Subgroups — #2248×C₂₂₃ #43251263

Order 28544.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₃₂×C₂₂₃	C₂	1	4	yes	maximal, complemented
2	#494×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#493×C₂₂₃	C₂	1	1	yes	maximal, characteristic
4	C₄×C₁₆×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₁₆×C₂₂₃	—	2	1	no	—
6	C₂×Q₁₆×C₂₂₃	C₂²	1	2	yes	—
7	C₂×D₈×C₂₂₃	—	2	1	no	—
8	C₂×D₈×C₂₂₃	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₂₂₃	—	2	1	no	—
10	#127×C₂₂₃	—	2	1	no	—
11	SD₃₂×C₂₂₃	—	2	8	no	—
12	C₂×C₁₆×C₂₂₃	C₂²	1	2	yes	—
13	C₄×C₈×C₂₂₃	C₂²	1	1	yes	characteristic
14	C₂×Q₈×C₂₂₃	—	2	2	no	—
15	C₂×D₄×C₂₂₃	—	2	2	no	—
16	C₂×D₄×C₂₂₃	—	4	2	no	—
17	Q₁₆×C₂₂₃	—	2	4	no	—
18	Q₁₆×C₂₂₃	—	4	2	no	—
19	D₈×C₂₂₃	—	2	4	no	—
20	D₈×C₂₂₃	—	4	2	no	—
21	C₂×C₈×C₂₂₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₂₂₃	—	4	2	no	—
24	C₄²×C₂₂₃	D₄	1	1	yes	characteristic
25	C₁₆×C₂₂₃	D₄	1	4	yes	complemented
26	C₂³×C₂₂₃	—	4	2	no	—
27	Q₈×C₂₂₃	—	4	2	no	—
28	Q₈×C₂₂₃	—	4	2	no	—
29	D₄×C₂₂₃	—	4	2	no	—
30	D₄×C₂₂₃	—	4	2	no	—
31	D₄×C₂₂₃	—	8	4	no	—
32	C₂×C₄×C₂₂₃	—	4	2	no	—
33	C₂×C₄×C₂₂₃	D₈	1	2	yes	—
34	C₂×C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
35	C₈×C₂₂₃	D₈	1	2	yes	—
36	C₈×C₂₂₃	C₂×D₄	1	2	yes	—
37	C₂²×C₂₂₃	—	8	2	no	—
38	C₂²×C₂₂₃	—	8	4	no	—
39	C₂²×C₂₂₃	C₂×D₈	1	1	yes	characteristic
40	C₄×C₂₂₃	—	8	2	no	—
41	C₄×C₂₂₃	SD₃₂	1	4	yes	complemented
42	C₄×C₂₂₃	#127	1	1	yes	characteristic
43	C₄×C₂₂₃	C₂×D₈	1	1	yes	characteristic
44	C₂×C₂₂₃	—	16	2	no	—
45	C₂×C₂₂₃	#493	1	1	yes	characteristic
46	C₂×C₂₂₃	C₂×SD₃₂	1	2	yes	—
47	C₂₂₃	#2248	1	1	yes	characteristic, direct product factor, Sylow
48	#2248	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×SD₃₂	C₂×C₂₂₃	1	4	yes	complemented
50	#494	C₂×C₂₂₃	1	1	yes	characteristic, complemented
51	#493	C₂×C₂₂₃	1	1	yes	characteristic
52	C₄×C₁₆	C₂×C₂₂₃	1	1	yes	characteristic, complemented
53	C₂×Q₁₆	—	2	1	no	—
54	C₂×Q₁₆	C₂²×C₂₂₃	1	2	yes	—
55	C₂×D₈	—	2	1	no	—
56	C₂×D₈	C₂²×C₂₂₃	1	2	yes	—
57	C₄⋊Q₈	—	2	1	no	—
58	#127	—	2	1	no	—
59	SD₃₂	—	2	8	no	—
60	C₂×C₁₆	C₂²×C₂₂₃	1	2	yes	—
61	C₄×C₈	C₂²×C₂₂₃	1	1	yes	characteristic
62	C₂×Q₈	—	2	2	no	—
63	C₂×D₄	—	2	2	no	—
64	C₂×D₄	—	4	2	no	—
65	Q₁₆	—	2	4	no	—
66	Q₁₆	—	4	2	no	—
67	D₈	—	2	4	no	—
68	D₈	—	4	2	no	—
69	C₂×C₈	D₄×C₂₂₃	1	1	yes	characteristic
70	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄²	D₄×C₂₂₃	1	1	yes	characteristic
73	C₁₆	D₄×C₂₂₃	1	4	yes	complemented
74	C₂³	—	4	2	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	4	2	no	—
77	D₄	—	4	2	no	—
78	D₄	—	4	2	no	—
79	D₄	—	8	4	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	D₈×C₂₂₃	1	2	yes	—
82	C₂×C₄	C₂×D₄×C₂₂₃	1	1	yes	characteristic
83	C₈	D₈×C₂₂₃	1	2	yes	—
84	C₈	C₂×D₄×C₂₂₃	1	2	yes	—
85	C₂²	—	8	2	no	—
86	C₂²	—	8	4	no	—
87	C₂²	C₂×D₈×C₂₂₃	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	SD₃₂×C₂₂₃	1	4	yes	complemented
90	C₄	#127×C₂₂₃	1	1	yes	characteristic
91	C₄	C₂×D₈×C₂₂₃	1	1	yes	characteristic
92	C₂	—	16	2	no	—
93	C₂	#493×C₂₂₃	1	1	yes	characteristic
94	C₂	C₂×SD₃₂×C₂₂₃	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₂₂₃
prime radical	#493×C₂₂₃	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₄²×C₂₂₃ ◁ C₄×C₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×SD₃₂×C₂₂₃	C₂	semidirect
2. #494×C₂₂₃	C₂	semidirect
4. C₄×C₁₆×C₂₂₃	C₂	semidirect
25. C₁₆×C₂₂₃	D₄	semidirect
41. C₄×C₂₂₃	SD₃₂	semidirect
47. C₂₂₃	#2248	direct (N × H)
48. #2248	C₂₂₃	direct (N × H)
49. C₂×SD₃₂	C₂×C₂₂₃	semidirect
50. #494	C₂×C₂₂₃	semidirect
52. C₄×C₁₆	C₂×C₂₂₃	semidirect
73. C₁₆	D₄×C₂₂₃	semidirect
89. C₄	SD₃₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₃₂×C₂₂₃	2
#494×C₂₂₃	2
#493×C₂₂₃	2
C₄×C₁₆×C₂₂₃	2
#2248	223

Subgroups — #2248×C223 #43251263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2248×C₂₂₃ #43251263