Subgroups — C₂₃×#8197 #19155823

Order 5888.

468 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2965	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×#104	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1476	C₂	1	1	yes	maximal, characteristic
5	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1276	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#446	C₄	1	2	yes	complemented
10	C₂₃×#436	C₄	1	2	yes	complemented
11	C₂₃×#435	—	2	2	no	—
12	C₂₃×#431	—	2	1	no	—
13	C₂₃×#431	C₂²	1	1	yes	characteristic
14	C₂₃×#431	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#104	C₄	1	2	yes	—
16	C₄×M₁₆×C₂₃	—	2	1	no	—
17	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₄³×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#339	C₄	1	2	yes	—
22	C₂₃×#336	—	2	2	no	—
23	C₂₃×Q₈⋊C₈	C₄	1	2	yes	complemented
24	C₂₃×D₄⋊C₈	C₄	1	2	yes	complemented
25	C₂₃×#322	—	2	1	no	—
26	C₈²×C₂₃	—	2	1	no	—
27	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₂₃	—	2	1	no	—
31	C₂²×C₈×C₂₃	D₄	1	2	yes	—
32	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₄×D₄×C₂₃	—	2	1	no	—
35	C₄×D₄×C₂₃	—	2	1	no	—
36	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂₃×#117	—	2	1	no	—
39	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	—	2	1	no	—
41	C₂×C₄²×C₂₃	—	2	1	no	—
42	C₂×C₄²×C₂₃	—	2	1	no	—
43	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
45	C₂₃×C₄⋊C₈	—	2	2	no	—
46	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₂₃×#104	—	2	4	no	—
49	C₂₃×C₂²⋊C₈	—	2	1	no	—
50	C₂₃×C₂²⋊C₈	—	2	1	no	—
51	C₂₃×C₂²⋊C₈	—	2	2	no	—
52	C₂₃×#97	—	2	2	no	—
53	C₂₃×#97	—	2	1	no	—
54	C₄×C₈×C₂₃	—	2	1	no	—
55	C₄×C₈×C₂₃	—	2	1	no	—
56	C₄×C₈×C₂₃	—	2	1	no	—
57	C₄×C₈×C₂₃	—	2	1	no	—
58	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
60	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
62	#41×C₂₃	—	2	2	no	—
63	#41×C₂₃	—	2	2	no	—
64	C₂×Q₈×C₂₃	C₄²	1	1	yes	characteristic
65	C₂×D₄×C₂₃	—	2	1	no	—
66	C₂×D₄×C₂₃	C₄²	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	—	2	2	no	—
68	C₂²×C₄×C₂₃	—	2	1	no	—
69	C₂²×C₄×C₂₃	—	2	1	no	—
70	C₂²×C₄×C₂₃	—	2	1	no	—
71	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
74	M₁₆×C₂₃	—	2	2	no	—
75	M₁₆×C₂₃	—	4	4	no	—
76	M₁₆×C₂₃	M₁₆	1	4	yes	complemented
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	—	4	2	no	—
82	C₂×C₈×C₂₃	—	4	2	no	—
83	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
85	C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₄⋊C₄×C₂₃	C₄²	1	1	yes	characteristic
88	C₄⋊C₄×C₂₃	C₄²	1	1	yes	characteristic
89	C₂²⋊C₄×C₂₃	—	2	1	no	—
90	C₂²⋊C₄×C₂₃	—	2	1	no	—
91	C₂²⋊C₄×C₂₃	—	2	1	no	—
92	C₄²×C₂₃	—	2	4	no	—
93	C₄²×C₂₃	—	2	1	no	—
94	C₄²×C₂₃	—	2	1	no	—
95	C₄²×C₂₃	—	2	1	no	—
96	C₄²×C₂₃	—	2	2	no	—
97	C₄²×C₂₃	—	2	2	no	—
98	C₄²×C₂₃	—	4	2	no	—
99	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₂₃	#41	1	1	yes	characteristic
103	#8197	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	—	2	1	no	—
105	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
106	Q₈×C₂₃	—	2	1	no	—
107	Q₈×C₂₃	—	2	1	no	—
108	D₄×C₂₃	—	2	1	no	—
109	D₄×C₂₃	—	2	1	no	—
110	D₄×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	4	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	4	2	no	—
118	C₂×C₄×C₂₃	—	4	2	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	2	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₂₃	#131	1	2	yes	—
130	C₈×C₂₃	—	2	2	no	—
131	C₈×C₂₃	—	2	2	no	—
132	C₈×C₂₃	—	2	4	no	—
133	C₈×C₂₃	—	4	2	no	—
134	C₈×C₂₃	—	4	2	no	—
135	C₈×C₂₃	—	4	2	no	—
136	#2965	C₂×C₂₃	1	1	yes	characteristic
137	C₄×#104	C₂×C₂₃	1	1	yes	characteristic
138	#1744	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#1476	C₂×C₂₃	1	1	yes	characteristic
140	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
141	#1278	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#1276	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂²×C₂₃	—	2	1	no	—
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	4	1	no	—
146	C₂²×C₂₃	—	4	1	no	—
147	C₂²×C₂₃	—	4	1	no	—
148	C₂²×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₂₃	#436	1	2	yes	—
150	C₄×C₂₃	—	2	1	no	—
151	C₄×C₂₃	—	2	1	no	—
152	C₄×C₂₃	—	4	4	no	—
153	C₄×C₂₃	—	4	1	no	—
154	C₄×C₂₃	—	4	1	no	—
155	C₄×C₂₃	—	4	1	no	—
156	C₄×C₂₃	C₄×M₁₆	1	1	yes	characteristic
157	C₄×C₂₃	#405	1	1	yes	characteristic
158	C₄×C₂₃	#421	1	2	yes	—
159	C₄×C₂₃	#435	1	2	yes	—
160	C₄×C₂₃	#443	1	2	yes	—
161	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
162	#446	C₄×C₂₃	1	2	yes	complemented
163	#436	C₄×C₂₃	1	2	yes	complemented
164	#435	—	2	2	no	—
165	#431	—	2	1	no	—
166	#431	C₂²×C₂₃	1	1	yes	characteristic
167	#431	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×#104	C₄×C₂₃	1	2	yes	—
169	C₄×M₁₆	—	2	1	no	—
170	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
173	C₄³	C₂²×C₂₃	1	1	yes	characteristic
174	#339	C₄×C₂₃	1	2	yes	—
175	#336	—	2	2	no	—
176	Q₈⋊C₈	C₄×C₂₃	1	2	yes	complemented
177	D₄⋊C₈	C₄×C₂₃	1	2	yes	complemented
178	#322	—	2	1	no	—
179	C₈²	—	2	1	no	—
180	C₂×C₂₃	—	2	2	no	—
181	C₂×C₂₃	—	8	1	no	—
182	C₂×C₂₃	#1753	1	1	yes	characteristic
183	C₂×C₂₃	#1759	1	1	yes	characteristic
184	C₂×C₂₃	#1764	1	1	yes	characteristic
185	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	D₄×C₂₃	1	2	yes	—
190	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
191	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
192	C₄×D₄	—	2	1	no	—
193	C₄×D₄	—	2	1	no	—
194	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
195	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
196	#117	—	2	1	no	—
197	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	2	1	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
202	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
203	C₄⋊C₈	—	2	2	no	—
204	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
205	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
206	#104	—	2	4	no	—
207	C₂²⋊C₈	—	2	1	no	—
208	C₂²⋊C₈	—	2	1	no	—
209	C₂²⋊C₈	—	2	2	no	—
210	#97	—	2	2	no	—
211	#97	—	2	1	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	2	1	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	—	2	1	no	—
216	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
217	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
218	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
219	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
220	C₂₃	#8197	1	1	yes	characteristic, direct product factor, Sylow
221	#41	—	2	2	no	—
222	#41	—	2	2	no	—
223	C₂×Q₈	C₄²×C₂₃	1	1	yes	characteristic
224	C₂×D₄	—	2	1	no	—
225	C₂×D₄	C₄²×C₂₃	1	1	yes	characteristic
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
232	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
233	M₁₆	—	2	2	no	—
234	M₁₆	—	4	4	no	—
235	M₁₆	M₁₆×C₂₃	1	4	yes	complemented
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	2	no	—
242	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
243	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	C₄²×C₂₃	1	1	yes	characteristic
247	C₄⋊C₄	C₄²×C₂₃	1	1	yes	characteristic
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
259	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
261	C₄²	#41×C₂₃	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
264	Q₈	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	D₄	—	2	1	no	—
267	D₄	—	2	1	no	—
268	D₄	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
281	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
282	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
283	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
284	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
285	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
286	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
287	C₂×C₄	C₂₃×#131	1	2	yes	—
288	C₈	—	2	2	no	—
289	C₈	—	2	2	no	—
290	C₈	—	2	4	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	4	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
300	C₂²	C₂₃×#436	1	2	yes	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	4	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	C₄×M₁₆×C₂₃	1	1	yes	characteristic
308	C₄	C₂₃×#405	1	1	yes	characteristic
309	C₄	C₂₃×#421	1	2	yes	—
310	C₄	C₂₃×#435	1	2	yes	—
311	C₄	C₂₃×#443	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	8	1	no	—
314	C₂	C₂₃×#1753	1	1	yes	characteristic
315	C₂	C₂₃×#1759	1	1	yes	characteristic
316	C₂	C₂₃×#1764	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1744	C₂	semidirect
5. C₈×M₁₆×C₂₃	C₂	semidirect
6. C₂₃×#1278	C₂	semidirect
7. C₂₃×#1276	C₂	semidirect
9. C₂₃×#446	C₄	semidirect
10. C₂₃×#436	C₄	semidirect
23. C₂₃×Q₈⋊C₈	C₄	semidirect
24. C₂₃×D₄⋊C₈	C₄	semidirect
76. M₁₆×C₂₃	M₁₆	semidirect
103. #8197	C₂₃	direct (N × H)
138. #1744	C₂×C₂₃	semidirect
140. C₈×M₁₆	C₂×C₂₃	semidirect
141. #1278	C₂×C₂₃	semidirect
142. #1276	C₂×C₂₃	semidirect
162. #446	C₄×C₂₃	semidirect
163. #436	C₄×C₂₃	semidirect
176. Q₈⋊C₈	C₄×C₂₃	semidirect
177. D₄⋊C₈	C₄×C₂₃	semidirect
220. C₂₃	#8197	direct (N × H)
235. M₁₆	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2965	2
C₄×C₂₃×#104	2
C₂₃×#1744	2
C₂₃×#1476	2
C₈×M₁₆×C₂₃	2
C₂₃×#1278	2
C₂₃×#1276	2
#8197	23

Subgroups — C23×#8197 #19155823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8197 #19155823