Subgroups — #28383693

Order 14272.

132 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	C₄×D₈₉₂	C₂	1	2	yes	maximal, complemented
3	#20297236	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297226	C₂	1	2	yes	maximal, complemented
5	#97×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₉₂	C₂²	1	2	yes	—
7	C₂×C₄×D₂₂₃	—	2	2	no	—
8	D₁₇₈₄	C₄	1	4	yes	complemented
9	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
10	(C₂×D₂₂₃)⋊C₄	—	2	2	no	—
11	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂²×D₂₂₃	—	2	2	no	—
15	D₈₉₂	—	2	2	no	—
16	D₈₉₂	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₂₂₃	—	2	2	no	—
18	C₄×D₂₂₃	—	4	4	no	—
19	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
20	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
21	C₈×C₂₂₃	—	2	1	no	—
22	C₈×C₂₂₃	C₂×C₄	1	2	yes	complemented
23	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
24	C₂×D₂₂₃	—	2	4	no	—
25	C₂×D₂₂₃	—	4	2	no	—
26	C₂×D₂₂₃	—	4	2	no	—
27	C₄×C₂₂₃	—	2	2	no	—
28	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₂₂₃	#41	1	1	yes	characteristic
30	Dic₂₂₃	—	4	2	no	—
31	C₂×C₂₂₃	C₄×D₄	1	1	yes	characteristic
32	C₂×C₂₂₃	AGL(1,ℤ/8)	1	2	yes	—
33	D₂₂₃	—	4	4	no	—
34	C₂₂₃	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	223	1	no	maximal, Sylow
36	C₂×D₈	—	223	1	no	—
37	C₄×D₄	—	223	2	no	—
38	#106	—	223	1	no	—
39	#102	—	223	2	no	—
40	#97	D₂₂₃	1	1	yes	characteristic, complemented
41	C₂×D₄	—	223	2	no	—
42	C₂²×C₄	—	446	2	no	—
43	D₈	—	223	4	no	—
44	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
45	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
46	C₄⋊C₄	—	223	2	no	—
47	C₂²⋊C₄	—	446	2	no	—
48	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
49	C₂³	—	446	2	no	—
50	D₄	—	446	2	no	—
51	D₄	—	223	4	no	—
52	C₂×C₄	—	892	4	no	—
53	C₂×C₄	—	446	2	no	—
54	C₂×C₄	D₈₉₂	1	2	yes	—
55	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	C₄×D₂₂₃	1	2	yes	complemented
58	C₂²	—	892	2	no	—
59	C₂²	—	892	2	no	—
60	C₂²	—	446	4	no	—
61	C₂²	C₂×D₈₉₂	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	—	892	2	no	—
64	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
65	C₄	#16813170	1	1	yes	characteristic
66	C₂	—	892	4	no	—
67	C₂	C₄×D₈₉₂	1	1	yes	characteristic
68	C₂	#20297332	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₉₂
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#97×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂×D₁₇₈₄	C₂
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	#97	D₂₂₃
hypocenter	C₂₂₃	#442
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. C₄×D₈₉₂	C₂	semidirect
3. #20297236	C₂	semidirect
4. #20297226	C₂	semidirect
5. #97×C₂₂₃	C₂	semidirect
8. D₁₇₈₄	C₄	semidirect
16. D₈₉₂	C₂×C₄	semidirect
22. C₈×C₂₂₃	C₂×C₄	semidirect
34. C₂₂₃	#442	semidirect
40. #97	D₂₂₃	semidirect
57. C₈	C₄×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
C₄×D₈₉₂	2
#20297236	2
#20297226	2
#97×C₂₂₃	2
#442	223