Subgroups — #36342351

Order 21568.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24925047	C₂	1	2	yes	maximal, complemented
2	#24925029	C₂	1	1	yes	maximal, characteristic, complemented
3	#24925024	C₂	1	2	yes	maximal, complemented
4	#24925013	C₂	1	1	yes	maximal, characteristic, complemented
5	#24924982	C₂	1	1	yes	maximal, characteristic
6	#19129195	—	2	2	no	—
7	#19129195	C₂²	1	2	yes	—
8	#19129193	—	2	2	no	—
9	#19129193	C₂²	1	1	yes	characteristic
10	C₄×#13062438	C₂²	1	1	yes	characteristic
11	#19129186	C₂²	1	2	yes	—
12	#19129172	C₄	1	2	yes	complemented
13	D₂₆₉₆	C₄	1	1	yes	characteristic, complemented
14	C₈×D₃₃₇	C₂²	1	1	yes	characteristic
15	Q₁₆×C₃₃₇	C₄	1	1	yes	characteristic, complemented
16	C₂×#13062438	—	2	1	no	—
17	#14582113	—	2	2	no	—
18	#14582113	D₄	1	2	yes	—
19	D₁₃₄₈	—	2	2	no	—
20	D₁₃₄₈	C₂×C₄	1	2	yes	—
21	C₄×D₃₃₇	—	2	2	no	—
22	C₄×D₃₃₇	C₂³	1	1	yes	characteristic
23	#14582106	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₃₃₇	C₂×C₄	1	2	yes	—
25	C₈×C₃₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂×D₃₃₇	—	2	2	no	—
27	C₂×D₃₃₇	#41	1	1	yes	characteristic
28	#13062438	—	4	1	no	—
29	C₄×C₃₃₇	—	2	2	no	—
30	C₄×C₃₃₇	C₂²×C₄	1	1	yes	characteristic
31	Dic₃₃₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃₃₇	C₄×D₄	1	1	yes	characteristic
33	D₃₃₇	—	2	1	no	—
34	D₃₃₇	—	4	2	no	—
35	C₃₃₇	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	337	1	no	maximal, Sylow
37	#135	—	337	1	no	—
38	#131	—	337	2	no	—
39	#108	—	337	1	no	—
40	#104	—	337	2	no	—
41	#97	—	337	1	no	—
42	#41	—	337	2	no	—
43	Q₁₆	#13062438	1	1	yes	characteristic, complemented
44	SD₁₆	—	337	2	no	—
45	D₈	—	337	1	no	—
46	M₁₆	—	337	2	no	—
47	M₁₆	—	674	2	no	—
48	C₂×C₈	—	337	1	no	—
49	C₂×C₈	—	337	1	no	—
50	C₂×C₈	—	674	2	no	—
51	C₄²	—	337	1	no	—
52	Q₈	C₂×#13062438	1	2	yes	—
53	D₄	—	337	2	no	—
54	D₄	—	674	2	no	—
55	C₂×C₄	—	337	1	no	—
56	C₂×C₄	—	674	2	no	—
57	C₂×C₄	—	674	1	no	—
58	C₈	—	337	2	no	—
59	C₈	—	337	1	no	—
60	C₈	—	674	2	no	—
61	C₈	C₂×#13062438	1	1	yes	characteristic
62	C₂²	—	337	1	no	—
63	C₂²	—	674	2	no	—
64	C₄	—	2	2	no	—
65	C₄	—	1348	1	no	—
66	C₄	—	337	1	no	—
67	C₄	C₂²×#13062438	1	1	yes	characteristic
68	C₂	—	1348	2	no	—
69	C₂	—	674	1	no	—
70	C₂	D₄×#13062438	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#13062438
commutator	C₄×C₃₃₇	C₂²×C₄
Frattini	C₄	C₂²×#13062438
Fitting	Q₁₆×C₃₃₇	C₄
socle	C₂×C₃₃₇	C₄×D₄
prime radical	#24924982	C₂
cosocle	C₄×D₃₃₇	C₂³
hypercenter	Q₁₆	#13062438
hypocenter	C₃₃₇	#444
LCS	G ▷ C₄×C₃₃₇ ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #24925047	C₂	semidirect
2. #24925029	C₂	semidirect
3. #24925024	C₂	semidirect
4. #24925013	C₂	semidirect
12. #19129172	C₄	semidirect
13. D₂₆₉₆	C₄	semidirect
15. Q₁₆×C₃₃₇	C₄	semidirect
35. C₃₃₇	#444	semidirect
43. Q₁₆	#13062438	semidirect

Maximal subgroups

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

subgroup	index
#24925047	2
#24925029	2
#24925024	2
#24925013	2
#24924982	2
#444	337