Subgroups — C₂₃×#8033 #19155659

Order 5888.

722 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#382	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1295	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	—	2	1	no	—
6	C₂²×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#432	—	2	2	no	—
9	C₂₃×#432	C₄	1	4	yes	complemented
10	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
11	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
12	C₄×M₁₆×C₂₃	—	2	2	no	—
13	C₄×M₁₆×C₂₃	C₄	1	4	yes	complemented
14	C₂×C₂₃×#97	C₂²	1	2	yes	—
15	C₂×C₂₃×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₂₃	—	2	1	no	—
17	C₂₃×#382	—	2	1	no	—
18	C₂₃×#382	—	2	1	no	—
19	C₂₃×#382	C₄	1	2	yes	—
20	C₂₃×#382	C₂²	1	2	yes	—
21	C₂×C₂₃×#95	—	2	1	no	—
22	C₂×C₂₃×#95	—	2	1	no	—
23	C₂³×C₄×C₂₃	—	2	2	no	—
24	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
27	C₂×M₁₆×C₂₃	—	2	8	no	—
28	C₂×M₁₆×C₂₃	—	2	4	no	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	D₄	1	2	yes	—
40	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
41	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
42	C₂₃×C₄⋊C₈	—	2	4	no	—
43	C₂₃×C₂²⋊C₈	—	2	4	no	—
44	C₂₃×#97	—	2	2	no	—
45	C₂₃×#97	—	2	2	no	—
46	C₂₃×#97	C₂×C₄	1	4	yes	complemented
47	C₂₃×#97	C₂×C₄	1	4	yes	complemented
48	C₄×C₈×C₂₃	—	2	4	no	—
49	C₂₃×#95	—	2	2	no	—
50	C₂₃×#95	—	2	2	no	—
51	C₂₃×#95	—	2	2	no	—
52	C₂₃×#95	—	2	2	no	—
53	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	—	4	2	no	—
59	C₂²×C₄×C₂₃	—	4	4	no	—
60	C₂²×C₄×C₂₃	—	4	4	no	—
61	C₂²×C₄×C₂₃	—	4	4	no	—
62	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
74	M₁₆×C₂₃	—	2	16	no	—
75	C₂×C₈×C₂₃	—	2	8	no	—
76	C₂×C₈×C₂₃	—	2	4	no	—
77	C₂×C₈×C₂₃	—	2	4	no	—
78	C₂×C₈×C₂₃	—	4	4	no	—
79	C₄⋊C₄×C₂₃	—	4	4	no	—
80	C₄⋊C₄×C₂₃	—	4	4	no	—
81	C₄²×C₂₃	C₄²	1	2	yes	—
82	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
83	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
84	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
86	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
87	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
91	#8033	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	—	2	1	no	—
93	C₂³×C₂₃	—	2	1	no	—
94	C₂³×C₂₃	—	2	2	no	—
95	C₂³×C₂₃	#95	1	2	yes	—
96	C₂³×C₂₃	#104	1	4	yes	—
97	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	2	4	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	4	no	—
107	C₂×C₄×C₂₃	—	2	4	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	—	4	4	no	—
111	C₂×C₄×C₂₃	—	4	4	no	—
112	C₂×C₄×C₂₃	—	4	4	no	—
113	C₂×C₄×C₂₃	—	8	8	no	—
114	C₂×C₄×C₂₃	#95	1	2	yes	—
115	C₂×C₄×C₂₃	#95	1	2	yes	—
116	C₂×C₄×C₂₃	#95	1	2	yes	—
117	C₂×C₄×C₂₃	#100	1	2	yes	—
118	C₂×C₄×C₂₃	#101	1	2	yes	—
119	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₈×C₂₃	—	2	8	no	—
126	C₈×C₂₃	—	4	4	no	—
127	C₂×#432	C₂×C₂₃	1	1	yes	characteristic
128	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
129	C₂×#382	C₂×C₂₃	1	1	yes	characteristic
130	#1295	C₂×C₂₃	1	4	yes	complemented
131	C₂²×C₂₃	—	2	2	no	—
132	C₂²×C₂₃	—	2	2	no	—
133	C₂²×C₂₃	—	2	4	no	—
134	C₂²×C₂₃	—	2	4	no	—
135	C₂²×C₂₃	#339	1	4	yes	—
136	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
137	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
138	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
139	C₂²×C₂₃	C₂×#104	1	2	yes	—
140	C₂²×C₂₃	#421	1	2	yes	—
141	C₄×C₂₃	—	2	1	no	—
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	2	1	no	—
145	C₄×C₂₃	—	2	2	no	—
146	C₄×C₂₃	—	2	1	no	—
147	C₄×C₂₃	—	2	1	no	—
148	C₄×C₂₃	—	2	1	no	—
149	C₄×C₂₃	—	8	4	no	—
150	C₄×C₂₃	#343	1	4	yes	—
151	C₂²×M₁₆	—	2	1	no	—
152	C₂²×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
153	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
154	#432	—	2	2	no	—
155	#432	C₄×C₂₃	1	4	yes	complemented
156	C₂×C₄⋊C₈	—	2	1	no	—
157	C₂×C₂²⋊C₈	—	2	1	no	—
158	C₄×M₁₆	—	2	2	no	—
159	C₄×M₁₆	C₄×C₂₃	1	4	yes	complemented
160	C₂×#97	C₂²×C₂₃	1	2	yes	—
161	C₂×#97	C₂²×C₂₃	1	2	yes	—
162	C₂×C₄×C₈	—	2	1	no	—
163	#382	—	2	1	no	—
164	#382	—	2	1	no	—
165	#382	C₄×C₂₃	1	2	yes	—
166	#382	C₂²×C₂₃	1	2	yes	—
167	C₂×#95	—	2	1	no	—
168	C₂×#95	—	2	1	no	—
169	C₂×C₂₃	—	2	4	no	—
170	C₂×C₂₃	#1527	1	2	yes	—
171	C₂×C₂₃	#1528	1	2	yes	—
172	C₂×C₂₃	C₂×#339	1	1	yes	characteristic
173	C₂×C₂₃	#1733	1	1	yes	characteristic
174	C₂×C₂₃	C₂×#343	1	1	yes	characteristic
175	C₂³×C₄	—	2	2	no	—
176	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
178	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
179	C₂×M₁₆	—	2	8	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	4	4	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
186	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
187	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
188	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
189	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
190	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
191	C₂×C₄²	D₄×C₂₃	1	2	yes	—
192	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
193	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
194	C₄⋊C₈	—	2	4	no	—
195	C₂²⋊C₈	—	2	4	no	—
196	#97	—	2	2	no	—
197	#97	—	2	2	no	—
198	#97	C₂×C₄×C₂₃	1	4	yes	complemented
199	#97	C₂×C₄×C₂₃	1	4	yes	complemented
200	C₄×C₈	—	2	4	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	C₂₃	#8033	1	1	yes	characteristic, direct product factor, Sylow
206	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	4	4	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
216	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
220	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
222	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
223	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
227	M₁₆	—	2	16	no	—
228	C₂×C₈	—	2	8	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	4	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄²	C₄²×C₂₃	1	2	yes	—
235	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
236	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
237	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
238	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
239	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
240	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
241	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
243	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	2	no	—
247	C₂³	C₂₃×#95	1	2	yes	—
248	C₂³	C₂₃×#104	1	4	yes	—
249	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	8	8	no	—
266	C₂×C₄	C₂₃×#95	1	2	yes	—
267	C₂×C₄	C₂₃×#95	1	2	yes	—
268	C₂×C₄	C₂₃×#95	1	2	yes	—
269	C₂×C₄	C₂₃×#100	1	2	yes	—
270	C₂×C₄	C₂₃×#101	1	2	yes	—
271	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
272	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
273	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
277	C₈	—	2	8	no	—
278	C₈	—	4	4	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	C₂₃×#339	1	4	yes	—
284	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
285	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
286	C₂²	C₂×C₂₃×#101	1	1	yes	characteristic
287	C₂²	C₂×C₂₃×#104	1	2	yes	—
288	C₂²	C₂₃×#421	1	2	yes	—
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	8	4	no	—
298	C₄	C₂₃×#343	1	4	yes	—
299	C₂	—	2	4	no	—
300	C₂	C₂₃×#1527	1	2	yes	—
301	C₂	C₂₃×#1528	1	2	yes	—
302	C₂	C₂×C₂₃×#339	1	1	yes	characteristic
303	C₂	C₂₃×#1733	1	1	yes	characteristic
304	C₂	C₂×C₂₃×#343	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1295	C₂	semidirect
9. C₂₃×#432	C₄	semidirect
13. C₄×M₁₆×C₂₃	C₄	semidirect
46. C₂₃×#97	C₂×C₄	semidirect
47. C₂₃×#97	C₂×C₄	semidirect
91. #8033	C₂₃	direct (N × H)
130. #1295	C₂×C₂₃	semidirect
155. #432	C₄×C₂₃	semidirect
159. C₄×M₁₆	C₄×C₂₃	semidirect
198. #97	C₂×C₄×C₂₃	semidirect
199. #97	C₂×C₄×C₂₃	semidirect
205. C₂₃	#8033	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×C₂₃×#432	2
C₂×C₄×M₁₆×C₂₃	2
C₂×C₂₃×#382	2
C₂₃×#1295	2
#8033	23

Subgroups — C23×#8033 #19155659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8033 #19155659