Subgroups — #43252827

Order 28544.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383949	C₂	1	1	yes	maximal, characteristic
2	#28383358	C₂	1	1	yes	maximal, characteristic
3	#347×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297304	C₄	1	1	yes	characteristic
5	#20297296	C₄	1	1	yes	characteristic
6	#20297244	—	2	1	no	—
7	#20297229	—	2	1	no	—
8	#20297229	—	2	1	no	—
9	M₃₂×C₂₂₃	—	2	1	no	—
10	#97×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₄₆	—	2	1	no	—
12	C₄⋊Dic₂₂₃	—	2	1	no	—
13	C₄⋊Dic₂₂₃	—	4	1	no	—
14	Dic₂₂₃⋊C₄	—	4	1	no	—
15	C₂₂₃⋊C₁₆	—	2	2	no	—
16	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
17	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
18	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₂₂₃	—	2	2	no	—
20	Dic₄₄₆	—	4	1	no	—
21	Dic₄₄₆	—	4	1	no	—
22	C₂×Dic₂₂₃	—	4	1	no	—
23	C₂×Dic₂₂₃	—	4	1	no	—
24	C₂×C₄×C₂₂₃	—	2	1	no	—
25	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₂₃	—	2	1	no	—
27	C₈×C₂₂₃	—	2	1	no	—
28	C₂²×C₂₂₃	#100	1	1	yes	characteristic
29	C₄×C₂₂₃	—	4	1	no	—
30	C₄×C₂₂₃	#104	1	1	yes	characteristic
31	C₄×C₂₂₃	#104	1	1	yes	characteristic
32	Dic₂₂₃	—	8	1	no	—
33	Dic₂₂₃	—	8	1	no	—
34	C₂×C₂₂₃	—	2	1	no	—
35	C₂×C₂₂₃	#329	1	1	yes	characteristic
36	C₂₂₃	#1356	1	1	yes	characteristic, complemented, Sylow
37	#1356	—	223	1	no	maximal, Sylow
38	#491	—	223	1	no	—
39	#347	—	223	1	no	—
40	#347	D₂₂₃	1	1	yes	characteristic
41	C₄⋊Q₈	—	223	1	no	—
42	#125	—	223	1	no	—
43	M₃₂	—	2	1	no	—
44	M₃₂	—	446	1	no	—
45	#103	—	446	1	no	—
46	#103	—	446	1	no	—
47	#97	C₂×D₂₂₃	1	1	yes	characteristic
48	C₂×Q₈	—	446	1	no	—
49	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
50	C₂×C₈	D₈₉₂	1	1	yes	characteristic
51	C₄⋊C₄	—	892	1	no	—
52	C₄⋊C₄	—	892	1	no	—
53	C₄⋊C₄	—	446	1	no	—
54	C₄²	C₄×D₂₂₃	1	1	yes	characteristic
55	C₁₆	—	2	2	no	—
56	C₁₆	—	446	2	no	—
57	Q₈	—	892	1	no	—
58	Q₈	—	892	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	892	1	no	—
61	C₂×C₄	—	892	1	no	—
62	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	2	1	no	—
65	C₂²	#20297220	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	—	1784	1	no	—
68	C₄	—	1784	1	no	—
69	C₄	#20297230	1	1	yes	characteristic
70	C₄	#20297232	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#28383319	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28383319
commutator	C₄²×C₂₂₃	C₂×C₄
Frattini	#97	C₂×D₂₂₃
Fitting	#347×C₂₂₃	C₂
socle	C₂×C₂₂₃	#329
prime radical	C₂²×C₂₂₃	#100
cosocle	#97×C₂₂₃	C₂²
hypercenter	#347	D₂₂₃
hypocenter	C₂₂₃	#1356
LCS	G ▷ C₄²×C₂₂₃ ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
36. C₂₂₃	#1356	semidirect

Maximal subgroups

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

subgroup	index
#28383949	2
#28383358	2
#347×C₂₂₃	2
#1356	223