Subgroups — #126×D₂₂₃ #28384103

Order 14272.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297299	C₂	1	3	yes	maximal, complemented
2	#20297298	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297297	C₂	1	3	yes	maximal, complemented
4	C₄⋊C₄×D₂₂₃	C₂	1	3	yes	maximal, complemented
5	C₂²⋊C₄×D₂₂₃	C₂	1	3	yes	maximal
6	C₄²×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#126×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₂₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₂₃	C₂²	1	3	yes	—
10	C₂×C₄×D₂₂₃	C₂²	1	3	yes	—
11	C₄⋊Dic₂₂₃	C₂²	1	3	yes	complemented
12	Dic₂₂₃⋊C₄	C₂²	1	3	yes	complemented
13	Dic₂₂₃⋊C₄	C₂²	1	3	yes	complemented
14	C₂²⋊Dic₂₂₃	C₂²	1	3	yes	—
15	(C₂×D₂₂₃)⋊C₄	C₂²	1	3	yes	—
16	(C₂×D₂₂₃)⋊C₄	C₂²	1	3	yes	—
17	C₄×Dic₂₂₃	C₂²	1	3	yes	complemented
18	C₄⋊C₄×C₂₂₃	C₂²	1	3	yes	complemented
19	C₂²⋊C₄×C₂₂₃	C₂²	1	3	yes	—
20	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₂₂₃	—	2	6	no	—
22	C₂²×D₂₂₃	C₂³	1	1	yes	characteristic
23	C₂²×D₂₂₃	C₂³	1	1	yes	characteristic
24	C₂×Dic₂₂₃	C₂³	1	3	yes	—
25	C₂×Dic₂₂₃	C₂³	1	3	yes	—
26	C₄×D₂₂₃	—	2	6	no	—
27	C₄×D₂₂₃	—	2	6	no	—
28	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
30	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
31	C₂²×C₂₂₃	—	2	3	no	—
32	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
33	C₂×D₂₂₃	—	2	3	no	—
34	C₂×D₂₂₃	—	4	4	no	—
35	C₂×D₂₂₃	#41	1	6	yes	—
36	C₄×C₂₂₃	—	2	3	no	—
37	C₄×C₂₂₃	—	2	3	no	—
38	Dic₂₂₃	—	2	3	no	—
39	Dic₂₂₃	—	2	3	no	—
40	C₂×C₂₂₃	—	4	1	no	—
41	C₂×C₂₂₃	C₂×#41	1	3	yes	—
42	D₂₂₃	—	4	1	no	—
43	D₂₂₃	#126	1	4	yes	direct product factor
44	C₂₂₃	C₂×#126	1	1	yes	characteristic, complemented, Sylow
45	C₂×#126	—	223	1	no	maximal, Sylow
46	#126	—	223	3	no	—
47	#126	—	223	3	no	—
48	#126	—	223	1	no	—
49	#126	D₂₂₃	1	1	yes	characteristic, direct product factor
50	C₂×C₄⋊C₄	—	223	3	no	—
51	C₂×C₂²⋊C₄	—	223	3	no	—
52	C₂×C₄²	—	223	1	no	—
53	C₂⁴	—	223	1	no	—
54	C₂²×C₄	—	223	3	no	—
55	C₂²×C₄	—	223	3	no	—
56	C₄⋊C₄	—	223	3	no	—
57	C₄⋊C₄	—	223	3	no	—
58	C₄⋊C₄	—	223	3	no	—
59	C₄⋊C₄	C₂×D₂₂₃	1	3	yes	complemented
60	C₂²⋊C₄	—	223	3	no	—
61	C₂²⋊C₄	—	223	3	no	—
62	C₂²⋊C₄	—	223	3	no	—
63	C₂²⋊C₄	C₂×D₂₂₃	1	3	yes	—
64	C₄²	—	223	3	no	—
65	C₄²	C₂×D₂₂₃	1	1	yes	characteristic, complemented
66	C₂³	—	446	6	no	—
67	C₂³	—	223	1	no	—
68	C₂³	—	223	1	no	—
69	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
70	C₂×C₄	—	446	6	no	—
71	C₂×C₄	—	446	6	no	—
72	C₂×C₄	—	223	3	no	—
73	C₂×C₄	—	223	3	no	—
74	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
75	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
76	C₂²	—	2	3	no	—
77	C₂²	—	892	4	no	—
78	C₂²	—	446	3	no	—
79	C₂²	—	223	6	no	—
80	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
81	C₄	—	2	3	no	—
82	C₄	—	2	3	no	—
83	C₄	—	446	3	no	—
84	C₄	—	446	3	no	—
85	C₂	—	4	1	no	—
86	C₂	—	892	1	no	—
87	C₂	—	223	4	no	—
88	C₂	#41×D₂₂₃	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	#126×C₂₂₃	C₂
prime radical	C₂³×D₂₂₃	C₂²
hypercenter	#126	D₂₂₃
hypocenter	C₂₂₃	C₂×#126
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ #126

Complements

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

N	H ≅ G/N	type
1. #20297299	C₂	semidirect
2. #20297298	C₂	semidirect
3. #20297297	C₂	semidirect
4. C₄⋊C₄×D₂₂₃	C₂	semidirect
6. C₄²×D₂₂₃	C₂	semidirect
7. #126×C₂₂₃	C₂	semidirect
11. C₄⋊Dic₂₂₃	C₂²	semidirect
12. Dic₂₂₃⋊C₄	C₂²	semidirect
13. Dic₂₂₃⋊C₄	C₂²	semidirect
17. C₄×Dic₂₂₃	C₂²	semidirect
18. C₄⋊C₄×C₂₂₃	C₂²	semidirect
20. C₄²×C₂₂₃	C₂²	semidirect
43. D₂₂₃	#126	direct (N × H)
44. C₂₂₃	C₂×#126	semidirect
49. #126	D₂₂₃	direct (N × H)
59. C₄⋊C₄	C₂×D₂₂₃	semidirect
65. C₄²	C₂×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297299	2
#20297298	2
#20297297	2
C₄⋊C₄×D₂₂₃	2
C₂²⋊C₄×D₂₂₃	2
C₄²×D₂₂₃	2
#126×C₂₂₃	2
C₂×#126	223

Subgroups — #126×D223 #28384103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #126×D₂₂₃ #28384103