Subgroups — C₂₃×#18084 #19165710

Order 5888.

652 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2979	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1589	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1586	C₂	1	4	yes	maximal, complemented
5	C₈×D₈×C₂₃	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1484	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#532	C₂²	1	2	yes	—
10	C₂₃×#500	C₄	1	1	yes	characteristic
11	C₂₃×#493	C₄	1	1	yes	characteristic
12	C₂₃×#486	C₄	1	2	yes	—
13	C₂₃×#466	C₄	1	2	yes	—
14	C₂₃×#466	C₄	1	2	yes	—
15	C₄×D₈×C₂₃	C₂²	1	2	yes	—
16	C₄×D₈×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#436	—	2	2	no	—
18	C₂₃×#436	C₂²	1	4	yes	—
19	C₈×D₄×C₂₃	—	2	2	no	—
20	C₈×D₄×C₂₃	C₂²	1	4	yes	—
21	C₂₃×#431	C₂²	1	2	yes	—
22	C₂₃×#431	C₂²	1	1	yes	characteristic
23	C₂₃×#408	—	2	4	no	—
24	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#335	C₂²	1	1	yes	characteristic
27	C₂₃×#335	C₂²	1	1	yes	characteristic
28	C₂₃×#334	C₂²	1	2	yes	—
29	C₂₃×D₄⋊C₈	C₂²	1	4	yes	—
30	C₂₃×D₄⋊C₈	C₂²	1	4	yes	—
31	C₂₃×#322	C₂²	1	2	yes	—
32	C₈²×C₂₃	C₂²	1	2	yes	—
33	C₂×#41×C₂₃	—	2	2	no	—
34	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₂₃	—	2	1	no	—
37	C₂×M₁₆×C₂₃	—	2	4	no	—
38	C₂²×C₈×C₂₃	—	2	4	no	—
39	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
42	C₂₃×#127	C₂×C₄	1	2	yes	—
43	C₂₃×#124	—	2	2	no	—
44	C₂₃×#121	—	2	2	no	—
45	C₂₃×#121	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₂₃	—	2	2	no	—
47	C₄×D₄×C₂₃	C₂³	1	4	yes	—
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×#107	C₂×C₄	1	2	yes	—
50	C₂₃×#107	C₂×C₄	1	2	yes	—
51	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
52	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
53	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
54	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
55	C₂₃×#102	C₂×C₄	1	4	yes	—
56	C₂₃×#102	C₂×C₄	1	4	yes	—
57	C₂₃×C₂²⋊C₈	—	2	2	no	—
58	C₂₃×C₂²⋊C₈	—	2	4	no	—
59	C₂₃×C₂²⋊C₈	—	2	4	no	—
60	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
61	C₂₃×#97	—	2	1	no	—
62	C₄×C₈×C₂₃	—	2	2	no	—
63	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₂₃	C₂³	1	2	yes	—
66	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₂₃	C₂³	1	2	yes	—
69	#41×C₂₃	—	8	2	no	—
70	#41×C₂₃	—	8	2	no	—
71	C₂×Q₈×C₂₃	—	2	2	no	—
72	C₂×D₄×C₂₃	—	2	4	no	—
73	C₂×D₄×C₂₃	—	2	2	no	—
74	C₂×D₄×C₂₃	—	2	2	no	—
75	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
76	C₂²×C₄×C₂₃	—	2	4	no	—
77	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
80	D₈×C₂₃	—	2	4	no	—
81	D₈×C₂₃	—	4	2	no	—
82	M₁₆×C₂₃	—	2	4	no	—
83	M₁₆×C₂₃	—	8	4	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	—	2	2	no	—
87	C₂×C₈×C₂₃	—	2	2	no	—
88	C₂×C₈×C₂₃	—	4	1	no	—
89	C₂×C₈×C₂₃	—	8	4	no	—
90	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
96	C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
98	C₂²⋊C₄×C₂₃	—	2	4	no	—
99	C₂²⋊C₄×C₂₃	—	2	4	no	—
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	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
104	#18084	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂³×C₂₃	—	2	4	no	—
106	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
107	Q₈×C₂₃	—	8	2	no	—
108	D₄×C₂₃	—	2	4	no	—
109	D₄×C₂₃	—	2	4	no	—
110	D₄×C₂₃	—	4	4	no	—
111	D₄×C₂₃	—	8	2	no	—
112	D₄×C₂₃	—	8	4	no	—
113	C₂×C₄×C₂₃	—	2	1	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	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	4	no	—
119	C₂×C₄×C₂₃	—	4	4	no	—
120	C₂×C₄×C₂₃	—	4	4	no	—
121	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
127	C₈×C₂₃	—	2	4	no	—
128	C₈×C₂₃	—	2	1	no	—
129	C₈×C₂₃	—	2	1	no	—
130	C₈×C₂₃	—	2	1	no	—
131	C₈×C₂₃	—	2	2	no	—
132	C₈×C₂₃	—	2	2	no	—
133	C₈×C₂₃	—	4	2	no	—
134	C₈×C₂₃	—	4	2	no	—
135	C₈×C₂₃	#131	1	2	yes	—
136	#3169	C₂×C₂₃	1	1	yes	characteristic
137	#2979	C₂×C₂₃	1	2	yes	complemented
138	#1589	C₂×C₂₃	1	2	yes	complemented
139	#1586	C₂×C₂₃	1	4	yes	complemented
140	C₈×D₈	C₂×C₂₃	1	2	yes	complemented
141	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#1484	C₂×C₂₃	1	2	yes	complemented
143	C₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	—	2	1	no	—
147	C₂²×C₂₃	—	4	4	no	—
148	C₂²×C₂₃	—	4	4	no	—
149	C₂²×C₂₃	—	4	4	no	—
150	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₂₃	—	2	1	no	—
152	C₄×C₂₃	—	2	1	no	—
153	C₄×C₂₃	—	2	1	no	—
154	C₄×C₂₃	—	4	4	no	—
155	C₄×C₂₃	#443	1	2	yes	—
156	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
157	C₄×C₂₃	#568	1	1	yes	characteristic
158	C₄×C₂₃	#576	1	1	yes	characteristic
159	C₄×C₂₃	#576	1	1	yes	characteristic
160	#532	C₂²×C₂₃	1	2	yes	—
161	#500	C₄×C₂₃	1	1	yes	characteristic
162	#493	C₄×C₂₃	1	1	yes	characteristic
163	#486	C₄×C₂₃	1	2	yes	—
164	#466	C₄×C₂₃	1	2	yes	—
165	#466	C₄×C₂₃	1	2	yes	—
166	C₄×D₈	C₂²×C₂₃	1	2	yes	—
167	C₄×D₈	C₂²×C₂₃	1	2	yes	—
168	#436	—	2	2	no	—
169	#436	C₂²×C₂₃	1	4	yes	—
170	C₈×D₄	—	2	2	no	—
171	C₈×D₄	C₂²×C₂₃	1	4	yes	—
172	#431	C₂²×C₂₃	1	2	yes	—
173	#431	C₂²×C₂₃	1	1	yes	characteristic
174	#408	—	2	4	no	—
175	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
176	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
177	#335	C₂²×C₂₃	1	1	yes	characteristic
178	#335	C₂²×C₂₃	1	1	yes	characteristic
179	#334	C₂²×C₂₃	1	2	yes	—
180	D₄⋊C₈	C₂²×C₂₃	1	4	yes	—
181	D₄⋊C₈	C₂²×C₂₃	1	4	yes	—
182	#322	C₂²×C₂₃	1	2	yes	—
183	C₈²	C₂²×C₂₃	1	2	yes	—
184	C₂×C₂₃	—	4	1	no	—
185	C₂×C₂₃	—	8	4	no	—
186	C₂×C₂₃	#2935	1	1	yes	characteristic
187	C₂×C₂₃	#2948	1	1	yes	characteristic
188	C₂×C₂₃	C₂×#443	1	1	yes	characteristic
189	C₂×#41	—	2	2	no	—
190	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
191	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
192	C₂×M₁₆	—	2	1	no	—
193	C₂×M₁₆	—	2	4	no	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
196	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
197	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
198	#127	C₂×C₄×C₂₃	1	2	yes	—
199	#124	—	2	2	no	—
200	#121	—	2	2	no	—
201	#121	C₂×C₄×C₂₃	1	4	yes	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	C₂³×C₂₃	1	4	yes	—
204	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
205	#107	C₂×C₄×C₂₃	1	2	yes	—
206	#107	C₂×C₄×C₂₃	1	2	yes	—
207	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
208	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
209	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
210	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
211	#102	C₂×C₄×C₂₃	1	4	yes	—
212	#102	C₂×C₄×C₂₃	1	4	yes	—
213	C₂²⋊C₈	—	2	2	no	—
214	C₂²⋊C₈	—	2	4	no	—
215	C₂²⋊C₈	—	2	4	no	—
216	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
217	#97	—	2	1	no	—
218	C₄×C₈	—	2	2	no	—
219	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
220	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
221	C₄×C₈	C₂³×C₂₃	1	2	yes	—
222	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
223	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
224	C₄×C₈	C₂³×C₂₃	1	2	yes	—
225	C₂₃	#18084	1	1	yes	characteristic, direct product factor, Sylow
226	#41	—	8	2	no	—
227	#41	—	8	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
236	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
237	D₈	—	2	4	no	—
238	D₈	—	4	2	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	8	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	8	4	no	—
247	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
248	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
250	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
252	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
255	C₂²⋊C₄	—	2	4	no	—
256	C₂²⋊C₄	—	2	4	no	—
257	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
259	C₄²	#41×C₂₃	1	1	yes	characteristic
260	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
261	C₂³	—	2	4	no	—
262	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
263	Q₈	—	8	2	no	—
264	D₄	—	2	4	no	—
265	D₄	—	2	4	no	—
266	D₄	—	4	4	no	—
267	D₄	—	8	2	no	—
268	D₄	—	8	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
279	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
280	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
281	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
282	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
283	C₈	—	2	4	no	—
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	2	2	no	—
288	C₈	—	2	2	no	—
289	C₈	—	4	2	no	—
290	C₈	—	4	2	no	—
291	C₈	C₂₃×#131	1	2	yes	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	4	4	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	4	4	no	—
298	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	4	no	—
303	C₄	C₂₃×#443	1	2	yes	—
304	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
305	C₄	C₂₃×#568	1	1	yes	characteristic
306	C₄	C₂₃×#576	1	1	yes	characteristic
307	C₄	C₂₃×#576	1	1	yes	characteristic
308	C₂	—	4	1	no	—
309	C₂	—	8	4	no	—
310	C₂	C₂₃×#2935	1	1	yes	characteristic
311	C₂	C₂₃×#2948	1	1	yes	characteristic
312	C₂	C₂×C₂₃×#443	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#3169	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2979	C₂	semidirect
3. C₂₃×#1589	C₂	semidirect
4. C₂₃×#1586	C₂	semidirect
5. C₈×D₈×C₂₃	C₂	semidirect
6. C₂₃×#1564	C₂	semidirect
7. C₂₃×#1484	C₂	semidirect
8. C₈×M₁₆×C₂₃	C₂	semidirect
104. #18084	C₂₃	direct (N × H)
137. #2979	C₂×C₂₃	semidirect
138. #1589	C₂×C₂₃	semidirect
139. #1586	C₂×C₂₃	semidirect
140. C₈×D₈	C₂×C₂₃	semidirect
141. #1564	C₂×C₂₃	semidirect
142. #1484	C₂×C₂₃	semidirect
143. C₈×M₁₆	C₂×C₂₃	semidirect
225. C₂₃	#18084	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×#3169	2
C₂₃×#2979	2
C₂₃×#1589	2
C₂₃×#1586	2
C₈×D₈×C₂₃	2
C₂₃×#1564	2
C₂₃×#1484	2
C₈×M₁₆×C₂₃	2
#18084	23

Subgroups — C23×#18084 #19165710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18084 #19165710