Subgroups — C₂₃×#17410 #19165036

Order 5888.

848 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#541	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2983	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1473	C₂	1	4	yes	maximal
4	C₂₃×#1470	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1467	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1464	C₂	1	2	yes	maximal
7	C₂₃×#541	—	2	8	no	—
8	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
9	C₂×C₂₃×#126	C₄	1	2	yes	—
10	C₂×C₂₃×#126	C₂²	1	2	yes	—
11	C₂×C₂₃×#125	C₄	1	1	yes	characteristic
12	C₂×C₂₃×#125	C₄	1	2	yes	—
13	C₂×C₂₃×#125	C₂²	1	2	yes	—
14	C₂×C₂₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
15	C₂×C₂₃×C₂²⋊Q₈	C₄	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
19	C₂₃×#446	—	2	8	no	—
20	C₂₃×#432	C₂²	1	4	yes	—
21	C₂₃×#432	C₂²	1	4	yes	—
22	C₂₃×#431	C₂²	1	4	yes	—
23	C₂₃×#324	C₂²	1	4	yes	—
24	C₂₃×#324	C₂²	1	8	yes	—
25	C₂₃×#323	C₂²	1	4	yes	—
26	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂₃×C₄⋊Q₈	—	2	4	no	—
30	C₂₃×#126	—	2	8	no	—
31	C₂₃×#126	—	2	4	no	—
32	C₂₃×#126	—	2	4	no	—
33	C₂₃×#125	—	2	4	no	—
34	C₂₃×#125	—	2	8	no	—
35	C₂₃×#125	—	2	4	no	—
36	C₂₃×#125	—	2	4	no	—
37	C₂₃×C₂²⋊Q₈	—	2	4	no	—
38	C₂₃×C₂²⋊Q₈	—	2	4	no	—
39	C₄×Q₈×C₂₃	—	2	4	no	—
40	C₄×Q₈×C₂₃	D₄	1	8	yes	—
41	C₂₃×#117	—	2	4	no	—
42	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
54	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
57	C₂×C₂²⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
61	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
62	C₂₃×C₄⋊C₈	—	2	4	no	—
63	C₂₃×C₄⋊C₈	—	2	4	no	—
64	C₂₃×C₄⋊C₈	—	2	4	no	—
65	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
66	C₂₃×#97	—	2	4	no	—
67	C₂₃×#97	—	2	4	no	—
68	C₄×C₈×C₂₃	—	2	4	no	—
69	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×Q₈×C₂₃	—	2	2	no	—
71	C₂×Q₈×C₂₃	—	4	2	no	—
72	C₂×Q₈×C₂₃	—	4	2	no	—
73	C₂×Q₈×C₂₃	C₂²⋊C₄	1	4	yes	—
74	C₂²×C₄×C₂₃	—	2	2	no	—
75	C₂²×C₄×C₂₃	—	4	2	no	—
76	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
86	C₂×C₈×C₂₃	—	2	8	no	—
87	C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	—	2	4	no	—
90	C₄⋊C₄×C₂₃	—	2	4	no	—
91	C₄⋊C₄×C₂₃	—	2	4	no	—
92	C₄⋊C₄×C₂₃	—	2	4	no	—
93	C₄⋊C₄×C₂₃	—	2	4	no	—
94	C₄⋊C₄×C₂₃	—	2	4	no	—
95	C₄⋊C₄×C₂₃	—	2	2	no	—
96	C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
99	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
100	C₂²⋊C₄×C₂₃	—	2	2	no	—
101	C₂²⋊C₄×C₂₃	—	2	4	no	—
102	C₂²⋊C₄×C₂₃	—	2	4	no	—
103	C₂²⋊C₄×C₂₃	—	2	2	no	—
104	C₄²×C₂₃	—	2	2	no	—
105	C₄²×C₂₃	—	2	4	no	—
106	C₄²×C₂₃	C₂×D₄	1	4	yes	—
107	C₄²×C₂₃	C₂×D₄	1	4	yes	—
108	C₄²×C₂₃	C₂×D₄	1	4	yes	—
109	#17410	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₂₃	—	2	1	no	—
111	C₂³×C₂₃	—	2	2	no	—
112	C₂³×C₂₃	—	4	1	no	—
113	C₂³×C₂₃	—	4	1	no	—
114	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
115	Q₈×C₂₃	—	4	4	no	—
116	Q₈×C₂₃	—	4	4	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	4	no	—
119	C₂×C₄×C₂₃	—	2	1	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	1	no	—
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	2	1	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	2	4	no	—
126	C₂×C₄×C₂₃	—	4	1	no	—
127	C₂×C₄×C₂₃	—	4	2	no	—
128	C₂×C₄×C₂₃	—	4	1	no	—
129	C₂×C₄×C₂₃	—	4	4	no	—
130	C₂×C₄×C₂₃	—	4	1	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	—	4	1	no	—
133	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
134	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
135	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
136	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
137	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
138	C₈×C₂₃	—	4	8	no	—
139	C₂×#541	C₂×C₂₃	1	1	yes	characteristic
140	#2983	C₂×C₂₃	1	2	yes	complemented
141	#1473	C₂×C₂₃	1	4	yes	—
142	#1470	C₂×C₂₃	1	4	yes	complemented
143	#1467	C₂×C₂₃	1	2	yes	complemented
144	#1464	C₂×C₂₃	1	2	yes	—
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	—	2	1	no	—
147	C₂²×C₂₃	—	4	1	no	—
148	C₂²×C₂₃	—	4	1	no	—
149	C₂²×C₂₃	—	4	1	no	—
150	C₂²×C₂₃	—	8	2	no	—
151	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₂₃	#568	1	2	yes	—
153	C₄×C₂₃	—	2	2	no	—
154	C₄×C₂₃	—	2	4	no	—
155	C₄×C₂₃	—	2	2	no	—
156	C₄×C₂₃	—	2	2	no	—
157	C₄×C₂₃	—	2	4	no	—
158	C₄×C₂₃	—	4	1	no	—
159	C₄×C₂₃	—	4	2	no	—
160	C₄×C₂₃	—	4	1	no	—
161	C₄×C₂₃	—	4	2	no	—
162	C₄×C₂₃	—	4	1	no	—
163	#541	—	2	8	no	—
164	C₂×C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
165	C₂×#126	C₄×C₂₃	1	2	yes	—
166	C₂×#126	C₂²×C₂₃	1	2	yes	—
167	C₂×#125	C₄×C₂₃	1	1	yes	characteristic
168	C₂×#125	C₄×C₂₃	1	2	yes	—
169	C₂×#125	C₂²×C₂₃	1	2	yes	—
170	C₂×C₂²⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
171	C₂×C₂²⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
172	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
174	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
175	#446	—	2	8	no	—
176	#432	C₂²×C₂₃	1	4	yes	—
177	#432	C₂²×C₂₃	1	4	yes	—
178	#431	C₂²×C₂₃	1	4	yes	—
179	#324	C₂²×C₂₃	1	4	yes	—
180	#324	C₂²×C₂₃	1	8	yes	—
181	#323	C₂²×C₂₃	1	4	yes	—
182	C₂×C₂₃	—	2	2	no	—
183	C₂×C₂₃	—	8	1	no	—
184	C₂×C₂₃	#2879	1	1	yes	characteristic
185	C₂×C₂₃	#2884	1	1	yes	characteristic
186	C₂×C₂₃	#2889	1	1	yes	characteristic
187	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
188	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
189	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
190	C₄⋊Q₈	—	2	4	no	—
191	#126	—	2	8	no	—
192	#126	—	2	4	no	—
193	#126	—	2	4	no	—
194	#125	—	2	4	no	—
195	#125	—	2	8	no	—
196	#125	—	2	4	no	—
197	#125	—	2	4	no	—
198	C₂²⋊Q₈	—	2	4	no	—
199	C₂²⋊Q₈	—	2	4	no	—
200	C₄×Q₈	—	2	4	no	—
201	C₄×Q₈	D₄×C₂₃	1	8	yes	—
202	#117	—	2	4	no	—
203	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
205	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
206	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
207	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
208	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
210	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
211	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
214	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
215	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
217	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
218	C₂×C₂²⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
219	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
220	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
221	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	4	no	—
225	C₄⋊C₈	—	2	4	no	—
226	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
227	#97	—	2	4	no	—
228	#97	—	2	4	no	—
229	C₄×C₈	—	2	4	no	—
230	C₂₃	#17410	1	1	yes	characteristic, direct product factor, Sylow
231	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	4	2	no	—
234	C₂×Q₈	—	4	2	no	—
235	C₂×Q₈	C₂²⋊C₄×C₂₃	1	4	yes	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	2	no	—
238	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
242	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
244	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
246	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
247	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
248	C₂×C₈	—	2	8	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
261	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	8	yes	—
262	C₂²⋊C₄	—	2	2	no	—
263	C₂²⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	—	2	4	no	—
265	C₂²⋊C₄	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	4	no	—
268	C₄²	C₂×D₄×C₂₃	1	4	yes	—
269	C₄²	C₂×D₄×C₂₃	1	4	yes	—
270	C₄²	C₂×D₄×C₂₃	1	4	yes	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
276	Q₈	—	4	4	no	—
277	Q₈	—	4	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
295	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
296	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
297	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
298	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
299	C₈	—	4	8	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	8	2	no	—
306	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
307	C₂²	C₂₃×#568	1	2	yes	—
308	C₄	—	2	2	no	—
309	C₄	—	2	4	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	4	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	1	no	—
318	C₂	—	2	2	no	—
319	C₂	—	8	1	no	—
320	C₂	C₂₃×#2879	1	1	yes	characteristic
321	C₂	C₂₃×#2884	1	1	yes	characteristic
322	C₂	C₂₃×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2983	C₂	semidirect
4. C₂₃×#1470	C₂	semidirect
5. C₂₃×#1467	C₂	semidirect
109. #17410	C₂₃	direct (N × H)
140. #2983	C₂×C₂₃	semidirect
142. #1470	C₂×C₂₃	semidirect
143. #1467	C₂×C₂₃	semidirect
230. C₂₃	#17410	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#541	2
C₂₃×#2983	2
C₂₃×#1473	2
C₂₃×#1470	2
C₂₃×#1467	2
C₂₃×#1464	2
#17410	23

Subgroups — C23×#17410 #19165036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17410 #19165036