Subgroups — #470×C₂₂₃ #28383182

Order 14272.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊Q₈×C₂₂₃	C₂	1	2	yes	maximal
4	#106×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#103×C₂₂₃	C₂	1	2	yes	maximal, complemented
6	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
7	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
8	Q₁₆×C₂₂₃	—	2	2	no	—
9	M₁₆×C₂₂₃	—	2	2	no	—
10	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₂₂₃	—	2	2	no	—
13	C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
14	C₂²⋊C₄×C₂₂₃	—	2	2	no	—
15	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
16	Q₈×C₂₂₃	—	2	2	no	—
17	Q₈×C₂₂₃	—	2	2	no	—
18	C₂×C₄×C₂₂₃	—	2	1	no	—
19	C₂×C₄×C₂₂₃	—	2	1	no	—
20	C₂×C₄×C₂₂₃	—	2	2	no	—
21	C₂×C₄×C₂₂₃	—	2	2	no	—
22	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
24	C₈×C₂₂₃	—	2	1	no	—
25	C₈×C₂₂₃	D₄	1	2	yes	—
26	C₂²×C₂₂₃	—	2	1	no	—
27	C₂²×C₂₂₃	—	2	2	no	—
28	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
29	C₄×C₂₂₃	—	2	1	no	—
30	C₄×C₂₂₃	—	4	2	no	—
31	C₄×C₂₂₃	—	4	2	no	—
32	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₂₃	#41	1	1	yes	characteristic
34	C₂×C₂₂₃	—	4	1	no	—
35	C₂×C₂₂₃	#121	1	1	yes	characteristic
36	C₂×C₂₂₃	#137	1	2	yes	—
37	C₂₂₃	#470	1	1	yes	characteristic, direct product factor, Sylow
38	#470	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×Q₁₆	C₂×C₂₂₃	1	1	yes	characteristic, complemented
40	C₂×M₁₆	C₂×C₂₂₃	1	1	yes	characteristic
41	C₂²⋊Q₈	C₂×C₂₂₃	1	2	yes	—
42	#106	C₂×C₂₂₃	1	1	yes	characteristic, complemented
43	#103	C₂×C₂₂₃	1	2	yes	complemented
44	C₂×Q₈	C₂²×C₂₂₃	1	2	yes	—
45	C₂²×C₄	C₂²×C₂₂₃	1	1	yes	characteristic
46	Q₁₆	—	2	2	no	—
47	M₁₆	—	2	2	no	—
48	C₂×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
49	C₂×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
50	C₄⋊C₄	—	2	2	no	—
51	C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
52	C₂²⋊C₄	—	2	2	no	—
53	C₂³	D₄×C₂₂₃	1	1	yes	characteristic
54	Q₈	—	2	2	no	—
55	Q₈	—	2	2	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	D₄×C₂₂₃	1	1	yes	characteristic
61	C₂×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
62	C₈	—	2	1	no	—
63	C₈	D₄×C₂₂₃	1	2	yes	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	2	no	—
66	C₂²	C₂×D₄×C₂₂₃	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	4	2	no	—
69	C₄	—	4	2	no	—
70	C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
71	C₄	#41×C₂₂₃	1	1	yes	characteristic
72	C₂	—	4	1	no	—
73	C₂	#121×C₂₂₃	1	1	yes	characteristic
74	C₂	#137×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₂₂₃
prime radical	C₂³×C₂₂₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₂²×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₁₆×C₂₂₃	C₂	semidirect
4. #106×C₂₂₃	C₂	semidirect
5. #103×C₂₂₃	C₂	semidirect
37. C₂₂₃	#470	direct (N × H)
38. #470	C₂₂₃	direct (N × H)
39. C₂×Q₁₆	C₂×C₂₂₃	semidirect
42. #106	C₂×C₂₂₃	semidirect
43. #103	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₁₆×C₂₂₃	2
C₂×M₁₆×C₂₂₃	2
C₂²⋊Q₈×C₂₂₃	2
#106×C₂₂₃	2
#103×C₂₂₃	2
#470	223

Subgroups — #470×C223 #28383182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #470×C₂₂₃ #28383182