Subgroups — C₂₃×#17721 #19165347

Order 5888.

1172 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3499	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1547	C₂	1	3	yes	maximal, complemented
5	C₂₃×#1546	C₂	1	3	yes	maximal, complemented
6	C₂₃×#1540	C₂	1	3	yes	maximal, complemented
7	C₂×C₂₃×#332	C₂	1	3	yes	maximal, complemented
8	C₂×C₂₃×2₋¹⁺⁴	—	2	1	no	—
9	C₂×C₂₃×#131	—	2	1	no	—
10	D₄×Q₈×C₂₃	—	2	1	no	—
11	D₄×Q₈×C₂₃	C₄	1	2	yes	—
12	C₂₃×#547	—	2	3	no	—
13	C₂₃×#547	C₄	1	6	yes	—
14	C₂₃×#536	—	2	3	no	—
15	C₂₃×#532	C₂²	1	3	yes	—
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	3	yes	—
17	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
18	C₈×Q₈×C₂₃	—	2	1	no	—
19	C₈×D₄×C₂₃	—	2	3	no	—
20	C₂₃×#431	C₂²	1	3	yes	—
21	C₂₃×#424	—	2	3	no	—
22	C₂₃×#423	C₂²	1	3	yes	—
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	—
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	—
25	C₂₃×#332	C₂²	1	3	yes	—
26	C₂₃×#332	C₂²	1	6	yes	—
27	C₂₃×#332	C₂²	1	3	yes	—
28	C₂₃×#331	C₂²	1	1	yes	characteristic
29	C₂₃×#331	C₂²	1	3	yes	—
30	C₂₃×2₋¹⁺⁴	—	4	4	no	—
31	C₂₃×2₋¹⁺⁴	—	4	4	no	—
32	C₂×#41×C₂₃	—	2	6	no	—
33	C₂×#41×C₂₃	—	2	3	no	—
34	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
35	C₂²×Q₈×C₂₃	—	2	2	no	—
36	C₂²×Q₈×C₂₃	—	2	3	no	—
37	C₂₃×#131	—	4	4	no	—
38	C₂×M₁₆×C₂₃	—	2	3	no	—
39	C₂²×C₈×C₂₃	—	2	3	no	—
40	C₂²×C₈×C₂₃	—	2	3	no	—
41	C₂₃×C₄⋊Q₈	—	2	3	no	—
42	C₂₃×C₄⋊Q₈	—	2	3	no	—
43	C₂₃×C₄⋊Q₈	C₂×C₄	1	6	yes	—
44	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
45	C₂₃×#127	C₂³	1	1	yes	characteristic
46	C₂₃×#124	—	2	3	no	—
47	C₂₃×#123	—	2	12	no	—
48	C₂₃×C₂²⋊Q₈	—	2	6	no	—
49	C₂₃×C₂²⋊Q₈	—	2	6	no	—
50	C₂₃×#121	—	2	3	no	—
51	C₂₃×#121	C₂×C₄	1	6	yes	—
52	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
54	C₄×D₄×C₂₃	—	2	3	no	—
55	C₄×D₄×C₂₃	C₂×C₄	1	6	yes	—
56	C₄×D₄×C₂₃	C₂×C₄	1	3	yes	—
57	C₄×D₄×C₂₃	D₄	1	3	yes	—
58	C₂₃×#117	D₄	1	3	yes	—
59	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
60	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	6	yes	—
61	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
62	C₂₃×C₄⋊C₈	—	2	3	no	—
63	C₂₃×C₄⋊C₈	—	2	3	no	—
64	C₂₃×C₄⋊C₈	—	2	3	no	—
65	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
66	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
67	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
68	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
69	C₂₃×C₂²⋊C₈	—	2	3	no	—
70	C₄×C₈×C₂₃	—	2	3	no	—
71	#41×C₂₃	—	4	24	no	—
72	#41×C₂₃	—	4	6	no	—
73	#41×C₂₃	—	4	6	no	—
74	#41×C₂₃	D₈	1	4	yes	—
75	#41×C₂₃	SD₁₆	1	4	yes	—
76	C₂×Q₈×C₂₃	—	2	6	no	—
77	C₂×Q₈×C₂₃	—	2	3	no	—
78	C₂×Q₈×C₂₃	—	4	8	no	—
79	C₂×Q₈×C₂₃	—	4	6	no	—
80	C₂×Q₈×C₂₃	—	4	6	no	—
81	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×D₄×C₂₃	—	2	6	no	—
83	C₂×D₄×C₂₃	—	2	1	no	—
84	C₂×D₄×C₂₃	—	2	1	no	—
85	C₂×D₄×C₂₃	C₂²⋊C₄	1	3	yes	—
86	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₂₃	—	2	6	no	—
88	C₂²×C₄×C₂₃	—	2	6	no	—
89	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
90	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
91	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
92	M₁₆×C₂₃	—	4	6	no	—
93	C₂×C₈×C₂₃	—	2	1	no	—
94	C₂×C₈×C₂₃	—	2	3	no	—
95	C₂×C₈×C₂₃	—	2	3	no	—
96	C₂×C₈×C₂₃	—	2	1	no	—
97	C₂×C₈×C₂₃	—	4	6	no	—
98	C₂×C₈×C₂₃	—	8	3	no	—
99	C₄⋊C₄×C₂₃	—	2	6	no	—
100	C₄⋊C₄×C₂₃	—	2	6	no	—
101	C₄⋊C₄×C₂₃	—	2	3	no	—
102	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	3	yes	—
103	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	3	yes	—
104	C₄⋊C₄×C₂₃	C₂²×C₄	1	6	yes	—
105	C₂²⋊C₄×C₂₃	—	2	6	no	—
106	C₂²⋊C₄×C₂₃	—	2	6	no	—
107	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	6	yes	—
108	C₄²×C₂₃	C₂²×C₄	1	3	yes	—
109	C₄²×C₂₃	C₂×D₄	1	3	yes	—
110	C₄²×C₂₃	C₂×D₄	1	3	yes	—
111	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
112	#17721	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₂₃	—	2	2	no	—
114	C₂³×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
115	Q₈×C₂₃	—	4	3	no	—
116	Q₈×C₂₃	—	4	3	no	—
117	Q₈×C₂₃	—	4	12	no	—
118	Q₈×C₂₃	#102	1	4	yes	—
119	D₄×C₂₃	—	4	2	no	—
120	D₄×C₂₃	—	4	12	no	—
121	D₄×C₂₃	—	4	1	no	—
122	D₄×C₂₃	—	4	1	no	—
123	D₄×C₂₃	—	4	1	no	—
124	D₄×C₂₃	—	4	1	no	—
125	D₄×C₂₃	#102	1	12	yes	—
126	C₂×C₄×C₂₃	—	2	3	no	—
127	C₂×C₄×C₂₃	—	2	3	no	—
128	C₂×C₄×C₂₃	—	2	6	no	—
129	C₂×C₄×C₂₃	—	4	3	no	—
130	C₂×C₄×C₂₃	—	4	12	no	—
131	C₂×C₄×C₂₃	—	4	12	no	—
132	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
133	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
134	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
135	C₂×C₄×C₂₃	C₂×D₈	1	6	yes	—
136	C₂×C₄×C₂₃	C₂×SD₁₆	1	6	yes	—
137	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₂₃	—	4	1	no	—
141	C₈×C₂₃	—	4	3	no	—
142	C₈×C₂₃	—	4	3	no	—
143	C₈×C₂₃	—	4	1	no	—
144	#3499	C₂×C₂₃	1	1	yes	characteristic
145	#2988	C₂×C₂₃	1	1	yes	characteristic, complemented
146	#2906	C₂×C₂₃	1	1	yes	characteristic, complemented
147	#1547	C₂×C₂₃	1	3	yes	complemented
148	#1546	C₂×C₂₃	1	3	yes	complemented
149	#1540	C₂×C₂₃	1	3	yes	complemented
150	C₂×#332	C₂×C₂₃	1	3	yes	complemented
151	C₂²×C₂₃	—	2	3	no	—
152	C₂²×C₂₃	—	2	3	no	—
153	C₂²×C₂₃	—	4	2	no	—
154	C₂²×C₂₃	—	4	2	no	—
155	C₂²×C₂₃	—	4	2	no	—
156	C₂²×C₂₃	C₂×#102	1	6	yes	—
157	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
158	C₄×C₂₃	—	2	1	no	—
159	C₄×C₂₃	—	2	3	no	—
160	C₄×C₂₃	—	2	3	no	—
161	C₄×C₂₃	—	4	6	no	—
162	C₄×C₂₃	C₂×#102	1	6	yes	—
163	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
164	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
165	C₄×C₂₃	#577	1	1	yes	characteristic
166	C₄×C₂₃	#578	1	1	yes	characteristic
167	C₂×2₋¹⁺⁴	—	2	1	no	—
168	C₂×#131	—	2	1	no	—
169	D₄×Q₈	—	2	1	no	—
170	D₄×Q₈	C₄×C₂₃	1	2	yes	—
171	#547	—	2	3	no	—
172	#547	C₄×C₂₃	1	6	yes	—
173	#536	—	2	3	no	—
174	#532	C₂²×C₂₃	1	3	yes	—
175	C₂×C₄⋊Q₈	C₂²×C₂₃	1	3	yes	—
176	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
177	C₈×Q₈	—	2	1	no	—
178	C₈×D₄	—	2	3	no	—
179	#431	C₂²×C₂₃	1	3	yes	—
180	#424	—	2	3	no	—
181	#423	C₂²×C₂₃	1	3	yes	—
182	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	—
183	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	—
184	#332	C₂²×C₂₃	1	3	yes	—
185	#332	C₂²×C₂₃	1	6	yes	—
186	#332	C₂²×C₂₃	1	3	yes	—
187	#331	C₂²×C₂₃	1	1	yes	characteristic
188	#331	C₂²×C₂₃	1	3	yes	—
189	C₂×C₂₃	—	2	6	no	—
190	C₂×C₂₃	—	8	2	no	—
191	C₂×C₂₃	#2889	1	1	yes	characteristic
192	C₂×C₂₃	C₂²×#102	1	1	yes	characteristic
193	C₂×C₂₃	#2898	1	1	yes	characteristic
194	2₋¹⁺⁴	—	4	4	no	—
195	2₋¹⁺⁴	—	4	4	no	—
196	C₂×#41	—	2	6	no	—
197	C₂×#41	—	2	3	no	—
198	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
199	C₂²×Q₈	—	2	2	no	—
200	C₂²×Q₈	—	2	3	no	—
201	#131	—	4	4	no	—
202	C₂×M₁₆	—	2	3	no	—
203	C₂²×C₈	—	2	3	no	—
204	C₂²×C₈	—	2	3	no	—
205	C₄⋊Q₈	—	2	3	no	—
206	C₄⋊Q₈	—	2	3	no	—
207	C₄⋊Q₈	C₂×C₄×C₂₃	1	6	yes	—
208	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
209	#127	C₂³×C₂₃	1	1	yes	characteristic
210	#124	—	2	3	no	—
211	#123	—	2	12	no	—
212	C₂²⋊Q₈	—	2	6	no	—
213	C₂²⋊Q₈	—	2	6	no	—
214	#121	—	2	3	no	—
215	#121	C₂×C₄×C₂₃	1	6	yes	—
216	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
217	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
218	C₄×D₄	—	2	3	no	—
219	C₄×D₄	C₂×C₄×C₂₃	1	6	yes	—
220	C₄×D₄	C₂×C₄×C₂₃	1	3	yes	—
221	C₄×D₄	D₄×C₂₃	1	3	yes	—
222	#117	D₄×C₂₃	1	3	yes	—
223	C₂×C₄⋊C₄	—	2	3	no	—
224	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	6	yes	—
225	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
226	C₄⋊C₈	—	2	3	no	—
227	C₄⋊C₈	—	2	3	no	—
228	C₄⋊C₈	—	2	3	no	—
229	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
230	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
231	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
232	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
233	C₂²⋊C₈	—	2	3	no	—
234	C₄×C₈	—	2	3	no	—
235	C₂₃	#17721	1	1	yes	characteristic, direct product factor, Sylow
236	#41	—	4	24	no	—
237	#41	—	4	6	no	—
238	#41	—	4	6	no	—
239	#41	D₈×C₂₃	1	4	yes	—
240	#41	SD₁₆×C₂₃	1	4	yes	—
241	C₂×Q₈	—	2	6	no	—
242	C₂×Q₈	—	2	3	no	—
243	C₂×Q₈	—	4	8	no	—
244	C₂×Q₈	—	4	6	no	—
245	C₂×Q₈	—	4	6	no	—
246	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
247	C₂×D₄	—	2	6	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	C₂²⋊C₄×C₂₃	1	3	yes	—
251	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
252	C₂²×C₄	—	2	6	no	—
253	C₂²×C₄	—	2	6	no	—
254	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
255	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
256	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
257	M₁₆	—	4	6	no	—
258	C₂×C₈	—	2	1	no	—
259	C₂×C₈	—	2	3	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	3	no	—
262	C₂×C₈	—	4	6	no	—
263	C₂×C₈	—	8	3	no	—
264	C₄⋊C₄	—	2	6	no	—
265	C₄⋊C₄	—	2	6	no	—
266	C₄⋊C₄	—	2	3	no	—
267	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	3	yes	—
268	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	3	yes	—
269	C₄⋊C₄	C₂²×C₄×C₂₃	1	6	yes	—
270	C₂²⋊C₄	—	2	6	no	—
271	C₂²⋊C₄	—	2	6	no	—
272	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	6	yes	—
273	C₄²	C₂²×C₄×C₂₃	1	3	yes	—
274	C₄²	C₂×D₄×C₂₃	1	3	yes	—
275	C₄²	C₂×D₄×C₂₃	1	3	yes	—
276	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
277	C₂³	—	2	2	no	—
278	C₂³	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
279	Q₈	—	4	3	no	—
280	Q₈	—	4	3	no	—
281	Q₈	—	4	12	no	—
282	Q₈	C₂₃×#102	1	4	yes	—
283	D₄	—	4	2	no	—
284	D₄	—	4	12	no	—
285	D₄	—	4	1	no	—
286	D₄	—	4	1	no	—
287	D₄	—	4	1	no	—
288	D₄	—	4	1	no	—
289	D₄	C₂₃×#102	1	12	yes	—
290	C₂×C₄	—	2	3	no	—
291	C₂×C₄	—	2	3	no	—
292	C₂×C₄	—	2	6	no	—
293	C₂×C₄	—	4	3	no	—
294	C₂×C₄	—	4	12	no	—
295	C₂×C₄	—	4	12	no	—
296	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
297	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
298	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
299	C₂×C₄	C₂×D₈×C₂₃	1	6	yes	—
300	C₂×C₄	C₂×SD₁₆×C₂₃	1	6	yes	—
301	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
304	C₈	—	4	1	no	—
305	C₈	—	4	3	no	—
306	C₈	—	4	3	no	—
307	C₈	—	4	1	no	—
308	C₂²	—	2	3	no	—
309	C₂²	—	2	3	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	—	4	2	no	—
313	C₂²	C₂×C₂₃×#102	1	6	yes	—
314	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	3	no	—
317	C₄	—	2	3	no	—
318	C₄	—	4	6	no	—
319	C₄	C₂×C₂₃×#102	1	6	yes	—
320	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
321	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
322	C₄	C₂₃×#577	1	1	yes	characteristic
323	C₄	C₂₃×#578	1	1	yes	characteristic
324	C₂	—	2	6	no	—
325	C₂	—	8	2	no	—
326	C₂	C₂₃×#2889	1	1	yes	characteristic
327	C₂	C₂²×C₂₃×#102	1	1	yes	characteristic
328	C₂	C₂₃×#2898	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3499	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2988	C₂	semidirect
3. C₂₃×#2906	C₂	semidirect
4. C₂₃×#1547	C₂	semidirect
5. C₂₃×#1546	C₂	semidirect
6. C₂₃×#1540	C₂	semidirect
7. C₂×C₂₃×#332	C₂	semidirect
112. #17721	C₂₃	direct (N × H)
145. #2988	C₂×C₂₃	semidirect
146. #2906	C₂×C₂₃	semidirect
147. #1547	C₂×C₂₃	semidirect
148. #1546	C₂×C₂₃	semidirect
149. #1540	C₂×C₂₃	semidirect
150. C₂×#332	C₂×C₂₃	semidirect
235. C₂₃	#17721	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×#3499	2
C₂₃×#2988	2
C₂₃×#2906	2
C₂₃×#1547	2
C₂₃×#1546	2
C₂₃×#1540	2
C₂×C₂₃×#332	2
#17721	23

Subgroups — C23×#17721 #19165347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17721 #19165347