Subgroups — C₂₃×#20479 #19168105

Order 5888.

1034 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1771	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1752	C₂	1	2	yes	maximal
6	C₈×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
7	C₂₃×#1745	C₂	1	2	yes	maximal, complemented
8	C₄×C₂₃×#97	C₂	1	2	yes	maximal, complemented
9	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
10	C₂×C₂₃×#117	C₄	1	2	yes	—
11	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#432	C₄	1	8	yes	complemented
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×#97	C₄	1	8	yes	complemented
16	C₂×C₂₃×#97	C₂²	1	4	yes	—
17	C₂×C₂₃×#97	C₂²	1	2	yes	—
18	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
22	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
23	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₄×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
25	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#376	C₄	1	2	yes	—
28	C₄³×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#336	C₂²	1	4	yes	—
30	C₂₃×#336	C₂²	1	4	yes	—
31	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	—	2	8	no	—
35	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
38	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂₃×#117	—	2	8	no	—
40	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
48	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
53	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
54	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
55	C₂₃×#97	—	2	8	no	—
56	C₂₃×#97	C₂×C₄	1	4	yes	—
57	C₂₃×#97	C₂×C₄	1	8	yes	—
58	C₂₃×#97	C₂×C₄	1	4	yes	—
59	C₂₃×#97	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₂₃	—	2	8	no	—
61	C₂₃×#95	C₂×C₄	1	4	yes	—
62	C₂₃×#95	C₂×C₄	1	2	yes	—
63	C₂₃×#95	C₂×C₄	1	2	yes	—
64	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	—	2	4	no	—
66	C₂²×C₄×C₂₃	—	2	4	no	—
67	C₂²×C₄×C₂₃	—	4	2	no	—
68	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
69	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
70	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	—	2	4	no	—
82	C₂×C₈×C₂₃	—	2	4	no	—
83	C₂×C₈×C₂₃	—	2	16	no	—
84	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
85	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₂₃	#41	1	8	yes	—
88	C₄⋊C₄×C₂₃	—	2	8	no	—
89	C₂²⋊C₄×C₂₃	—	2	4	no	—
90	C₂²⋊C₄×C₂₃	M₁₆	1	8	yes	—
91	C₄²×C₂₃	—	2	8	no	—
92	C₄²×C₂₃	—	2	4	no	—
93	C₄²×C₂₃	—	2	4	no	—
94	C₄²×C₂₃	—	4	4	no	—
95	C₄²×C₂₃	C₄²	1	4	yes	—
96	C₄²×C₂₃	C₄²	1	4	yes	—
97	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
98	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
101	#20479	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₂₃	—	2	2	no	—
103	C₂³×C₂₃	—	2	2	no	—
104	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
105	C₂³×C₂₃	C₂×M₁₆	1	4	yes	—
106	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
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	2	no	—
114	C₂×C₄×C₂₃	—	2	8	no	—
115	C₂×C₄×C₂₃	—	2	4	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	2	4	no	—
118	C₂×C₄×C₂₃	—	2	4	no	—
119	C₂×C₄×C₂₃	—	2	4	no	—
120	C₂×C₄×C₂₃	—	4	4	no	—
121	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
123	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₂₃	C₂×M₁₆	1	8	yes	—
127	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
130	C₈×C₂₃	—	2	4	no	—
131	C₈×C₂₃	—	2	8	no	—
132	C₈×C₂₃	#117	1	8	yes	complemented
133	C₂×#432	C₂×C₂₃	1	1	yes	characteristic
134	C₂²×#97	C₂×C₂₃	1	1	yes	characteristic
135	C₄×#117	C₂×C₂₃	1	1	yes	characteristic
136	#1771	C₂×C₂₃	1	4	yes	complemented
137	#1752	C₂×C₂₃	1	2	yes	—
138	C₈×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
139	#1745	C₂×C₂₃	1	2	yes	complemented
140	C₄×#97	C₂×C₂₃	1	2	yes	complemented
141	C₂²×C₂₃	—	2	2	no	—
142	C₂²×C₂₃	—	2	8	no	—
143	C₂²×C₂₃	—	2	2	no	—
144	C₂²×C₂₃	C₄×M₁₆	1	4	yes	—
145	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
146	C₂²×C₂₃	C₄×#41	1	2	yes	—
147	C₂²×C₂₃	C₂²×M₁₆	1	2	yes	—
148	C₂²×C₂₃	#568	1	2	yes	—
149	C₄×C₂₃	—	2	2	no	—
150	C₄×C₂₃	—	2	2	no	—
151	C₄×C₂₃	—	2	2	no	—
152	C₄×C₂₃	—	2	2	no	—
153	C₄×C₂₃	—	2	8	no	—
154	C₄×C₂₃	—	2	2	no	—
155	C₄×C₂₃	—	4	4	no	—
156	C₄×C₂₃	C₂×#117	1	2	yes	—
157	C₄×C₂₃	C₄×#41	1	2	yes	—
158	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
159	C₂×#117	C₄×C₂₃	1	2	yes	—
160	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
161	#432	C₄×C₂₃	1	8	yes	complemented
162	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
163	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
164	C₂×#97	C₄×C₂₃	1	8	yes	complemented
165	C₂×#97	C₂²×C₂₃	1	4	yes	—
166	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
167	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×#97	C₂²×C₂₃	1	2	yes	—
169	C₂×#97	C₂²×C₂₃	1	2	yes	—
170	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
171	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
172	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
173	C₄×C₂²⋊C₄	C₄×C₂₃	1	2	yes	—
174	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
175	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
176	#376	C₄×C₂₃	1	2	yes	—
177	C₄³	C₂²×C₂₃	1	2	yes	—
178	#336	C₂²×C₂₃	1	4	yes	—
179	#336	C₂²×C₂₃	1	4	yes	—
180	C₂×C₂₃	—	2	4	no	—
181	C₂×C₂₃	C₄×#117	1	1	yes	characteristic
182	C₂×C₂₃	C₂×C₄×M₁₆	1	1	yes	characteristic
183	C₂×C₂₃	#2873	1	1	yes	characteristic
184	C₂×C₂₃	#2970	1	4	yes	—
185	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
186	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
188	C₂²×C₈	—	2	8	no	—
189	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
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	#117	—	2	8	no	—
194	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
195	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
196	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
198	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
199	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
200	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
201	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
202	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
203	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
204	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
205	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
207	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
208	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
209	#97	—	2	8	no	—
210	#97	C₂×C₄×C₂₃	1	4	yes	—
211	#97	C₂×C₄×C₂₃	1	8	yes	—
212	#97	C₂×C₄×C₂₃	1	4	yes	—
213	#97	C₂×C₄×C₂₃	1	8	yes	—
214	C₄×C₈	—	2	8	no	—
215	#95	C₂×C₄×C₂₃	1	4	yes	—
216	#95	C₂×C₄×C₂₃	1	2	yes	—
217	#95	C₂×C₄×C₂₃	1	2	yes	—
218	C₂₃	#20479	1	1	yes	characteristic, direct product factor, Sylow
219	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
224	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
226	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
234	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	16	no	—
239	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
241	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
242	C₂×C₈	#41×C₂₃	1	8	yes	—
243	C₄⋊C₄	—	2	8	no	—
244	C₂²⋊C₄	—	2	4	no	—
245	C₂²⋊C₄	M₁₆×C₂₃	1	8	yes	—
246	C₄²	—	2	8	no	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	4	4	no	—
250	C₄²	C₄²×C₂₃	1	4	yes	—
251	C₄²	C₄²×C₂₃	1	4	yes	—
252	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
253	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
254	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
255	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
259	C₂³	C₂×M₁₆×C₂₃	1	4	yes	—
260	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	4	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	—	2	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
276	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
277	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
278	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
279	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
280	C₂×C₄	C₂×M₁₆×C₂₃	1	8	yes	—
281	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
282	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
283	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
284	C₈	—	2	4	no	—
285	C₈	—	2	8	no	—
286	C₈	C₂₃×#117	1	8	yes	complemented
287	C₂²	—	2	8	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	C₄×M₁₆×C₂₃	1	4	yes	—
291	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
292	C₂²	C₄×#41×C₂₃	1	2	yes	—
293	C₂²	C₂²×M₁₆×C₂₃	1	2	yes	—
294	C₂²	C₂₃×#568	1	2	yes	—
295	C₄	—	2	8	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	4	no	—
302	C₄	C₂×C₂₃×#117	1	2	yes	—
303	C₄	C₄×#41×C₂₃	1	2	yes	—
304	C₂	—	2	4	no	—
305	C₂	C₄×C₂₃×#117	1	1	yes	characteristic
306	C₂	C₂×C₄×M₁₆×C₂₃	1	1	yes	characteristic
307	C₂	C₂₃×#2873	1	1	yes	characteristic
308	C₂	C₂₃×#2970	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
7. C₂₃×#1745	C₂	semidirect
8. C₄×C₂₃×#97	C₂	semidirect
12. C₂₃×#432	C₄	semidirect
15. C₂×C₂₃×#97	C₄	semidirect
101. #20479	C₂₃	direct (N × H)
132. C₈×C₂₃	#117	semidirect
136. #1771	C₂×C₂₃	semidirect
139. #1745	C₂×C₂₃	semidirect
140. C₄×#97	C₂×C₂₃	semidirect
161. #432	C₄×C₂₃	semidirect
164. C₂×#97	C₄×C₂₃	semidirect
218. C₂₃	#20479	direct (N × H)
286. C₈	C₂₃×#117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#432	2
C₂²×C₂₃×#97	2
C₄×C₂₃×#117	2
C₂₃×#1771	2
C₂₃×#1752	2
C₈×C₂²⋊C₄×C₂₃	2
C₂₃×#1745	2
C₄×C₂₃×#97	2
#20479	23

Subgroups — C23×#20479 #19168105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20479 #19168105