Subgroups — #28383394

Order 14272.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₇₈₄	C₂	1	1	yes	maximal, characteristic
2	#20297240	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₉₂	—	2	1	no	—
5	D₁₇₈₄	—	2	1	no	—
6	D₁₇₈₄	C₄	1	2	yes	—
7	#16813154	—	2	1	no	—
8	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₁₆×C₂₂₃	—	2	1	no	—
10	C₂²×D₂₂₃	—	4	1	no	—
11	D₈₉₂	—	2	2	no	—
12	D₈₉₂	—	4	1	no	—
13	C₂₂₃⋊C₈	—	4	1	no	—
14	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
15	C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₈×C₂₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₂₂₃	SD₁₆	1	1	yes	characteristic
18	C₂×D₂₂₃	—	4	2	no	—
19	C₂×D₂₂₃	—	8	2	no	—
20	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
21	C₄×C₂₂₃	D₈	1	1	yes	characteristic
22	C₂×C₂₂₃	—	2	1	no	—
23	C₂×C₂₂₃	#102	1	1	yes	characteristic
24	D₂₂₃	—	8	2	no	—
25	C₂₂₃	#361	1	1	yes	characteristic, complemented, Sylow
26	#361	—	223	1	no	maximal, Sylow
27	C₂×D₈	—	223	1	no	—
28	M₃₂	D₂₂₃	1	1	yes	characteristic, complemented
29	#108	—	223	1	no	—
30	C₂×D₄	—	446	1	no	—
31	D₈	—	446	1	no	—
32	D₈	—	223	2	no	—
33	M₁₆	—	446	1	no	—
34	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
35	C₁₆	—	2	1	no	—
36	C₂³	—	892	1	no	—
37	D₄	—	892	1	no	—
38	D₄	—	446	2	no	—
39	C₂×C₄	D₈₉₂	1	1	yes	characteristic
40	C₈	—	892	1	no	—
41	C₈	C₄×D₂₂₃	1	1	yes	characteristic
42	C₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
43	C₂²	—	892	2	no	—
44	C₂²	—	1784	2	no	—
45	C₂²	#16813159	1	1	yes	characteristic
46	C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
47	C₄	D₁₇₈₄	1	1	yes	characteristic
48	C₂	—	2	1	no	—
49	C₂	—	1784	2	no	—
50	C₂	#20297226	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20297226
commutator	C₈×C₂₂₃	C₂×C₄
Frattini	C₂×C₈	C₂×D₂₂₃
Fitting	M₃₂×C₂₂₃	C₂
socle	C₂×C₂₂₃	#102
prime radical	C₂×D₁₇₈₄	C₂
cosocle	C₂×C₈×C₂₂₃	C₂²
hypercenter	M₃₂	D₂₂₃
hypocenter	C₂₂₃	#361
LCS	G ▷ C₈×C₂₂₃ ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #20297240	C₂	semidirect
3. M₃₂×C₂₂₃	C₂	semidirect
25. C₂₂₃	#361	semidirect
28. M₃₂	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₇₈₄	2
#20297240	2
M₃₂×C₂₂₃	2
#361	223