Subgroups — C₂×#576×C₂₂₃ #43252596

Order 28544.

1512 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)×C₂₂₃	C₂	1	2	yes	maximal, complemented
2	#576×C₂₂₃	C₂	1	16	yes	maximal, direct product factor
3	C₂×AGL(1,ℤ/8)×C₂₂₃	C₂	1	6	yes	maximal, complemented
4	C₂×#135×C₂₂₃	C₂	1	3	yes	maximal, complemented
5	C₂²×D₈×C₂₂₃	C₂	1	3	yes	maximal, complemented
6	C₂×#131×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₂₂₃	—	2	8	no	—
8	H(5,𝔽₂)×C₂₂₃	C₂²	1	16	yes	complemented
9	C₂×#41×C₂₂₃	—	2	2	no	—
10	C₂×#41×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂×#41×C₂₂₃	C₂²	1	6	yes	complemented
12	C₂²×D₄×C₂₂₃	—	2	6	no	—
13	C₂²×D₄×C₂₂₃	C₂²	1	6	yes	complemented
14	AGL(1,ℤ/8)×C₂₂₃	C₂²	1	48	yes	complemented
15	#135×C₂₂₃	C₂²	1	24	yes	complemented
16	C₂×Q₁₆×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
17	C₂×SD₁₆×C₂₂₃	C₂²	1	6	yes	complemented
18	C₂×D₈×C₂₂₃	C₂²	1	3	yes	complemented
19	C₂×D₈×C₂₂₃	C₂²	1	6	yes	complemented
20	C₂×D₈×C₂₂₃	C₂²	1	24	yes	complemented
21	#131×C₂₂₃	C₂²	1	8	yes	complemented
22	C₂×M₁₆×C₂₂₃	C₂²	1	3	yes	complemented
23	C₂²×C₈×C₂₂₃	C₂²	1	3	yes	complemented
24	C₂⁴×C₂₂₃	—	2	6	no	—
25	#41×C₂₂₃	—	2	16	no	—
26	#41×C₂₂₃	—	2	12	no	—
27	#41×C₂₂₃	D₄	1	4	yes	—
28	#41×C₂₂₃	C₂³	1	4	yes	—
29	#41×C₂₂₃	C₂³	1	24	yes	complemented
30	C₂×Q₈×C₂₂₃	D₄	1	1	yes	characteristic
31	C₂×Q₈×C₂₂₃	C₂³	1	2	yes	—
32	C₂×D₄×C₂₂₃	—	2	6	no	—
33	C₂×D₄×C₂₂₃	—	2	48	no	—
34	C₂×D₄×C₂₂₃	—	2	6	no	—
35	C₂×D₄×C₂₂₃	—	2	12	no	—
36	C₂×D₄×C₂₂₃	D₄	1	3	yes	—
37	C₂×D₄×C₂₂₃	C₂³	1	6	yes	—
38	C₂×D₄×C₂₂₃	C₂³	1	24	yes	complemented
39	C₂²×C₄×C₂₂₃	—	2	6	no	—
40	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
41	Q₁₆×C₂₂₃	C₂³	1	4	yes	complemented
42	SD₁₆×C₂₂₃	C₂³	1	24	yes	complemented
43	D₈×C₂₂₃	C₂³	1	12	yes	complemented
44	D₈×C₂₂₃	C₂³	1	24	yes	complemented
45	M₁₆×C₂₂₃	C₂³	1	12	yes	complemented
46	C₂×C₈×C₂₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₂₂₃	C₂³	1	3	yes	—
48	C₂×C₈×C₂₂₃	C₂³	1	12	yes	complemented
49	C₂³×C₂₂₃	—	2	6	no	—
50	C₂³×C₂₂₃	—	2	24	no	—
51	C₂³×C₂₂₃	—	8	12	no	—
52	C₂³×C₂₂₃	C₂×D₄	1	3	yes	—
53	Q₈×C₂₂₃	—	2	2	no	—
54	Q₈×C₂₂₃	C₂×D₄	1	4	yes	—
55	Q₈×C₂₂₃	C₂⁴	1	4	yes	—
56	D₄×C₂₂₃	—	2	24	no	—
57	D₄×C₂₂₃	—	2	6	no	—
58	D₄×C₂₂₃	—	2	24	no	—
59	D₄×C₂₂₃	C₂×D₄	1	12	yes	—
60	D₄×C₂₂₃	C₂⁴	1	12	yes	—
61	C₂×C₄×C₂₂₃	—	2	2	no	—
62	C₂×C₄×C₂₂₃	—	2	24	no	—
63	C₂×C₄×C₂₂₃	C₂×D₄	1	6	yes	—
64	C₂×C₄×C₂₂₃	C₂×D₄	1	3	yes	—
65	C₂×C₄×C₂₂₃	C₂⁴	1	6	yes	—
66	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
67	C₈×C₂₂₃	C₂⁴	1	2	yes	—
68	C₈×C₂₂₃	C₂⁴	1	6	yes	—
69	C₂²×C₂₂₃	—	2	6	no	—
70	C₂²×C₂₂₃	—	2	12	no	—
71	C₂²×C₂₂₃	—	4	12	no	—
72	C₂²×C₂₂₃	—	8	24	no	—
73	C₂²×C₂₂₃	C₂²×D₄	1	6	yes	—
74	C₂²×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
75	C₄×C₂₂₃	—	2	4	no	—
76	C₄×C₂₂₃	C₂²×D₄	1	6	yes	—
77	C₄×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
78	C₄×C₂₂₃	C₂⁵	1	1	yes	characteristic
79	C₂×C₂₂₃	—	2	6	no	—
80	C₂×C₂₂₃	—	4	12	no	—
81	C₂×C₂₂₃	#576	1	2	yes	direct product factor
82	C₂×C₂₂₃	C₂³×D₄	1	1	yes	characteristic
83	C₂₂₃	C₂×#576	1	1	yes	characteristic, direct product factor, Sylow
84	C₂×#576	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂×H(5,𝔽₂)	C₂×C₂₂₃	1	2	yes	complemented
86	#576	C₂×C₂₂₃	1	16	yes	direct product factor
87	C₂×AGL(1,ℤ/8)	C₂×C₂₂₃	1	6	yes	complemented
88	C₂×#135	C₂×C₂₂₃	1	3	yes	complemented
89	C₂²×D₈	C₂×C₂₂₃	1	3	yes	complemented
90	C₂×#131	C₂×C₂₂₃	1	1	yes	characteristic, complemented
91	H(5,𝔽₂)	—	2	8	no	—
92	H(5,𝔽₂)	C₂²×C₂₂₃	1	16	yes	complemented
93	C₂×#41	—	2	2	no	—
94	C₂×#41	C₂²×C₂₂₃	1	1	yes	characteristic
95	C₂×#41	C₂²×C₂₂₃	1	6	yes	complemented
96	C₂²×D₄	—	2	6	no	—
97	C₂²×D₄	C₂²×C₂₂₃	1	6	yes	complemented
98	AGL(1,ℤ/8)	C₂²×C₂₂₃	1	48	yes	complemented
99	#135	C₂²×C₂₂₃	1	24	yes	complemented
100	C₂×Q₁₆	C₂²×C₂₂₃	1	1	yes	characteristic, complemented
101	C₂×SD₁₆	C₂²×C₂₂₃	1	6	yes	complemented
102	C₂×D₈	C₂²×C₂₂₃	1	3	yes	complemented
103	C₂×D₈	C₂²×C₂₂₃	1	6	yes	complemented
104	C₂×D₈	C₂²×C₂₂₃	1	24	yes	complemented
105	#131	C₂²×C₂₂₃	1	8	yes	complemented
106	C₂×M₁₆	C₂²×C₂₂₃	1	3	yes	complemented
107	C₂²×C₈	C₂²×C₂₂₃	1	3	yes	complemented
108	C₂⁴	—	2	6	no	—
109	#41	—	2	16	no	—
110	#41	—	2	12	no	—
111	#41	D₄×C₂₂₃	1	4	yes	—
112	#41	C₂³×C₂₂₃	1	4	yes	—
113	#41	C₂³×C₂₂₃	1	24	yes	complemented
114	C₂×Q₈	D₄×C₂₂₃	1	1	yes	characteristic
115	C₂×Q₈	C₂³×C₂₂₃	1	2	yes	—
116	C₂×D₄	—	2	6	no	—
117	C₂×D₄	—	2	48	no	—
118	C₂×D₄	—	2	6	no	—
119	C₂×D₄	—	2	12	no	—
120	C₂×D₄	D₄×C₂₂₃	1	3	yes	—
121	C₂×D₄	C₂³×C₂₂₃	1	6	yes	—
122	C₂×D₄	C₂³×C₂₂₃	1	24	yes	complemented
123	C₂²×C₄	—	2	6	no	—
124	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
125	Q₁₆	C₂³×C₂₂₃	1	4	yes	complemented
126	SD₁₆	C₂³×C₂₂₃	1	24	yes	complemented
127	D₈	C₂³×C₂₂₃	1	12	yes	complemented
128	D₈	C₂³×C₂₂₃	1	24	yes	complemented
129	M₁₆	C₂³×C₂₂₃	1	12	yes	complemented
130	C₂×C₈	C₂³×C₂₂₃	1	1	yes	characteristic
131	C₂×C₈	C₂³×C₂₂₃	1	3	yes	—
132	C₂×C₈	C₂³×C₂₂₃	1	12	yes	complemented
133	C₂³	—	2	6	no	—
134	C₂³	—	2	24	no	—
135	C₂³	—	8	12	no	—
136	C₂³	C₂×D₄×C₂₂₃	1	3	yes	—
137	Q₈	—	2	2	no	—
138	Q₈	C₂×D₄×C₂₂₃	1	4	yes	—
139	Q₈	C₂⁴×C₂₂₃	1	4	yes	—
140	D₄	—	2	24	no	—
141	D₄	—	2	6	no	—
142	D₄	—	2	24	no	—
143	D₄	C₂×D₄×C₂₂₃	1	12	yes	—
144	D₄	C₂⁴×C₂₂₃	1	12	yes	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	24	no	—
147	C₂×C₄	C₂×D₄×C₂₂₃	1	3	yes	—
148	C₂×C₄	C₂×D₄×C₂₂₃	1	6	yes	—
149	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
150	C₂×C₄	C₂⁴×C₂₂₃	1	6	yes	—
151	C₈	C₂⁴×C₂₂₃	1	2	yes	—
152	C₈	C₂⁴×C₂₂₃	1	6	yes	—
153	C₂²	—	2	6	no	—
154	C₂²	—	2	12	no	—
155	C₂²	—	4	12	no	—
156	C₂²	—	8	24	no	—
157	C₂²	C₂²×D₄×C₂₂₃	1	6	yes	—
158	C₂²	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
159	C₄	—	2	4	no	—
160	C₄	C₂²×D₄×C₂₂₃	1	6	yes	—
161	C₄	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
162	C₄	C₂⁵×C₂₂₃	1	1	yes	characteristic
163	C₂	—	2	6	no	—
164	C₂	—	4	12	no	—
165	C₂	#576×C₂₂₃	1	2	yes	direct product factor
166	C₂	C₂³×D₄×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₂₂₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₂₃ ◁ C₂×#41×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×H(5,𝔽₂)×C₂₂₃	C₂	semidirect
2. #576×C₂₂₃	C₂	direct (N × H)
3. C₂×AGL(1,ℤ/8)×C₂₂₃	C₂	semidirect
4. C₂×#135×C₂₂₃	C₂	semidirect
5. C₂²×D₈×C₂₂₃	C₂	semidirect
6. C₂×#131×C₂₂₃	C₂	semidirect
8. H(5,𝔽₂)×C₂₂₃	C₂²	semidirect
11. C₂×#41×C₂₂₃	C₂²	semidirect
13. C₂²×D₄×C₂₂₃	C₂²	semidirect
14. AGL(1,ℤ/8)×C₂₂₃	C₂²	semidirect
15. #135×C₂₂₃	C₂²	semidirect
16. C₂×Q₁₆×C₂₂₃	C₂²	semidirect
17. C₂×SD₁₆×C₂₂₃	C₂²	semidirect
18. C₂×D₈×C₂₂₃	C₂²	semidirect
19. C₂×D₈×C₂₂₃	C₂²	semidirect
20. C₂×D₈×C₂₂₃	C₂²	semidirect
21. #131×C₂₂₃	C₂²	semidirect
22. C₂×M₁₆×C₂₂₃	C₂²	semidirect
23. C₂²×C₈×C₂₂₃	C₂²	semidirect
29. #41×C₂₂₃	C₂³	semidirect
38. C₂×D₄×C₂₂₃	C₂³	semidirect
41. Q₁₆×C₂₂₃	C₂³	semidirect
42. SD₁₆×C₂₂₃	C₂³	semidirect
43. D₈×C₂₂₃	C₂³	semidirect
44. D₈×C₂₂₃	C₂³	semidirect
45. M₁₆×C₂₂₃	C₂³	semidirect
48. C₂×C₈×C₂₂₃	C₂³	semidirect
81. C₂×C₂₂₃	#576	direct (N × H)
83. C₂₂₃	C₂×#576	direct (N × H)
84. C₂×#576	C₂₂₃	direct (N × H)
85. C₂×H(5,𝔽₂)	C₂×C₂₂₃	semidirect
86. #576	C₂×C₂₂₃	direct (N × H)
87. C₂×AGL(1,ℤ/8)	C₂×C₂₂₃	semidirect
88. C₂×#135	C₂×C₂₂₃	semidirect
89. C₂²×D₈	C₂×C₂₂₃	semidirect
90. C₂×#131	C₂×C₂₂₃	semidirect
92. H(5,𝔽₂)	C₂²×C₂₂₃	semidirect
95. C₂×#41	C₂²×C₂₂₃	semidirect
97. C₂²×D₄	C₂²×C₂₂₃	semidirect
98. AGL(1,ℤ/8)	C₂²×C₂₂₃	semidirect
99. #135	C₂²×C₂₂₃	semidirect
100. C₂×Q₁₆	C₂²×C₂₂₃	semidirect
101. C₂×SD₁₆	C₂²×C₂₂₃	semidirect
102. C₂×D₈	C₂²×C₂₂₃	semidirect
103. C₂×D₈	C₂²×C₂₂₃	semidirect
104. C₂×D₈	C₂²×C₂₂₃	semidirect
105. #131	C₂²×C₂₂₃	semidirect
106. C₂×M₁₆	C₂²×C₂₂₃	semidirect
107. C₂²×C₈	C₂²×C₂₂₃	semidirect
113. #41	C₂³×C₂₂₃	semidirect
122. C₂×D₄	C₂³×C₂₂₃	semidirect
125. Q₁₆	C₂³×C₂₂₃	semidirect
126. SD₁₆	C₂³×C₂₂₃	semidirect
127. D₈	C₂³×C₂₂₃	semidirect
128. D₈	C₂³×C₂₂₃	semidirect
129. M₁₆	C₂³×C₂₂₃	semidirect
132. C₂×C₈	C₂³×C₂₂₃	semidirect
165. C₂	#576×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
C₂×H(5,𝔽₂)×C₂₂₃	2
#576×C₂₂₃	2
C₂×AGL(1,ℤ/8)×C₂₂₃	2
C₂×#135×C₂₂₃	2
C₂²×D₈×C₂₂₃	2
C₂×#131×C₂₂₃	2
C₂×#576	223

Subgroups — C2×#576×C223 #43252596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#576×C₂₂₃ #43252596