Subgroups — #43252885

Order 28544.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20297238	C₂	1	2	yes	maximal
2	C₂²×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊Dic₂₂₃	—	2	2	no	—
4	#20297238	—	2	2	no	—
5	#20297238	C₄	1	4	yes	complemented
6	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₆×C₂₂₃	—	2	2	no	—
8	C₂×C₁₆×C₂₂₃	C₄	1	2	yes	complemented
9	C₂²×Dic₂₂₃	—	4	2	no	—
10	C₄⋊Dic₂₂₃	—	2	4	no	—
11	C₄⋊Dic₂₂₃	—	4	2	no	—
12	C₂²×C₄×C₂₂₃	D₄	1	1	yes	characteristic
13	C₂×C₈×C₂₂₃	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
16	C₂×C₈×C₂₂₃	Q₈	1	1	yes	characteristic
17	C₁₆×C₂₂₃	—	2	2	no	—
18	C₂×Dic₂₂₃	—	4	4	no	—
19	C₂×Dic₂₂₃	—	8	2	no	—
20	C₂×Dic₂₂₃	—	8	2	no	—
21	C₂³×C₂₂₃	D₈	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
23	C₂×C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	SD₁₆	1	2	yes	—
25	C₂×C₄×C₂₂₃	Q₁₆	1	1	yes	characteristic
26	C₈×C₂₂₃	C₄²	1	1	yes	characteristic
27	C₈×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₈×C₂₂₃	C₄⋊C₄	1	2	yes	—
29	C₂²×C₂₂₃	#102	1	2	yes	—
30	C₂²×C₂₂₃	#107	1	1	yes	characteristic
31	C₂²×C₂₂₃	D₁₆	1	1	yes	characteristic
32	C₂²×C₂₂₃	SD₃₂	1	2	yes	—
33	C₂²×C₂₂₃	Q₃₂	1	1	yes	characteristic
34	C₄×C₂₂₃	#95	1	1	yes	characteristic
35	C₄×C₂₂₃	#103	1	2	yes	—
36	C₄×C₂₂₃	#106	1	1	yes	characteristic
37	Dic₂₂₃	—	8	4	no	—
38	C₂×C₂₂₃	#340	1	1	yes	characteristic
39	C₂×C₂₂₃	#357	1	2	yes	—
40	C₂×C₂₂₃	#358	1	2	yes	—
41	C₂×C₂₂₃	#366	1	1	yes	characteristic
42	C₂×C₂₂₃	#367	1	1	yes	characteristic
43	C₂₂₃	#1380	1	1	yes	characteristic, complemented, Sylow
44	#1380	—	223	1	no	maximal, Sylow
45	C₂²×C₁₆	D₂₂₃	1	1	yes	characteristic
46	C₂×#107	—	223	2	no	—
47	C₂²×C₈	C₂×D₂₂₃	1	1	yes	characteristic
48	C₂×C₄⋊C₄	—	446	2	no	—
49	C₂×C₁₆	—	2	2	no	—
50	C₂×C₁₆	Dic₂₂₃	1	2	yes	complemented
51	#107	—	446	2	no	—
52	#107	—	223	4	no	—
53	C₂²×C₄	—	892	2	no	—
54	C₂²×C₄	D₈₉₂	1	1	yes	characteristic
55	C₂×C₈	C₄×D₂₂₃	1	2	yes	—
56	C₂×C₈	C₂×Dic₂₂₃	1	1	yes	characteristic
57	C₂×C₈	C₂²⋊D₂₂₃	1	2	yes	—
58	C₂×C₈	Dic₄₄₆	1	1	yes	characteristic
59	C₄⋊C₄	—	892	2	no	—
60	C₄⋊C₄	—	446	4	no	—
61	C₁₆	—	2	2	no	—
62	C₂³	D₁₇₈₄	1	1	yes	characteristic
63	C₂×C₄	—	892	4	no	—
64	C₂×C₄	—	1784	2	no	—
65	C₂×C₄	—	1784	2	no	—
66	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	2	yes	—
67	C₂×C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
68	C₂×C₄	#16813159	1	2	yes	—
69	C₂×C₄	Dic₈₉₂	1	1	yes	characteristic
70	C₈	C₄×Dic₂₂₃	1	1	yes	characteristic
71	C₈	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
72	C₈	Dic₂₂₃⋊C₄	1	2	yes	—
73	C₂²	#20297226	1	2	yes	—
74	C₂²	#20297238	1	1	yes	characteristic
75	C₂²	D₃₅₆₈	1	1	yes	characteristic
76	C₂²	#20297249	1	2	yes	—
77	C₂²	Dic₁₇₈₄	1	1	yes	characteristic
78	C₄	—	1784	4	no	—
79	C₄	C₂₂₃⋊#95	1	1	yes	characteristic
80	C₄	#20297229	1	2	yes	—
81	C₄	#20297236	1	1	yes	characteristic
82	C₂	#28383343	1	1	yes	characteristic
83	C₂	#28383382	1	2	yes	—
84	C₂	#28383385	1	2	yes	—
85	C₂	#28383406	1	1	yes	characteristic
86	C₂	#28383408	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₇₈₄
commutator	C₈×C₂₂₃	C₄²
Frattini	C₂²×C₈	C₂×D₂₂₃
Fitting	C₂²×C₁₆×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	D₈
cosocle	C₂²×C₈×C₂₂₃	C₂²
hypercenter	C₂²×C₁₆	D₂₂₃
hypocenter	C₂₂₃	#1380
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
5. #20297238	C₄	semidirect
8. C₂×C₁₆×C₂₂₃	C₄	semidirect
43. C₂₂₃	#1380	semidirect
50. C₂×C₁₆	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20297238	2
C₂²×C₁₆×C₂₂₃	2
#1380	223