Subgroups — #28383342

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	C₂×C₄⋊Dic₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₂₃	—	2	1	no	—
5	C₂×C₂₂₃⋊C₈	—	2	1	no	—
6	C₂×C₂₂₃⋊C₈	—	2	1	no	—
7	C₂×C₂₂₃⋊C₈	C₄	1	2	yes	complemented
8	C₄⋊Dic₂₂₃	—	2	1	no	—
9	C₄⋊Dic₂₂₃	C₄	1	2	yes	complemented
10	C₂²×C₄×C₂₂₃	—	2	1	no	—
11	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₂₃	—	2	1	no	—
13	C₄⋊C₄×C₂₂₃	C₄	1	2	yes	complemented
14	C₂×Dic₂₂₃	—	2	2	no	—
15	C₂×Dic₂₂₃	—	4	1	no	—
16	C₂×Dic₂₂₃	—	4	1	no	—
17	C₂₂₃⋊C₈	—	2	2	no	—
18	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	—	2	2	no	—
20	C₂×C₄×C₂₂₃	—	4	1	no	—
21	C₂×C₄×C₂₂₃	—	4	1	no	—
22	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
28	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂²×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₂₂₃	D₈	1	1	yes	characteristic
32	C₂²×C₂₂₃	SD₁₆	1	1	yes	characteristic
33	C₂²×C₂₂₃	SD₁₆	1	1	yes	characteristic
34	C₂²×C₂₂₃	Q₁₆	1	1	yes	characteristic
35	C₄×C₂₂₃	—	4	2	no	—
36	C₄×C₂₂₃	C₄²	1	1	yes	characteristic
37	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
39	C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
40	Dic₂₂₃	—	4	2	no	—
41	C₂×C₂₂₃	#95	1	1	yes	characteristic
42	C₂×C₂₂₃	#102	1	1	yes	characteristic
43	C₂×C₂₂₃	#102	1	1	yes	characteristic
44	C₂×C₂₂₃	#103	1	1	yes	characteristic
45	C₂×C₂₂₃	#103	1	1	yes	characteristic
46	C₂×C₂₂₃	#106	1	1	yes	characteristic
47	C₂×C₂₂₃	#107	1	1	yes	characteristic
48	C₂₂₃	#340	1	1	yes	characteristic, complemented, Sylow
49	#340	—	223	1	no	maximal, Sylow
50	C₂²×C₈	—	223	1	no	—
51	C₂×C₄⋊C₄	—	223	1	no	—
52	C₂×C₄⋊C₄	D₂₂₃	1	1	yes	characteristic
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	—	446	1	no	—
55	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
56	C₂×C₈	—	446	1	no	—
57	C₂×C₈	—	446	1	no	—
58	C₂×C₈	—	223	2	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	—	446	1	no	—
61	C₄⋊C₄	—	223	2	no	—
62	C₄⋊C₄	Dic₂₂₃	1	2	yes	complemented
63	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	1	no	—
66	C₂×C₄	—	4	1	no	—
67	C₂×C₄	—	892	1	no	—
68	C₂×C₄	—	892	1	no	—
69	C₂×C₄	—	446	2	no	—
70	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
71	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
72	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
73	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
74	C₂×C₄	D₈₉₂	1	1	yes	characteristic
75	C₂×C₄	Dic₄₄₆	1	1	yes	characteristic
76	C₈	—	446	2	no	—
77	C₂²	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
78	C₂²	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
79	C₂²	Dic₂₂₃⋊C₄	1	1	yes	characteristic
80	C₂²	#16813155	1	1	yes	characteristic
81	C₂²	#16813157	1	1	yes	characteristic
82	C₂²	#16813158	1	1	yes	characteristic
83	C₂²	#16813160	1	1	yes	characteristic
84	C₄	—	4	2	no	—
85	C₄	—	892	2	no	—
86	C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
87	C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
88	C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
89	C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
90	C₂	C₂₂₃⋊#95	1	1	yes	characteristic
91	C₂	#20297224	1	1	yes	characteristic
92	C₂	#20297225	1	1	yes	characteristic
93	C₂	#20297227	1	1	yes	characteristic
94	C₂	#20297228	1	1	yes	characteristic
95	C₂	#20297235	1	1	yes	characteristic
96	C₂	#20297237	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₄²
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting	C₂×C₄⋊C₄×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	D₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	C₂×C₄⋊C₄	D₂₂₃
hypocenter	C₂₂₃	#340
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
7. C₂×C₂₂₃⋊C₈	C₄	semidirect
9. C₄⋊Dic₂₂₃	C₄	semidirect
13. C₄⋊C₄×C₂₂₃	C₄	semidirect
48. C₂₂₃	#340	semidirect
62. C₄⋊C₄	Dic₂₂₃	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
C₂×C₄⋊Dic₂₂₃	2
C₂×C₄⋊C₄×C₂₂₃	2
#340	223