Subgroups — #28383914

Order 14272.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297280	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×D₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#20297238	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297237	C₂	1	1	yes	maximal, characteristic, complemented
5	#20297228	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₂₂₃)⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#103×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₂₂₃	—	2	1	no	—
9	C₂×C₄×D₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
13	Dic₂₂₃⋊C₄	—	2	1	no	—
14	Dic₂₂₃⋊C₄	—	2	1	no	—
15	(C₂×D₂₂₃)⋊C₄	—	2	1	no	—
16	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₂²×D₂₂₃	D₄	1	1	yes	characteristic
20	C₂×Dic₂₂₃	—	2	1	no	—
21	C₂×Dic₂₂₃	—	2	1	no	—
22	C₂×Dic₂₂₃	D₄	1	1	yes	characteristic
23	C₄×D₂₂₃	—	2	1	no	—
24	C₄×D₂₂₃	—	2	1	no	—
25	C₄×D₂₂₃	—	4	2	no	—
26	C₂₂₃⋊C₈	—	2	1	no	—
27	Q₈×C₂₂₃	—	2	1	no	—
28	Q₈×C₂₂₃	—	2	1	no	—
29	C₂×C₄×C₂₂₃	—	2	1	no	—
30	C₂×C₄×C₂₂₃	—	2	1	no	—
31	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
32	C₈×C₂₂₃	—	2	1	no	—
33	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
34	C₂×D₂₂₃	—	2	1	no	—
35	C₂×D₂₂₃	—	2	1	no	—
36	C₂×D₂₂₃	Q₁₆	1	2	yes	—
37	C₄×C₂₂₃	—	4	1	no	—
38	C₄×C₂₂₃	—	4	1	no	—
39	C₄×C₂₂₃	#41	1	1	yes	characteristic
40	C₄×C₂₂₃	#41	1	1	yes	characteristic
41	Dic₂₂₃	—	2	1	no	—
42	Dic₂₂₃	—	4	1	no	—
43	Dic₂₂₃	—	4	1	no	—
44	C₂×C₂₂₃	#123	1	1	yes	characteristic
45	C₂×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
46	C₂×C₂₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
47	D₂₂₃	—	2	2	no	—
48	C₂₂₃	#484	1	1	yes	characteristic, complemented, Sylow
49	#484	—	223	1	no	maximal, Sylow
50	C₂²⋊Q₈	—	223	1	no	—
51	C₂×C₄⋊C₄	—	223	1	no	—
52	#107	—	223	1	no	—
53	#107	—	223	1	no	—
54	#103	—	223	1	no	—
55	#103	D₂₂₃	1	1	yes	characteristic, complemented
56	C₂²⋊C₈	—	223	1	no	—
57	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
58	C₂²×C₄	—	446	1	no	—
59	C₂²×C₄	—	223	1	no	—
60	C₂×C₈	—	223	1	no	—
61	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
62	C₄⋊C₄	—	446	1	no	—
63	C₄⋊C₄	—	446	1	no	—
64	C₄⋊C₄	—	223	1	no	—
65	C₄⋊C₄	—	223	1	no	—
66	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
67	C₂²⋊C₄	—	446	1	no	—
68	C₂³	—	223	1	no	—
69	Q₈	—	2	1	no	—
70	Q₈	—	2	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	892	2	no	—
74	C₂×C₄	—	446	1	no	—
75	C₂×C₄	—	446	1	no	—
76	C₂×C₄	—	446	1	no	—
77	C₂×C₄	—	446	1	no	—
78	C₂×C₄	—	223	1	no	—
79	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	446	1	no	—
82	C₂²	—	446	1	no	—
83	C₂²	—	446	1	no	—
84	C₂²	—	223	2	no	—
85	C₂²	D₄×D₂₂₃	1	1	yes	characteristic
86	C₄	—	4	1	no	—
87	C₄	—	4	1	no	—
88	C₄	—	892	1	no	—
89	C₄	—	892	1	no	—
90	C₄	—	446	1	no	—
91	C₄	#16813169	1	1	yes	characteristic
92	C₄	#16813170	1	1	yes	characteristic
93	C₂	—	446	2	no	—
94	C₂	#20297286	1	1	yes	characteristic
95	C₂	Q₁₆×D₂₂₃	1	1	yes	characteristic
96	C₂	#20297331	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₂₃
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#103×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂²×D₂₂₃	D₄
hypercenter	#103	D₂₂₃
hypocenter	C₂₂₃	#484
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
3. #20297238	C₂	semidirect
4. #20297237	C₂	semidirect
5. #20297228	C₂	semidirect
7. #103×C₂₂₃	C₂	semidirect
48. C₂₂₃	#484	semidirect
55. #103	D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297280	2
C₄⋊C₄×D₂₂₃	2
#20297238	2
#20297237	2
#20297228	2
(C₂×D₂₂₃)⋊C₈	2
#103×C₂₂₃	2
#484	223