Subgroups — C₂₃×#20552 #19168178

Order 5888.

890 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#433	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2298	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1771	C₂	1	2	yes	maximal, complemented
5	C₈×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1768	C₂	1	2	yes	maximal, complemented
7	C₄×C₂₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1753	C₂	1	2	yes	maximal
9	C₈×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
10	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
13	C₂₃×#433	C₄	1	8	yes	complemented
14	C₂₃×#424	C₄	1	8	yes	complemented
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₂₃×#97	C₂²	1	2	yes	—
20	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#388	C₄	1	4	yes	—
26	C₂₃×#387	C₄	1	4	yes	—
27	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#376	C₂²	1	1	yes	characteristic
30	C₄³×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#336	C₂²	1	2	yes	—
32	C₂₃×#336	C₂²	1	2	yes	—
33	C₂₃×#336	C₂²	1	2	yes	—
34	C₂₃×#336	C₂²	1	2	yes	—
35	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₂₃	—	4	2	no	—
37	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
38	C₂²×C₈×C₂₃	—	4	2	no	—
39	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
40	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
44	C₂₃×#117	C₂×C₄	1	4	yes	—
45	C₂₃×#117	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
47	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
49	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
52	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
55	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
56	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
57	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
58	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
59	C₂₃×#97	—	4	2	no	—
60	C₂₃×#97	C₂×C₄	1	4	yes	—
61	C₄×C₈×C₂₃	—	4	2	no	—
62	C₄×C₈×C₂₃	—	4	2	no	—
63	C₄×C₈×C₂₃	—	4	2	no	—
64	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
65	C₂₃×#95	C₂×C₄	1	4	yes	—
66	C₂₃×#95	C₂×C₄	1	4	yes	—
67	C₂₃×#95	C₂×C₄	1	4	yes	—
68	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	—	2	2	no	—
70	C₂²×C₄×C₂₃	—	2	2	no	—
71	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
79	M₁₆×C₂₃	—	4	4	no	—
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	—	2	2	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	4	no	—
89	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
93	C₄⋊C₄×C₂₃	—	2	8	no	—
94	C₄⋊C₄×C₂₃	C₄²	1	8	yes	—
95	C₂²⋊C₄×C₂₃	—	2	8	no	—
96	C₂²⋊C₄×C₂₃	C₄²	1	8	yes	—
97	C₄²×C₂₃	—	2	8	no	—
98	C₄²×C₂₃	—	2	2	no	—
99	C₄²×C₂₃	—	2	2	no	—
100	C₄²×C₂₃	—	2	2	no	—
101	C₄²×C₂₃	—	2	2	no	—
102	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
103	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
105	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
106	#20552	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂³×C₂₃	—	2	2	no	—
108	C₂³×C₂₃	—	2	4	no	—
109	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
110	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	—	2	4	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	4	no	—
114	C₂×C₄×C₂₃	—	2	4	no	—
115	C₂×C₄×C₂₃	—	2	4	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	4	no	—
119	C₂×C₄×C₂₃	—	2	4	no	—
120	C₂×C₄×C₂₃	—	2	4	no	—
121	C₂×C₄×C₂₃	—	2	4	no	—
122	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
123	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
124	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₂₃	#131	1	8	yes	—
126	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
129	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
130	C₈×C₂₃	—	2	4	no	—
131	C₈×C₂₃	—	2	4	no	—
132	C₈×C₂₃	—	2	4	no	—
133	C₈×C₂₃	—	2	4	no	—
134	C₂×#433	C₂×C₂₃	1	1	yes	characteristic
135	C₂×#424	C₂×C₂₃	1	1	yes	characteristic
136	#2298	C₂×C₂₃	1	1	yes	characteristic
137	#1771	C₂×C₂₃	1	2	yes	complemented
138	C₈×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
139	#1768	C₂×C₂₃	1	2	yes	complemented
140	C₄×C₄⋊C₈	C₂×C₂₃	1	2	yes	complemented
141	#1753	C₂×C₂₃	1	2	yes	—
142	C₈×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
143	C₂²×C₂₃	—	2	4	no	—
144	C₂²×C₂₃	—	2	2	no	—
145	C₂²×C₂₃	—	4	4	no	—
146	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
147	C₂²×C₂₃	#520	1	2	yes	—
148	C₂²×C₂₃	C₂×#131	1	4	yes	—
149	C₄×C₂₃	—	2	4	no	—
150	C₄×C₂₃	—	2	4	no	—
151	C₄×C₂₃	—	2	8	no	—
152	C₄×C₂₃	—	2	2	no	—
153	C₄×C₂₃	—	4	4	no	—
154	C₄×C₂₃	#518	1	1	yes	characteristic
155	C₄×C₂₃	#519	1	1	yes	characteristic
156	C₄×C₂₃	#520	1	2	yes	—
157	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
158	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
159	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
160	#433	C₄×C₂₃	1	8	yes	complemented
161	#424	C₄×C₂₃	1	8	yes	complemented
162	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
163	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
164	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
165	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
166	C₂×#97	C₂²×C₂₃	1	2	yes	—
167	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
170	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
172	#388	C₄×C₂₃	1	4	yes	—
173	#387	C₄×C₂₃	1	4	yes	—
174	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
175	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
176	#376	C₂²×C₂₃	1	1	yes	characteristic
177	C₄³	C₂²×C₂₃	1	1	yes	characteristic
178	#336	C₂²×C₂₃	1	2	yes	—
179	#336	C₂²×C₂₃	1	2	yes	—
180	#336	C₂²×C₂₃	1	2	yes	—
181	#336	C₂²×C₂₃	1	2	yes	—
182	C₂×C₂₃	—	4	2	no	—
183	C₂×C₂₃	#2274	1	1	yes	characteristic
184	C₂×C₂₃	C₄×#131	1	2	yes	—
185	C₂×C₂₃	#2994	1	4	yes	—
186	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
187	C₂×M₁₆	—	4	2	no	—
188	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
189	C₂²×C₈	—	4	2	no	—
190	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
191	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
192	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
193	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
194	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
195	#117	C₂×C₄×C₂₃	1	4	yes	—
196	#117	C₂×C₄×C₂₃	1	4	yes	—
197	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
198	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
199	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
200	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
201	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
202	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
203	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
204	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
206	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
207	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
208	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
209	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
210	#97	—	4	2	no	—
211	#97	C₂×C₄×C₂₃	1	4	yes	—
212	C₄×C₈	—	4	2	no	—
213	C₄×C₈	—	4	2	no	—
214	C₄×C₈	—	4	2	no	—
215	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
216	#95	C₂×C₄×C₂₃	1	4	yes	—
217	#95	C₂×C₄×C₂₃	1	4	yes	—
218	#95	C₂×C₄×C₂₃	1	4	yes	—
219	C₂₃	#20552	1	1	yes	characteristic, direct product factor, Sylow
220	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
224	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
230	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
231	M₁₆	—	4	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
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	4	no	—
241	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
242	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
243	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
244	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	C₄²×C₂₃	1	8	yes	—
247	C₂²⋊C₄	—	2	8	no	—
248	C₂²⋊C₄	C₄²×C₂₃	1	8	yes	—
249	C₄²	—	2	8	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
255	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
256	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
257	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
258	C₂³	—	2	4	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
261	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
274	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
275	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
276	C₂×C₄	C₂₃×#131	1	8	yes	—
277	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
279	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
280	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	2	4	no	—
284	C₈	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	4	4	no	—
288	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
289	C₂²	C₂₃×#520	1	2	yes	—
290	C₂²	C₂×C₂₃×#131	1	4	yes	—
291	C₄	—	2	8	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	4	no	—
294	C₄	—	2	4	no	—
295	C₄	—	4	4	no	—
296	C₄	C₂₃×#518	1	1	yes	characteristic
297	C₄	C₂₃×#519	1	1	yes	characteristic
298	C₄	C₂₃×#520	1	2	yes	—
299	C₂	—	4	2	no	—
300	C₂	C₂₃×#2274	1	1	yes	characteristic
301	C₂	C₄×C₂₃×#131	1	2	yes	—
302	C₂	C₂₃×#2994	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1771	C₂	semidirect
5. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
6. C₂₃×#1768	C₂	semidirect
7. C₄×C₂₃×C₄⋊C₈	C₂	semidirect
13. C₂₃×#433	C₄	semidirect
14. C₂₃×#424	C₄	semidirect
106. #20552	C₂₃	direct (N × H)
137. #1771	C₂×C₂₃	semidirect
138. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
139. #1768	C₂×C₂₃	semidirect
140. C₄×C₄⋊C₈	C₂×C₂₃	semidirect
160. #433	C₄×C₂₃	semidirect
161. #424	C₄×C₂₃	semidirect
219. C₂₃	#20552	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×#433	2
C₂×C₂₃×#424	2
C₂₃×#2298	2
C₂₃×#1771	2
C₈×C₄⋊C₄×C₂₃	2
C₂₃×#1768	2
C₄×C₂₃×C₄⋊C₈	2
C₂₃×#1753	2
C₈×C₂²⋊C₄×C₂₃	2
#20552	23

Subgroups — C23×#20552 #19168178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20552 #19168178