Subgroups — #43252734

Order 28544.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28383606	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₂₂₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#16813154	C₄	1	1	yes	characteristic
5	#20297263	C₄	1	1	yes	characteristic
6	C₂×C₂₂₃⋊C₁₆	—	2	1	no	—
7	Dic₂₂₃⋊C₈	—	2	1	no	—
8	Dic₂₂₃⋊C₈	—	2	1	no	—
9	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₂₂₃	—	2	1	no	—
11	#16813154	—	4	1	no	—
12	C₂×C₂₂₃⋊C₈	—	2	1	no	—
13	C₄⋊Dic₂₂₃	C₈	1	1	yes	characteristic
14	C₄⋊Dic₂₂₃	C₈	1	1	yes	characteristic
15	C₂²⋊Dic₂₂₃	—	2	1	no	—
16	C₄×Dic₂₂₃	—	2	1	no	—
17	C₂₂₃⋊C₁₆	—	4	1	no	—
18	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₂₂₃	—	4	1	no	—
20	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
21	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
22	C₁₆×C₂₂₃	—	4	1	no	—
23	C₂×Dic₂₂₃	—	2	1	no	—
24	C₂×Dic₂₂₃	—	2	1	no	—
25	C₂₂₃⋊C₈	—	4	1	no	—
26	C₂³×C₂₂₃	M₁₆	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	—	2	1	no	—
28	C₂×C₄×C₂₂₃	—	2	1	no	—
29	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
31	C₈×C₂₂₃	—	2	1	no	—
32	C₈×C₂₂₃	—	2	1	no	—
33	C₂²×C₂₂₃	—	2	1	no	—
34	C₂²×C₂₂₃	—	2	1	no	—
35	C₂²×C₂₂₃	—	2	1	no	—
36	C₂²×C₂₂₃	C₂²⋊C₈	1	1	yes	characteristic
37	C₄×C₂₂₃	—	2	1	no	—
38	C₄×C₂₂₃	#99	1	1	yes	characteristic
39	C₄×C₂₂₃	#101	1	1	yes	characteristic
40	Dic₂₂₃	—	4	1	no	—
41	Dic₂₂₃	—	4	1	no	—
42	C₂×C₂₂₃	—	4	1	no	—
43	C₂×C₂₂₃	#324	1	1	yes	characteristic
44	C₂×C₂₂₃	#350	1	1	yes	characteristic
45	C₂×C₂₂₃	#350	1	1	yes	characteristic
46	C₂₂₃	#1323	1	1	yes	characteristic, complemented, Sylow
47	#1323	—	223	1	no	maximal, Sylow
48	#424	—	223	1	no	—
49	C₂²⋊C₁₆	—	223	1	no	—
50	C₂²⋊C₁₆	D₂₂₃	1	1	yes	characteristic
51	C₂×M₁₆	—	223	1	no	—
52	C₂²×C₈	C₂×D₂₂₃	1	1	yes	characteristic
53	#117	—	223	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	446	1	no	—
56	C₄⋊C₈	—	446	1	no	—
57	C₄⋊C₈	—	446	1	no	—
58	C₂²×C₄	C₄×D₂₂₃	1	1	yes	characteristic
59	M₁₆	—	892	1	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	446	1	no	—
62	C₂×C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
63	C₂×C₈	D₈₉₂	1	1	yes	characteristic
64	C₄⋊C₄	—	223	1	no	—
65	C₄⋊C₄	—	223	1	no	—
66	C₂²⋊C₄	—	446	1	no	—
67	C₄²	—	446	1	no	—
68	C₁₆	—	4	1	no	—
69	C₁₆	—	892	1	no	—
70	C₂³	#16813153	1	1	yes	characteristic
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	446	1	no	—
74	C₂×C₄	—	446	1	no	—
75	C₂×C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
76	C₂×C₄	C₈×D₂₂₃	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	1	no	—
79	C₈	—	892	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	1	no	—
83	C₂²	(C₂×D₂₂₃)⋊C₈	1	1	yes	characteristic
84	C₄	—	2	1	no	—
85	C₄	—	892	1	no	—
86	C₄	—	892	1	no	—
87	C₄	#20297218	1	1	yes	characteristic
88	C₄	#20297222	1	1	yes	characteristic
89	C₂	—	4	1	no	—
90	C₂	#28383305	1	1	yes	characteristic
91	C₂	#28383363	1	1	yes	characteristic
92	C₂	#28383365	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₂₂₃)⋊C₄
commutator	C₂×C₄×C₂₂₃	C₂×C₈
Frattini	C₂²×C₈	C₂×D₂₂₃
Fitting	C₂²⋊C₁₆×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₈
prime radical	C₂³×C₂₂₃	M₁₆
cosocle	C₂²×C₈×C₂₂₃	C₂²
hypercenter	C₂²⋊C₁₆	D₂₂₃
hypocenter	C₂₂₃	#1323
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
46. C₂₂₃	#1323	semidirect

Maximal subgroups

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

subgroup	index
#28383606	2
C₂²⋊(C₂₂₃⋊C₁₆)	2
C₂²⋊C₁₆×C₂₂₃	2
#1323	223