Subgroups — #2884×C₂₂₃ #43251899

Order 28544.

716 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×2₋¹⁺⁴×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#519×C₂₂₃	C₂	1	2	yes	maximal, complemented
3	#410×C₂₂₃	C₂	1	12	yes	maximal, complemented
4	2₋¹⁺⁴×C₂₂₃	—	2	8	no	—
5	C₂×#41×C₂₂₃	C₄	1	4	yes	complemented
6	C₂×#41×C₂₂₃	C₂²	1	6	yes	—
7	C₂²×Q₈×C₂₂₃	C₄	1	4	yes	complemented
8	C₂²×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
9	C₄×Q₈×C₂₂₃	—	2	8	no	—
10	#117×C₂₂₃	C₂²	1	12	yes	—
11	#99×C₂₂₃	C₂²	1	16	yes	—
12	#41×C₂₂₃	—	2	16	no	—
13	#41×C₂₂₃	—	2	24	no	—
14	C₂×Q₈×C₂₂₃	—	2	16	no	—
15	C₂×Q₈×C₂₂₃	C₂×C₄	1	6	yes	—
16	C₂×Q₈×C₂₂₃	C₂×C₄	1	4	yes	—
17	C₂×Q₈×C₂₂₃	D₄	1	8	yes	—
18	C₂×D₄×C₂₂₃	C₂×C₄	1	6	yes	—
19	C₂×D₄×C₂₂₃	C₂³	1	4	yes	—
20	C₂²×C₄×C₂₂₃	C₂×C₄	1	12	yes	—
21	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
22	C₄⋊C₄×C₂₂₃	—	2	12	no	—
23	C₂²⋊C₄×C₂₂₃	C₂³	1	8	yes	—
24	C₄²×C₂₂₃	—	2	12	no	—
25	C₂³×C₂₂₃	C₂²×C₄	1	4	yes	—
26	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
27	Q₈×C₂₂₃	—	2	12	no	—
28	Q₈×C₂₂₃	C₂²⋊C₄	1	16	yes	—
29	D₄×C₂₂₃	—	2	8	no	—
30	D₄×C₂₂₃	—	2	12	no	—
31	C₂×C₄×C₂₂₃	—	2	24	no	—
32	C₂×C₄×C₂₂₃	—	2	8	no	—
33	C₂×C₄×C₂₂₃	C₂²×C₄	1	4	yes	—
34	C₂×C₄×C₂₂₃	C₂²×C₄	1	6	yes	—
35	C₂×C₄×C₂₂₃	C₂×D₄	1	12	yes	—
36	C₂²×C₂₂₃	—	2	4	no	—
37	C₂²×C₂₂₃	—	4	1	no	—
38	C₂²×C₂₂₃	—	4	4	no	—
39	C₂²×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
40	C₂²×C₂₂₃	C₂²×D₄	1	2	yes	—
41	C₄×C₂₂₃	—	2	4	no	—
42	C₄×C₂₂₃	—	4	8	no	—
43	C₄×C₂₂₃	C₂×C₂²⋊C₄	1	12	yes	—
44	C₂×C₂₂₃	—	2	2	no	—
45	C₂×C₂₂₃	—	2	1	no	—
46	C₂×C₂₂₃	—	4	4	no	—
47	C₂×C₂₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
48	C₂₂₃	#2884	1	1	yes	characteristic, direct product factor, Sylow
49	#2884	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂×2₋¹⁺⁴	C₂×C₂₂₃	1	1	yes	characteristic
51	#519	C₂×C₂₂₃	1	2	yes	complemented
52	#410	C₂×C₂₂₃	1	12	yes	complemented
53	2₋¹⁺⁴	—	2	8	no	—
54	C₂×#41	C₄×C₂₂₃	1	4	yes	complemented
55	C₂×#41	C₂²×C₂₂₃	1	6	yes	—
56	C₂²×Q₈	C₄×C₂₂₃	1	4	yes	complemented
57	C₂²×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
58	C₄×Q₈	—	2	8	no	—
59	#117	C₂²×C₂₂₃	1	12	yes	—
60	#99	C₂²×C₂₂₃	1	16	yes	—
61	#41	—	2	16	no	—
62	#41	—	2	24	no	—
63	C₂×Q₈	—	2	16	no	—
64	C₂×Q₈	C₂×C₄×C₂₂₃	1	6	yes	—
65	C₂×Q₈	C₂×C₄×C₂₂₃	1	4	yes	—
66	C₂×Q₈	D₄×C₂₂₃	1	8	yes	—
67	C₂×D₄	C₂×C₄×C₂₂₃	1	6	yes	—
68	C₂×D₄	C₂³×C₂₂₃	1	4	yes	—
69	C₂²×C₄	C₂×C₄×C₂₂₃	1	12	yes	—
70	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
71	C₄⋊C₄	—	2	12	no	—
72	C₂²⋊C₄	C₂³×C₂₂₃	1	8	yes	—
73	C₄²	—	2	12	no	—
74	C₂³	C₂²×C₄×C₂₂₃	1	4	yes	—
75	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
76	Q₈	—	2	12	no	—
77	Q₈	C₂²⋊C₄×C₂₂₃	1	16	yes	—
78	D₄	—	2	8	no	—
79	D₄	—	2	12	no	—
80	C₂×C₄	—	2	24	no	—
81	C₂×C₄	—	2	8	no	—
82	C₂×C₄	C₂²×C₄×C₂₂₃	1	4	yes	—
83	C₂×C₄	C₂²×C₄×C₂₂₃	1	6	yes	—
84	C₂×C₄	C₂×D₄×C₂₂₃	1	12	yes	—
85	C₂²	—	2	4	no	—
86	C₂²	—	4	1	no	—
87	C₂²	—	4	4	no	—
88	C₂²	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
89	C₂²	C₂²×D₄×C₂₂₃	1	2	yes	—
90	C₄	—	2	4	no	—
91	C₄	—	4	8	no	—
92	C₄	C₂×C₂²⋊C₄×C₂₂₃	1	12	yes	—
93	C₂	—	2	2	no	—
94	C₂	—	2	1	no	—
95	C₂	—	4	4	no	—
96	C₂	C₂²×C₂²⋊C₄×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₂₃	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₄×C₂₂₃
Frattini, cosocle	C₂³	C₂⁴×C₂₂₃
prime radical	C₂×2₋¹⁺⁴×C₂₂₃	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₂₃ ◁ C₂²×Q₈×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. #519×C₂₂₃	C₂	semidirect
3. #410×C₂₂₃	C₂	semidirect
5. C₂×#41×C₂₂₃	C₄	semidirect
7. C₂²×Q₈×C₂₂₃	C₄	semidirect
48. C₂₂₃	#2884	direct (N × H)
49. #2884	C₂₂₃	direct (N × H)
51. #519	C₂×C₂₂₃	semidirect
52. #410	C₂×C₂₂₃	semidirect
54. C₂×#41	C₄×C₂₂₃	semidirect
56. C₂²×Q₈	C₄×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×2₋¹⁺⁴×C₂₂₃	2
#519×C₂₂₃	2
#410×C₂₂₃	2
#2884	223

Subgroups — #2884×C223 #43251899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2884×C₂₂₃ #43251899