Subgroups — C₂×#20297228 #28383571

Order 14272.

162 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	#20297228	C₂	1	4	yes	maximal, direct product factor
4	C₂²×Q₈×C₂₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₂₃	—	2	1	no	—
6	C₂×C₂₂₃⋊C₈	—	2	2	no	—
7	C₂×C₂₂₃⋊C₈	C₂²	1	2	yes	—
8	C₄⋊Dic₂₂₃	—	2	1	no	—
9	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
10	C₂×Q₈×C₂₂₃	—	2	2	no	—
11	C₂×Q₈×C₂₂₃	—	2	1	no	—
12	C₂×Q₈×C₂₂₃	C₄	1	4	yes	complemented
13	C₂×Q₈×C₂₂₃	C₂²	1	2	yes	—
14	C₂²×C₄×C₂₂₃	—	2	1	no	—
15	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₂₃	—	2	2	no	—
17	C₂×Dic₂₂₃	—	4	2	no	—
18	C₂₂₃⋊C₈	—	2	2	no	—
19	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
20	Q₈×C₂₂₃	—	2	4	no	—
21	Q₈×C₂₂₃	—	2	2	no	—
22	Q₈×C₂₂₃	C₂×C₄	1	4	yes	complemented
23	C₂×C₄×C₂₂₃	—	2	4	no	—
24	C₂×C₄×C₂₂₃	—	2	2	no	—
25	C₂×C₄×C₂₂₃	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	D₄	1	2	yes	—
28	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
29	C₂²×C₂₂₃	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₂₂₃	SD₁₆	1	2	yes	—
31	C₂²×C₂₂₃	Q₁₆	1	2	yes	—
32	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₂₂₃	—	2	4	no	—
34	C₄×C₂₂₃	C₂²⋊C₄	1	2	yes	—
35	C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
36	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
37	Dic₂₂₃	—	4	2	no	—
38	C₂×C₂₂₃	#103	1	4	yes	complemented
39	C₂×C₂₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₂₂₃	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₂₂₃	C₂×Q₁₆	1	1	yes	characteristic
42	C₂₂₃	C₂×#103	1	1	yes	characteristic, complemented, Sylow
43	C₂×#103	—	223	1	no	maximal, Sylow
44	C₂²×Q₈	D₂₂₃	1	1	yes	characteristic
45	C₂²×C₈	—	223	1	no	—
46	C₂×C₄⋊C₄	—	223	1	no	—
47	#103	—	223	4	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	—	2	1	no	—
50	C₂×Q₈	Dic₂₂₃	1	4	yes	complemented
51	C₂×Q₈	C₂×D₂₂₃	1	2	yes	—
52	C₂²×C₄	—	2	1	no	—
53	C₂²×C₄	—	446	1	no	—
54	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
55	C₂×C₈	—	446	2	no	—
56	C₂×C₈	—	223	2	no	—
57	C₄⋊C₄	—	446	1	no	—
58	C₄⋊C₄	—	223	2	no	—
59	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
60	Q₈	—	2	4	no	—
61	Q₈	—	2	2	no	—
62	Q₈	C₂×Dic₂₂₃	1	4	yes	complemented
63	C₂×C₄	—	2	4	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	892	2	no	—
66	C₂×C₄	—	446	2	no	—
67	C₂×C₄	C₂×Dic₂₂₃	1	2	yes	—
68	C₂×C₄	C₂²⋊D₂₂₃	1	1	yes	characteristic
69	C₂×C₄	C₂²⋊D₂₂₃	1	2	yes	—
70	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
71	C₈	—	446	2	no	—
72	C₂²	C₂²⋊Dic₂₂₃	1	2	yes	—
73	C₂²	#16813157	1	2	yes	—
74	C₂²	#16813160	1	2	yes	—
75	C₂²	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	892	2	no	—
78	C₄	C₂²⋊Dic₂₂₃	1	2	yes	—
79	C₄	C₂²×Dic₂₂₃	1	1	yes	characteristic
80	C₄	C₂×C₂²⋊D₂₂₃	1	1	yes	characteristic
81	C₂	#20297228	1	4	yes	direct product factor
82	C₂	C₂×C₂²⋊Dic₂₂₃	1	1	yes	characteristic
83	C₂	C₂×#16813157	1	1	yes	characteristic
84	C₂	C₂×#16813160	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₂₂₃
commutator	C₄×C₂₂₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	C₂²×Q₈×C₂₂₃	C₂
socle, prime radical	C₂³×C₂₂₃	D₄
cosocle	C₂×C₄×C₂₂₃	C₂³
hypercenter	C₂²×Q₈	D₂₂₃
hypocenter	C₂₂₃	C₂×#103
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
3. #20297228	C₂	direct (N × H)
12. C₂×Q₈×C₂₂₃	C₄	semidirect
22. Q₈×C₂₂₃	C₂×C₄	semidirect
38. C₂×C₂₂₃	#103	semidirect
42. C₂₂₃	C₂×#103	semidirect
50. C₂×Q₈	Dic₂₂₃	semidirect
62. Q₈	C₂×Dic₂₂₃	semidirect
81. C₂	#20297228	direct (N × H)

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
#20297228	2
C₂²×Q₈×C₂₂₃	2
C₂×#103	223

Subgroups — C2×#20297228 #28383571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20297228 #28383571