Subgroups — #28383696

Order 14272.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297326	C₂	1	1	yes	maximal, characteristic
2	#20297312	C₂	1	2	yes	maximal, complemented
3	#20297230	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#108×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813170	C₂²	1	2	yes	—
7	Dic₈₉₂	C₄	1	1	yes	characteristic, complemented
8	#16813159	C₄	1	2	yes	complemented
9	D₁₇₈₄	C₄	1	1	yes	characteristic, complemented
10	#16813153	—	2	2	no	—
11	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₈×D₂₂₃	—	2	2	no	—
13	C₄×Dic₂₂₃	C₂²	1	1	yes	characteristic
14	M₁₆×C₂₂₃	C₂²	1	2	yes	—
15	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
16	Dic₄₄₆	C₂×C₄	1	2	yes	—
17	D₈₉₂	C₂×C₄	1	2	yes	—
18	C₂²⋊D₂₂₃	—	2	2	no	—
19	C₂×Dic₂₂₃	—	2	1	no	—
20	C₄×D₂₂₃	—	2	2	no	—
21	C₂₂₃⋊C₈	D₄	1	1	yes	characteristic
22	C₂₂₃⋊C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	C₈×C₂₂₃	—	2	2	no	—
25	C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
26	C₈×C₂₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₂₂₃	#41	1	1	yes	characteristic
28	C₂×D₂₂₃	—	2	2	no	—
29	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
31	Dic₂₂₃	—	2	2	no	—
32	Dic₂₂₃	—	2	2	no	—
33	C₂×C₂₂₃	—	2	1	no	—
34	C₂×C₂₂₃	C₄×D₄	1	1	yes	characteristic
35	D₂₂₃	—	4	2	no	—
36	C₂₂₃	#443	1	1	yes	characteristic, complemented, Sylow
37	#443	—	223	1	no	maximal, Sylow
38	#135	—	223	1	no	—
39	#131	—	223	2	no	—
40	#108	D₂₂₃	1	1	yes	characteristic, complemented
41	#104	—	223	2	no	—
42	C₄×C₈	—	223	1	no	—
43	#41	—	223	2	no	—
44	Q₁₆	—	223	1	no	—
45	SD₁₆	—	223	2	no	—
46	D₈	—	223	1	no	—
47	M₁₆	—	446	2	no	—
48	M₁₆	C₂×D₂₂₃	1	2	yes	—
49	C₂×C₈	—	446	2	no	—
50	C₂×C₈	—	223	1	no	—
51	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
52	C₄²	—	223	1	no	—
53	Q₈	—	223	2	no	—
54	D₄	—	446	2	no	—
55	D₄	—	223	2	no	—
56	C₂×C₄	—	446	1	no	—
57	C₂×C₄	—	446	2	no	—
58	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	—	223	1	no	—
61	C₈	—	223	1	no	—
62	C₈	C₄×D₂₂₃	1	1	yes	characteristic
63	C₈	C₄×D₂₂₃	1	1	yes	characteristic
64	C₂²	—	446	2	no	—
65	C₂²	#16813171	1	1	yes	characteristic
66	C₄	—	446	2	no	—
67	C₄	—	446	2	no	—
68	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
69	C₄	D₄×D₂₂₃	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	892	2	no	—
72	C₂	#20297266	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#108×C₂₂₃	C₂
socle	C₂×C₂₂₃	C₄×D₄
prime radical	#20297326	C₂
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	#108	D₂₂₃
hypocenter	C₂₂₃	#443
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
2. #20297312	C₂	semidirect
3. #20297230	C₂	semidirect
4. C₈×Dic₂₂₃	C₂	semidirect
5. #108×C₂₂₃	C₂	semidirect
7. Dic₈₉₂	C₄	semidirect
8. #16813159	C₄	semidirect
9. D₁₇₈₄	C₄	semidirect
36. C₂₂₃	#443	semidirect
40. #108	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297326	2
#20297312	2
#20297230	2
C₈×Dic₂₂₃	2
#108×C₂₂₃	2
#443	223