Subgroups — #43252969

Order 28544.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383934	C₂	1	1	yes	maximal, characteristic
2	#28383375	C₂	1	1	yes	maximal, characteristic
3	#355×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297296	C₄	1	1	yes	characteristic, complemented
5	#20297228	—	2	1	no	—
6	#20297225	—	2	1	no	—
7	#20297219	—	2	1	no	—
8	C₄×C₂₂₃⋊C₈	C₄	1	1	yes	characteristic, complemented
9	#124×C₂₂₃	C₂²	1	1	yes	characteristic
10	#101×C₂₂₃	—	2	1	no	—
11	C₂×C₂₂₃⋊C₈	—	2	1	no	—
12	C₄⋊Dic₂₂₃	—	2	1	no	—
13	Dic₂₂₃⋊C₄	—	4	1	no	—
14	C₂²⋊Dic₂₂₃	—	4	1	no	—
15	C₂×Q₈×C₂₂₃	D₄	1	1	yes	characteristic
16	C₂×D₄×C₂₂₃	D₄	1	1	yes	characteristic
17	M₁₆×C₂₂₃	—	4	1	no	—
18	C₂²⋊C₄×C₂₂₃	—	4	1	no	—
19	C₄²×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₂₃	—	4	1	no	—
21	C₂×Dic₂₂₃	—	8	1	no	—
22	C₂₂₃⋊C₈	—	4	1	no	—
23	C₂³×C₂₂₃	—	2	1	no	—
24	Q₈×C₂₂₃	—	4	1	no	—
25	D₄×C₂₂₃	—	4	1	no	—
26	C₂×C₄×C₂₂₃	—	2	1	no	—
27	C₂×C₄×C₂₂₃	—	2	1	no	—
28	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₂₂₃	—	8	1	no	—
30	C₂²×C₂₂₃	—	4	1	no	—
31	C₂²×C₂₂₃	—	8	1	no	—
32	C₂²×C₂₂₃	#99	1	1	yes	characteristic
33	C₄×C₂₂₃	—	2	1	no	—
34	C₄×C₂₂₃	—	4	1	no	—
35	C₄×C₂₂₃	—	4	1	no	—
36	Dic₂₂₃	—	8	1	no	—
37	Dic₂₂₃	—	16	1	no	—
38	C₂×C₂₂₃	—	2	1	no	—
39	C₂×C₂₂₃	—	8	1	no	—
40	C₂×C₂₂₃	#354	1	1	yes	characteristic
41	C₂₂₃	#1414	1	1	yes	characteristic, complemented, Sylow
42	#1414	—	223	1	no	maximal, Sylow
43	#488	—	223	1	no	—
44	#355	D₂₂₃	1	1	yes	characteristic
45	#354	—	223	1	no	—
46	#125	—	223	1	no	—
47	#124	C₂×D₂₂₃	1	1	yes	characteristic
48	#103	—	446	1	no	—
49	#102	—	446	1	no	—
50	#101	—	2	1	no	—
51	#99	—	446	1	no	—
52	C₄×C₈	—	223	1	no	—
53	C₂×Q₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
54	C₂×D₄	D₈₉₂	1	1	yes	characteristic
55	M₁₆	—	4	1	no	—
56	C₂×C₈	—	446	1	no	—
57	C₄⋊C₄	—	892	1	no	—
58	C₄⋊C₄	—	446	1	no	—
59	C₂²⋊C₄	—	4	1	no	—
60	C₂²⋊C₄	—	892	1	no	—
61	C₄²	C₄×D₂₂₃	1	1	yes	characteristic
62	C₂³	—	2	1	no	—
63	Q₈	—	4	1	no	—
64	D₄	—	4	1	no	—
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	892	1	no	—
68	C₂×C₄	—	1784	1	no	—
69	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
70	C₈	—	8	1	no	—
71	C₈	—	892	1	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	8	1	no	—
74	C₂²	#20297218	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	4	1	no	—
78	C₄	—	3568	1	no	—
79	C₄	—	1784	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	8	1	no	—
82	C₂	#28383374	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28383374
commutator	C₄²×C₂₂₃	C₂×C₄
Frattini	#124	C₂×D₂₂₃
Fitting	#355×C₂₂₃	C₂
socle	C₂×C₂₂₃	#354
prime radical	C₂×D₄×C₂₂₃	D₄
cosocle	#124×C₂₂₃	C₂²
hypercenter	#355	D₂₂₃
hypocenter	C₂₂₃	#1414
LCS	G ▷ C₄²×C₂₂₃ ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #355

Complements

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

N	H ≅ G/N	type
4. #20297296	C₄	semidirect
8. C₄×C₂₂₃⋊C₈	C₄	semidirect
41. C₂₂₃	#1414	semidirect

Maximal subgroups

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

subgroup	index
#28383934	2
#28383375	2
#355×C₂₂₃	2
#1414	223