Subgroups — C₂×#20297261 #28384034

Order 14272.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#20297261	C₂	1	8	yes	maximal, direct product factor
3	C₂×Dic₂₂₃⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂×(C₂×D₂₂₃)⋊C₄	C₂	1	2	yes	maximal
5	C₂×C₄×Dic₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄²×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₂₂₃	C₂²	1	1	yes	characteristic
8	C₂²×Dic₂₂₃	C₂²	1	2	yes	—
9	C₂²×Dic₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₂₂₃	C₄	1	8	yes	complemented
11	C₂×C₄×D₂₂₃	C₂²	1	4	yes	—
12	Dic₂₂₃⋊C₄	C₂²	1	8	yes	complemented
13	(C₂×D₂₂₃)⋊C₄	C₂²	1	8	yes	—
14	C₄×Dic₂₂₃	C₂²	1	4	yes	complemented
15	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	—
16	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
17	C₄²×C₂₂₃	C₂²	1	4	yes	complemented
18	C₂²×D₂₂₃	—	2	4	no	—
19	C₂²×D₂₂₃	C₂×C₄	1	4	yes	—
20	C₂²×D₂₂₃	C₂³	1	2	yes	—
21	C₂×Dic₂₂₃	—	2	4	no	—
22	C₂×Dic₂₂₃	C₂×C₄	1	4	yes	—
23	C₂×Dic₂₂₃	C₂³	1	4	yes	—
24	C₂×Dic₂₂₃	C₂³	1	2	yes	—
25	C₄×D₂₂₃	C₂×C₄	1	16	yes	complemented
26	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₂₂₃	—	2	4	no	—
28	C₂×C₄×C₂₂₃	C₂×C₄	1	4	yes	—
29	C₂×C₄×C₂₂₃	C₂³	1	4	yes	—
30	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
31	C₂²×C₂₂₃	C₂²×C₄	1	2	yes	—
32	C₂²×C₂₂₃	#41	1	4	yes	—
33	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
34	C₂×D₂₂₃	—	2	8	no	—
35	C₂×D₂₂₃	—	2	4	no	—
36	C₂×D₂₂₃	C₂²×C₄	1	4	yes	—
37	C₄×C₂₂₃	—	2	4	no	—
38	C₄×C₂₂₃	C₂²×C₄	1	4	yes	—
39	Dic₂₂₃	—	2	4	no	—
40	Dic₂₂₃	C₂²×C₄	1	4	yes	—
41	C₂×C₂₂₃	#117	1	4	yes	complemented
42	C₂×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
43	C₂×C₂₂₃	C₂×#41	1	2	yes	—
44	D₂₂₃	—	2	4	no	—
45	C₂₂₃	C₂×#117	1	1	yes	characteristic, complemented, Sylow
46	C₂×#117	—	223	1	no	maximal, Sylow
47	C₂³×C₄	—	223	1	no	—
48	#117	—	223	8	no	—
49	C₂×C₄⋊C₄	—	223	2	no	—
50	C₂×C₂²⋊C₄	—	223	2	no	—
51	C₂×C₄²	—	223	1	no	—
52	C₂×C₄²	D₂₂₃	1	1	yes	characteristic, complemented
53	C₂⁴	—	223	1	no	—
54	C₂²×C₄	—	223	2	no	—
55	C₂²×C₄	—	223	1	no	—
56	C₂²×C₄	—	223	8	no	—
57	C₂²×C₄	—	223	4	no	—
58	C₂²×C₄	C₂×D₂₂₃	1	2	yes	—
59	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
60	C₄⋊C₄	—	223	8	no	—
61	C₂²⋊C₄	—	223	8	no	—
62	C₄²	—	223	4	no	—
63	C₄²	C₂×D₂₂₃	1	4	yes	complemented
64	C₂³	—	446	4	no	—
65	C₂³	—	223	4	no	—
66	C₂³	—	223	2	no	—
67	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
68	C₂×C₄	—	2	4	no	—
69	C₂×C₄	—	446	4	no	—
70	C₂×C₄	—	223	4	no	—
71	C₂×C₄	—	223	4	no	—
72	C₂×C₄	—	223	2	no	—
73	C₂×C₄	—	223	16	no	—
74	C₂×C₄	C₄×D₂₂₃	1	4	yes	—
75	C₂×C₄	C₂²×D₂₂₃	1	4	yes	—
76	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
77	C₂²	—	446	8	no	—
78	C₂²	—	446	4	no	—
79	C₂²	—	223	4	no	—
80	C₂²	C₂×C₄×D₂₂₃	1	2	yes	—
81	C₂²	#16813170	1	4	yes	—
82	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
83	C₄	—	2	4	no	—
84	C₄	—	446	4	no	—
85	C₄	—	223	4	no	—
86	C₄	C₂×C₄×D₂₂₃	1	4	yes	—
87	C₂	—	446	4	no	—
88	C₂	#20297261	1	4	yes	direct product factor
89	C₂	C₂²×C₄×D₂₂₃	1	1	yes	characteristic
90	C₂	C₂×#16813170	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₂₃
commutator	C₂×C₂₂₃	C₂³×C₄
Frattini	C₂²	C₂³×D₂₂₃
Fitting	C₂×C₄²×C₂₂₃	C₂
socle	C₂³×C₂₂₃	C₂³
prime radical	C₂³×D₂₂₃	C₂²
cosocle	C₂²×C₂₂₃	C₂⁴
hypercenter	C₂×C₄²	D₂₂₃
hypocenter	C₂₂₃	C₂×#117
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #20297261	C₂	direct (N × H)
3. C₂×Dic₂₂₃⋊C₄	C₂	semidirect
5. C₂×C₄×Dic₂₂₃	C₂	semidirect
6. C₂×C₄²×C₂₂₃	C₂	semidirect
10. C₂×C₄×D₂₂₃	C₄	semidirect
12. Dic₂₂₃⋊C₄	C₂²	semidirect
14. C₄×Dic₂₂₃	C₂²	semidirect
17. C₄²×C₂₂₃	C₂²	semidirect
25. C₄×D₂₂₃	C₂×C₄	semidirect
41. C₂×C₂₂₃	#117	semidirect
45. C₂₂₃	C₂×#117	semidirect
52. C₂×C₄²	D₂₂₃	semidirect
63. C₄²	C₂×D₂₂₃	semidirect
88. C₂	#20297261	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×C₄×D₂₂₃	2
#20297261	2
C₂×Dic₂₂₃⋊C₄	2
C₂×(C₂×D₂₂₃)⋊C₄	2
C₂×C₄×Dic₂₂₃	2
C₂×C₄²×C₂₂₃	2
C₂×#117	223

Subgroups — C2×#20297261 #28384034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20297261 #28384034