Subgroups — #28383336

Order 14272.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16813154	C₂	1	1	yes	maximal, characteristic
2	#20297263	C₂	1	1	yes	maximal, characteristic
3	#117×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	#16813154	—	2	1	no	—
5	#16813154	—	2	1	no	—
6	C₂×C₂₂₃⋊C₈	C₄	1	2	yes	complemented
7	C₄⋊Dic₂₂₃	—	2	1	no	—
8	C₂²⋊Dic₂₂₃	—	2	1	no	—
9	C₄×Dic₂₂₃	C₄	1	2	yes	complemented
10	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×C₂₂₃	—	2	1	no	—
12	C₂²⋊C₄×C₂₂₃	—	2	1	no	—
13	C₄²×C₂₂₃	C₄	1	2	yes	complemented
14	C₂×Dic₂₂₃	—	2	2	no	—
15	C₂₂₃⋊C₈	—	2	2	no	—
16	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
17	C₂×C₄×C₂₂₃	—	2	2	no	—
18	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
22	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	Q₈	1	1	yes	characteristic
24	C₂²×C₂₂₃	—	4	1	no	—
25	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
28	C₄×C₂₂₃	—	4	2	no	—
29	C₄×C₂₂₃	C₄²	1	1	yes	characteristic
30	C₄×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
32	C₄×C₂₂₃	C₄⋊C₄	1	1	yes	characteristic
33	Dic₂₂₃	—	4	2	no	—
34	C₂×C₂₂₃	—	2	1	no	—
35	C₂×C₂₂₃	—	2	1	no	—
36	C₂×C₂₂₃	—	2	1	no	—
37	C₂×C₂₂₃	#95	1	1	yes	characteristic
38	C₂₂₃	H(3,ℤ/4)	1	1	yes	characteristic, complemented, Sylow
39	H(3,ℤ/4)	—	223	1	no	maximal, Sylow
40	C₂×M₁₆	—	223	1	no	—
41	#117	—	223	1	no	—
42	#117	D₂₂₃	1	1	yes	characteristic
43	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
44	M₁₆	—	446	1	no	—
45	M₁₆	—	446	1	no	—
46	C₂×C₈	—	223	2	no	—
47	C₄⋊C₄	—	2	1	no	—
48	C₄⋊C₄	—	446	1	no	—
49	C₂²⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	—	446	1	no	—
51	C₄²	—	223	2	no	—
52	C₄²	Dic₂₂₃	1	2	yes	complemented
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	1	yes	characteristic
57	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
58	C₂×C₄	C₂×Dic₂₂₃	1	1	yes	characteristic
59	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
60	C₂×C₄	D₈₉₂	1	1	yes	characteristic
61	C₂×C₄	Dic₄₄₆	1	1	yes	characteristic
62	C₈	—	446	2	no	—
63	C₂²	—	4	1	no	—
64	C₂²	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
65	C₂²	C₂²⋊Dic₂₂₃	1	1	yes	characteristic
66	C₂²	Dic₂₂₃⋊C₄	1	1	yes	characteristic
67	C₄	—	4	2	no	—
68	C₄	—	892	2	no	—
69	C₄	C₄×Dic₂₂₃	1	1	yes	characteristic
70	C₄	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
71	C₄	Dic₂₂₃⋊C₄	1	1	yes	characteristic
72	C₄	C₄⋊Dic₂₂₃	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	2	1	no	—
75	C₂	—	2	1	no	—
76	C₂	C₂₂₃⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×Dic₂₂₃
commutator	C₄×C₂₂₃	C₄²
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting	#117×C₂₂₃	C₂
socle	C₂×C₂₂₃	#95
prime radical	C₂³×C₂₂₃	D₄
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	#117	D₂₂₃
hypocenter	C₂₂₃	H(3,ℤ/4)
LCS	G ▷ C₄×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
6. C₂×C₂₂₃⋊C₈	C₄	semidirect
9. C₄×Dic₂₂₃	C₄	semidirect
13. C₄²×C₂₂₃	C₄	semidirect
38. C₂₂₃	H(3,ℤ/4)	semidirect
52. C₄²	Dic₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16813154	2
#20297263	2
#117×C₂₂₃	2
H(3,ℤ/4)	223