₄×C₆₉₁	C₂³
Frattini	C₄	C₂²×D₆₉₁
Fitting	SD₁₆×C₆₉₁	C₂
socle	C₂×C₆₉₁	C₂×D₄
prime radical	#26010865	C₂
hypercenter	SD₁₆	D₆₉₁
hypocenter	C₆₉₁	#137
LCS	G ▷ C₄×C₆₉₁ ▷ C₂×C₆₉₁ ▷ C₆₉₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₆₉₁	C₂	semidirect
3. Dic₂₇₆₄	C₂	semidirect
4. #26010856	C₂	semidirect
5. #26010854	C₂	semidirect
6. #26010849	C₂	semidirect
7. SD₁₆×C₆₉₁	C₂	semidirect
29. C₆₉₁	#137	semidirect
36. SD₁₆	D₆₉₁	semidirect

Maximal subgroups

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

subgroup	index
#26010865	2
Q₈×D₆₉₁	2
Dic₂₇₆₄	2
#26010856	2
#26010854	2
#26010849	2
SD₁₆×C₆₉₁	2
#137	691