Subgroups — C₂×#568×C₂₂₃ #43252587

Order 28544.

1460 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	#568×C₂₂₃	C₂	1	16	yes	maximal, direct product factor
3	C₂×#131×C₂₂₃	C₂	1	8	yes	maximal, complemented
4	C₂²×M₁₆×C₂₂₃	C₂	1	6	yes	maximal, complemented
5	C₂×#41×C₂₂₃	C₄	1	4	yes	—
6	C₂×#41×C₂₂₃	C₄	1	8	yes	—
7	C₂×#41×C₂₂₃	C₂²	1	4	yes	—
8	C₂×#41×C₂₂₃	C₂²	1	8	yes	—
9	C₂²×Q₈×C₂₂₃	C₄	1	1	yes	characteristic
10	C₂²×D₄×C₂₂₃	C₄	1	3	yes	—
11	C₂³×C₄×C₂₂₃	C₂²	1	3	yes	—
12	#131×C₂₂₃	C₂²	1	64	yes	complemented
13	C₂×M₁₆×C₂₂₃	C₂²	1	4	yes	—
14	C₂×M₁₆×C₂₂₃	C₂²	1	12	yes	complemented
15	C₂×M₁₆×C₂₂₃	C₂²	1	48	yes	complemented
16	C₂²×C₈×C₂₂₃	C₂²	1	12	yes	complemented
17	C₂⁴×C₂₂₃	C₂×C₄	1	3	yes	—
18	#41×C₂₂₃	C₂×C₄	1	16	yes	—
19	#41×C₂₂₃	C₂×C₄	1	32	yes	—
20	#41×C₂₂₃	C₂³	1	16	yes	—
21	C₂×Q₈×C₂₂₃	C₂×C₄	1	4	yes	—
22	C₂×Q₈×C₂₂₃	C₂×C₄	1	8	yes	—
23	C₂×D₄×C₂₂₃	C₂×C₄	1	12	yes	—
24	C₂×D₄×C₂₂₃	C₂×C₄	1	24	yes	—
25	C₂²×C₄×C₂₂₃	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₂₂₃	C₂×C₄	1	6	yes	—
27	C₂²×C₄×C₂₂₃	C₂×C₄	1	12	yes	—
28	C₂²×C₄×C₂₂₃	C₂³	1	12	yes	—
29	C₂²×C₄×C₂₂₃	C₂³	1	6	yes	—
30	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
31	M₁₆×C₂₂₃	C₂³	1	16	yes	—
32	M₁₆×C₂₂₃	C₂³	1	48	yes	complemented
33	C₂×C₈×C₂₂₃	C₂³	1	8	yes	—
34	C₂×C₈×C₂₂₃	C₂³	1	48	yes	complemented
35	C₂³×C₂₂₃	—	4	6	no	—
36	C₂³×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
37	C₂³×C₂₂₃	C₂²×C₄	1	6	yes	—
38	C₂³×C₂₂₃	C₂²×C₄	1	12	yes	—
39	Q₈×C₂₂₃	C₂²×C₄	1	16	yes	—
40	D₄×C₂₂₃	C₂²×C₄	1	48	yes	—
41	C₂×C₄×C₂₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₂₃	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₂₂₃	C₂²×C₄	1	12	yes	—
44	C₂×C₄×C₂₂₃	C₂²×C₄	1	6	yes	—
45	C₂×C₄×C₂₂₃	C₂²×C₄	1	24	yes	—
46	C₂×C₄×C₂₂₃	C₂²×C₄	1	12	yes	—
47	C₂×C₄×C₂₂₃	C₂⁴	1	12	yes	—
48	C₂×C₄×C₂₂₃	C₂⁴	1	2	yes	—
49	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
50	C₈×C₂₂₃	C₂⁴	1	16	yes	—
51	C₂²×C₂₂₃	—	2	2	no	—
52	C₂²×C₂₂₃	—	2	12	no	—
53	C₂²×C₂₂₃	—	4	12	no	—
54	C₂²×C₂₂₃	C₂³×C₄	1	2	yes	—
55	C₂²×C₂₂₃	C₂³×C₄	1	12	yes	—
56	C₂²×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
57	C₄×C₂₂₃	C₂³×C₄	1	2	yes	—
58	C₄×C₂₂₃	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₂₂₃	C₂³×C₄	1	12	yes	—
60	C₄×C₂₂₃	C₂⁵	1	1	yes	characteristic
61	C₂×C₂₂₃	—	2	2	no	—
62	C₂×C₂₂₃	—	2	12	no	—
63	C₂×C₂₂₃	#568	1	2	yes	direct product factor
64	C₂×C₂₂₃	C₂⁴×C₄	1	1	yes	characteristic
65	C₂₂₃	C₂×#568	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×#568	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂²×#41	C₂×C₂₂₃	1	1	yes	characteristic
68	#568	C₂×C₂₂₃	1	16	yes	direct product factor
69	C₂×#131	C₂×C₂₂₃	1	8	yes	complemented
70	C₂²×M₁₆	C₂×C₂₂₃	1	6	yes	complemented
71	C₂×#41	C₄×C₂₂₃	1	4	yes	—
72	C₂×#41	C₄×C₂₂₃	1	8	yes	—
73	C₂×#41	C₂²×C₂₂₃	1	4	yes	—
74	C₂×#41	C₂²×C₂₂₃	1	8	yes	—
75	C₂²×Q₈	C₄×C₂₂₃	1	1	yes	characteristic
76	C₂²×D₄	C₄×C₂₂₃	1	3	yes	—
77	C₂³×C₄	C₂²×C₂₂₃	1	3	yes	—
78	#131	C₂²×C₂₂₃	1	64	yes	complemented
79	C₂×M₁₆	C₂²×C₂₂₃	1	4	yes	—
80	C₂×M₁₆	C₂²×C₂₂₃	1	12	yes	complemented
81	C₂×M₁₆	C₂²×C₂₂₃	1	48	yes	complemented
82	C₂²×C₈	C₂²×C₂₂₃	1	12	yes	complemented
83	C₂⁴	C₂×C₄×C₂₂₃	1	3	yes	—
84	#41	C₂×C₄×C₂₂₃	1	16	yes	—
85	#41	C₂×C₄×C₂₂₃	1	32	yes	—
86	#41	C₂³×C₂₂₃	1	16	yes	—
87	C₂×Q₈	C₂×C₄×C₂₂₃	1	4	yes	—
88	C₂×Q₈	C₂×C₄×C₂₂₃	1	8	yes	—
89	C₂×D₄	C₂×C₄×C₂₂₃	1	12	yes	—
90	C₂×D₄	C₂×C₄×C₂₂₃	1	24	yes	—
91	C₂²×C₄	C₂×C₄×C₂₂₃	1	3	yes	—
92	C₂²×C₄	C₂×C₄×C₂₂₃	1	6	yes	—
93	C₂²×C₄	C₂×C₄×C₂₂₃	1	12	yes	—
94	C₂²×C₄	C₂³×C₂₂₃	1	6	yes	—
95	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
96	C₂²×C₄	C₂³×C₂₂₃	1	12	yes	—
97	M₁₆	C₂³×C₂₂₃	1	16	yes	—
98	M₁₆	C₂³×C₂₂₃	1	48	yes	complemented
99	C₂×C₈	C₂³×C₂₂₃	1	8	yes	—
100	C₂×C₈	C₂³×C₂₂₃	1	48	yes	complemented
101	C₂³	—	4	6	no	—
102	C₂³	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
103	C₂³	C₂²×C₄×C₂₂₃	1	6	yes	—
104	C₂³	C₂²×C₄×C₂₂₃	1	12	yes	—
105	Q₈	C₂²×C₄×C₂₂₃	1	16	yes	—
106	D₄	C₂²×C₄×C₂₂₃	1	48	yes	—
107	C₂×C₄	C₂²×C₄×C₂₂₃	1	2	yes	—
108	C₂×C₄	C₂²×C₄×C₂₂₃	1	1	yes	characteristic
109	C₂×C₄	C₂²×C₄×C₂₂₃	1	12	yes	—
110	C₂×C₄	C₂²×C₄×C₂₂₃	1	6	yes	—
111	C₂×C₄	C₂²×C₄×C₂₂₃	1	12	yes	—
112	C₂×C₄	C₂²×C₄×C₂₂₃	1	24	yes	—
113	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
114	C₂×C₄	C₂⁴×C₂₂₃	1	12	yes	—
115	C₂×C₄	C₂⁴×C₂₂₃	1	2	yes	—
116	C₈	C₂⁴×C₂₂₃	1	16	yes	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	12	no	—
119	C₂²	—	4	12	no	—
120	C₂²	C₂³×C₄×C₂₂₃	1	2	yes	—
121	C₂²	C₂³×C₄×C₂₂₃	1	12	yes	—
122	C₂²	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
123	C₄	C₂³×C₄×C₂₂₃	1	1	yes	characteristic
124	C₄	C₂³×C₄×C₂₂₃	1	2	yes	—
125	C₄	C₂³×C₄×C₂₂₃	1	12	yes	—
126	C₄	C₂⁵×C₂₂₃	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	12	no	—
129	C₂	#568×C₂₂₃	1	2	yes	direct product factor
130	C₂	C₂⁴×C₄×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₂₃	C₂⁴
commutator	C₂	C₂⁴×C₄×C₂₂₃
Frattini, cosocle	C₄	C₂⁵×C₂₂₃
socle	C₂²×C₂₂₃	C₂³×C₄
prime radical	C₂²×#41×C₂₂₃	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. #568×C₂₂₃	C₂	direct (N × H)
3. C₂×#131×C₂₂₃	C₂	semidirect
4. C₂²×M₁₆×C₂₂₃	C₂	semidirect
12. #131×C₂₂₃	C₂²	semidirect
14. C₂×M₁₆×C₂₂₃	C₂²	semidirect
15. C₂×M₁₆×C₂₂₃	C₂²	semidirect
16. C₂²×C₈×C₂₂₃	C₂²	semidirect
32. M₁₆×C₂₂₃	C₂³	semidirect
34. C₂×C₈×C₂₂₃	C₂³	semidirect
63. C₂×C₂₂₃	#568	direct (N × H)
65. C₂₂₃	C₂×#568	direct (N × H)
66. C₂×#568	C₂₂₃	direct (N × H)
68. #568	C₂×C₂₂₃	direct (N × H)
69. C₂×#131	C₂×C₂₂₃	semidirect
70. C₂²×M₁₆	C₂×C₂₂₃	semidirect
78. #131	C₂²×C₂₂₃	semidirect
80. C₂×M₁₆	C₂²×C₂₂₃	semidirect
81. C₂×M₁₆	C₂²×C₂₂₃	semidirect
82. C₂²×C₈	C₂²×C₂₂₃	semidirect
98. M₁₆	C₂³×C₂₂₃	semidirect
100. C₂×C₈	C₂³×C₂₂₃	semidirect
129. C₂	#568×C₂₂₃	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×#41×C₂₂₃	2
#568×C₂₂₃	2
C₂×#131×C₂₂₃	2
C₂²×M₁₆×C₂₂₃	2
C₂×#568	223

Subgroups — C2×#568×C223 #43252587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#568×C₂₂₃ #43252587