Subgroups — C₂₃×#20437 #19168063

Order 5888.

1166 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#405	C₂	1	2	yes	maximal
3	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1770	C₂	1	4	yes	maximal, complemented
6	C₈×C₄⋊C₄×C₂₃	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1729	C₂	1	2	yes	maximal
8	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#117	C₂²	1	2	yes	—
12	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#423	C₄	1	8	yes	complemented
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
15	C₂₃×#405	—	2	8	no	—
16	C₂×C₂₃×#97	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₂₃	C₄	1	8	yes	complemented
18	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
21	C₂₃×#384	C₄	1	2	yes	—
22	C₂₃×#382	C₄	1	2	yes	—
23	C₂₃×#380	C₄	1	4	yes	—
24	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
25	C₂₃×#336	C₂²	1	8	yes	—
26	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
27	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₂₃	—	2	4	no	—
29	C₂×M₁₆×C₂₃	—	2	4	no	—
30	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₂₃	—	2	2	no	—
32	C₂²×C₈×C₂₃	—	2	2	no	—
33	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
35	C₂²×C₈×C₂₃	D₄	1	4	yes	—
36	C₂²×C₈×C₂₃	Q₈	1	4	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
39	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×#117	—	2	4	no	—
42	C₂₃×#117	—	2	4	no	—
43	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
51	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
52	C₂₃×C₄⋊C₈	C₂×C₄	1	16	yes	—
53	C₂₃×#97	—	2	8	no	—
54	C₄×C₈×C₂₃	—	2	8	no	—
55	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	—
57	C₂₃×#95	C₂×C₄	1	8	yes	—
58	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	—	2	8	no	—
60	C₂²×C₄×C₂₃	C₄²	1	8	yes	—
61	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
62	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
69	C₂²×C₄×C₂₃	#41	1	2	yes	—
70	C₂²×C₄×C₂₃	#41	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₂₃	—	2	16	no	—
73	C₂×C₈×C₂₃	—	2	8	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	4	no	—
78	C₂×C₈×C₂₃	C₄⋊C₄	1	16	yes	complemented
79	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
80	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
86	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
87	C₄⋊C₄×C₂₃	—	2	8	no	—
88	C₄⋊C₄×C₂₃	—	2	16	no	—
89	C₂²⋊C₄×C₂₃	—	2	8	no	—
90	C₄²×C₂₃	—	2	4	no	—
91	C₄²×C₂₃	—	2	4	no	—
92	C₄²×C₂₃	C₄²	1	4	yes	—
93	C₄²×C₂₃	C₄²	1	4	yes	—
94	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
95	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
96	#20437	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₃	—	2	4	no	—
98	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
99	C₂³×C₂₃	C₄×D₄	1	2	yes	—
100	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
101	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	16	no	—
104	C₂×C₄×C₂₃	—	2	8	no	—
105	C₂×C₄×C₂₃	—	2	16	no	—
106	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
107	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
109	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
110	C₂×C₄×C₂₃	#117	1	4	yes	—
111	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
112	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₂₃	C₄×Q₈	1	4	yes	—
114	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₂₃	#131	1	8	yes	—
116	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
122	C₈×C₂₃	—	2	4	no	—
123	C₈×C₂₃	—	2	8	no	—
124	C₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
127	C₂×#405	C₂×C₂₃	1	2	yes	—
128	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
129	#2285	C₂×C₂₃	1	1	yes	characteristic
130	#1770	C₂×C₂₃	1	4	yes	complemented
131	C₈×C₄⋊C₄	C₂×C₂₃	1	4	yes	complemented
132	#1729	C₂×C₂₃	1	2	yes	—
133	C₂²×C₂₃	—	2	4	no	—
134	C₂²×C₂₃	—	2	4	no	—
135	C₂²×C₂₃	—	2	4	no	—
136	C₂²×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
137	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
138	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
139	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₂²×C₂₃	C₂×#131	1	4	yes	—
141	C₄×C₂₃	—	2	4	no	—
142	C₄×C₂₃	—	4	8	no	—
143	C₄×C₂₃	C₄×C₄⋊C₄	1	4	yes	—
144	C₄×C₂₃	#405	1	8	yes	complemented
145	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
147	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
150	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
151	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×#117	C₂²×C₂₃	1	2	yes	—
153	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
154	#423	C₄×C₂₃	1	8	yes	complemented
155	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
156	#405	—	2	8	no	—
157	C₂×#97	C₂²×C₂₃	1	4	yes	—
158	C₂×C₄×C₈	C₄×C₂₃	1	8	yes	complemented
159	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
160	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
162	#384	C₄×C₂₃	1	2	yes	—
163	#382	C₄×C₂₃	1	2	yes	—
164	#380	C₄×C₂₃	1	4	yes	—
165	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
166	#336	C₂²×C₂₃	1	8	yes	—
167	C₂×C₂₃	—	2	4	no	—
168	C₂×C₂₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₂₃	C₂×#405	1	2	yes	—
170	C₂×C₂₃	#2931	1	2	yes	—
171	C₂×C₂₃	#2960	1	2	yes	—
172	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
173	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
174	C₂×M₁₆	—	2	4	no	—
175	C₂×M₁₆	—	2	4	no	—
176	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
180	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
181	C₂²×C₈	D₄×C₂₃	1	4	yes	—
182	C₂²×C₈	Q₈×C₂₃	1	4	yes	—
183	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
184	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
185	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
187	#117	—	2	4	no	—
188	#117	—	2	4	no	—
189	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
190	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
191	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
192	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
195	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
196	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
197	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
198	C₄⋊C₈	C₂×C₄×C₂₃	1	16	yes	—
199	#97	—	2	8	no	—
200	C₄×C₈	—	2	8	no	—
201	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
202	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	—
203	#95	C₂×C₄×C₂₃	1	8	yes	—
204	C₂₃	#20437	1	1	yes	characteristic, direct product factor, Sylow
205	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	C₄²×C₂₃	1	8	yes	—
208	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
209	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
210	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
212	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
213	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
215	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
216	C₂²×C₄	#41×C₂₃	1	2	yes	—
217	C₂²×C₄	#41×C₂₃	1	2	yes	—
218	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
219	M₁₆	—	2	16	no	—
220	C₂×C₈	—	2	8	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	C₄⋊C₄×C₂₃	1	16	yes	complemented
226	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
227	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
228	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
229	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
230	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
231	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
232	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
233	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	16	no	—
236	C₂²⋊C₄	—	2	8	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	4	no	—
239	C₄²	C₄²×C₂₃	1	4	yes	—
240	C₄²	C₄²×C₂₃	1	4	yes	—
241	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
242	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
243	C₂³	—	2	4	no	—
244	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
245	C₂³	C₄×D₄×C₂₃	1	2	yes	—
246	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
247	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	2	16	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
253	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
254	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
255	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
256	C₂×C₄	C₂₃×#117	1	4	yes	—
257	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
258	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
259	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
260	C₂×C₄	C₄×Q₈×C₂₃	1	4	yes	—
261	C₂×C₄	C₂₃×#131	1	8	yes	—
262	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
263	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
267	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
268	C₈	—	2	4	no	—
269	C₈	—	2	8	no	—
270	C₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
271	C₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
276	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
277	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
278	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
279	C₂²	C₂×C₂₃×#131	1	4	yes	—
280	C₄	—	2	4	no	—
281	C₄	—	4	8	no	—
282	C₄	C₄×C₄⋊C₄×C₂₃	1	4	yes	—
283	C₄	C₂₃×#405	1	8	yes	complemented
284	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
285	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
286	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
287	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
288	C₂	—	2	4	no	—
289	C₂	C₂×C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
290	C₂	C₂×C₂₃×#405	1	2	yes	—
291	C₂	C₂₃×#2931	1	2	yes	—
292	C₂	C₂₃×#2960	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₂₃	C₂³
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×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₂₃×#1770	C₂	semidirect
6. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
13. C₂₃×#423	C₄	semidirect
17. C₂×C₄×C₈×C₂₃	C₄	semidirect
78. C₂×C₈×C₂₃	C₄⋊C₄	semidirect
96. #20437	C₂₃	direct (N × H)
130. #1770	C₂×C₂₃	semidirect
131. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
144. C₄×C₂₃	#405	semidirect
154. #423	C₄×C₂₃	semidirect
158. C₂×C₄×C₈	C₄×C₂₃	semidirect
204. C₂₃	#20437	direct (N × H)
225. C₂×C₈	C₄⋊C₄×C₂₃	semidirect
283. C₄	C₂₃×#405	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#423	2
C₂×C₂₃×#405	2
C₂²×C₄×C₈×C₂₃	2
C₂₃×#2285	2
C₂₃×#1770	2
C₈×C₄⋊C₄×C₂₃	2
C₂₃×#1729	2
#20437	23

Subgroups — C23×#20437 #19168063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20437 #19168063