Subgroups — #28383534

Order 14272.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813153	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#20297261	C₂	1	1	yes	maximal, characteristic
4	#20297215	C₂	1	1	yes	maximal, characteristic, complemented
5	#20297214	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₂₃	C₂²	1	1	yes	characteristic
9	#16813153	C₄	1	4	yes	complemented
10	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₈×D₂₂₃	C₄	1	4	yes	complemented
13	Dic₂₂₃⋊C₄	C₄	1	2	yes	—
14	(C₂×D₂₂₃)⋊C₄	C₄	1	2	yes	—
15	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₂²×D₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₂₃	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₂₃	C₂×C₄	1	1	yes	characteristic
22	C₄×D₂₂₃	C₂×C₄	1	2	yes	—
23	C₄×D₂₂₃	C₂×C₄	1	2	yes	—
24	C₂₂₃⋊C₈	C₂×C₄	1	2	yes	—
25	C₂₂₃⋊C₈	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
28	C₈×C₂₂₃	C₂×C₄	1	2	yes	—
29	C₈×C₂₂₃	C₂×C₄	1	2	yes	—
30	C₂²×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×D₂₂₃	—	2	2	no	—
32	C₂×D₂₂₃	C₄²	1	2	yes	—
33	C₄×C₂₂₃	—	2	2	no	—
34	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
36	Dic₂₂₃	—	2	2	no	—
37	Dic₂₂₃	C₄²	1	2	yes	—
38	C₂×C₂₂₃	C₂×C₄²	1	1	yes	characteristic
39	C₂×C₂₂₃	#131	1	2	yes	—
40	D₂₂₃	—	2	2	no	—
41	C₂₂₃	#405	1	1	yes	characteristic, complemented, Sylow
42	#405	—	223	1	no	maximal, Sylow
43	C₂×M₁₆	—	223	1	no	—
44	C₂²×C₈	—	223	1	no	—
45	#117	—	223	1	no	—
46	#97	—	223	1	no	—
47	#97	—	223	1	no	—
48	C₄×C₈	—	223	1	no	—
49	C₄×C₈	D₂₂₃	1	1	yes	characteristic, complemented
50	C₂²×C₄	—	223	1	no	—
51	M₁₆	—	223	4	no	—
52	C₂×C₈	—	223	1	no	—
53	C₂×C₈	—	223	1	no	—
54	C₂×C₈	—	223	4	no	—
55	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
56	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
57	C₄⋊C₄	—	223	2	no	—
58	C₂²⋊C₄	—	223	2	no	—
59	C₄²	—	223	1	no	—
60	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
61	C₂³	—	223	1	no	—
62	C₂×C₄	—	223	2	no	—
63	C₂×C₄	—	223	1	no	—
64	C₂×C₄	—	223	2	no	—
65	C₂×C₄	—	223	2	no	—
66	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
67	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
68	C₈	—	223	2	no	—
69	C₈	—	223	2	no	—
70	C₈	C₄×D₂₂₃	1	2	yes	—
71	C₈	C₄×D₂₂₃	1	2	yes	—
72	C₂²	—	446	2	no	—
73	C₂²	—	223	2	no	—
74	C₂²	C₂×C₄×D₂₂₃	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	446	2	no	—
77	C₄	—	223	2	no	—
78	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
79	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
80	C₂	—	446	2	no	—
81	C₂	C₄²×D₂₂₃	1	1	yes	characteristic
82	C₂	#20297310	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₄×C₈×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²×C₄
prime radical	C₂²×D₂₂₃	C₂×C₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₄×C₈	D₂₂₃
hypocenter	C₂₂₃	#405
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #20297215	C₂	semidirect
5. #20297214	C₂	semidirect
6. C₈×Dic₂₂₃	C₂	semidirect
7. C₄×C₈×C₂₂₃	C₂	semidirect
9. #16813153	C₄	semidirect
12. C₈×D₂₂₃	C₄	semidirect
41. C₂₂₃	#405	semidirect
49. C₄×C₈	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
C₂×C₈×D₂₂₃	2
#20297261	2
#20297215	2
#20297214	2
C₈×Dic₂₂₃	2
C₄×C₈×C₂₂₃	2
#405	223