Subgroups — C₂₃×#29142 #19176768

Order 5888.

770 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2862	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2024	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2023	C₂	1	2	yes	maximal
5	C₂₃×#1958	C₂	1	2	yes	maximal
6	C₂₃×#1835	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1805	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1804	C₂	1	2	yes	maximal
9	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×#103	C₂²	1	4	yes	—
16	C₂×C₂₃×#102	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#391	—	2	2	no	—
19	C₂₃×#391	—	2	2	no	—
20	C₂₃×#391	C₂²	1	1	yes	characteristic
21	C₂₃×#390	C₂²	1	1	yes	characteristic
22	C₂₃×#388	—	2	2	no	—
23	C₂₃×#382	C₂²	1	2	yes	—
24	C₂₃×#382	C₂²	1	1	yes	characteristic
25	C₂₃×#382	C₂²	1	2	yes	—
26	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
27	C₄³×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#340	C₂²	1	4	yes	—
29	C₂₃×#340	C₂²	1	4	yes	—
30	C₂₃×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₂₃	—	2	2	no	—
32	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
33	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₂₃	—	2	1	no	—
35	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
36	C₂₃×C₄⋊Q₈	—	4	4	no	—
37	C₂₃×#124	—	2	2	no	—
38	C₂₃×#124	Q₈	1	4	yes	complemented
39	C₄×Q₈×C₂₃	—	2	4	no	—
40	C₄×D₄×C₂₃	—	2	4	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
46	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
49	C₂×C₄²×C₂₃	—	2	1	no	—
50	C₂×C₄²×C₂₃	D₄	1	2	yes	—
51	C₂×C₄²×C₂₃	D₄	1	2	yes	—
52	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
55	C₂₃×C₄⋊C₈	—	2	4	no	—
56	C₂₃×#103	—	4	4	no	—
57	C₂₃×#102	—	4	4	no	—
58	C₄×C₈×C₂₃	—	2	4	no	—
59	C₂₃×#95	—	2	2	no	—
60	C₂₃×#95	—	2	2	no	—
61	C₂₃×#95	—	2	4	no	—
62	C₂₃×#95	—	2	2	no	—
63	C₂₃×#95	—	2	2	no	—
64	C₂⁴×C₂₃	—	2	1	no	—
65	C₂×Q₈×C₂₃	—	2	1	no	—
66	C₂×Q₈×C₂₃	—	2	2	no	—
67	C₂×Q₈×C₂₃	—	2	2	no	—
68	C₂×Q₈×C₂₃	—	4	4	no	—
69	C₂×Q₈×C₂₃	—	4	4	no	—
70	C₂×Q₈×C₂₃	—	4	4	no	—
71	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
72	C₂×D₄×C₂₃	—	2	2	no	—
73	C₂×D₄×C₂₃	—	2	1	no	—
74	C₂×D₄×C₂₃	—	2	2	no	—
75	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₂₃	—	2	1	no	—
77	C₂²×C₄×C₂₃	—	2	1	no	—
78	C₂²×C₄×C₂₃	—	2	1	no	—
79	C₂²×C₄×C₂₃	—	2	4	no	—
80	C₂²×C₄×C₂₃	—	4	4	no	—
81	C₂²×C₄×C₂₃	—	4	2	no	—
82	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
87	C₂×C₈×C₂₃	—	2	4	no	—
88	C₂×C₈×C₂₃	—	2	4	no	—
89	C₂×C₈×C₂₃	—	2	4	no	—
90	C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₄⋊C₄×C₂₃	—	2	2	no	—
92	C₄⋊C₄×C₂₃	—	2	4	no	—
93	C₄⋊C₄×C₂₃	—	2	4	no	—
94	C₄⋊C₄×C₂₃	—	4	4	no	—
95	C₄⋊C₄×C₂₃	—	4	2	no	—
96	C₂²⋊C₄×C₂₃	—	2	8	no	—
97	C₂²⋊C₄×C₂₃	—	4	2	no	—
98	C₂²⋊C₄×C₂₃	—	4	4	no	—
99	C₄²×C₂₃	—	2	4	no	—
100	C₄²×C₂₃	—	2	2	no	—
101	C₄²×C₂₃	—	4	2	no	—
102	C₄²×C₂₃	—	4	2	no	—
103	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
104	C₄²×C₂₃	#41	1	2	yes	—
105	C₄²×C₂₃	#41	1	4	yes	—
106	#29142	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₂₃	—	2	2	no	—
108	C₂³×C₂₃	—	4	1	no	—
109	C₂³×C₂₃	—	4	1	no	—
110	C₂³×C₂₃	—	4	2	no	—
111	C₂³×C₂₃	—	4	2	no	—
112	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
113	Q₈×C₂₃	—	2	4	no	—
114	Q₈×C₂₃	—	4	2	no	—
115	Q₈×C₂₃	—	8	8	no	—
116	D₄×C₂₃	—	2	4	no	—
117	D₄×C₂₃	—	4	2	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	2	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	2	no	—
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	2	2	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	4	1	no	—
126	C₂×C₄×C₂₃	—	4	1	no	—
127	C₂×C₄×C₂₃	—	4	1	no	—
128	C₂×C₄×C₂₃	—	4	1	no	—
129	C₂×C₄×C₂₃	—	4	4	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	—	4	4	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	4	4	no	—
134	C₂×C₄×C₂₃	—	8	4	no	—
135	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
136	C₂×C₄×C₂₃	#124	1	2	yes	—
137	C₂×C₄×C₂₃	#124	1	2	yes	—
138	C₂×C₄×C₂₃	#135	1	4	yes	—
139	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
141	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
143	C₈×C₂₃	—	4	4	no	—
144	#2862	C₂×C₂₃	1	1	yes	characteristic, complemented
145	C₄×#124	C₂×C₂₃	1	1	yes	characteristic
146	#2024	C₂×C₂₃	1	2	yes	complemented
147	#2023	C₂×C₂₃	1	2	yes	—
148	#1958	C₂×C₂₃	1	2	yes	—
149	#1835	C₂×C₂₃	1	2	yes	complemented
150	#1805	C₂×C₂₃	1	2	yes	complemented
151	#1804	C₂×C₂₃	1	2	yes	—
152	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
153	C₂²×C₂₃	—	4	2	no	—
154	C₂²×C₂₃	—	4	4	no	—
155	C₂²×C₂₃	—	8	1	no	—
156	C₂²×C₂₃	—	8	1	no	—
157	C₂²×C₂₃	—	8	2	no	—
158	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
160	C₂²×C₂₃	#535	1	1	yes	characteristic
161	C₂²×C₂₃	C₂×#135	1	2	yes	—
162	C₂²×C₂₃	#575	1	2	yes	—
163	C₄×C₂₃	—	2	4	no	—
164	C₄×C₂₃	—	2	2	no	—
165	C₄×C₂₃	—	2	2	no	—
166	C₄×C₂₃	—	2	4	no	—
167	C₄×C₂₃	—	4	2	no	—
168	C₄×C₂₃	—	8	1	no	—
169	C₄×C₂₃	—	8	1	no	—
170	C₄×C₂₃	—	8	4	no	—
171	C₄×C₂₃	#554	1	2	yes	—
172	C₄×C₂₃	#555	1	2	yes	—
173	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
174	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
175	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
176	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
177	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
178	C₂×#103	C₂²×C₂₃	1	4	yes	—
179	C₂×#102	C₂²×C₂₃	1	4	yes	—
180	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
181	#391	—	2	2	no	—
182	#391	—	2	2	no	—
183	#391	C₂²×C₂₃	1	1	yes	characteristic
184	#390	C₂²×C₂₃	1	1	yes	characteristic
185	#388	—	2	2	no	—
186	#382	C₂²×C₂₃	1	2	yes	—
187	#382	C₂²×C₂₃	1	1	yes	characteristic
188	#382	C₂²×C₂₃	1	2	yes	—
189	C₄×C₂²⋊C₄	—	2	2	no	—
190	C₄³	C₂²×C₂₃	1	1	yes	characteristic
191	#340	C₂²×C₂₃	1	4	yes	—
192	#340	C₂²×C₂₃	1	4	yes	—
193	#336	C₂²×C₂₃	1	2	yes	—
194	C₂×C₂₃	—	8	2	no	—
195	C₂×C₂₃	#2563	1	1	yes	characteristic
196	C₂×C₂₃	#3076	1	2	yes	—
197	C₂×C₂₃	#3131	1	1	yes	characteristic
198	C₂×C₂₃	#3140	1	1	yes	characteristic
199	C₂×C₂₃	#3211	1	2	yes	—
200	C₂²×Q₈	—	2	2	no	—
201	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
202	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
203	C₂³×C₄	—	2	1	no	—
204	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
205	C₄⋊Q₈	—	4	4	no	—
206	#124	—	2	2	no	—
207	#124	Q₈×C₂₃	1	4	yes	complemented
208	C₄×Q₈	—	2	4	no	—
209	C₄×D₄	—	2	4	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₂²⋊C₄	—	2	1	no	—
217	C₂×C₂²⋊C₄	—	2	2	no	—
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	D₄×C₂₃	1	2	yes	—
220	C₂×C₄²	D₄×C₂₃	1	2	yes	—
221	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
223	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
224	C₄⋊C₈	—	2	4	no	—
225	#103	—	4	4	no	—
226	#102	—	4	4	no	—
227	C₄×C₈	—	2	4	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	4	no	—
231	#95	—	2	2	no	—
232	#95	—	2	2	no	—
233	C₂₃	#29142	1	1	yes	characteristic, direct product factor, Sylow
234	C₂⁴	—	2	1	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	4	no	—
239	C₂×Q₈	—	4	4	no	—
240	C₂×Q₈	—	4	4	no	—
241	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	4	4	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
256	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	2	8	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	4	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	—	4	2	no	—
273	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
274	C₄²	#41×C₂₃	1	2	yes	—
275	C₄²	#41×C₂₃	1	4	yes	—
276	C₂³	—	2	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	1	no	—
280	C₂³	—	4	1	no	—
281	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
282	Q₈	—	2	4	no	—
283	Q₈	—	4	2	no	—
284	Q₈	—	8	8	no	—
285	D₄	—	2	4	no	—
286	D₄	—	4	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	8	4	no	—
304	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
305	C₂×C₄	C₂₃×#124	1	2	yes	—
306	C₂×C₄	C₂₃×#124	1	2	yes	—
307	C₂×C₄	C₂₃×#135	1	4	yes	—
308	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
310	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
311	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
312	C₈	—	4	4	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	8	1	no	—
318	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
319	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
320	C₂²	C₂₃×#535	1	1	yes	characteristic
321	C₂²	C₂×C₂₃×#135	1	2	yes	—
322	C₂²	C₂₃×#575	1	2	yes	—
323	C₄	—	2	4	no	—
324	C₄	—	2	4	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	2	no	—
327	C₄	—	4	2	no	—
328	C₄	—	8	4	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	1	no	—
331	C₄	C₂₃×#554	1	2	yes	—
332	C₄	C₂₃×#555	1	2	yes	—
333	C₂	—	8	2	no	—
334	C₂	C₂₃×#2563	1	1	yes	characteristic
335	C₂	C₂₃×#3076	1	2	yes	—
336	C₂	C₂₃×#3131	1	1	yes	characteristic
337	C₂	C₂₃×#3140	1	1	yes	characteristic
338	C₂	C₂₃×#3211	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2862	C₂	semidirect
3. C₂₃×#2024	C₂	semidirect
6. C₂₃×#1835	C₂	semidirect
7. C₂₃×#1805	C₂	semidirect
9. C₄×C₂₃×C₄⋊C₈	C₂	semidirect
38. C₂₃×#124	Q₈	semidirect
106. #29142	C₂₃	direct (N × H)
144. #2862	C₂×C₂₃	semidirect
146. #2024	C₂×C₂₃	semidirect
149. #1835	C₂×C₂₃	semidirect
150. #1805	C₂×C₂₃	semidirect
152. C₄×C₄⋊C₈	C₂×C₂₃	semidirect
207. #124	Q₈×C₂₃	semidirect
233. C₂₃	#29142	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2862	2
C₄×C₂₃×#124	2
C₂₃×#2024	2
C₂₃×#2023	2
C₂₃×#1958	2
C₂₃×#1835	2
C₂₃×#1805	2
C₂₃×#1804	2
C₄×C₂₃×C₄⋊C₈	2
#29142	23

Subgroups — C23×#29142 #19176768

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29142 #19176768