Subgroups — #25017651

Order 11008.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136904	C₂	1	2	yes	maximal, complemented
2	Dic₁₃₇₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#19132811	C₂	1	2	yes	maximal, complemented
4	D₂₇₅₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583331	—	2	2	no	—
7	Dic₆₈₈	C₂²	1	2	yes	—
8	#14582738	—	2	2	no	—
9	D₁₃₇₆	C₂²	1	2	yes	—
10	C₂×C₃₂×C₄₃	C₂²	1	1	yes	characteristic
11	C₄₃×C₆₄	C₂²	1	1	yes	characteristic
12	C₄₃×C₆₄	C₂²	1	1	yes	characteristic
13	#13063347	—	4	2	no	—
14	Dic₃₄₄	—	2	2	no	—
15	#13063270	—	4	2	no	—
16	D₆₈₈	—	2	2	no	—
17	C₂×C₁₆×C₄₃	D₄	1	1	yes	characteristic
18	C₃₂×C₄₃	D₄	1	1	yes	characteristic
19	C₃₂×C₄₃	C₂³	1	1	yes	characteristic
20	#10621898	—	8	2	no	—
21	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
22	Dic₁₇₂	—	4	2	no	—
23	D₃₄₄	—	4	2	no	—
24	#10621873	—	8	2	no	—
25	C₁₆×C₄₃	D₈	1	1	yes	characteristic
26	C₁₆×C₄₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₄₃	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₄₃	—	16	2	no	—
29	D₁₇₂	—	8	2	no	—
30	C₄×D₄₃	—	16	2	no	—
31	Dic₈₆	—	8	2	no	—
32	C₈×C₄₃	D₁₆	1	1	yes	characteristic
33	C₈×C₄₃	C₂×D₈	1	1	yes	characteristic
34	#13741	—	43	1	no	maximal, Sylow
35	C₂²×C₄₃	D₃₂	1	1	yes	characteristic
36	C₂×D₄₃	—	16	2	no	—
37	C₄×C₄₃	D₃₂	1	1	yes	characteristic
38	C₄×C₄₃	C₂×D₁₆	1	1	yes	characteristic
39	Dic₄₃	—	16	2	no	—
40	#2262	—	43	2	no	—
41	Q₁₂₈	—	43	1	no	—
42	SD₁₂₈	—	43	2	no	—
43	D₆₄	—	43	1	no	—
44	C₂×C₆₄	D₄₃	1	1	yes	characteristic, complemented
45	C₂×C₄₃	—	2	1	no	—
46	C₂×C₄₃	C₂×D₃₂	1	1	yes	characteristic
47	D₄₃	—	32	2	no	—
48	#508	—	86	2	no	—
49	Q₆₄	—	43	2	no	—
50	SD₆₄	—	86	2	no	—
51	D₃₂	—	43	2	no	—
52	C₂×C₃₂	C₂×D₄₃	1	1	yes	characteristic
53	C₆₄	C₂×D₄₃	1	1	yes	characteristic
54	C₆₄	C₂×D₄₃	1	1	yes	characteristic
55	C₄₃	#13741	1	1	yes	characteristic, complemented, Sylow
56	#135	—	172	2	no	—
57	Q₃₂	—	86	2	no	—
58	SD₃₂	—	172	2	no	—
59	D₁₆	—	86	2	no	—
60	C₂×C₁₆	D₁₇₂	1	1	yes	characteristic
61	C₃₂	D₁₇₂	1	1	yes	characteristic
62	C₃₂	C₂²×D₄₃	1	1	yes	characteristic
63	#41	—	344	2	no	—
64	Q₁₆	—	172	2	no	—
65	SD₁₆	—	344	2	no	—
66	D₈	—	172	2	no	—
67	C₂×C₈	D₃₄₄	1	1	yes	characteristic
68	C₁₆	D₃₄₄	1	1	yes	characteristic
69	C₁₆	C₂×D₁₇₂	1	1	yes	characteristic
70	Q₈	—	344	2	no	—
71	D₄	—	344	2	no	—
72	D₄	—	688	2	no	—
73	C₂×C₄	—	688	2	no	—
74	C₂×C₄	D₆₈₈	1	1	yes	characteristic
75	C₈	D₆₈₈	1	1	yes	characteristic
76	C₈	C₂×D₃₄₄	1	1	yes	characteristic
77	C₂²	—	688	2	no	—
78	C₂²	D₁₃₇₆	1	1	yes	characteristic
79	C₄	—	688	2	no	—
80	C₄	D₁₃₇₆	1	1	yes	characteristic
81	C₄	C₂×D₆₈₈	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	1376	2	no	—
84	C₂	C₂×D₁₃₇₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₃₇₆
commutator, cosocle	C₃₂×C₄₃	C₂³
Frattini	C₃₂	C₂²×D₄₃
Fitting	C₂×C₄₃×C₆₄	C₂
socle	C₂×C₄₃	C₂×D₃₂
hypercenter	C₂×C₆₄	D₄₃
hypocenter	C₄₃	#13741
LCS	G ▷ C₃₂×C₄₃ ▷ C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ 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
1. #19136904	C₂	semidirect
2. Dic₁₃₇₆	C₂	semidirect
3. #19132811	C₂	semidirect
4. D₂₇₅₂	C₂	semidirect
5. C₂×C₄₃×C₆₄	C₂	semidirect
44. C₂×C₆₄	D₄₃	semidirect
55. C₄₃	#13741	semidirect

Maximal subgroups

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

subgroup	index
#19136904	2
Dic₁₃₇₆	2
#19132811	2
D₂₇₅₂	2
C₂×C₄₃×C₆₄	2
#13741	43