Subgroups — #28383619

Order 14272.

118 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	#20297262	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×D₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#20297238	C₂	1	1	yes	maximal, characteristic, complemented
5	#20297237	C₂	1	1	yes	maximal, characteristic, complemented
6	#20297235	C₂	1	1	yes	maximal, characteristic, complemented
7	#106×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	#16813153	C₄	1	4	yes	complemented
11	C₂×C₂₂₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
14	Dic₂₂₃⋊C₄	—	2	1	no	—
15	(C₂×D₂₂₃)⋊C₄	—	2	1	no	—
16	C₄×Dic₂₂₃	—	2	1	no	—
17	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₂₂₃	C₂²	1	1	yes	characteristic
20	C₂²×D₂₂₃	D₄	1	1	yes	characteristic
21	C₂×Dic₂₂₃	—	2	1	no	—
22	C₂×Dic₂₂₃	—	2	1	no	—
23	C₂×Dic₂₂₃	D₄	1	1	yes	characteristic
24	C₄×D₂₂₃	—	4	2	no	—
25	C₄×D₂₂₃	C₂×C₄	1	2	yes	—
26	C₄×D₂₂₃	Q₈	1	2	yes	—
27	C₂₂₃⋊C₈	C₂×C₄	1	2	yes	complemented
28	C₂×C₄×C₂₂₃	—	2	1	no	—
29	C₂×C₄×C₂₂₃	—	2	1	no	—
30	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
31	C₈×C₂₂₃	C₂×C₄	1	2	yes	complemented
32	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×D₂₂₃	—	2	1	no	—
34	C₂×D₂₂₃	—	2	1	no	—
35	C₂×D₂₂₃	C₄⋊C₄	1	2	yes	—
36	C₄×C₂₂₃	—	4	1	no	—
37	C₄×C₂₂₃	—	4	1	no	—
38	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₂₂₃	C₂×Q₈	1	1	yes	characteristic
40	Dic₂₂₃	—	4	1	no	—
41	Dic₂₂₃	—	4	1	no	—
42	Dic₂₂₃	C₄⋊C₄	1	2	yes	—
43	C₂×C₂₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₂₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
45	C₂×C₂₂₃	#137	1	1	yes	characteristic
46	D₂₂₃	—	2	2	no	—
47	C₂₂₃	#428	1	1	yes	characteristic, complemented, Sylow
48	#428	—	223	1	no	maximal, Sylow
49	C₂×M₁₆	—	223	1	no	—
50	#117	—	223	1	no	—
51	C₂×C₄⋊C₄	—	223	1	no	—
52	#107	—	223	1	no	—
53	#107	—	223	1	no	—
54	#106	—	223	1	no	—
55	#106	D₂₂₃	1	1	yes	characteristic, complemented
56	C₂²×C₄	—	446	1	no	—
57	C₂²×C₄	—	223	1	no	—
58	M₁₆	—	223	4	no	—
59	C₂×C₈	—	223	1	no	—
60	C₂×C₈	C₂×D₂₂₃	1	1	yes	characteristic
61	C₄⋊C₄	—	446	1	no	—
62	C₄⋊C₄	—	223	1	no	—
63	C₄⋊C₄	—	223	1	no	—
64	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
65	C₄⋊C₄	C₂×D₂₂₃	1	1	yes	characteristic
66	C₂²⋊C₄	—	446	1	no	—
67	C₄²	—	446	1	no	—
68	C₂³	—	223	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	892	2	no	—
72	C₂×C₄	—	446	1	no	—
73	C₂×C₄	—	446	1	no	—
74	C₂×C₄	—	223	1	no	—
75	C₂×C₄	—	223	2	no	—
76	C₂×C₄	—	223	2	no	—
77	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
78	C₈	—	223	2	no	—
79	C₈	C₄×D₂₂₃	1	2	yes	complemented
80	C₂²	—	446	1	no	—
81	C₂²	—	446	1	no	—
82	C₂²	—	223	2	no	—
83	C₂²	D₄×D₂₂₃	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	892	1	no	—
87	C₄	—	892	1	no	—
88	C₄	—	223	2	no	—
89	C₄	C₂×C₄×D₂₂₃	1	1	yes	characteristic
90	C₄	Q₈×D₂₂₃	1	1	yes	characteristic
91	C₂	—	446	2	no	—
92	C₂	C₄⋊C₄×D₂₂₃	1	1	yes	characteristic
93	C₂	#20297331	1	1	yes	characteristic
94	C₂	#20297333	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#106×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂×D₄
prime radical	C₂²×D₂₂₃	D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	#106	D₂₂₃
hypocenter	C₂₂₃	#428
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
4. #20297238	C₂	semidirect
5. #20297237	C₂	semidirect
6. #20297235	C₂	semidirect
7. #106×C₂₂₃	C₂	semidirect
10. #16813153	C₄	semidirect
27. C₂₂₃⋊C₈	C₂×C₄	semidirect
31. C₈×C₂₂₃	C₂×C₄	semidirect
47. C₂₂₃	#428	semidirect
55. #106	D₂₂₃	semidirect
79. 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
#20297262	2
C₄⋊C₄×D₂₂₃	2
#20297238	2
#20297237	2
#20297235	2
#106×C₂₂₃	2
#428	223