Subgroups — #43252932

Order 28544.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20297243	C₂	1	1	yes	maximal, characteristic
2	#28383414	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×D₂₂₃	C₄	1	1	yes	characteristic
5	#20297243	—	2	1	no	—
6	#20297243	—	2	1	no	—
7	C₂×C₂₂₃⋊C₁₆	C₄	1	1	yes	characteristic
8	C₂₂₃⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₂₂₃	C₂²	1	1	yes	characteristic
10	C₃₂×C₂₂₃	—	2	1	no	—
11	C₂×C₄×D₂₂₃	C₈	1	1	yes	characteristic
12	C₂×C₂₂₃⋊C₈	C₈	1	1	yes	characteristic
13	C₈×D₂₂₃	—	2	1	no	—
14	C₈×D₂₂₃	—	2	1	no	—
15	C₂₂₃⋊C₁₆	—	2	1	no	—
16	C₂×C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₂₂₃	D₄	1	1	yes	characteristic
18	C₁₆×C₂₂₃	D₄	1	1	yes	characteristic
19	C₂²×D₂₂₃	C₁₆	1	1	yes	characteristic
20	C₂×Dic₂₂₃	C₁₆	1	1	yes	characteristic
21	C₄×D₂₂₃	—	2	1	no	—
22	C₄×D₂₂₃	—	2	1	no	—
23	C₂₂₃⋊C₈	—	2	1	no	—
24	C₂×C₄×C₂₂₃	C₂×C₈	1	1	yes	characteristic
25	C₈×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₂₃	M₁₆	1	1	yes	characteristic
27	C₂²×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
28	C₂×D₂₂₃	—	2	1	no	—
29	C₂×D₂₂₃	—	4	1	no	—
30	C₄×C₂₂₃	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₂₂₃	M₃₂	1	1	yes	characteristic
32	Dic₂₂₃	—	2	1	no	—
33	C₂×C₂₂₃	—	2	1	no	—
34	C₂×C₂₂₃	C₂²⋊C₁₆	1	1	yes	characteristic
35	D₂₂₃	—	4	1	no	—
36	C₂₂₃	#1400	1	1	yes	characteristic, complemented, Sylow
37	#1400	—	223	1	no	maximal, Sylow
38	C₂×M₃₂	—	223	1	no	—
39	M₆₄	—	223	1	no	—
40	M₆₄	D₂₂₃	1	1	yes	characteristic, complemented
41	C₂²×C₈	—	223	1	no	—
42	M₃₂	—	446	1	no	—
43	M₃₂	—	446	1	no	—
44	C₂×C₁₆	—	223	1	no	—
45	C₂×C₁₆	C₂×D₂₂₃	1	1	yes	characteristic
46	C₃₂	—	2	1	no	—
47	C₃₂	—	446	1	no	—
48	C₂²×C₄	—	223	1	no	—
49	C₂×C₈	—	446	1	no	—
50	C₂×C₈	—	446	1	no	—
51	C₂×C₈	—	223	1	no	—
52	C₂×C₈	C₄×D₂₂₃	1	1	yes	characteristic
53	C₁₆	—	446	1	no	—
54	C₁₆	C₂²⋊D₂₂₃	1	1	yes	characteristic
55	C₁₆	D₈₉₂	1	1	yes	characteristic
56	C₂³	—	223	1	no	—
57	C₂×C₄	—	446	1	no	—
58	C₂×C₄	—	446	1	no	—
59	C₂×C₄	—	223	1	no	—
60	C₂×C₄	C₈×D₂₂₃	1	1	yes	characteristic
61	C₈	—	446	1	no	—
62	C₈	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
63	C₈	#16813153	1	1	yes	characteristic
64	C₂²	—	892	1	no	—
65	C₂²	—	446	1	no	—
66	C₂²	C₁₆×D₂₂₃	1	1	yes	characteristic
67	C₄	—	446	1	no	—
68	C₄	(C₂×D₂₂₃)⋊C₈	1	1	yes	characteristic
69	C₄	#20297243	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	892	1	no	—
72	C₂	(C₂×D₂₂₃)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₈	(C₂×D₂₂₃)⋊C₄
commutator	C₂²×C₂₂₃	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₂₂₃
Fitting	M₆₄×C₂₂₃	C₂
socle	C₂×C₂₂₃	C₂²⋊C₁₆
prime radical	C₂²×D₂₂₃	C₁₆
cosocle	C₂×C₁₆×C₂₂₃	C₂²
hypercenter	M₆₄	D₂₂₃
hypocenter	C₂₂₃	#1400
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #28383414	C₂	semidirect
3. M₆₄×C₂₂₃	C₂	semidirect
36. C₂₂₃	#1400	semidirect
40. M₆₄	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20297243	2
#28383414	2
M₆₄×C₂₂₃	2
#1400	223