Subgroups — #28383616

Order 14272.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#20297263	C₂	1	1	yes	maximal, characteristic
3	#20297237	C₂	1	2	yes	maximal, complemented
4	#20297235	C₂	1	2	yes	maximal, complemented
5	#117×C₂₂₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₂₃⋊C₈	C₄	1	4	yes	complemented
7	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
10	C₂²⋊Dic₂₂₃	—	2	1	no	—
11	C₄×Dic₂₂₃	—	2	1	no	—
12	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
15	C₄²×C₂₂₃	—	2	1	no	—
16	C₂×Dic₂₂₃	—	2	2	no	—
17	C₂₂₃⋊C₈	C₂×C₄	1	2	yes	—
18	C₂₂₃⋊C₈	C₂×C₄	1	2	yes	—
19	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	—	2	2	no	—
21	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	Q₈	1	2	yes	—
24	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₂₂₃	—	2	2	no	—
26	C₂²×C₂₂₃	C₄⋊C₄	1	2	yes	—
27	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
28	C₄×C₂₂₃	—	4	2	no	—
29	C₄×C₂₂₃	C₄⋊C₄	1	2	yes	—
30	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
32	Dic₂₂₃	—	4	2	no	—
33	C₂×C₂₂₃	—	2	2	no	—
34	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₂₂₃	#135	1	2	yes	—
36	C₂₂₃	#427	1	1	yes	characteristic, complemented, Sylow
37	#427	—	223	1	no	maximal, Sylow
38	C₂²×C₈	—	223	1	no	—
39	#117	—	223	1	no	—
40	#117	D₂₂₃	1	1	yes	characteristic
41	#107	—	223	2	no	—
42	#106	—	223	2	no	—
43	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
44	C₂×C₈	—	223	4	no	—
45	C₂×C₈	—	223	1	no	—
46	C₂×C₈	—	223	1	no	—
47	C₄⋊C₄	—	223	2	no	—
48	C₄⋊C₄	C₂×D₂₂₃	1	2	yes	—
49	C₂²⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	—	446	1	no	—
51	C₄²	—	2	1	no	—
52	C₄²	—	446	1	no	—
53	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	—	446	2	no	—
56	C₂×C₄	C₄×D₂₂₃	1	2	yes	—
57	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
58	C₂×C₄	Dic₄₄₆	1	2	yes	—
59	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
60	C₈	—	223	2	no	—
61	C₈	—	223	2	no	—
62	C₂²	—	2	2	no	—
63	C₂²	Dic₂₂₃⋊C₄	1	2	yes	—
64	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
65	C₄	—	4	2	no	—
66	C₄	—	892	2	no	—
67	C₄	Dic₂₂₃⋊C₄	1	2	yes	—
68	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
69	C₄	C₂×Dic₄₄₆	1	1	yes	characteristic
70	C₂	—	2	2	no	—
71	C₂	C₂×Dic₂₂₃⋊C₄	1	1	yes	characteristic
72	C₂	#20297324	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#117×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂³×C₂₂₃	D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	#117	D₂₂₃
hypocenter	C₂₂₃	#427
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
3. #20297237	C₂	semidirect
4. #20297235	C₂	semidirect
6. C₂×C₂₂₃⋊C₈	C₄	semidirect
36. C₂₂₃	#427	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₂₃⋊C₈	2
#20297263	2
#20297237	2
#20297235	2
#117×C₂₂₃	2
#427	223