Subgroups — #24984474

Order 11008.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₃₇₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₃⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#1423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₆₈₈	—	2	1	no	—
5	D₁₃₇₆	—	2	1	no	—
6	D₁₃₇₆	C₄	1	2	yes	complemented
7	C₄₃⋊C₆₄	—	2	1	no	—
8	C₂×C₃₂×C₄₃	C₂²	1	1	yes	characteristic
9	C₄₃×#366	—	2	1	no	—
10	C₂×D₃₄₄	—	4	1	no	—
11	D₆₈₈	—	2	2	no	—
12	D₆₈₈	—	4	1	no	—
13	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
14	#107×C₄₃	—	4	1	no	—
15	C₃₂×C₄₃	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₄₃	D₄	1	1	yes	characteristic
17	C₂×D₁₇₂	—	8	1	no	—
18	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
19	C₄⋊C₄×C₄₃	—	8	1	no	—
20	D₃₄₄	—	4	2	no	—
21	D₃₄₄	—	8	1	no	—
22	C₁₆×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₄₃	SD₁₆	1	1	yes	characteristic
24	C₂²×D₄₃	—	16	1	no	—
25	C₂×C₄×C₄₃	—	16	1	no	—
26	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
27	D₁₇₂	—	8	2	no	—
28	D₁₇₂	—	16	1	no	—
29	C₈×C₄₃	#102	1	1	yes	characteristic
30	C₈×C₄₃	SD₃₂	1	1	yes	characteristic
31	#7537	—	43	1	no	maximal, Sylow
32	C₂²×C₄₃	D₃₂	1	1	yes	characteristic
33	C₂×D₄₃	—	16	2	no	—
34	C₂×D₄₃	—	32	1	no	—
35	C₂×D₄₃	—	32	1	no	—
36	C₄×C₄₃	—	32	1	no	—
37	C₄×C₄₃	#357	1	1	yes	characteristic
38	C₄×C₄₃	SD₆₄	1	1	yes	characteristic
39	C₂×D₃₂	—	43	1	no	—
40	C₂×C₆₄	—	43	1	no	—
41	#1423	D₄₃	1	1	yes	characteristic, complemented
42	C₂×C₄₃	#1415	1	1	yes	characteristic
43	C₂×C₄₃	D₆₄	1	1	yes	characteristic
44	C₂×C₄₃	SD₁₂₈	1	1	yes	characteristic
45	D₄₃	—	32	2	no	—
46	C₂×D₁₆	—	86	1	no	—
47	D₃₂	—	43	2	no	—
48	D₃₂	—	86	1	no	—
49	C₂×C₃₂	C₂×D₄₃	1	1	yes	characteristic
50	#366	—	2	1	no	—
51	C₆₄	—	86	1	no	—
52	C₄₃	#7537	1	1	yes	characteristic, complemented, Sylow
53	C₂×D₈	—	172	1	no	—
54	D₁₆	—	86	2	no	—
55	D₁₆	—	172	1	no	—
56	C₂×C₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
57	#107	—	4	1	no	—
58	C₃₂	C₄×D₄₃	1	1	yes	characteristic
59	C₃₂	D₁₇₂	1	1	yes	characteristic
60	C₂×D₄	—	344	1	no	—
61	D₈	—	344	1	no	—
62	D₈	—	172	2	no	—
63	C₂×C₈	#10621882	1	1	yes	characteristic
64	C₄⋊C₄	—	8	1	no	—
65	C₁₆	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
66	C₁₆	#10621884	1	1	yes	characteristic
67	C₂³	—	688	1	no	—
68	D₄	—	344	2	no	—
69	D₄	—	688	1	no	—
70	C₂×C₄	—	16	1	no	—
71	C₂×C₄	#13063266	1	1	yes	characteristic
72	C₈	#13063245	1	1	yes	characteristic
73	C₈	#13063268	1	1	yes	characteristic
74	C₂²	—	1376	1	no	—
75	C₂²	—	1376	1	no	—
76	C₂²	—	688	2	no	—
77	C₂²	#14582734	1	1	yes	characteristic
78	C₄	—	32	1	no	—
79	C₄	#14582699	1	1	yes	characteristic
80	C₄	#14582736	1	1	yes	characteristic
81	C₂	—	1376	2	no	—
82	C₂	#19132772	1	1	yes	characteristic
83	C₂	#19132807	1	1	yes	characteristic
84	C₂	#19132809	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14582734
commutator	C₃₂×C₄₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₄₃
Fitting	C₄₃×#1423	C₂
socle	C₂²×C₄₃	D₃₂
prime radical	C₂×D₁₃₇₆	C₂
cosocle	C₂×C₃₂×C₄₃	C₂²
hypercenter	#1423	D₄₃
hypocenter	C₄₃	#7537
LCS	G ▷ C₃₂×C₄₃ ▷ C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ #1423

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₃⋊C₆₄	C₂	semidirect
3. C₄₃×#1423	C₂	semidirect
6. D₁₃₇₆	C₄	semidirect
41. #1423	D₄₃	semidirect
52. C₄₃	#7537	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₃₇₆	2
C₂×C₄₃⋊C₆₄	2
C₄₃×#1423	2
#7537	43