Subgroups — #43252833

Order 28544.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383953	C₂	1	1	yes	maximal, characteristic
2	#28383401	C₂	1	1	yes	maximal, characteristic, complemented
3	#364×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16813159	—	2	1	no	—
5	C₂×#16813159	—	2	1	no	—
6	#20297304	C₄	1	1	yes	characteristic
7	#20297301	C₄	1	1	yes	characteristic
8	#20297244	—	2	1	no	—
9	M₃₂×C₂₂₃	—	2	1	no	—
10	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₄₆	—	2	1	no	—
12	C₂×D₈₉₂	—	2	1	no	—
13	C₂×D₈₉₂	—	4	1	no	—
14	#16813159	—	4	1	no	—
15	#16813159	—	4	1	no	—
16	#16813159	—	4	1	no	—
17	#16813159	—	4	1	no	—
18	C₄⋊Dic₂₂₃	—	4	1	no	—
19	C₂₂₃⋊C₁₆	—	4	1	no	—
20	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
22	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₁₆×C₂₂₃	—	4	1	no	—
24	C₂²×D₂₂₃	—	4	1	no	—
25	Dic₄₄₆	—	4	1	no	—
26	Dic₄₄₆	—	4	1	no	—
27	D₈₉₂	—	4	1	no	—
28	D₈₉₂	—	4	1	no	—
29	D₈₉₂	—	8	2	no	—
30	C₂×Dic₂₂₃	—	4	1	no	—
31	C₂×C₄×C₂₂₃	—	2	1	no	—
32	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₈×C₂₂₃	D₈	1	1	yes	characteristic
34	C₈×C₂₂₃	D₈	1	1	yes	characteristic
35	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
36	C₈×C₂₂₃	SD₁₆	1	1	yes	characteristic
37	C₂²×C₂₂₃	#100	1	1	yes	characteristic
38	C₂×D₂₂₃	—	8	2	no	—
39	C₂×D₂₂₃	—	8	1	no	—
40	C₄×C₂₂₃	—	2	2	no	—
41	C₄×C₂₂₃	#102	1	1	yes	characteristic
42	C₄×C₂₂₃	#102	1	1	yes	characteristic
43	Dic₂₂₃	—	8	1	no	—
44	C₂×C₂₂₃	—	2	1	no	—
45	C₂×C₂₂₃	#331	1	1	yes	characteristic
46	D₂₂₃	—	16	1	no	—
47	C₂₂₃	#1359	1	1	yes	characteristic, complemented, Sylow
48	#1359	—	223	1	no	maximal, Sylow
49	#492	—	223	1	no	—
50	#364	—	223	1	no	—
51	#364	D₂₂₃	1	1	yes	characteristic, complemented
52	C₂×SD₁₆	—	446	1	no	—
53	C₂×SD₁₆	—	446	1	no	—
54	C₄⋊Q₈	—	223	1	no	—
55	#127	—	223	1	no	—
56	M₃₂	—	2	1	no	—
57	M₃₂	—	446	1	no	—
58	C₄×C₈	C₂×D₂₂₃	1	1	yes	characteristic
59	C₂×Q₈	—	446	1	no	—
60	C₂×D₄	—	892	1	no	—
61	C₂×D₄	—	446	1	no	—
62	SD₁₆	—	892	1	no	—
63	SD₁₆	—	892	1	no	—
64	SD₁₆	—	892	1	no	—
65	SD₁₆	—	892	1	no	—
66	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
67	C₂×C₈	D₈₉₂	1	1	yes	characteristic
68	C₄⋊C₄	—	892	1	no	—
69	C₄²	C₄×D₂₂₃	1	1	yes	characteristic
70	C₁₆	—	4	1	no	—
71	C₁₆	—	892	1	no	—
72	C₂³	—	892	1	no	—
73	Q₈	—	892	1	no	—
74	Q₈	—	892	1	no	—
75	D₄	—	892	1	no	—
76	D₄	—	892	1	no	—
77	D₄	—	1784	2	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	892	1	no	—
80	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
81	C₈	#16813155	1	1	yes	characteristic
82	C₈	D₁₇₈₄	1	1	yes	characteristic
83	C₈	#16813157	1	1	yes	characteristic
84	C₈	#16813159	1	1	yes	characteristic
85	C₂²	—	1784	2	no	—
86	C₂²	—	1784	1	no	—
87	C₂²	#20297220	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	1784	1	no	—
90	C₄	#20297224	1	1	yes	characteristic
91	C₄	#20297226	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	3568	1	no	—
94	C₂	#28383323	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28383323
commutator	C₄²×C₂₂₃	C₂×C₄
Frattini	C₄×C₈	C₂×D₂₂₃
Fitting	#364×C₂₂₃	C₂
socle	C₂×C₂₂₃	#331
prime radical	#20297301	C₄
cosocle	C₄×C₈×C₂₂₃	C₂²
hypercenter	#364	D₂₂₃
hypocenter	C₂₂₃	#1359
LCS	G ▷ C₄²×C₂₂₃ ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
2. #28383401	C₂	semidirect
3. #364×C₂₂₃	C₂	semidirect
47. C₂₂₃	#1359	semidirect
51. #364	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#28383953	2
#28383401	2
#364×C₂₂₃	2
#1359	223