Subgroups — #3586×C₂₂₃ #43252601

Order 28544.

1412 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#585×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×2₋¹⁺⁴×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#578×C₂₂₃	C₂	1	8	yes	maximal, complemented
4	#577×C₂₂₃	C₂	1	8	yes	maximal, complemented
5	#575×C₂₂₃	C₂	1	6	yes	maximal, complemented
6	C₂×#137×C₂₂₃	C₂	1	6	yes	maximal, complemented
7	#568×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	2₋¹⁺⁴×C₂₂₃	—	2	4	no	—
9	2₋¹⁺⁴×C₂₂₃	—	2	1	no	—
10	2₋¹⁺⁴×C₂₂₃	C₂²	1	8	yes	complemented
11	2₋¹⁺⁴×C₂₂₃	C₂²	1	4	yes	complemented
12	H(5,𝔽₂)×C₂₂₃	—	2	3	no	—
13	H(5,𝔽₂)×C₂₂₃	C₂²	1	4	yes	complemented
14	C₂×#41×C₂₂₃	—	2	4	no	—
15	C₂×#41×C₂₂₃	—	2	3	no	—
16	C₂×#41×C₂₂₃	C₂²	1	6	yes	—
17	C₂×#41×C₂₂₃	C₂²	1	1	yes	characteristic
18	C₂×#41×C₂₂₃	C₂²	1	3	yes	complemented
19	C₂²×Q₈×C₂₂₃	—	2	1	no	—
20	C₂²×Q₈×C₂₂₃	C₂²	1	3	yes	complemented
21	#137×C₂₂₃	C₂²	1	12	yes	complemented
22	#137×C₂₂₃	C₂²	1	24	yes	complemented
23	#137×C₂₂₃	C₂²	1	12	yes	complemented
24	AGL(1,ℤ/8)×C₂₂₃	C₂²	1	4	yes	complemented
25	AGL(1,ℤ/8)×C₂₂₃	C₂²	1	12	yes	complemented
26	#135×C₂₂₃	C₂²	1	24	yes	complemented
27	C₂×Q₁₆×C₂₂₃	C₂²	1	12	yes	complemented
28	C₂×SD₁₆×C₂₂₃	C₂²	1	12	yes	complemented
29	#131×C₂₂₃	C₂²	1	8	yes	complemented
30	C₂×M₁₆×C₂₂₃	C₂²	1	6	yes	complemented
31	#41×C₂₂₃	—	2	4	no	—
32	#41×C₂₂₃	—	2	12	no	—
33	#41×C₂₂₃	—	2	24	no	—
34	#41×C₂₂₃	—	2	6	no	—
35	#41×C₂₂₃	—	2	6	no	—
36	#41×C₂₂₃	—	2	6	no	—
37	#41×C₂₂₃	D₄	1	4	yes	—
38	#41×C₂₂₃	C₂³	1	12	yes	—
39	#41×C₂₂₃	C₂³	1	12	yes	—
40	#41×C₂₂₃	C₂³	1	4	yes	—
41	#41×C₂₂₃	C₂³	1	4	yes	—
42	C₂×Q₈×C₂₂₃	—	2	8	no	—
43	C₂×Q₈×C₂₂₃	—	2	4	no	—
44	C₂×Q₈×C₂₂₃	—	2	3	no	—
45	C₂×Q₈×C₂₂₃	—	2	6	no	—
46	C₂×Q₈×C₂₂₃	D₄	1	1	yes	characteristic
47	C₂×Q₈×C₂₂₃	C₂³	1	12	yes	—
48	C₂×Q₈×C₂₂₃	C₂³	1	6	yes	—
49	C₂×Q₈×C₂₂₃	C₂³	1	3	yes	—
50	C₂×D₄×C₂₂₃	—	2	3	no	—
51	C₂×D₄×C₂₂₃	—	2	6	no	—
52	C₂×D₄×C₂₂₃	—	2	12	no	—
53	C₂×D₄×C₂₂₃	D₄	1	3	yes	—
54	C₂×D₄×C₂₂₃	C₂³	1	6	yes	—
55	C₂×D₄×C₂₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₂₃	—	2	3	no	—
57	C₂²×C₄×C₂₂₃	—	2	3	no	—
58	C₂²×C₄×C₂₂₃	—	2	6	no	—
59	C₂²×C₄×C₂₂₃	C₂³	1	3	yes	—
60	Q₁₆×C₂₂₃	C₂³	1	24	yes	—
61	SD₁₆×C₂₂₃	C₂³	1	24	yes	—
62	SD₁₆×C₂₂₃	C₂³	1	8	yes	—
63	D₈×C₂₂₃	C₂³	1	8	yes	—
64	M₁₆×C₂₂₃	C₂³	1	4	yes	—
65	M₁₆×C₂₂₃	C₂³	1	12	yes	—
66	C₂×C₈×C₂₂₃	C₂³	1	12	yes	—
67	C₂³×C₂₂₃	—	2	1	no	—
68	C₂³×C₂₂₃	—	2	6	no	—
69	C₂³×C₂₂₃	C₂×D₄	1	3	yes	—
70	Q₈×C₂₂₃	—	2	3	no	—
71	Q₈×C₂₂₃	—	2	12	no	—
72	Q₈×C₂₂₃	—	2	6	no	—
73	Q₈×C₂₂₃	C₂×D₄	1	4	yes	—
74	Q₈×C₂₂₃	C₂⁴	1	6	yes	—
75	Q₈×C₂₂₃	C₂⁴	1	4	yes	—
76	D₄×C₂₂₃	—	2	4	no	—
77	D₄×C₂₂₃	—	2	12	no	—
78	D₄×C₂₂₃	—	2	12	no	—
79	D₄×C₂₂₃	—	2	6	no	—
80	D₄×C₂₂₃	—	2	1	no	—
81	D₄×C₂₂₃	C₂×D₄	1	12	yes	—
82	D₄×C₂₂₃	C₂⁴	1	2	yes	—
83	D₄×C₂₂₃	C₂⁴	1	4	yes	—
84	C₂×C₄×C₂₂₃	—	2	3	no	—
85	C₂×C₄×C₂₂₃	—	2	6	no	—
86	C₂×C₄×C₂₂₃	—	2	12	no	—
87	C₂×C₄×C₂₂₃	—	2	4	no	—
88	C₂×C₄×C₂₂₃	—	2	12	no	—
89	C₂×C₄×C₂₂₃	—	2	12	no	—
90	C₂×C₄×C₂₂₃	C₂×D₄	1	3	yes	—
91	C₂×C₄×C₂₂₃	C₂×D₄	1	6	yes	—
92	C₂×C₄×C₂₂₃	C₂⁴	1	6	yes	—
93	C₂×C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
94	C₈×C₂₂₃	C₂⁴	1	8	yes	—
95	C₂²×C₂₂₃	—	2	2	no	—
96	C₂²×C₂₂₃	—	2	4	no	—
97	C₂²×C₂₂₃	—	4	3	no	—
98	C₂²×C₂₂₃	—	4	2	no	—
99	C₂²×C₂₂₃	—	8	6	no	—
100	C₂²×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₂₂₃	C₂²×D₄	1	6	yes	—
102	C₄×C₂₂₃	—	2	4	no	—
103	C₄×C₂₂₃	—	2	6	no	—
104	C₄×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₂₂₃	C₂²×D₄	1	6	yes	—
106	C₄×C₂₂₃	C₂⁵	1	1	yes	characteristic
107	C₂×C₂₂₃	—	2	1	no	—
108	C₂×C₂₂₃	—	2	6	no	—
109	C₂×C₂₂₃	—	4	2	no	—
110	C₂×C₂₂₃	—	4	4	no	—
111	C₂×C₂₂₃	C₂³×D₄	1	1	yes	characteristic
112	C₂₂₃	#3586	1	1	yes	characteristic, direct product factor, Sylow
113	#3586	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	#585	C₂×C₂₂₃	1	1	yes	characteristic, complemented
115	C₂×2₋¹⁺⁴	C₂×C₂₂₃	1	1	yes	characteristic, complemented
116	#578	C₂×C₂₂₃	1	8	yes	complemented
117	#577	C₂×C₂₂₃	1	8	yes	complemented
118	#575	C₂×C₂₂₃	1	6	yes	complemented
119	C₂×#137	C₂×C₂₂₃	1	6	yes	complemented
120	#568	C₂×C₂₂₃	1	1	yes	characteristic, complemented
121	2₋¹⁺⁴	—	2	4	no	—
122	2₋¹⁺⁴	—	2	1	no	—
123	2₋¹⁺⁴	C₂²×C₂₂₃	1	8	yes	complemented
124	2₋¹⁺⁴	C₂²×C₂₂₃	1	4	yes	complemented
125	H(5,𝔽₂)	—	2	3	no	—
126	H(5,𝔽₂)	C₂²×C₂₂₃	1	4	yes	complemented
127	C₂×#41	—	2	4	no	—
128	C₂×#41	—	2	3	no	—
129	C₂×#41	C₂²×C₂₂₃	1	6	yes	—
130	C₂×#41	C₂²×C₂₂₃	1	1	yes	characteristic
131	C₂×#41	C₂²×C₂₂₃	1	3	yes	complemented
132	C₂²×Q₈	—	2	1	no	—
133	C₂²×Q₈	C₂²×C₂₂₃	1	3	yes	complemented
134	#137	C₂²×C₂₂₃	1	12	yes	complemented
135	#137	C₂²×C₂₂₃	1	24	yes	complemented
136	#137	C₂²×C₂₂₃	1	12	yes	complemented
137	AGL(1,ℤ/8)	C₂²×C₂₂₃	1	4	yes	complemented
138	AGL(1,ℤ/8)	C₂²×C₂₂₃	1	12	yes	complemented
139	#135	C₂²×C₂₂₃	1	24	yes	complemented
140	C₂×Q₁₆	C₂²×C₂₂₃	1	12	yes	complemented
141	C₂×SD₁₆	C₂²×C₂₂₃	1	12	yes	complemented
142	#131	C₂²×C₂₂₃	1	8	yes	complemented
143	C₂×M₁₆	C₂²×C₂₂₃	1	6	yes	complemented
144	#41	—	2	4	no	—
145	#41	—	2	12	no	—
146	#41	—	2	24	no	—
147	#41	—	2	6	no	—
148	#41	—	2	6	no	—
149	#41	—	2	6	no	—
150	#41	D₄×C₂₂₃	1	4	yes	—
151	#41	C₂³×C₂₂₃	1	12	yes	—
152	#41	C₂³×C₂₂₃	1	12	yes	—
153	#41	C₂³×C₂₂₃	1	4	yes	—
154	#41	C₂³×C₂₂₃	1	4	yes	—
155	C₂×Q₈	—	2	8	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	—	2	3	no	—
158	C₂×Q₈	—	2	6	no	—
159	C₂×Q₈	D₄×C₂₂₃	1	1	yes	characteristic
160	C₂×Q₈	C₂³×C₂₂₃	1	12	yes	—
161	C₂×Q₈	C₂³×C₂₂₃	1	6	yes	—
162	C₂×Q₈	C₂³×C₂₂₃	1	3	yes	—
163	C₂×D₄	—	2	3	no	—
164	C₂×D₄	—	2	6	no	—
165	C₂×D₄	—	2	12	no	—
166	C₂×D₄	D₄×C₂₂₃	1	3	yes	—
167	C₂×D₄	C₂³×C₂₂₃	1	6	yes	—
168	C₂×D₄	C₂³×C₂₂₃	1	1	yes	characteristic
169	C₂²×C₄	—	2	3	no	—
170	C₂²×C₄	—	2	3	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	C₂³×C₂₂₃	1	3	yes	—
173	Q₁₆	C₂³×C₂₂₃	1	24	yes	—
174	SD₁₆	C₂³×C₂₂₃	1	24	yes	—
175	SD₁₆	C₂³×C₂₂₃	1	8	yes	—
176	D₈	C₂³×C₂₂₃	1	8	yes	—
177	M₁₆	C₂³×C₂₂₃	1	4	yes	—
178	M₁₆	C₂³×C₂₂₃	1	12	yes	—
179	C₂×C₈	C₂³×C₂₂₃	1	12	yes	—
180	C₂³	—	2	1	no	—
181	C₂³	—	2	6	no	—
182	C₂³	C₂×D₄×C₂₂₃	1	3	yes	—
183	Q₈	—	2	3	no	—
184	Q₈	—	2	12	no	—
185	Q₈	—	2	6	no	—
186	Q₈	C₂×D₄×C₂₂₃	1	4	yes	—
187	Q₈	C₂⁴×C₂₂₃	1	6	yes	—
188	Q₈	C₂⁴×C₂₂₃	1	4	yes	—
189	D₄	—	2	4	no	—
190	D₄	—	2	12	no	—
191	D₄	—	2	12	no	—
192	D₄	—	2	6	no	—
193	D₄	—	2	1	no	—
194	D₄	C₂×D₄×C₂₂₃	1	12	yes	—
195	D₄	C₂⁴×C₂₂₃	1	2	yes	—
196	D₄	C₂⁴×C₂₂₃	1	4	yes	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	2	12	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	12	no	—
202	C₂×C₄	—	2	12	no	—
203	C₂×C₄	C₂×D₄×C₂₂₃	1	3	yes	—
204	C₂×C₄	C₂×D₄×C₂₂₃	1	6	yes	—
205	C₂×C₄	C₂⁴×C₂₂₃	1	6	yes	—
206	C₂×C₄	C₂⁴×C₂₂₃	1	1	yes	characteristic
207	C₈	C₂⁴×C₂₂₃	1	8	yes	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	4	no	—
210	C₂²	—	4	3	no	—
211	C₂²	—	4	2	no	—
212	C₂²	—	8	6	no	—
213	C₂²	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
214	C₂²	C₂²×D₄×C₂₂₃	1	6	yes	—
215	C₄	—	2	4	no	—
216	C₄	—	2	6	no	—
217	C₄	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
218	C₄	C₂²×D₄×C₂₂₃	1	6	yes	—
219	C₄	C₂⁵×C₂₂₃	1	1	yes	characteristic
220	C₂	—	2	1	no	—
221	C₂	—	2	6	no	—
222	C₂	—	4	2	no	—
223	C₂	—	4	4	no	—
224	C₂	C₂³×D₄×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₂₂₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₂₃ ◁ C₂×#41×C₂₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. #585×C₂₂₃	C₂	semidirect
2. C₂×2₋¹⁺⁴×C₂₂₃	C₂	semidirect
3. #578×C₂₂₃	C₂	semidirect
4. #577×C₂₂₃	C₂	semidirect
5. #575×C₂₂₃	C₂	semidirect
6. C₂×#137×C₂₂₃	C₂	semidirect
7. #568×C₂₂₃	C₂	semidirect
10. 2₋¹⁺⁴×C₂₂₃	C₂²	semidirect
11. 2₋¹⁺⁴×C₂₂₃	C₂²	semidirect
13. H(5,𝔽₂)×C₂₂₃	C₂²	semidirect
18. C₂×#41×C₂₂₃	C₂²	semidirect
20. C₂²×Q₈×C₂₂₃	C₂²	semidirect
21. #137×C₂₂₃	C₂²	semidirect
22. #137×C₂₂₃	C₂²	semidirect
23. #137×C₂₂₃	C₂²	semidirect
24. AGL(1,ℤ/8)×C₂₂₃	C₂²	semidirect
25. AGL(1,ℤ/8)×C₂₂₃	C₂²	semidirect
26. #135×C₂₂₃	C₂²	semidirect
27. C₂×Q₁₆×C₂₂₃	C₂²	semidirect
28. C₂×SD₁₆×C₂₂₃	C₂²	semidirect
29. #131×C₂₂₃	C₂²	semidirect
30. C₂×M₁₆×C₂₂₃	C₂²	semidirect
112. C₂₂₃	#3586	direct (N × H)
113. #3586	C₂₂₃	direct (N × H)
114. #585	C₂×C₂₂₃	semidirect
115. C₂×2₋¹⁺⁴	C₂×C₂₂₃	semidirect
116. #578	C₂×C₂₂₃	semidirect
117. #577	C₂×C₂₂₃	semidirect
118. #575	C₂×C₂₂₃	semidirect
119. C₂×#137	C₂×C₂₂₃	semidirect
120. #568	C₂×C₂₂₃	semidirect
123. 2₋¹⁺⁴	C₂²×C₂₂₃	semidirect
124. 2₋¹⁺⁴	C₂²×C₂₂₃	semidirect
126. H(5,𝔽₂)	C₂²×C₂₂₃	semidirect
131. C₂×#41	C₂²×C₂₂₃	semidirect
133. C₂²×Q₈	C₂²×C₂₂₃	semidirect
134. #137	C₂²×C₂₂₃	semidirect
135. #137	C₂²×C₂₂₃	semidirect
136. #137	C₂²×C₂₂₃	semidirect
137. AGL(1,ℤ/8)	C₂²×C₂₂₃	semidirect
138. AGL(1,ℤ/8)	C₂²×C₂₂₃	semidirect
139. #135	C₂²×C₂₂₃	semidirect
140. C₂×Q₁₆	C₂²×C₂₂₃	semidirect
141. C₂×SD₁₆	C₂²×C₂₂₃	semidirect
142. #131	C₂²×C₂₂₃	semidirect
143. C₂×M₁₆	C₂²×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#585×C₂₂₃	2
C₂×2₋¹⁺⁴×C₂₂₃	2
#578×C₂₂₃	2
#577×C₂₂₃	2
#575×C₂₂₃	2
C₂×#137×C₂₂₃	2
#568×C₂₂₃	2
#3586	223

Subgroups — #3586×C223 #43252601

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3586×C₂₂₃ #43252601