Subgroups — #24984479

Order 11008.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₆₈₈	C₂	1	1	yes	maximal, characteristic
2	#19132795	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₃×C₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃₄₄	—	2	1	no	—
5	Dic₆₈₈	—	2	1	no	—
6	Dic₆₈₈	C₄	1	2	yes	complemented
7	#14582725	—	2	1	no	—
8	C₂×C₃₂×C₄₃	C₂²	1	1	yes	characteristic
9	C₄₃×C₆₄	—	2	1	no	—
10	C₂×Dic₁₇₂	—	4	1	no	—
11	Dic₃₄₄	—	2	2	no	—
12	Dic₃₄₄	—	4	1	no	—
13	#13063259	—	4	1	no	—
14	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
15	C₃₂×C₄₃	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₄₃	D₄	1	1	yes	characteristic
17	C₂×Dic₈₆	—	8	1	no	—
18	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
19	C₄⋊Dic₄₃	—	8	1	no	—
20	Dic₁₇₂	—	4	2	no	—
21	Dic₁₇₂	—	8	1	no	—
22	C₁₆×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₄₃	SD₁₆	1	1	yes	characteristic
24	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
25	C₂×Dic₄₃	—	16	1	no	—
26	C₂×Dic₄₃	—	16	1	no	—
27	Dic₈₆	—	8	2	no	—
28	Dic₈₆	—	16	1	no	—
29	C₈×C₄₃	#102	1	1	yes	characteristic
30	C₈×C₄₃	SD₃₂	1	1	yes	characteristic
31	#7538	—	43	1	no	maximal, Sylow
32	C₂²×C₄₃	D₃₂	1	1	yes	characteristic
33	C₄×C₄₃	#357	1	1	yes	characteristic
34	C₄×C₄₃	SD₆₄	1	1	yes	characteristic
35	Dic₄₃	—	16	2	no	—
36	Dic₄₃	—	32	1	no	—
37	C₂×Q₆₄	—	43	1	no	—
38	C₂×C₆₄	D₄₃	1	1	yes	characteristic
39	#1423	—	43	1	no	—
40	C₂×C₄₃	#1415	1	1	yes	characteristic
41	C₂×C₄₃	SD₁₂₈	1	1	yes	characteristic
42	C₂×C₄₃	Q₁₂₈	1	1	yes	characteristic
43	C₂×Q₃₂	—	86	1	no	—
44	Q₆₄	—	43	2	no	—
45	Q₆₄	—	86	1	no	—
46	C₂×C₃₂	C₂×D₄₃	1	1	yes	characteristic
47	#366	—	86	1	no	—
48	C₆₄	—	2	1	no	—
49	C₄₃	#7538	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₁₆	—	172	1	no	—
51	Q₃₂	—	86	2	no	—
52	Q₃₂	—	172	1	no	—
53	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
54	#107	—	172	1	no	—
55	C₃₂	C₄×D₄₃	1	1	yes	characteristic
56	C₃₂	C₂²⋊D₄₃	1	1	yes	characteristic
57	C₂×Q₈	—	344	1	no	—
58	Q₁₆	—	344	1	no	—
59	Q₁₆	—	172	2	no	—
60	C₂×C₈	D₃₄₄	1	1	yes	characteristic
61	C₄⋊C₄	—	344	1	no	—
62	C₁₆	#10621873	1	1	yes	characteristic
63	C₁₆	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
64	Q₈	—	344	2	no	—
65	Q₈	—	688	1	no	—
66	C₂×C₄	—	688	1	no	—
67	C₂×C₄	—	688	1	no	—
68	C₂×C₄	D₆₈₈	1	1	yes	characteristic
69	C₈	#13063247	1	1	yes	characteristic
70	C₈	#13063270	1	1	yes	characteristic
71	C₂²	D₁₃₇₆	1	1	yes	characteristic
72	C₄	—	1376	1	no	—
73	C₄	—	688	2	no	—
74	C₄	#14582701	1	1	yes	characteristic
75	C₄	#14582738	1	1	yes	characteristic
76	C₂	#19132774	1	1	yes	characteristic
77	C₂	#19132811	1	1	yes	characteristic
78	C₂	Dic₁₃₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₃₇₆
commutator	C₃₂×C₄₃	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₄₃
Fitting	C₂×C₄₃×C₆₄	C₂
socle, prime radical	C₂²×C₄₃	D₃₂
cosocle	C₂×C₃₂×C₄₃	C₂²
hypercenter	C₂×C₆₄	D₄₃
hypocenter	C₄₃	#7538
LCS	G ▷ C₃₂×C₄₃ ▷ C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
6. Dic₆₈₈	C₄	semidirect
49. C₄₃	#7538	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₆₈₈	2
#19132795	2
C₂×C₄₃×C₆₄	2
#7538	43