Subgroups — #19136472

Order 5504.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₆₈₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₃₄₄	C₂	1	2	yes	maximal, complemented
3	#14582727	C₂	1	1	yes	maximal, characteristic, complemented
4	#14582701	C₂	1	2	yes	maximal, complemented
5	C₄₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₃₄₄	C₂²	1	2	yes	—
7	C₄×D₁₇₂	—	2	2	no	—
8	D₆₈₈	C₄	1	4	yes	complemented
9	#13063259	C₂²	1	2	yes	—
10	#13063247	—	2	2	no	—
11	C₂×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×D₁₇₂	—	2	2	no	—
15	C₂×C₄×D₄₃	—	4	2	no	—
16	C₂×C₈×C₄₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
18	C₄²×C₄₃	D₄	1	1	yes	characteristic
19	(C₂×D₄₃)⋊C₄	—	4	2	no	—
20	C₄⋊Dic₄₃	—	2	2	no	—
21	D₃₄₄	—	2	2	no	—
22	D₃₄₄	C₂×C₄	1	4	yes	complemented
23	C₁₆×C₄₃	—	2	1	no	—
24	C₁₆×C₄₃	C₂×C₄	1	2	yes	complemented
25	C₂²×D₄₃	—	4	2	no	—
26	C₂×C₄×C₄₃	D₈	1	2	yes	—
27	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₄₃	—	4	2	no	—
29	D₁₇₂	—	2	4	no	—
30	D₁₇₂	—	4	2	no	—
31	C₄×D₄₃	—	8	4	no	—
32	C₈×C₄₃	—	2	1	no	—
33	C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₄₃	#41	1	1	yes	characteristic
35	C₂²×C₄₃	C₂×D₈	1	1	yes	characteristic
36	C₂×D₄₃	—	4	4	no	—
37	C₂×D₄₃	—	8	2	no	—
38	C₂×D₄₃	—	8	2	no	—
39	C₄×C₄₃	—	2	2	no	—
40	C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
41	C₄×C₄₃	#135	1	1	yes	characteristic
42	Dic₄₃	—	8	2	no	—
43	#2177	—	43	1	no	maximal, Sylow
44	C₂×C₄₃	C₄×D₈	1	1	yes	characteristic
45	C₂×C₄₃	#509	1	2	yes	—
46	D₄₃	—	8	4	no	—
47	C₂×D₁₆	—	43	1	no	—
48	C₄×D₈	—	43	2	no	—
49	#367	—	43	1	no	—
50	#357	—	43	2	no	—
51	#346	D₄₃	1	1	yes	characteristic, complemented
52	C₄₃	#2177	1	1	yes	characteristic, complemented, Sylow
53	C₂×D₈	—	43	2	no	—
54	C₄×D₄	—	86	2	no	—
55	D₁₆	—	43	4	no	—
56	C₂×C₁₆	C₂×D₄₃	1	1	yes	characteristic
57	C₂×C₁₆	C₂×D₄₃	1	1	yes	characteristic
58	#107	—	43	2	no	—
59	#102	—	86	2	no	—
60	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
61	C₂×D₄	—	86	2	no	—
62	C₂²×C₄	—	172	2	no	—
63	D₈	—	43	4	no	—
64	D₈	—	86	2	no	—
65	C₂×C₈	D₁₇₂	1	1	yes	characteristic
66	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
67	C₄⋊C₄	—	86	2	no	—
68	C₂²⋊C₄	—	172	2	no	—
69	C₄²	D₁₇₂	1	1	yes	characteristic
70	C₁₆	—	2	1	no	—
71	C₁₆	C₄×D₄₃	1	2	yes	complemented
72	C₂³	—	172	2	no	—
73	D₄	—	86	4	no	—
74	D₄	—	172	2	no	—
75	C₂×C₄	—	344	4	no	—
76	C₂×C₄	—	172	2	no	—
77	C₂×C₄	D₃₄₄	1	2	yes	—
78	C₂×C₄	C₂×D₁₇₂	1	1	yes	characteristic
79	C₈	—	2	1	no	—
80	C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
81	C₈	#10621898	1	1	yes	characteristic
82	C₂²	—	344	2	no	—
83	C₂²	—	344	2	no	—
84	C₂²	—	172	4	no	—
85	C₂²	C₂×D₃₄₄	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	344	2	no	—
88	C₄	C₄×D₁₇₂	1	1	yes	characteristic
89	C₄	#13063347	1	1	yes	characteristic
90	C₂	—	344	4	no	—
91	C₂	C₄×D₃₄₄	1	1	yes	characteristic
92	C₂	#14583337	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₃₄₄
commutator	C₈×C₄₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₄₃
Fitting	C₄₃×#346	C₂
socle	C₂²×C₄₃	C₂×D₈
hypercenter	#346	D₄₃
hypocenter	C₄₃	#2177
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
2. C₄×D₃₄₄	C₂	semidirect
3. #14582727	C₂	semidirect
4. #14582701	C₂	semidirect
5. C₄₃×#346	C₂	semidirect
8. D₆₈₈	C₄	semidirect
22. D₃₄₄	C₂×C₄	semidirect
24. C₁₆×C₄₃	C₂×C₄	semidirect
51. #346	D₄₃	semidirect
52. C₄₃	#2177	semidirect
71. C₁₆	C₄×D₄₃	semidirect