Subgroups — C₂²×C₂₃×#510 #19210256

Order 5888.

2078 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₂³×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#510	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂₃×Q₃₂	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂₃×SD₃₂	C₂	1	2	yes	maximal, complemented
6	C₂²×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₂₃	—	2	1	no	—
8	C₂³×Q₈×C₂₃	—	2	1	no	—
9	C₂×C₂₃×#135	—	2	6	no	—
10	C₂×C₂₃×#135	C₂²	1	12	yes	—
11	C₂²×Q₁₆×C₂₃	—	2	1	no	—
12	C₂²×Q₁₆×C₂₃	—	2	6	no	—
13	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆×C₂₃	C₂²	1	12	yes	complemented
16	C₂²×SD₁₆×C₂₃	—	2	1	no	—
17	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#510	C₂²	1	64	yes	direct product factor
20	C₂×C₂₃×Q₃₂	C₂²	1	24	yes	complemented
21	C₂×C₂₃×SD₃₂	C₂²	1	24	yes	complemented
22	C₂×C₂₃×M₃₂	C₂²	1	12	yes	complemented
23	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
24	C₂×#41×C₂₃	—	2	12	no	—
25	C₂×#41×C₂₃	—	4	6	no	—
26	C₂²×Q₈×C₂₃	—	2	1	no	—
27	C₂²×Q₈×C₂₃	—	2	2	no	—
28	C₂²×Q₈×C₂₃	—	2	12	no	—
29	C₂²×Q₈×C₂₃	—	4	6	no	—
30	C₂²×Q₈×C₂₃	—	4	1	no	—
31	C₂²×D₄×C₂₃	—	2	1	no	—
32	C₂²×D₄×C₂₃	—	4	1	no	—
33	C₂³×C₄×C₂₃	—	4	1	no	—
34	C₂³×C₄×C₂₃	—	4	1	no	—
35	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
36	C₂₃×#135	—	2	24	no	—
37	C₂₃×#135	C₂³	1	16	yes	—
38	C₂×Q₁₆×C₂₃	—	2	6	no	—
39	C₂×Q₁₆×C₂₃	—	2	12	no	—
40	C₂×Q₁₆×C₂₃	—	2	24	no	—
41	C₂×Q₁₆×C₂₃	—	2	3	no	—
42	C₂×Q₁₆×C₂₃	C₂³	1	12	yes	—
43	C₂×Q₁₆×C₂₃	C₂³	1	6	yes	—
44	C₂×Q₁₆×C₂₃	C₂³	1	16	yes	complemented
45	C₂×SD₁₆×C₂₃	—	2	12	no	—
46	C₂×D₈×C₂₃	—	2	3	no	—
47	C₂×D₈×C₂₃	C₂³	1	6	yes	—
48	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₂₃	D₄	1	6	yes	—
50	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₂₃	C₂³	1	6	yes	—
52	C₂₃×Q₃₂	C₂³	1	32	yes	complemented
53	C₂₃×SD₃₂	C₂³	1	32	yes	complemented
54	C₂₃×M₃₂	C₂³	1	16	yes	complemented
55	C₂×C₁₆×C₂₃	C₂³	1	12	yes	—
56	C₂⁴×C₂₃	—	4	1	no	—
57	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
58	#41×C₂₃	—	2	16	no	—
59	#41×C₂₃	—	4	24	no	—
60	C₂×Q₈×C₂₃	—	2	6	no	—
61	C₂×Q₈×C₂₃	—	2	12	no	—
62	C₂×Q₈×C₂₃	—	2	16	no	—
63	C₂×Q₈×C₂₃	—	4	12	no	—
64	C₂×Q₈×C₂₃	—	4	6	no	—
65	C₂×Q₈×C₂₃	—	4	24	no	—
66	C₂×Q₈×C₂₃	—	4	3	no	—
67	C₂×D₄×C₂₃	—	2	6	no	—
68	C₂×D₄×C₂₃	—	4	12	no	—
69	C₂×D₄×C₂₃	—	4	3	no	—
70	C₂²×C₄×C₂₃	—	4	12	no	—
71	C₂²×C₄×C₂₃	—	4	1	no	—
72	C₂²×C₄×C₂₃	—	4	2	no	—
73	C₂²×C₄×C₂₃	—	4	12	no	—
74	C₂²×C₄×C₂₃	D₈	1	6	yes	—
75	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂×D₄	1	6	yes	—
77	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₂₃	—	2	12	no	—
79	Q₁₆×C₂₃	—	2	16	no	—
80	Q₁₆×C₂₃	—	2	6	no	—
81	Q₁₆×C₂₃	C₂⁴	1	8	yes	—
82	Q₁₆×C₂₃	C₂⁴	1	4	yes	—
83	SD₁₆×C₂₃	—	2	16	no	—
84	D₈×C₂₃	—	2	6	no	—
85	D₈×C₂₃	C₂⁴	1	4	yes	—
86	C₂×C₈×C₂₃	C₂×D₄	1	3	yes	—
87	C₂×C₈×C₂₃	C₂×D₄	1	6	yes	—
88	C₂×C₈×C₂₃	C₂×D₄	1	12	yes	—
89	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
90	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
91	C₁₆×C₂₃	C₂⁴	1	8	yes	—
92	C₂²×#510	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₃	—	2	4	no	—
94	C₂³×C₂₃	—	4	6	no	—
95	C₂³×C₂₃	—	8	4	no	—
96	C₂³×C₂₃	C₂×D₈	1	6	yes	—
97	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
98	Q₈×C₂₃	—	2	4	no	—
99	Q₈×C₂₃	—	2	8	no	—
100	Q₈×C₂₃	—	4	16	no	—
101	Q₈×C₂₃	—	4	12	no	—
102	Q₈×C₂₃	—	4	6	no	—
103	D₄×C₂₃	—	2	4	no	—
104	D₄×C₂₃	—	4	16	no	—
105	D₄×C₂₃	—	4	6	no	—
106	C₂×C₄×C₂₃	—	4	6	no	—
107	C₂×C₄×C₂₃	—	4	16	no	—
108	C₂×C₄×C₂₃	—	4	12	no	—
109	C₂×C₄×C₂₃	—	4	16	no	—
110	C₂×C₄×C₂₃	C₂×D₈	1	6	yes	—
111	C₂×C₄×C₂₃	C₂×D₈	1	3	yes	—
112	C₂×C₄×C₂₃	C₂×D₈	1	12	yes	—
113	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
114	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
115	C₈×C₂₃	C₂²×D₄	1	3	yes	—
116	C₈×C₂₃	C₂²×D₄	1	4	yes	—
117	C₈×C₂₃	C₂⁵	1	1	yes	characteristic
118	C₂²×#135	C₂×C₂₃	1	1	yes	characteristic
119	C₂³×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₂×#510	C₂×C₂₃	1	24	yes	direct product factor
121	C₂²×Q₃₂	C₂×C₂₃	1	2	yes	complemented
122	C₂²×SD₃₂	C₂×C₂₃	1	2	yes	complemented
123	C₂²×M₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
124	C₂²×C₂₃	—	2	12	no	—
125	C₂²×C₂₃	—	4	4	no	—
126	C₂²×C₂₃	—	8	12	no	—
127	C₂²×C₂₃	#510	1	4	yes	direct product factor
128	C₂²×C₂₃	C₂²×D₈	1	4	yes	—
129	C₂²×C₂₃	C₂²×D₈	1	3	yes	—
130	C₄×C₂₃	—	4	4	no	—
131	C₄×C₂₃	—	4	8	no	—
132	C₄×C₂₃	C₂²×D₈	1	4	yes	—
133	C₄×C₂₃	C₂²×D₈	1	3	yes	—
134	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
135	C₂²×#41	—	2	1	no	—
136	C₂³×Q₈	—	2	1	no	—
137	C₂×#135	—	2	6	no	—
138	C₂×#135	C₂²×C₂₃	1	12	yes	—
139	C₂²×Q₁₆	—	2	1	no	—
140	C₂²×Q₁₆	—	2	6	no	—
141	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
142	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
143	C₂²×Q₁₆	C₂²×C₂₃	1	12	yes	complemented
144	C₂²×SD₁₆	—	2	1	no	—
145	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
146	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
147	#510	C₂²×C₂₃	1	64	yes	direct product factor
148	C₂×Q₃₂	C₂²×C₂₃	1	24	yes	complemented
149	C₂×SD₃₂	C₂²×C₂₃	1	24	yes	complemented
150	C₂×M₃₂	C₂²×C₂₃	1	12	yes	complemented
151	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
152	C₂×C₂₃	—	2	4	no	—
153	C₂×C₂₃	—	8	4	no	—
154	C₂×C₂₃	C₂×#510	1	6	yes	direct product factor
155	C₂×C₂₃	C₂³×D₈	1	1	yes	characteristic
156	C₂×#41	—	2	12	no	—
157	C₂×#41	—	4	6	no	—
158	C₂²×Q₈	—	2	1	no	—
159	C₂²×Q₈	—	2	2	no	—
160	C₂²×Q₈	—	2	12	no	—
161	C₂²×Q₈	—	4	1	no	—
162	C₂²×Q₈	—	4	6	no	—
163	C₂²×D₄	—	2	1	no	—
164	C₂²×D₄	—	4	1	no	—
165	C₂³×C₄	—	4	1	no	—
166	C₂³×C₄	—	4	1	no	—
167	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
168	#135	—	2	24	no	—
169	#135	C₂³×C₂₃	1	16	yes	—
170	C₂×Q₁₆	—	2	6	no	—
171	C₂×Q₁₆	—	2	12	no	—
172	C₂×Q₁₆	—	2	24	no	—
173	C₂×Q₁₆	—	2	3	no	—
174	C₂×Q₁₆	C₂³×C₂₃	1	12	yes	—
175	C₂×Q₁₆	C₂³×C₂₃	1	6	yes	—
176	C₂×Q₁₆	C₂³×C₂₃	1	16	yes	complemented
177	C₂×SD₁₆	—	2	12	no	—
178	C₂×D₈	—	2	3	no	—
179	C₂×D₈	C₂³×C₂₃	1	6	yes	—
180	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₈	D₄×C₂₃	1	6	yes	—
182	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
183	C₂²×C₈	C₂³×C₂₃	1	6	yes	—
184	Q₃₂	C₂³×C₂₃	1	32	yes	complemented
185	SD₃₂	C₂³×C₂₃	1	32	yes	complemented
186	M₃₂	C₂³×C₂₃	1	16	yes	complemented
187	C₂×C₁₆	C₂³×C₂₃	1	12	yes	—
188	C₂₃	C₂²×#510	1	1	yes	characteristic, direct product factor, Sylow
189	C₂⁴	—	4	1	no	—
190	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
191	#41	—	2	16	no	—
192	#41	—	4	24	no	—
193	C₂×Q₈	—	2	6	no	—
194	C₂×Q₈	—	2	12	no	—
195	C₂×Q₈	—	2	16	no	—
196	C₂×Q₈	—	4	12	no	—
197	C₂×Q₈	—	4	6	no	—
198	C₂×Q₈	—	4	24	no	—
199	C₂×Q₈	—	4	3	no	—
200	C₂×D₄	—	2	6	no	—
201	C₂×D₄	—	4	12	no	—
202	C₂×D₄	—	4	3	no	—
203	C₂²×C₄	—	4	1	no	—
204	C₂²×C₄	—	4	12	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	12	no	—
207	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	D₈×C₂₃	1	6	yes	—
209	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₂₃	1	6	yes	—
211	Q₁₆	—	2	12	no	—
212	Q₁₆	—	2	16	no	—
213	Q₁₆	—	2	6	no	—
214	Q₁₆	C₂⁴×C₂₃	1	8	yes	—
215	Q₁₆	C₂⁴×C₂₃	1	4	yes	—
216	SD₁₆	—	2	16	no	—
217	D₈	—	2	6	no	—
218	D₈	C₂⁴×C₂₃	1	4	yes	—
219	C₂×C₈	C₂×D₄×C₂₃	1	3	yes	—
220	C₂×C₈	C₂×D₄×C₂₃	1	6	yes	—
221	C₂×C₈	C₂×D₄×C₂₃	1	12	yes	—
222	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
223	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
224	C₁₆	C₂⁴×C₂₃	1	8	yes	—
225	C₂³	—	2	4	no	—
226	C₂³	—	4	6	no	—
227	C₂³	—	8	4	no	—
228	C₂³	C₂×D₈×C₂₃	1	6	yes	—
229	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
230	Q₈	—	2	4	no	—
231	Q₈	—	2	8	no	—
232	Q₈	—	4	16	no	—
233	Q₈	—	4	12	no	—
234	Q₈	—	4	6	no	—
235	D₄	—	2	4	no	—
236	D₄	—	4	16	no	—
237	D₄	—	4	6	no	—
238	C₂×C₄	—	4	6	no	—
239	C₂×C₄	—	4	16	no	—
240	C₂×C₄	—	4	12	no	—
241	C₂×C₄	—	4	16	no	—
242	C₂×C₄	C₂×D₈×C₂₃	1	6	yes	—
243	C₂×C₄	C₂×D₈×C₂₃	1	3	yes	—
244	C₂×C₄	C₂×D₈×C₂₃	1	12	yes	—
245	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
246	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
247	C₈	C₂²×D₄×C₂₃	1	3	yes	—
248	C₈	C₂²×D₄×C₂₃	1	4	yes	—
249	C₈	C₂⁵×C₂₃	1	1	yes	characteristic
250	C₂²	—	2	12	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	8	12	no	—
253	C₂²	C₂₃×#510	1	4	yes	direct product factor
254	C₂²	C₂²×D₈×C₂₃	1	4	yes	—
255	C₂²	C₂²×D₈×C₂₃	1	3	yes	—
256	C₄	—	4	4	no	—
257	C₄	—	4	8	no	—
258	C₄	C₂²×D₈×C₂₃	1	4	yes	—
259	C₄	C₂²×D₈×C₂₃	1	3	yes	—
260	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
261	C₂	—	2	4	no	—
262	C₂	—	8	4	no	—
263	C₂	C₂×C₂₃×#510	1	6	yes	direct product factor
264	C₂	C₂³×D₈×C₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₂₃
prime radical	C₂²×C₂₃×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×Q₁₆×C₂₃	C₂	semidirect
3. C₂×C₂₃×#510	C₂	direct (N × H)
4. C₂²×C₂₃×Q₃₂	C₂	semidirect
5. C₂²×C₂₃×SD₃₂	C₂	semidirect
6. C₂²×C₂₃×M₃₂	C₂	semidirect
15. C₂²×Q₁₆×C₂₃	C₂²	semidirect
19. C₂₃×#510	C₂²	direct (N × H)
20. C₂×C₂₃×Q₃₂	C₂²	semidirect
21. C₂×C₂₃×SD₃₂	C₂²	semidirect
22. C₂×C₂₃×M₃₂	C₂²	semidirect
44. C₂×Q₁₆×C₂₃	C₂³	semidirect
52. C₂₃×Q₃₂	C₂³	semidirect
53. C₂₃×SD₃₂	C₂³	semidirect
54. C₂₃×M₃₂	C₂³	semidirect
92. C₂²×#510	C₂₃	direct (N × H)
119. C₂³×Q₁₆	C₂×C₂₃	semidirect
120. C₂×#510	C₂×C₂₃	direct (N × H)
121. C₂²×Q₃₂	C₂×C₂₃	semidirect
122. C₂²×SD₃₂	C₂×C₂₃	semidirect
123. C₂²×M₃₂	C₂×C₂₃	semidirect
127. C₂²×C₂₃	#510	direct (N × H)
143. C₂²×Q₁₆	C₂²×C₂₃	semidirect
147. #510	C₂²×C₂₃	direct (N × H)
148. C₂×Q₃₂	C₂²×C₂₃	semidirect
149. C₂×SD₃₂	C₂²×C₂₃	semidirect
150. C₂×M₃₂	C₂²×C₂₃	semidirect
154. C₂×C₂₃	C₂×#510	direct (N × H)
176. C₂×Q₁₆	C₂³×C₂₃	semidirect
184. Q₃₂	C₂³×C₂₃	semidirect
185. SD₃₂	C₂³×C₂₃	semidirect
186. M₃₂	C₂³×C₂₃	semidirect
188. C₂₃	C₂²×#510	direct (N × H)
253. C₂²	C₂₃×#510	direct (N × H)
263. C₂	C₂×C₂₃×#510	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×C₂₃×#135	2
C₂³×Q₁₆×C₂₃	2
C₂×C₂₃×#510	2
C₂²×C₂₃×Q₃₂	2
C₂²×C₂₃×SD₃₂	2
C₂²×C₂₃×M₃₂	2
C₂²×#510	23

Subgroups — C22×C23×#510 #19210256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#510 #19210256