Subgroups — #25011309

Order 11008.

398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19136845	C₂	1	4	yes	maximal, complemented
2	#19134210	C₂	1	2	yes	maximal, complemented
3	C₈×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₆₈₈	—	2	8	no	—
5	#14583276	—	2	4	no	—
6	#14583276	C₂²	1	4	yes	—
7	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
8	C₈²×C₄₃	C₂²	1	1	yes	characteristic
9	C₂×D₃₄₄	—	2	8	no	—
10	C₂×D₃₄₄	—	4	8	no	—
11	#13063322	—	2	4	no	—
12	D₆₈₈	—	4	16	no	—
13	C₂×C₁₆×C₄₃	D₄	1	4	yes	—
14	C₄×C₈×C₄₃	D₄	1	2	yes	—
15	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
16	C₂×D₁₇₂	—	4	4	no	—
17	C₂×D₁₇₂	—	4	8	no	—
18	C₂×C₈×C₄₃	D₈	1	4	yes	—
19	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
20	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
21	D₃₄₄	—	4	16	no	—
22	D₃₄₄	—	8	16	no	—
23	C₁₆×C₄₃	D₈	1	8	yes	complemented
24	C₂²×D₄₃	—	8	4	no	—
25	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
26	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
27	D₁₇₂	—	8	8	no	—
28	D₁₇₂	—	8	16	no	—
29	C₈×C₄₃	D₁₆	1	8	yes	complemented
30	C₈×C₄₃	C₂×D₈	1	4	yes	—
31	#12515	—	43	1	no	maximal, Sylow
32	C₂²×C₄₃	#493	1	1	yes	characteristic
33	C₂×D₄₃	—	16	4	no	—
34	C₂×D₄₃	—	16	8	no	—
35	C₄×C₄₃	#493	1	2	yes	—
36	C₄×C₄₃	C₂×D₁₆	1	4	yes	—
37	#2246	—	43	4	no	—
38	#1712	—	43	2	no	—
39	C₈×C₁₆	D₄₃	1	1	yes	characteristic, complemented
40	C₂×C₄₃	#1712	1	1	yes	characteristic
41	C₂×C₄₃	#2246	1	2	yes	—
42	D₄₃	—	32	4	no	—
43	C₂×D₁₆	—	86	8	no	—
44	#493	—	43	4	no	—
45	#493	—	86	4	no	—
46	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
47	C₈²	C₂×D₄₃	1	1	yes	characteristic
48	C₄₃	#12515	1	1	yes	characteristic, complemented, Sylow
49	C₂×D₈	—	86	8	no	—
50	C₂×D₈	—	172	8	no	—
51	#127	—	86	4	no	—
52	D₁₆	—	172	16	no	—
53	C₂×C₁₆	D₁₇₂	1	4	yes	—
54	C₄×C₈	D₁₇₂	1	2	yes	—
55	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
56	C₂×D₄	—	172	4	no	—
57	C₂×D₄	—	172	8	no	—
58	D₈	—	344	16	no	—
59	D₈	—	172	16	no	—
60	C₂×C₈	D₃₄₄	1	4	yes	—
61	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
62	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
63	C₁₆	D₃₄₄	1	8	yes	complemented
64	C₂³	—	344	4	no	—
65	D₄	—	344	16	no	—
66	D₄	—	344	8	no	—
67	C₂×C₄	#13063322	1	1	yes	characteristic
68	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
69	C₈	D₆₈₈	1	8	yes	complemented
70	C₈	C₂×D₃₄₄	1	4	yes	—
71	C₂²	—	688	4	no	—
72	C₂²	—	688	8	no	—
73	C₂²	#14583276	1	1	yes	characteristic
74	C₄	#14583276	1	2	yes	—
75	C₄	C₂×D₆₈₈	1	4	yes	—
76	C₂	—	1376	4	no	—
77	C₂	#19134210	1	1	yes	characteristic
78	C₂	#19136845	1	2	yes	—

	subgroup	G/H
center	C₂²	#14583276
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₈×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	#493
hypercenter	C₈×C₁₆	D₄₃
hypocenter	C₄₃	#12515
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #19136845	C₂	semidirect
2. #19134210	C₂	semidirect
3. C₈×C₁₆×C₄₃	C₂	semidirect
23. C₁₆×C₄₃	D₈	semidirect
29. C₈×C₄₃	D₁₆	semidirect
39. C₈×C₁₆	D₄₃	semidirect
48. C₄₃	#12515	semidirect
63. C₁₆	D₃₄₄	semidirect
69. C₈	D₆₈₈	semidirect

Maximal subgroups

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

subgroup	index
#19136845	2
#19134210	2
C₈×C₁₆×C₄₃	2
#12515	43