Subgroups — #43252825

Order 28544.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383945	C₂	1	1	yes	maximal, characteristic
2	#28383358	C₂	1	1	yes	maximal, characteristic, complemented
3	#347×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297301	C₄	1	1	yes	characteristic
5	#20297296	C₄	1	1	yes	characteristic
6	#20297244	—	2	1	no	—
7	#20297226	—	2	1	no	—
8	#20297226	—	2	1	no	—
9	M₃₂×C₂₂₃	—	2	1	no	—
10	#97×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×D₈₉₂	—	2	1	no	—
12	C₂×D₈₉₂	—	4	1	no	—
13	C₄⋊Dic₂₂₃	—	2	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	C₂²×D₂₂₃	—	4	1	no	—
21	D₈₉₂	—	4	1	no	—
22	D₈₉₂	—	4	1	no	—
23	D₈₉₂	—	8	2	no	—
24	C₂×Dic₂₂₃	—	4	1	no	—
25	C₂×C₄×C₂₂₃	—	2	1	no	—
26	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₈×C₂₂₃	—	2	1	no	—
28	C₈×C₂₂₃	—	2	1	no	—
29	C₂²×C₂₂₃	#100	1	1	yes	characteristic
30	C₂×D₂₂₃	—	8	2	no	—
31	C₂×D₂₂₃	—	8	1	no	—
32	C₄×C₂₂₃	—	4	1	no	—
33	C₄×C₂₂₃	#104	1	1	yes	characteristic
34	C₄×C₂₂₃	#104	1	1	yes	characteristic
35	Dic₂₂₃	—	8	1	no	—
36	C₂×C₂₂₃	—	2	1	no	—
37	C₂×C₂₂₃	#329	1	1	yes	characteristic
38	D₂₂₃	—	16	1	no	—
39	C₂₂₃	#1355	1	1	yes	characteristic, complemented, Sylow
40	#1355	—	223	1	no	maximal, Sylow
41	#490	—	223	1	no	—
42	#347	—	223	1	no	—
43	#347	D₂₂₃	1	1	yes	characteristic, complemented
44	#127	—	223	1	no	—
45	#125	—	223	1	no	—
46	M₃₂	—	2	1	no	—
47	M₃₂	—	446	1	no	—
48	#102	—	446	1	no	—
49	#102	—	446	1	no	—
50	#97	C₂×D₂₂₃	1	1	yes	characteristic
51	C₂×D₄	—	892	1	no	—
52	C₂×D₄	—	446	1	no	—
53	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
54	C₂×C₈	D₈₉₂	1	1	yes	characteristic
55	C₄⋊C₄	—	892	1	no	—
56	C₄⋊C₄	—	446	1	no	—
57	C₄²	C₄×D₂₂₃	1	1	yes	characteristic
58	C₁₆	—	2	2	no	—
59	C₁₆	—	446	2	no	—
60	C₂³	—	892	1	no	—
61	D₄	—	892	1	no	—
62	D₄	—	892	1	no	—
63	D₄	—	1784	2	no	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	892	1	no	—
66	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
67	C₈	—	2	1	no	—
68	C₈	—	2	1	no	—
69	C₂²	—	1784	2	no	—
70	C₂²	—	1784	1	no	—
71	C₂²	#20297220	1	1	yes	characteristic
72	C₄	—	4	1	no	—
73	C₄	—	1784	1	no	—
74	C₄	#20297230	1	1	yes	characteristic
75	C₄	#20297232	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	3568	1	no	—
78	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	#20297301	C₄
cosocle	#97×C₂₂₃	C₂²
hypercenter	#347	D₂₂₃
hypocenter	C₂₂₃	#1355
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
2. #28383358	C₂	semidirect
3. #347×C₂₂₃	C₂	semidirect
39. C₂₂₃	#1355	semidirect
43. #347	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#28383945	2
#28383358	2
#347×C₂₂₃	2
#1355	223