Subgroups — C₂₃×#33627 #19181253

Order 5888.

598 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2228	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2226	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2224	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2159	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2140	C₂	1	2	yes	maximal
9	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#533	—	2	1	no	—
11	C₂×C₄×D₄×C₂₃	—	2	1	no	—
12	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
13	C₂₃×#500	C₂²	1	1	yes	characteristic, complemented
14	C₂₃×#500	C₂²	1	1	yes	characteristic, complemented
15	C₂₃×#499	—	2	1	no	—
16	C₂₃×#499	C₂²	1	2	yes	complemented
17	C₂₃×#498	—	2	1	no	—
18	C₄×D₈×C₂₃	—	2	1	no	—
19	C₄×D₈×C₂₃	—	2	1	no	—
20	C₄×D₈×C₂₃	C₂²	1	2	yes	—
21	C₄×D₈×C₂₃	C₂²	1	2	yes	—
22	C₂₃×#427	—	2	1	no	—
23	C₂₃×#427	C₂²	1	2	yes	—
24	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#102	—	2	1	no	—
26	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#367	C₂²	1	4	yes	complemented
28	C₂₃×#366	C₂²	1	4	yes	complemented
29	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
30	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
31	C₂₃×#357	C₂²	1	4	yes	—
32	C₂₃×#357	C₂²	1	4	yes	—
33	C₂²×D₄×C₂₃	—	2	1	no	—
34	C₂³×C₄×C₂₃	—	4	1	no	—
35	C₂×D₈×C₂₃	—	2	1	no	—
36	C₂×D₈×C₂₃	—	2	2	no	—
37	C₂×D₈×C₂₃	Q₈	1	4	yes	complemented
38	C₂×D₈×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×C₄⋊Q₈	—	2	1	no	—
42	C₂₃×C₄⋊Q₈	—	2	1	no	—
43	C₂₃×#125	—	2	2	no	—
44	C₂₃×C₂²⋊Q₈	—	4	1	no	—
45	C₂₃×C₂²⋊Q₈	—	4	1	no	—
46	C₄×Q₈×C₂₃	—	4	1	no	—
47	C₄×Q₈×C₂₃	—	4	1	no	—
48	C₄×D₄×C₂₃	—	2	2	no	—
49	C₄×D₄×C₂₃	—	2	2	no	—
50	C₄×D₄×C₂₃	—	4	1	no	—
51	C₄×D₄×C₂₃	—	4	1	no	—
52	C₂₃×#117	—	2	2	no	—
53	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
55	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
56	C₂₃×M₃₂	—	2	4	no	—
57	C₂×C₁₆×C₂₃	C₂³	1	4	yes	—
58	C₂₃×#107	—	2	1	no	—
59	C₂₃×#107	—	2	1	no	—
60	C₂₃×#107	—	2	1	no	—
61	C₂₃×#107	C₂³	1	2	yes	—
62	C₂₃×#107	C₂³	1	2	yes	—
63	C₂₃×#107	C₂³	1	2	yes	—
64	C₂₃×#106	—	2	2	no	—
65	C₂₃×#106	—	2	2	no	—
66	C₂₃×#102	—	2	2	no	—
67	C₂₃×#102	—	2	2	no	—
68	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂⁴×C₂₃	—	4	1	no	—
73	C₂×Q₈×C₂₃	—	4	1	no	—
74	C₂×Q₈×C₂₃	—	4	1	no	—
75	C₂×D₄×C₂₃	—	2	4	no	—
76	C₂×D₄×C₂₃	—	2	2	no	—
77	C₂×D₄×C₂₃	—	4	1	no	—
78	C₂×D₄×C₂₃	—	4	2	no	—
79	C₂²×C₄×C₂₃	—	4	1	no	—
80	C₂²×C₄×C₂₃	—	4	2	no	—
81	C₂²×C₄×C₂₃	—	4	2	no	—
82	C₂²×C₄×C₂₃	—	8	4	no	—
83	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
86	D₈×C₂₃	—	2	4	no	—
87	D₈×C₂₃	—	2	2	no	—
88	D₈×C₂₃	C₂×Q₈	1	4	yes	—
89	C₂×C₈×C₂₃	—	2	2	no	—
90	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₂₃	#41	1	2	yes	—
95	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₄⋊C₄×C₂₃	—	4	1	no	—
100	C₄⋊C₄×C₂₃	—	4	2	no	—
101	C₄⋊C₄×C₂₃	—	4	2	no	—
102	C₄⋊C₄×C₂₃	—	4	1	no	—
103	C₄⋊C₄×C₂₃	—	4	1	no	—
104	C₂²⋊C₄×C₂₃	—	4	2	no	—
105	C₂²⋊C₄×C₂₃	—	4	2	no	—
106	C₂²⋊C₄×C₂₃	—	4	2	no	—
107	C₄²×C₂₃	—	4	2	no	—
108	C₄²×C₂₃	D₈	1	1	yes	characteristic
109	C₄²×C₂₃	D₈	1	1	yes	characteristic
110	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₂₃	—	2	4	no	—
113	#33627	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂³×C₂₃	—	4	2	no	—
115	C₂³×C₂₃	—	4	4	no	—
116	C₂³×C₂₃	—	8	2	no	—
117	C₂³×C₂₃	—	8	2	no	—
118	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
119	Q₈×C₂₃	—	8	2	no	—
120	Q₈×C₂₃	—	8	2	no	—
121	D₄×C₂₃	—	2	4	no	—
122	D₄×C₂₃	—	4	2	no	—
123	D₄×C₂₃	—	4	4	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	4	1	no	—
126	C₂×C₄×C₂₃	—	4	2	no	—
127	C₂×C₄×C₂₃	—	4	2	no	—
128	C₂×C₄×C₂₃	—	4	2	no	—
129	C₂×C₄×C₂₃	—	8	4	no	—
130	C₂×C₄×C₂₃	—	8	2	no	—
131	C₂×C₄×C₂₃	—	8	4	no	—
132	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
133	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₂₃	#137	1	2	yes	—
139	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₂₃	—	2	1	no	—
141	C₈×C₂₃	—	2	1	no	—
142	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
143	C₈×C₂₃	C₂²×Q₈	1	1	yes	characteristic
144	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
145	#3160	C₂×C₂₃	1	1	yes	characteristic, complemented
146	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic
147	#2228	C₂×C₂₃	1	4	yes	complemented
148	#2226	C₂×C₂₃	1	2	yes	complemented
149	#2224	C₂×C₂₃	1	2	yes	complemented
150	#2159	C₂×C₂₃	1	2	yes	complemented
151	#2150	C₂×C₂₃	1	1	yes	characteristic, complemented
152	#2140	C₂×C₂₃	1	2	yes	—
153	C₂²×C₂₃	—	2	1	no	—
154	C₂²×C₂₃	—	2	1	no	—
155	C₂²×C₂₃	—	4	4	no	—
156	C₂²×C₂₃	—	8	8	no	—
157	C₂²×C₂₃	—	8	2	no	—
158	C₂²×C₂₃	—	8	2	no	—
159	C₂²×C₂₃	#474	1	2	yes	—
160	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	2	1	no	—
163	C₄×C₂₃	—	2	1	no	—
164	C₄×C₂₃	—	8	2	no	—
165	C₄×C₂₃	—	8	2	no	—
166	C₄×C₂₃	—	8	2	no	—
167	C₄×C₂₃	#474	1	2	yes	—
168	C₄×C₂₃	#509	1	2	yes	—
169	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
170	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
171	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
172	#533	—	2	1	no	—
173	C₂×C₄×D₄	—	2	1	no	—
174	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
175	#500	C₂²×C₂₃	1	1	yes	characteristic, complemented
176	#500	C₂²×C₂₃	1	1	yes	characteristic, complemented
177	#499	—	2	1	no	—
178	#499	C₂²×C₂₃	1	2	yes	complemented
179	#498	—	2	1	no	—
180	C₄×D₈	—	2	1	no	—
181	C₄×D₈	—	2	1	no	—
182	C₄×D₈	C₂²×C₂₃	1	2	yes	—
183	C₄×D₈	C₂²×C₂₃	1	2	yes	—
184	#427	—	2	1	no	—
185	#427	C₂²×C₂₃	1	2	yes	—
186	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
187	C₂×#102	—	2	1	no	—
188	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
189	#367	C₂²×C₂₃	1	4	yes	complemented
190	#366	C₂²×C₂₃	1	4	yes	complemented
191	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	complemented
192	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	complemented
193	#357	C₂²×C₂₃	1	4	yes	—
194	#357	C₂²×C₂₃	1	4	yes	—
195	C₂×C₂₃	—	2	2	no	—
196	C₂×C₂₃	—	8	4	no	—
197	C₂×C₂₃	C₂×#474	1	1	yes	characteristic
198	C₂×C₂₃	C₂×#509	1	1	yes	characteristic
199	C₂×C₂₃	#3414	1	1	yes	characteristic
200	C₂²×D₄	—	2	1	no	—
201	C₂³×C₄	—	4	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	2	no	—
204	C₂×D₈	Q₈×C₂₃	1	4	yes	complemented
205	C₂×D₈	C₂³×C₂₃	1	2	yes	—
206	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	#125	—	2	2	no	—
211	C₂²⋊Q₈	—	4	1	no	—
212	C₂²⋊Q₈	—	4	1	no	—
213	C₄×Q₈	—	4	1	no	—
214	C₄×Q₈	—	4	1	no	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	4	1	no	—
218	C₄×D₄	—	4	1	no	—
219	#117	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	4	1	no	—
222	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
223	M₃₂	—	2	4	no	—
224	C₂×C₁₆	C₂³×C₂₃	1	4	yes	—
225	#107	—	2	1	no	—
226	#107	—	2	1	no	—
227	#107	—	2	1	no	—
228	#107	C₂³×C₂₃	1	2	yes	—
229	#107	C₂³×C₂₃	1	2	yes	—
230	#107	C₂³×C₂₃	1	2	yes	—
231	#106	—	2	2	no	—
232	#106	—	2	2	no	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
236	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
237	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
239	C₂₃	#33627	1	1	yes	characteristic, direct product factor, Sylow
240	C₂⁴	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	8	4	no	—
251	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
252	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
254	D₈	—	2	4	no	—
255	D₈	—	2	2	no	—
256	D₈	C₂×Q₈×C₂₃	1	4	yes	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
261	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
262	C₂×C₈	#41×C₂₃	1	2	yes	—
263	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	—	4	2	no	—
276	C₄²	D₈×C₂₃	1	1	yes	characteristic
277	C₄²	D₈×C₂₃	1	1	yes	characteristic
278	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
280	C₁₆	—	2	4	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	4	4	no	—
283	C₂³	—	8	2	no	—
284	C₂³	—	8	2	no	—
285	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
286	Q₈	—	8	2	no	—
287	Q₈	—	8	2	no	—
288	D₄	—	2	4	no	—
289	D₄	—	4	2	no	—
290	D₄	—	4	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	8	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
300	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
304	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
305	C₂×C₄	C₂₃×#137	1	2	yes	—
306	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
310	C₈	C₂²×Q₈×C₂₃	1	1	yes	characteristic
311	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	8	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	C₂₃×#474	1	2	yes	—
319	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₂₃×#474	1	2	yes	—
327	C₄	C₂₃×#509	1	2	yes	—
328	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
329	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	8	4	no	—
332	C₂	C₂×C₂₃×#474	1	1	yes	characteristic
333	C₂	C₂×C₂₃×#509	1	1	yes	characteristic
334	C₂	C₂₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3160	C₂	semidirect
3. C₂₃×#2228	C₂	semidirect
4. C₂₃×#2226	C₂	semidirect
5. C₂₃×#2224	C₂	semidirect
6. C₂₃×#2159	C₂	semidirect
7. C₂₃×#2150	C₂	semidirect
13. C₂₃×#500	C₂²	semidirect
14. C₂₃×#500	C₂²	semidirect
16. C₂₃×#499	C₂²	semidirect
27. C₂₃×#367	C₂²	semidirect
28. C₂₃×#366	C₂²	semidirect
29. C₂₃×C₄⋊C₁₆	C₂²	semidirect
30. C₂₃×C₄⋊C₁₆	C₂²	semidirect
37. C₂×D₈×C₂₃	Q₈	semidirect
113. #33627	C₂₃	direct (N × H)
145. #3160	C₂×C₂₃	semidirect
147. #2228	C₂×C₂₃	semidirect
148. #2226	C₂×C₂₃	semidirect
149. #2224	C₂×C₂₃	semidirect
150. #2159	C₂×C₂₃	semidirect
151. #2150	C₂×C₂₃	semidirect
175. #500	C₂²×C₂₃	semidirect
176. #500	C₂²×C₂₃	semidirect
178. #499	C₂²×C₂₃	semidirect
189. #367	C₂²×C₂₃	semidirect
190. #366	C₂²×C₂₃	semidirect
191. C₄⋊C₁₆	C₂²×C₂₃	semidirect
192. C₄⋊C₁₆	C₂²×C₂₃	semidirect
204. C₂×D₈	Q₈×C₂₃	semidirect
239. C₂₃	#33627	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×#3160	2
C₂×C₄×D₈×C₂₃	2
C₂₃×#2228	2
C₂₃×#2226	2
C₂₃×#2224	2
C₂₃×#2159	2
C₂₃×#2150	2
C₂₃×#2140	2
#33627	23

Subgroups — C23×#33627 #19181253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33627 #19181253