Subgroups — #14583101

Order 2752.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10621884	C₂	1	2	yes	maximal
2	C₄×D₁₇₂	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#13063249	C₂	1	2	yes	maximal, complemented
5	C₄⋊Q₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₄₃	C₂²	1	2	yes	—
7	C₂×D₁₇₂	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄₃	—	2	1	no	—
9	C₄⋊C₄×C₄₃	—	2	1	no	—
10	C₄⋊C₄×C₄₃	—	2	1	no	—
11	C₄²×C₄₃	C₂²	1	1	yes	characteristic
12	#10621884	—	2	4	no	—
13	(C₂×D₄₃)⋊C₄	—	2	1	no	—
14	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
15	C₂×C₄₃⋊C₈	C₂²	1	2	yes	—
16	C₂²×D₄₃	—	2	1	no	—
17	Q₈×C₄₃	—	2	2	no	—
18	Q₈×C₄₃	—	2	2	no	—
19	C₂×C₄×C₄₃	—	2	2	no	—
20	C₂×C₄×C₄₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₄₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
23	C₂×Dic₄₃	—	2	1	no	—
24	D₁₇₂	—	2	1	no	—
25	D₁₇₂	D₄	1	2	yes	—
26	C₄×D₄₃	—	4	2	no	—
27	C₄₃⋊C₈	—	2	2	no	—
28	C₂²×C₄₃	C₂×D₄	1	1	yes	characteristic
29	C₂×D₄₃	—	2	2	no	—
30	C₂×D₄₃	—	4	1	no	—
31	C₂×D₄₃	—	4	1	no	—
32	C₄×C₄₃	—	2	1	no	—
33	C₄×C₄₃	—	4	2	no	—
34	C₄×C₄₃	SD₁₆	1	2	yes	complemented
35	C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
36	C₄×C₄₃	#41	1	1	yes	characteristic
37	Dic₄₃	—	4	1	no	—
38	C₂×C₄₃	#121	1	1	yes	characteristic
39	C₂×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₄₃	#137	1	1	yes	characteristic
41	D₄₃	—	4	2	no	—
42	#461	—	43	1	no	maximal, Sylow
43	C₄₃	#461	1	1	yes	characteristic, complemented, Sylow
44	C₂×SD₁₆	—	43	2	no	—
45	C₄⋊Q₈	D₄₃	1	1	yes	characteristic, complemented
46	C₄×D₄	—	43	1	no	—
47	C₄⋊C₈	—	43	1	no	—
48	#103	—	43	2	no	—
49	C₂×Q₈	C₂×D₄₃	1	2	yes	—
50	C₂×D₄	—	43	1	no	—
51	C₂²×C₄	—	86	1	no	—
52	SD₁₆	—	86	4	no	—
53	C₂×C₈	—	43	2	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	—	43	1	no	—
57	C₂²⋊C₄	—	86	1	no	—
58	C₄²	C₂×D₄₃	1	1	yes	characteristic
59	C₂³	—	86	1	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	2	2	no	—
62	D₄	—	43	2	no	—
63	D₄	—	86	1	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	86	1	no	—
66	C₂×C₄	—	172	2	no	—
67	C₂×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
70	C₈	—	86	2	no	—
71	C₂²	—	86	2	no	—
72	C₂²	—	172	1	no	—
73	C₂²	—	172	1	no	—
74	C₂²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	2	no	—
77	C₄	—	172	1	no	—
78	C₄	#10621884	1	2	yes	complemented
79	C₄	D₄×D₄₃	1	1	yes	characteristic
80	C₄	#10621900	1	1	yes	characteristic
81	C₂	—	172	2	no	—
82	C₂	#13063296	1	1	yes	characteristic
83	C₂	C₂×#10621884	1	1	yes	characteristic
84	C₂	#13063356	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₃
commutator, cosocle	C₂×C₄×C₄₃	C₂³
Frattini	C₂×C₄	C₂²×D₄₃
Fitting	C₄⋊Q₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂×D₄
prime radical	C₂×D₁₇₂	C₂²
hypercenter	C₄⋊Q₈	D₄₃
hypocenter	C₄₃	#461
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₄₃⋊C₈)	C₂	semidirect
4. #13063249	C₂	semidirect
5. C₄⋊Q₈×C₄₃	C₂	semidirect
34. C₄×C₄₃	SD₁₆	semidirect
43. C₄₃	#461	semidirect
45. C₄⋊Q₈	D₄₃	semidirect
78. C₄	#10621884	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10621884	2
C₄×D₁₇₂	2
C₄⋊(C₄₃⋊C₈)	2
#13063249	2
C₄⋊Q₈×C₄₃	2
#461	43