Subgroups — #28383532

Order 14272.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813153	C₂	1	2	yes	maximal
2	C₄²×D₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#20297215	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×Dic₂₂₃	C₂	1	2	yes	maximal, complemented
5	#97×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₂₂₃	C₄	1	2	yes	—
7	C₂×C₄×D₂₂₃	C₂²	1	1	yes	characteristic
8	#16813153	C₄	1	8	yes	complemented
9	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₂₂₃	C₄	1	2	yes	—
11	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₂₃	C₂²	1	2	yes	—
13	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂²×D₂₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₂₂₃	C₂×C₄	1	2	yes	—
17	C₄×D₂₂₃	—	2	4	no	—
18	C₄×D₂₂₃	C₂×C₄	1	4	yes	—
19	C₂₂₃⋊C₈	C₂×C₄	1	4	yes	complemented
20	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
22	C₈×C₂₂₃	C₂×C₄	1	4	yes	complemented
23	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
24	C₂×D₂₂₃	—	2	2	no	—
25	C₂×D₂₂₃	C₄²	1	2	yes	—
26	C₄×C₂₂₃	—	2	2	no	—
27	C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
28	Dic₂₂₃	C₄²	1	2	yes	—
29	Dic₂₂₃	M₁₆	1	4	yes	complemented
30	C₂×C₂₂₃	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₂₂₃	C₂×M₁₆	1	2	yes	—
32	D₂₂₃	—	2	2	no	—
33	C₂₂₃	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₄×M₁₆	—	223	1	no	maximal, Sylow
35	C₂×M₁₆	—	223	2	no	—
36	C₂×C₄²	—	223	1	no	—
37	#97	—	223	1	no	—
38	#97	D₂₂₃	1	1	yes	characteristic, complemented
39	C₄×C₈	—	223	2	no	—
40	C₂²×C₄	—	223	2	no	—
41	C₂²×C₄	—	223	1	no	—
42	M₁₆	—	223	8	no	—
43	C₂×C₈	—	223	2	no	—
44	C₂×C₈	C₂×D₂₂₃	1	2	yes	—
45	C₄²	—	223	2	no	—
46	C₄²	—	223	1	no	—
47	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
48	C₂³	—	223	1	no	—
49	C₂×C₄	—	446	4	no	—
50	C₂×C₄	—	223	1	no	—
51	C₂×C₄	—	223	2	no	—
52	C₂×C₄	—	223	4	no	—
53	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
54	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
55	C₈	—	223	4	no	—
56	C₈	C₄×D₂₂₃	1	4	yes	complemented
57	C₂²	—	446	2	no	—
58	C₂²	—	223	2	no	—
59	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	223	2	no	—
62	C₄	—	223	4	no	—
63	C₄	C₂×C₄×D₂₂₃	1	2	yes	—
64	C₂	—	446	2	no	—
65	C₂	C₄²×D₂₂₃	1	1	yes	characteristic
66	C₂	M₁₆×D₂₂₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₂₃
commutator	C₂×C₂₂₃	C₂×C₄²
Fitting	#97×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²×C₄
prime radical	C₂²×D₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	#97	D₂₂₃
hypocenter	C₂₂₃	C₄×M₁₆
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
3. #20297215	C₂	semidirect
4. C₈×Dic₂₂₃	C₂	semidirect
5. #97×C₂₂₃	C₂	semidirect
8. #16813153	C₄	semidirect
19. C₂₂₃⋊C₈	C₂×C₄	semidirect
22. C₈×C₂₂₃	C₂×C₄	semidirect
29. Dic₂₂₃	M₁₆	semidirect
33. C₂₂₃	C₄×M₁₆	semidirect
38. #97	D₂₂₃	semidirect
56. 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×#16813153	2
C₄²×D₂₂₃	2
#20297215	2
C₈×Dic₂₂₃	2
#97×C₂₂₃	2
C₄×M₁₆	223