Subgroups — C₂₃×#20567 #19168193

Order 5888.

1106 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#436	C₂	1	2	yes	maximal
2	C₄²×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1753	C₂	1	4	yes	maximal, complemented
6	C₄×C₂₃×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1728	C₂	1	2	yes	maximal, complemented
9	C₂²×M₁₆×C₂₃	C₂²	1	4	yes	—
10	C₂×C₄×D₄×C₂₃	C₄	1	2	yes	—
11	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#436	C₄	1	16	yes	complemented
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
16	C₂×C₂₃×#97	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
19	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₄×C₂²⋊C₄×C₂₃	C₄	1	4	yes	—
22	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₄³×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#336	C₂²	1	1	yes	characteristic
25	C₂₃×#336	C₂²	1	2	yes	—
26	C₂₃×#336	C₂²	1	4	yes	—
27	C₂₃×#336	C₂²	1	1	yes	characteristic
28	C₂₃×#336	C₂²	1	2	yes	—
29	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
31	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
32	C₂×M₁₆×C₂₃	—	4	4	no	—
33	C₂×M₁₆×C₂₃	—	4	4	no	—
34	C₂×M₁₆×C₂₃	C₂×C₄	1	16	yes	complemented
35	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
38	C₄×D₄×C₂₃	—	2	8	no	—
39	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
40	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₂₃	—	4	4	no	—
45	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	complemented
53	C₂₃×C₂²⋊C₈	C₂×C₄	1	16	yes	complemented
54	C₂₃×#97	—	4	2	no	—
55	C₄×C₈×C₂₃	—	4	4	no	—
56	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	complemented
57	C₂₃×#95	C₂×C₄	1	2	yes	—
58	C₂₃×#95	C₂×C₄	1	4	yes	—
59	C₂⁴×C₂₃	C₂²×C₄	1	2	yes	—
60	C₂×D₄×C₂₃	—	2	4	no	—
61	C₂×D₄×C₂₃	C₄²	1	4	yes	—
62	C₂²×C₄×C₂₃	—	2	4	no	—
63	C₂²×C₄×C₂₃	—	2	2	no	—
64	C₂²×C₄×C₂₃	—	2	2	no	—
65	C₂²×C₄×C₂₃	—	2	8	no	—
66	C₂²×C₄×C₂₃	—	2	8	no	—
67	C₂²×C₄×C₂₃	—	4	4	no	—
68	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
76	M₁₆×C₂₃	—	4	16	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	4	no	—
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
84	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
86	C₄⋊C₄×C₂₃	C₄²	1	4	yes	—
87	C₄⋊C₄×C₂₃	M₁₆	1	8	yes	complemented
88	C₂²⋊C₄×C₂₃	—	2	8	no	—
89	C₂²⋊C₄×C₂₃	C₄²	1	8	yes	—
90	C₄²×C₂₃	—	2	4	no	—
91	C₄²×C₂₃	—	2	4	no	—
92	C₄²×C₂₃	—	2	2	no	—
93	C₄²×C₂₃	—	2	4	no	—
94	C₄²×C₂₃	—	2	1	no	—
95	C₄²×C₂₃	—	2	1	no	—
96	C₄²×C₂₃	—	4	4	no	—
97	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
98	#20567	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₂₃	—	2	2	no	—
100	C₂³×C₂₃	—	2	2	no	—
101	C₂³×C₂₃	—	2	4	no	—
102	C₂³×C₂₃	—	2	4	no	—
103	C₂³×C₂₃	C₂×C₄²	1	4	yes	—
104	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
105	D₄×C₂₃	—	2	8	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	4	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	4	no	—
113	C₂×C₄×C₂₃	—	2	8	no	—
114	C₂×C₄×C₂₃	—	2	8	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	4	16	no	—
119	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
120	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
121	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₂₃	C₂×M₁₆	1	8	yes	—
123	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
124	C₂×C₄×C₂₃	#131	1	4	yes	—
125	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
126	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
128	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
129	C₈×C₂₃	—	2	8	no	—
130	C₈×C₂₃	—	2	4	no	—
131	C₈×C₂₃	—	2	4	no	—
132	C₂×#436	C₂×C₂₃	1	2	yes	—
133	C₄²×D₄	C₂×C₂₃	1	1	yes	characteristic
134	C₈×C₄⋊C₄	C₂×C₂₃	1	2	yes	complemented
135	#1767	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#1753	C₂×C₂₃	1	4	yes	complemented
137	C₄×C₂²⋊C₈	C₂×C₂₃	1	2	yes	complemented
138	#1744	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#1728	C₂×C₂₃	1	2	yes	complemented
140	C₂²×C₂₃	—	2	4	no	—
141	C₂²×C₂₃	—	2	4	no	—
142	C₂²×C₂₃	—	2	4	no	—
143	C₂²×C₂₃	—	4	4	no	—
144	C₂²×C₂₃	—	4	4	no	—
145	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
146	C₂²×C₂₃	#518	1	1	yes	characteristic
147	C₂²×C₂₃	#520	1	1	yes	characteristic
148	C₂²×C₂₃	C₂²×M₁₆	1	2	yes	—
149	C₂²×C₂₃	C₂×#131	1	2	yes	—
150	C₄×C₂₃	—	2	2	no	—
151	C₄×C₂₃	—	2	4	no	—
152	C₄×C₂₃	—	2	8	no	—
153	C₄×C₂₃	—	2	4	no	—
154	C₄×C₂₃	—	4	2	no	—
155	C₄×C₂₃	C₄×M₁₆	1	4	yes	—
156	C₄×C₂₃	#518	1	2	yes	—
157	C₄×C₂₃	#520	1	2	yes	—
158	C₂²×M₁₆	C₂²×C₂₃	1	4	yes	—
159	C₂×C₄×D₄	C₄×C₂₃	1	2	yes	—
160	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
161	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
162	#436	C₄×C₂₃	1	16	yes	complemented
163	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
164	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
165	C₂×#97	C₂²×C₂₃	1	2	yes	—
166	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
167	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
168	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
169	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
170	C₄×C₂²⋊C₄	C₄×C₂₃	1	4	yes	—
171	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
172	C₄³	C₂²×C₂₃	1	1	yes	characteristic
173	#336	C₂²×C₂₃	1	1	yes	characteristic
174	#336	C₂²×C₂₃	1	2	yes	—
175	#336	C₂²×C₂₃	1	4	yes	—
176	#336	C₂²×C₂₃	1	1	yes	characteristic
177	#336	C₂²×C₂₃	1	2	yes	—
178	C₂×C₂₃	—	4	4	no	—
179	C₂×C₂₃	#2275	1	1	yes	characteristic
180	C₂×C₂₃	C₂×C₄×M₁₆	1	1	yes	characteristic
181	C₂×C₂₃	C₄×#131	1	1	yes	characteristic
182	C₂×C₂₃	#2974	1	2	yes	—
183	C₂×C₂₃	#2991	1	2	yes	—
184	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
185	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
186	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
187	C₂×M₁₆	—	4	4	no	—
188	C₂×M₁₆	—	4	4	no	—
189	C₂×M₁₆	C₂×C₄×C₂₃	1	16	yes	complemented
190	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
191	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
192	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
193	C₄×D₄	—	2	8	no	—
194	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
195	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
196	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
197	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
198	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
199	C₂×C₄²	—	4	4	no	—
200	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
201	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
202	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
203	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
206	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
207	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	complemented
208	C₂²⋊C₈	C₂×C₄×C₂₃	1	16	yes	complemented
209	#97	—	4	2	no	—
210	C₄×C₈	—	4	4	no	—
211	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	complemented
212	#95	C₂×C₄×C₂₃	1	2	yes	—
213	#95	C₂×C₄×C₂₃	1	4	yes	—
214	C₂₃	#20567	1	1	yes	characteristic, direct product factor, Sylow
215	C₂⁴	C₂²×C₄×C₂₃	1	2	yes	—
216	C₂×D₄	—	2	4	no	—
217	C₂×D₄	C₄²×C₂₃	1	4	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	4	4	no	—
224	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
225	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
228	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
229	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
231	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
232	M₁₆	—	4	16	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
241	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
242	C₄⋊C₄	C₄²×C₂₃	1	4	yes	—
243	C₄⋊C₄	M₁₆×C₂₃	1	8	yes	complemented
244	C₂²⋊C₄	—	2	8	no	—
245	C₂²⋊C₄	C₄²×C₂₃	1	8	yes	—
246	C₄²	—	2	4	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	2	1	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	1	no	—
252	C₄²	—	4	4	no	—
253	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
254	C₂³	—	2	4	no	—
255	C₂³	—	2	4	no	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×C₄²×C₂₃	1	4	yes	—
259	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
260	D₄	—	2	8	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	16	no	—
274	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
275	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
276	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
277	C₂×C₄	C₂×M₁₆×C₂₃	1	8	yes	—
278	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
279	C₂×C₄	C₂₃×#131	1	4	yes	—
280	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
281	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
282	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
283	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	2	8	no	—
286	C₈	—	2	4	no	—
287	C₂²	—	2	4	no	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
293	C₂²	C₂₃×#518	1	1	yes	characteristic
294	C₂²	C₂₃×#520	1	1	yes	characteristic
295	C₂²	C₂²×M₁₆×C₂₃	1	2	yes	—
296	C₂²	C₂×C₂₃×#131	1	2	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	2	8	no	—
299	C₄	—	2	4	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	2	no	—
302	C₄	C₄×M₁₆×C₂₃	1	4	yes	—
303	C₄	C₂₃×#518	1	2	yes	—
304	C₄	C₂₃×#520	1	2	yes	—
305	C₂	—	4	4	no	—
306	C₂	C₂₃×#2275	1	1	yes	characteristic
307	C₂	C₂×C₄×M₁₆×C₂₃	1	1	yes	characteristic
308	C₂	C₄×C₂₃×#131	1	1	yes	characteristic
309	C₂	C₂₃×#2974	1	2	yes	—
310	C₂	C₂₃×#2991	1	2	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	C₂²×D₄×C₂₃	C₂×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
3. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
4. C₂₃×#1767	C₂	semidirect
5. C₂₃×#1753	C₂	semidirect
6. C₄×C₂₃×C₂²⋊C₈	C₂	semidirect
7. C₂₃×#1744	C₂	semidirect
8. C₂₃×#1728	C₂	semidirect
13. C₂₃×#436	C₄	semidirect
34. C₂×M₁₆×C₂₃	C₂×C₄	semidirect
52. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
53. C₂₃×C₂²⋊C₈	C₂×C₄	semidirect
56. C₄×C₈×C₂₃	C₂×C₄	semidirect
87. C₄⋊C₄×C₂₃	M₁₆	semidirect
98. #20567	C₂₃	direct (N × H)
134. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
135. #1767	C₂×C₂₃	semidirect
136. #1753	C₂×C₂₃	semidirect
137. C₄×C₂²⋊C₈	C₂×C₂₃	semidirect
138. #1744	C₂×C₂₃	semidirect
139. #1728	C₂×C₂₃	semidirect
162. #436	C₄×C₂₃	semidirect
189. C₂×M₁₆	C₂×C₄×C₂₃	semidirect
207. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
208. C₂²⋊C₈	C₂×C₄×C₂₃	semidirect
211. C₄×C₈	C₂×C₄×C₂₃	semidirect
214. C₂₃	#20567	direct (N × H)
243. C₄⋊C₄	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#436	2
C₄²×D₄×C₂₃	2
C₈×C₄⋊C₄×C₂₃	2
C₂₃×#1767	2
C₂₃×#1753	2
C₄×C₂₃×C₂²⋊C₈	2
C₂₃×#1744	2
C₂₃×#1728	2
#20567	23

Subgroups — C23×#20567 #19168193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20567 #19168193