Subgroups — #43252733

Order 28544.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊(C₂₂₃⋊C₁₆)	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₂₃⋊C₁₆	—	2	2	no	—
4	C₂²×C₈×C₂₂₃	C₄	1	1	yes	characteristic
5	C₂²×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄²×C₂₂₃	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₂₂₃	—	2	2	no	—
8	C₂₂₃⋊C₁₆	—	4	2	no	—
9	C₂²×C₄×C₂₂₃	C₈	1	1	yes	characteristic
10	C₂²×C₄×C₂₂₃	C₈	1	1	yes	characteristic
11	C₂²×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₂₂₃	—	2	1	no	—
13	C₂×C₈×C₂₂₃	—	4	1	no	—
14	C₂×C₈×C₂₂₃	—	4	1	no	—
15	C₂×C₈×C₂₂₃	D₄	1	2	yes	—
16	C₄²×C₂₂₃	—	2	1	no	—
17	C₄²×C₂₂₃	—	2	1	no	—
18	C₂³×C₂₂₃	C₂×C₈	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	—	2	1	no	—
20	C₂×C₄×C₂₂₃	—	2	1	no	—
21	C₂×C₄×C₂₂₃	—	2	1	no	—
22	C₂×C₄×C₂₂₃	—	2	1	no	—
23	C₂×C₄×C₂₂₃	—	2	1	no	—
24	C₂×C₄×C₂₂₃	—	4	1	no	—
25	C₂×C₄×C₂₂₃	C₁₆	1	2	yes	complemented
26	C₂×C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	M₁₆	1	1	yes	characteristic
28	C₈×C₂₂₃	—	2	2	no	—
29	C₈×C₂₂₃	—	4	1	no	—
30	C₂²×C₂₂₃	—	2	1	no	—
31	C₂²×C₂₂₃	—	2	1	no	—
32	C₂²×C₂₂₃	C₂²⋊C₈	1	1	yes	characteristic
33	C₂²×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
34	C₂²×C₂₂₃	M₃₂	1	1	yes	characteristic
35	C₄×C₂₂₃	—	2	1	no	—
36	C₄×C₂₂₃	—	2	2	no	—
37	C₄×C₂₂₃	—	4	1	no	—
38	C₄×C₂₂₃	#99	1	1	yes	characteristic
39	C₄×C₂₂₃	#101	1	1	yes	characteristic
40	C₂×C₂₂₃	—	2	1	no	—
41	C₂×C₂₂₃	—	2	1	no	—
42	C₂×C₂₂₃	#324	1	1	yes	characteristic
43	C₂×C₂₂₃	C₂²⋊C₁₆	1	1	yes	characteristic
44	C₂×C₂₂₃	#349	1	1	yes	characteristic
45	C₂₂₃	#1322	1	1	yes	characteristic, complemented, Sylow
46	#1322	—	223	1	no	maximal, Sylow
47	C₂×C₄⋊C₈	D₂₂₃	1	1	yes	characteristic
48	C₂²⋊C₁₆	—	223	2	no	—
49	C₂²×C₈	Dic₂₂₃	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₂₂₃	1	1	yes	characteristic
51	C₂×C₄²	Dic₂₂₃	1	1	yes	characteristic
52	C₂×C₁₆	—	446	2	no	—
53	C₄⋊C₈	—	2	2	no	—
54	C₂²×C₄	C₂₂₃⋊C₈	1	1	yes	characteristic
55	C₂²×C₄	C₂₂₃⋊C₈	1	1	yes	characteristic
56	C₂²×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	4	1	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	C₂²⋊D₂₂₃	1	2	yes	—
61	C₄²	—	2	1	no	—
62	C₄²	—	2	1	no	—
63	C₁₆	—	892	2	no	—
64	C₂³	C₂×C₂₂₃⋊C₈	1	1	yes	characteristic
65	C₂×C₄	—	2	1	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	4	1	no	—
71	C₂×C₄	C₂₂₃⋊C₁₆	1	2	yes	complemented
72	C₂×C₄	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
73	C₂×C₄	#16813154	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	—	4	1	no	—
76	C₂²	—	2	1	no	—
77	C₂²	—	2	1	no	—
78	C₂²	C₂²⋊(C₂₂₃⋊C₈)	1	1	yes	characteristic
79	C₂²	C₂×C₂₂₃⋊C₁₆	1	1	yes	characteristic
80	C₂²	#20297244	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	#20297219	1	1	yes	characteristic
85	C₄	#20297223	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	1	no	—
88	C₂	#28383306	1	1	yes	characteristic
89	C₂	C₂²⋊(C₂₂₃⋊C₁₆)	1	1	yes	characteristic
90	C₂	#28383362	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₂₂₃
commutator	C₂²×C₂₂₃	C₂×C₁₆
Frattini	C₂²×C₈	C₂×D₂₂₃
Fitting	C₂×C₄⋊C₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₈
prime radical	C₂³×C₂₂₃	C₂×C₈
cosocle	C₂²×C₈×C₂₂₃	C₂²
hypercenter	C₂×C₄⋊C₈	D₂₂₃
hypocenter	C₂₂₃	#1322
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
25. C₂×C₄×C₂₂₃	C₁₆	semidirect
45. C₂₂₃	#1322	semidirect
71. C₂×C₄	C₂₂₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊(C₂₂₃⋊C₁₆)	2
C₂×C₄⋊C₈×C₂₂₃	2
#1322	223