Subgroups — #36342384

Order 21568.

258 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	#24925046	C₂	1	6	yes	maximal, complemented
3	C₂×#19129195	C₂	1	3	yes	maximal, complemented
4	D₃₃₇⋊M₁₆	C₂	1	3	yes	maximal, complemented
5	#41×D₃₃₇	C₂	1	1	yes	maximal, characteristic
6	#19129195	C₂²	1	1	yes	characteristic
7	#19129195	C₂²	1	3	yes	—
8	#19129195	C₂²	1	6	yes	complemented
9	#19129194	C₂²	1	6	yes	complemented
10	#19129193	C₂²	1	6	yes	complemented
11	C₂×#14582113	C₂²	1	6	yes	complemented
12	#19129186	C₂²	1	1	yes	characteristic
13	#19129185	C₄	1	3	yes	—
14	#19129184	C₂²	1	3	yes	—
15	Q₈×D₃₃₇	C₄	1	1	yes	characteristic
16	D₄×D₃₃₇	C₄	1	3	yes	—
17	C₂×C₄×D₃₃₇	C₂²	1	3	yes	—
18	#41×C₃₃₇	C₄	1	1	yes	characteristic
19	#14582113	C₂³	1	6	yes	—
20	#14582113	C₂³	1	2	yes	—
21	C₂²×D₃₃₇	C₂×C₄	1	3	yes	—
22	Dic₆₇₄	C₂×C₄	1	3	yes	—
23	D₁₃₄₈	C₂×C₄	1	3	yes	—
24	C₂²⋊D₃₃₇	C₂×C₄	1	6	yes	—
25	C₂×Dic₃₃₇	C₂³	1	3	yes	—
26	C₄×D₃₃₇	C₂×C₄	1	3	yes	—
27	C₄×D₃₃₇	C₂×C₄	1	3	yes	—
28	C₄×D₃₃₇	C₂³	1	1	yes	characteristic
29	C₄×D₃₃₇	C₂³	1	3	yes	—
30	Q₈×C₃₃₇	C₂×C₄	1	1	yes	characteristic
31	D₄×C₃₃₇	C₂×C₄	1	3	yes	—
32	C₂×C₄×C₃₃₇	C₂×C₄	1	3	yes	—
33	C₂²×C₃₃₇	C₂²×C₄	1	3	yes	—
34	C₂×D₃₃₇	—	4	3	no	—
35	C₂×D₃₃₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×D₃₃₇	C₂²×C₄	1	3	yes	—
37	C₄×C₃₃₇	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₃₃₇	C₂²×C₄	1	3	yes	—
39	Dic₃₃₇	C₂²×C₄	1	3	yes	—
40	Dic₃₃₇	C₂⁴	1	1	yes	characteristic
41	C₂×C₃₃₇	—	2	3	no	—
42	C₂×C₃₃₇	C₂³×C₄	1	1	yes	characteristic
43	D₃₃₇	—	2	1	no	—
44	D₃₃₇	—	2	3	no	—
45	C₃₃₇	#568	1	1	yes	characteristic, complemented, Sylow
46	#568	—	337	1	no	maximal, Sylow
47	C₂×#41	—	337	1	no	—
48	#131	—	337	6	no	—
49	#131	—	337	2	no	—
50	C₂×M₁₆	—	337	3	no	—
51	C₂×M₁₆	—	337	3	no	—
52	#41	—	337	1	no	—
53	#41	—	337	3	no	—
54	#41	—	337	3	no	—
55	#41	#13062438	1	1	yes	characteristic
56	C₂×Q₈	—	337	1	no	—
57	C₂×D₄	—	337	3	no	—
58	C₂²×C₄	—	337	3	no	—
59	M₁₆	—	337	6	no	—
60	M₁₆	—	337	1	no	—
61	M₁₆	—	337	3	no	—
62	M₁₆	—	337	6	no	—
63	C₂×C₈	—	337	6	no	—
64	C₂×C₈	—	337	6	no	—
65	C₂³	—	337	3	no	—
66	Q₈	—	337	3	no	—
67	Q₈	C₂×#13062438	1	1	yes	characteristic
68	D₄	—	337	6	no	—
69	D₄	—	337	3	no	—
70	D₄	C₂×#13062438	1	3	yes	—
71	C₂×C₄	—	337	3	no	—
72	C₂×C₄	—	337	3	no	—
73	C₂×C₄	—	337	1	no	—
74	C₂×C₄	—	337	3	no	—
75	C₂×C₄	—	337	3	no	—
76	C₂×C₄	C₂×#13062438	1	3	yes	—
77	C₈	—	337	6	no	—
78	C₈	—	337	2	no	—
79	C₂²	—	1348	3	no	—
80	C₂²	—	337	1	no	—
81	C₂²	—	337	3	no	—
82	C₂²	C₂²×#13062438	1	3	yes	—
83	C₄	—	337	1	no	—
84	C₄	—	337	3	no	—
85	C₄	C₂²×#13062438	1	1	yes	characteristic
86	C₄	C₂²×#13062438	1	3	yes	—
87	C₂	—	2	3	no	—
88	C₂	—	674	1	no	—
89	C₂	—	674	3	no	—
90	C₂	C₂³×#13062438	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×#13062438
commutator, socle	C₂×C₃₃₇	C₂³×C₄
Fitting	#41×C₃₃₇	C₄
prime radical	#41×D₃₃₇	C₂
cosocle	Dic₃₃₇	C₂⁴
hypercenter	#41	#13062438
hypocenter	C₃₃₇	#568
LCS	G ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #24925047	C₂	semidirect
2. #24925046	C₂	semidirect
3. C₂×#19129195	C₂	semidirect
4. D₃₃₇⋊M₁₆	C₂	semidirect
8. #19129195	C₂²	semidirect
9. #19129194	C₂²	semidirect
10. #19129193	C₂²	semidirect
11. C₂×#14582113	C₂²	semidirect
45. C₃₃₇	#568	semidirect

Maximal subgroups

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

subgroup	index
#24925047	2
#24925046	2
C₂×#19129195	2
D₃₃₇⋊M₁₆	2
#41×D₃₃₇	2
#568	337