Subgroups — C₂₃×#7728 #19155354

Order 5888.

494 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2857	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1817	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1797	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1271	C₂	1	4	yes	maximal, complemented
5	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
9	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
10	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
11	C₂₃×#391	—	2	1	no	—
12	C₂₃×#389	—	2	1	no	—
13	C₂₃×#389	C₂²	1	2	yes	—
14	C₂₃×#384	C₄	1	1	yes	characteristic
15	C₂₃×#380	—	2	1	no	—
16	C₂₃×#380	C₄	1	2	yes	—
17	C₂₃×#336	—	2	1	no	—
18	C₂₃×#336	C₂²	1	2	yes	—
19	C₂₃×#336	C₂²	1	2	yes	—
20	C₂²×Q₈×C₂₃	—	2	1	no	—
21	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂₃×C₄⋊Q₈	—	4	2	no	—
27	C₂₃×C₂²⋊Q₈	—	4	2	no	—
28	C₂₃×C₂²⋊Q₈	—	4	2	no	—
29	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
36	C₂×C₄²×C₂₃	—	4	2	no	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×C₄⋊C₈	—	4	2	no	—
42	C₂₃×C₂²⋊C₈	—	4	2	no	—
43	C₂₃×C₂²⋊C₈	—	4	2	no	—
44	C₂₃×#95	—	2	2	no	—
45	C₂₃×#95	—	2	2	no	—
46	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×Q₈×C₂₃	—	4	2	no	—
48	C₂×Q₈×C₂₃	—	4	2	no	—
49	C₂×Q₈×C₂₃	—	4	2	no	—
50	C₂²×C₄×C₂₃	—	2	2	no	—
51	C₂²×C₄×C₂₃	—	2	2	no	—
52	C₂²×C₄×C₂₃	—	2	2	no	—
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	4	1	no	—
57	C₂²×C₄×C₂₃	—	4	1	no	—
58	C₂²×C₄×C₂₃	—	4	1	no	—
59	C₂²×C₄×C₂₃	—	4	1	no	—
60	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	D₈	1	2	yes	—
64	C₂²×C₄×C₂₃	SD₁₆	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₂₃	—	4	2	no	—
69	C₂×C₈×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	4	2	no	—
71	C₂×C₈×C₂₃	—	4	2	no	—
72	C₂×C₈×C₂₃	—	4	2	no	—
73	C₂×C₈×C₂₃	—	4	2	no	—
74	C₄⋊C₄×C₂₃	—	2	4	no	—
75	C₄⋊C₄×C₂₃	—	4	2	no	—
76	C₄⋊C₄×C₂₃	—	4	2	no	—
77	C₄⋊C₄×C₂₃	—	4	2	no	—
78	C₄⋊C₄×C₂₃	—	4	2	no	—
79	C₄⋊C₄×C₂₃	—	4	2	no	—
80	C₄⋊C₄×C₂₃	—	4	2	no	—
81	C₂²⋊C₄×C₂₃	—	4	4	no	—
82	C₄²×C₂₃	—	2	2	no	—
83	C₄²×C₂₃	—	2	2	no	—
84	C₄²×C₂₃	—	4	1	no	—
85	C₄²×C₂₃	—	4	1	no	—
86	#7728	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₂₃	—	2	1	no	—
88	C₂³×C₂₃	—	2	1	no	—
89	C₂³×C₂₃	—	2	1	no	—
90	C₂³×C₂₃	—	2	1	no	—
91	C₂³×C₂₃	—	2	1	no	—
92	C₂³×C₂₃	—	2	1	no	—
93	C₂³×C₂₃	#102	1	2	yes	—
94	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	Q₈×C₂₃	—	8	4	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	1	no	—
105	C₂×C₄×C₂₃	—	2	1	no	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	—	4	4	no	—
111	C₂×C₄×C₂₃	—	4	4	no	—
112	C₂×C₄×C₂₃	—	4	2	no	—
113	C₂×C₄×C₂₃	—	4	2	no	—
114	C₂×C₄×C₂₃	—	4	2	no	—
115	C₂×C₄×C₂₃	—	4	2	no	—
116	C₂×C₄×C₂₃	—	4	2	no	—
117	C₂×C₄×C₂₃	—	4	2	no	—
118	C₂×C₄×C₂₃	#99	1	2	yes	—
119	C₂×C₄×C₂₃	#101	1	2	yes	—
120	C₂×C₄×C₂₃	#102	1	2	yes	—
121	C₂×C₄×C₂₃	#131	1	2	yes	—
122	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₂₃	#137	1	2	yes	—
125	C₈×C₂₃	—	8	2	no	—
126	C₈×C₂₃	—	8	2	no	—
127	#2857	C₂×C₂₃	1	1	yes	characteristic
128	#1817	C₂×C₂₃	1	1	yes	characteristic
129	#1797	C₂×C₂₃	1	1	yes	characteristic
130	#1271	C₂×C₂₃	1	4	yes	complemented
131	C₂²×C₂₃	—	2	2	no	—
132	C₂²×C₂₃	—	2	2	no	—
133	C₂²×C₂₃	—	2	2	no	—
134	C₂²×C₂₃	—	4	1	no	—
135	C₂²×C₂₃	—	4	1	no	—
136	C₂²×C₂₃	—	4	1	no	—
137	C₂²×C₂₃	—	4	1	no	—
138	C₂²×C₂₃	#408	1	1	yes	characteristic
139	C₂²×C₂₃	C₂×#99	1	1	yes	characteristic
140	C₂²×C₂₃	C₂×#101	1	1	yes	characteristic
141	C₂²×C₂₃	C₂×#102	1	1	yes	characteristic
142	C₂²×C₂₃	#419	1	1	yes	characteristic
143	C₂²×C₂₃	#421	1	1	yes	characteristic
144	C₂²×C₂₃	#421	1	1	yes	characteristic
145	C₄×C₂₃	—	2	2	no	—
146	C₄×C₂₃	—	2	2	no	—
147	C₄×C₂₃	—	2	2	no	—
148	C₄×C₂₃	—	2	2	no	—
149	C₄×C₂₃	—	2	1	no	—
150	C₄×C₂₃	—	2	2	no	—
151	C₄×C₂₃	—	2	1	no	—
152	C₄×C₂₃	—	8	2	no	—
153	C₄×C₂₃	—	8	2	no	—
154	C₂×C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic
155	C₂×C₂²⋊Q₈	—	2	1	no	—
156	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×C₄⋊C₈	—	2	1	no	—
158	C₂×C₂²⋊C₈	—	2	1	no	—
159	C₂×C₂²⋊C₈	—	2	1	no	—
160	#391	—	2	1	no	—
161	#389	—	2	1	no	—
162	#389	C₂²×C₂₃	1	2	yes	—
163	#384	C₄×C₂₃	1	1	yes	characteristic
164	#380	—	2	1	no	—
165	#380	C₄×C₂₃	1	2	yes	—
166	#336	—	2	1	no	—
167	#336	C₂²×C₂₃	1	2	yes	—
168	#336	C₂²×C₂₃	1	2	yes	—
169	C₂×C₂₃	—	4	2	no	—
170	C₂×C₂₃	#1472	1	1	yes	characteristic
171	C₂×C₂₃	#1480	1	1	yes	characteristic
172	C₂×C₂₃	#1497	1	1	yes	characteristic
173	C₂×C₂₃	#1511	1	1	yes	characteristic
174	C₂×C₂₃	#1519	1	1	yes	characteristic
175	C₂×C₂₃	#1537	1	1	yes	characteristic
176	C₂×C₂₃	#1546	1	1	yes	characteristic
177	C₂²×Q₈	—	2	1	no	—
178	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
179	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
180	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₄⋊Q₈	—	4	2	no	—
184	C₂²⋊Q₈	—	4	2	no	—
185	C₂²⋊Q₈	—	4	2	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₄²	—	4	2	no	—
194	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
196	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
198	C₄⋊C₈	—	4	2	no	—
199	C₂²⋊C₈	—	4	2	no	—
200	C₂²⋊C₈	—	4	2	no	—
201	#95	—	2	2	no	—
202	#95	—	2	2	no	—
203	C₂₃	#7728	1	1	yes	characteristic, direct product factor, Sylow
204	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	D₈×C₂₃	1	2	yes	—
222	C₂²×C₄	SD₁₆×C₂₃	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₂²⋊C₄	—	4	4	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	4	1	no	—
244	C₂³	—	2	1	no	—
245	C₂³	—	2	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	C₂₃×#102	1	2	yes	—
251	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
252	Q₈	—	8	4	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	C₂₃×#99	1	2	yes	—
276	C₂×C₄	C₂₃×#101	1	2	yes	—
277	C₂×C₄	C₂₃×#102	1	2	yes	—
278	C₂×C₄	C₂₃×#131	1	2	yes	—
279	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
280	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
281	C₂×C₄	C₂₃×#137	1	2	yes	—
282	C₈	—	8	2	no	—
283	C₈	—	8	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₂₃×#408	1	1	yes	characteristic
292	C₂²	C₂×C₂₃×#99	1	1	yes	characteristic
293	C₂²	C₂×C₂₃×#101	1	1	yes	characteristic
294	C₂²	C₂×C₂₃×#102	1	1	yes	characteristic
295	C₂²	C₂₃×#419	1	1	yes	characteristic
296	C₂²	C₂₃×#421	1	1	yes	characteristic
297	C₂²	C₂₃×#421	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	2	no	—
304	C₄	—	2	1	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₂	—	4	2	no	—
308	C₂	C₂₃×#1472	1	1	yes	characteristic
309	C₂	C₂₃×#1480	1	1	yes	characteristic
310	C₂	C₂₃×#1497	1	1	yes	characteristic
311	C₂	C₂₃×#1511	1	1	yes	characteristic
312	C₂	C₂₃×#1519	1	1	yes	characteristic
313	C₂	C₂₃×#1537	1	1	yes	characteristic
314	C₂	C₂₃×#1546	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1271	C₂	semidirect
86. #7728	C₂₃	direct (N × H)
130. #1271	C₂×C₂₃	semidirect
203. C₂₃	#7728	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×#2857	2
C₂₃×#1817	2
C₂₃×#1797	2
C₂₃×#1271	2
#7728	23

Subgroups — C23×#7728 #19155354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7728 #19155354