Subgroups — #14583110

Order 2752.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₇₂	C₂	1	2	yes	maximal
2	#13063325	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic
4	#13063250	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₆	C₂²	1	2	yes	—
7	C₂×Dic₈₆	C₂²	1	1	yes	characteristic
8	C₂×C₈×C₄₃	C₂²	1	2	yes	—
9	C₄²×C₄₃	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₄₃	—	2	1	no	—
11	C₄⋊Dic₄₃	—	2	1	no	—
12	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
13	Dic₄₃⋊C₄	—	2	1	no	—
14	C₄×Dic₄₃	—	2	1	no	—
15	Dic₁₇₂	—	2	4	no	—
16	C₂×C₄×C₄₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₄₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
19	C₂×Dic₄₃	—	2	1	no	—
20	C₂×Dic₄₃	—	2	1	no	—
21	C₂×Dic₄₃	—	2	2	no	—
22	Dic₈₆	—	2	1	no	—
23	Dic₈₆	—	2	2	no	—
24	Dic₈₆	—	2	2	no	—
25	Dic₈₆	D₄	1	2	yes	—
26	C₈×C₄₃	—	2	2	no	—
27	C₂²×C₄₃	C₂×D₄	1	1	yes	characteristic
28	C₄×C₄₃	—	2	1	no	—
29	C₄×C₄₃	Q₁₆	1	2	yes	complemented
30	C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
31	C₄×C₄₃	#41	1	1	yes	characteristic
32	Dic₄₃	—	2	2	no	—
33	Dic₄₃	—	4	2	no	—
34	Dic₄₃	—	4	1	no	—
35	C₂×C₄₃	#121	1	1	yes	characteristic
36	C₂×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₄₃	#137	1	1	yes	characteristic
38	#462	—	43	1	no	maximal, Sylow
39	C₄₃	#462	1	1	yes	characteristic, complemented, Sylow
40	C₂×Q₁₆	—	43	2	no	—
41	C₄⋊Q₈	—	43	1	no	—
42	C₄×Q₈	—	43	1	no	—
43	C₄⋊C₈	D₄₃	1	1	yes	characteristic
44	#103	—	43	2	no	—
45	C₂×Q₈	—	43	1	no	—
46	C₂×Q₈	—	43	2	no	—
47	Q₁₆	—	86	4	no	—
48	C₂×C₈	C₂×D₄₃	1	2	yes	—
49	C₄⋊C₄	—	43	1	no	—
50	C₄⋊C₄	—	86	1	no	—
51	C₄⋊C₄	—	86	1	no	—
52	C₄⋊C₄	—	86	1	no	—
53	C₄²	—	86	1	no	—
54	C₄²	C₂×D₄₃	1	1	yes	characteristic
55	Q₈	—	43	2	no	—
56	Q₈	—	86	1	no	—
57	Q₈	—	86	2	no	—
58	Q₈	—	86	2	no	—
59	C₂×C₄	—	86	1	no	—
60	C₂×C₄	—	86	1	no	—
61	C₂×C₄	—	86	2	no	—
62	C₂×C₄	D₁₇₂	1	1	yes	characteristic
63	C₂×C₄	D₁₇₂	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₂²	C₂×D₁₇₂	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	86	2	no	—
69	C₄	—	172	2	no	—
70	C₄	—	172	1	no	—
71	C₄	Dic₁₇₂	1	2	yes	complemented
72	C₄	D₄×D₄₃	1	1	yes	characteristic
73	C₄	#10621902	1	1	yes	characteristic
74	C₂	#13063300	1	1	yes	characteristic
75	C₂	C₂×Dic₁₇₂	1	1	yes	characteristic
76	C₂	#13063358	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂³
Frattini	C₂×C₄	C₂²×D₄₃
Fitting	C₄⋊C₈×C₄₃	C₂
socle, prime radical	C₂²×C₄₃	C₂×D₄
hypercenter	C₄⋊C₈	D₄₃
hypocenter	C₄₃	#462
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
29. C₄×C₄₃	Q₁₆	semidirect
39. C₄₃	#462	semidirect
71. C₄	Dic₁₇₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₇₂	2
#13063325	2
C₄×Dic₈₆	2
#13063250	2
C₄⋊C₈×C₄₃	2
#462	43