Subgroups — C₂₃×#17605 #19165231

Order 5888.

930 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3495	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2924	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1520	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1518	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1513	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1512	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1508	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1506	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1501	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1500	C₂	1	2	yes	maximal, complemented
12	C₂₃×#556	—	2	1	no	—
13	C₂₃×#554	—	2	1	no	—
14	C₂₃×#552	—	2	2	no	—
15	C₂₃×#552	—	2	1	no	—
16	C₂₃×#552	—	2	1	no	—
17	C₂₃×#551	—	2	2	no	—
18	C₂₃×#544	C₄	1	1	yes	characteristic, complemented
19	C₂₃×#542	C₄	1	2	yes	complemented
20	C₂₃×#541	C₄	1	2	yes	complemented
21	C₂₃×#536	C₄	1	1	yes	characteristic, complemented
22	C₂₃×#533	C₂²	1	2	yes	—
23	C₂₃×#529	C₂²	1	2	yes	—
24	C₂₃×#525	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂₃×#520	C₄	1	1	yes	characteristic, complemented
27	C₂₃×#519	C₄	1	1	yes	characteristic, complemented
28	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
29	C₂₃×#433	—	2	2	no	—
30	C₂₃×#432	C₂²	1	2	yes	—
31	C₂₃×#431	C₂²	1	2	yes	—
32	C₂₃×#424	—	2	1	no	—
33	C₂₃×#407	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
35	C₂₃×#405	—	2	1	no	—
36	C₂₃×#389	—	2	1	no	—
37	C₂₃×#388	—	2	2	no	—
38	C₂₃×#382	C₂²	1	2	yes	—
39	C₂₃×#380	C₂²	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
41	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
42	C₂₃×#376	C₂²	1	2	yes	—
43	C₂₃×#328	C₂²	1	2	yes	—
44	C₂₃×#328	C₂²	1	2	yes	—
45	C₂₃×#328	C₂²	1	1	yes	characteristic
46	C₂₃×#328	C₂²	1	1	yes	characteristic
47	C₂₃×#328	C₂²	1	2	yes	—
48	C₂₃×#328	C₂²	1	2	yes	—
49	C₂₃×#328	C₂²	1	2	yes	—
50	C₂₃×#327	C₂²	1	1	yes	characteristic
51	C₂₃×#327	C₂²	1	2	yes	—
52	C₂₃×#327	C₂²	1	1	yes	characteristic
53	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₂₃	—	2	1	no	—
57	C₂²×C₈×C₂₃	—	2	1	no	—
58	C₂₃×C₄⋊Q₈	—	2	1	no	—
59	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
60	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
61	C₂₃×#126	—	2	2	no	—
62	C₂₃×#126	—	2	2	no	—
63	C₂₃×#126	—	2	2	no	—
64	C₂₃×#126	—	2	2	no	—
65	C₂₃×#125	—	2	2	no	—
66	C₂₃×#125	—	2	1	no	—
67	C₂₃×#125	—	2	2	no	—
68	C₂₃×#125	—	2	2	no	—
69	C₂₃×#125	C₂×C₄	1	2	yes	—
70	C₂₃×#124	—	2	2	no	—
71	C₂₃×#124	C₂×C₄	1	2	yes	—
72	C₂₃×#123	—	2	2	no	—
73	C₂₃×#123	—	2	1	no	—
74	C₂₃×#123	—	2	2	no	—
75	C₂₃×#123	—	2	2	no	—
76	C₂₃×C₂²⋊Q₈	—	2	2	no	—
77	C₂₃×C₂²⋊Q₈	—	2	1	no	—
78	C₂₃×C₂²⋊Q₈	—	2	1	no	—
79	C₂₃×C₂²⋊Q₈	—	2	1	no	—
80	C₂₃×C₂²⋊Q₈	—	2	2	no	—
81	C₂₃×C₂²⋊Q₈	—	2	1	no	—
82	C₂₃×C₂²⋊Q₈	—	2	2	no	—
83	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
84	C₂₃×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
85	C₂₃×C₂²⋊Q₈	C₂³	1	2	yes	—
86	C₂₃×C₂²⋊Q₈	C₂³	1	1	yes	characteristic
87	C₂₃×#121	C₂×C₄	1	1	yes	characteristic
88	C₂₃×#121	C₂³	1	1	yes	characteristic
89	C₂₃×#120	—	2	1	no	—
90	C₄×Q₈×C₂₃	—	2	1	no	—
91	C₄×Q₈×C₂₃	—	2	1	no	—
92	C₄×Q₈×C₂₃	—	2	2	no	—
93	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
94	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
95	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
96	C₄×D₄×C₂₃	—	2	2	no	—
97	C₄×D₄×C₂₃	—	2	1	no	—
98	C₄×D₄×C₂₃	—	2	1	no	—
99	C₄×D₄×C₂₃	—	2	2	no	—
100	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
101	C₂₃×#117	—	2	2	no	—
102	C₂₃×#117	—	2	1	no	—
103	C₂₃×#117	—	2	2	no	—
104	C₂₃×#117	—	2	2	no	—
105	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
106	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
107	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
108	C₂₃×#117	C₂×C₄	1	2	yes	—
109	C₂₃×#117	D₄	1	1	yes	characteristic
110	C₂₃×#117	D₄	1	2	yes	—
111	C₂₃×#117	D₄	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
113	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
115	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
116	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
119	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
120	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
121	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
122	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
123	C₂×C₄²×C₂₃	—	2	2	no	—
124	C₂×C₄²×C₂₃	—	2	1	no	—
125	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
126	C₂₃×C₄⋊C₈	—	2	2	no	—
127	C₂₃×C₄⋊C₈	—	2	2	no	—
128	C₂₃×C₂²⋊C₈	—	2	1	no	—
129	C₂₃×C₂²⋊C₈	—	2	1	no	—
130	C₂₃×C₂²⋊C₈	C₂³	1	1	yes	characteristic
131	C₂₃×C₂²⋊C₈	C₂³	1	1	yes	characteristic
132	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
133	C₂₃×#97	—	2	2	no	—
134	C₄×C₈×C₂₃	—	2	2	no	—
135	C₂₃×#95	—	2	2	no	—
136	C₂₃×#95	—	2	2	no	—
137	C₂₃×#95	C₂³	1	1	yes	characteristic
138	C₂₃×#95	C₂³	1	2	yes	—
139	C₂₃×#95	C₂³	1	1	yes	characteristic
140	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
141	#41×C₂₃	—	4	2	no	—
142	C₂×Q₈×C₂₃	—	2	1	no	—
143	C₂×Q₈×C₂₃	—	2	2	no	—
144	C₂×Q₈×C₂₃	—	4	2	no	—
145	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
146	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
147	C₂×D₄×C₂₃	—	2	1	no	—
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	—	2	1	no	—
150	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	—	2	1	no	—
152	C₂²×C₄×C₂₃	—	2	1	no	—
153	C₂²×C₄×C₂₃	—	2	1	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	2	no	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	2	no	—
158	C₂²×C₄×C₂₃	—	4	2	no	—
159	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
167	M₁₆×C₂₃	—	4	2	no	—
168	C₂×C₈×C₂₃	—	2	1	no	—
169	C₂×C₈×C₂₃	—	2	1	no	—
170	C₂×C₈×C₂₃	—	2	2	no	—
171	C₂×C₈×C₂₃	—	4	2	no	—
172	C₄⋊C₄×C₂₃	—	2	1	no	—
173	C₄⋊C₄×C₂₃	—	2	2	no	—
174	C₄⋊C₄×C₂₃	—	2	2	no	—
175	C₄⋊C₄×C₂₃	—	2	2	no	—
176	C₄⋊C₄×C₂₃	—	2	2	no	—
177	C₄⋊C₄×C₂₃	—	2	2	no	—
178	C₄⋊C₄×C₂₃	—	2	2	no	—
179	C₄⋊C₄×C₂₃	—	2	2	no	—
180	C₄⋊C₄×C₂₃	—	2	2	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	2	no	—
183	C₄⋊C₄×C₂₃	—	2	2	no	—
184	C₄⋊C₄×C₂₃	—	2	1	no	—
185	C₄⋊C₄×C₂₃	—	4	2	no	—
186	C₄⋊C₄×C₂₃	—	8	2	no	—
187	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
188	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
189	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
190	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
191	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
192	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
193	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
194	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
195	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
196	C₂²⋊C₄×C₂₃	—	2	2	no	—
197	C₂²⋊C₄×C₂₃	—	2	1	no	—
198	C₂²⋊C₄×C₂₃	—	2	1	no	—
199	C₂²⋊C₄×C₂₃	—	2	2	no	—
200	C₂²⋊C₄×C₂₃	—	2	1	no	—
201	C₂²⋊C₄×C₂₃	—	2	2	no	—
202	C₂²⋊C₄×C₂₃	—	2	2	no	—
203	C₂²⋊C₄×C₂₃	—	2	1	no	—
204	C₂²⋊C₄×C₂₃	—	2	2	no	—
205	C₂²⋊C₄×C₂₃	—	2	1	no	—
206	C₂²⋊C₄×C₂₃	—	4	2	no	—
207	C₂²⋊C₄×C₂₃	—	8	1	no	—
208	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
209	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
210	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
211	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
212	C₄²×C₂₃	—	2	2	no	—
213	C₄²×C₂₃	—	2	2	no	—
214	C₄²×C₂₃	—	2	2	no	—
215	C₄²×C₂₃	—	2	1	no	—
216	C₄²×C₂₃	—	4	2	no	—
217	C₄²×C₂₃	—	4	4	no	—
218	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
219	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
220	C₄²×C₂₃	C₂×D₄	1	2	yes	—
221	C₄²×C₂₃	C₂×D₄	1	2	yes	—
222	#17605	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂³×C₂₃	—	2	1	no	—
224	C₂³×C₂₃	—	2	1	no	—
225	C₂³×C₂₃	—	2	1	no	—
226	C₂³×C₂₃	—	4	1	no	—
227	C₂³×C₂₃	—	4	1	no	—
228	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
229	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
230	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
231	Q₈×C₂₃	—	4	2	no	—
232	Q₈×C₂₃	—	4	1	no	—
233	Q₈×C₂₃	—	4	2	no	—
234	Q₈×C₂₃	—	8	2	no	—
235	D₄×C₂₃	—	4	1	no	—
236	D₄×C₂₃	—	4	2	no	—
237	D₄×C₂₃	—	4	1	no	—
238	D₄×C₂₃	—	4	1	no	—
239	D₄×C₂₃	—	8	1	no	—
240	C₂×C₄×C₂₃	—	2	2	no	—
241	C₂×C₄×C₂₃	—	2	2	no	—
242	C₂×C₄×C₂₃	—	2	2	no	—
243	C₂×C₄×C₂₃	—	2	2	no	—
244	C₂×C₄×C₂₃	—	2	2	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	2	no	—
249	C₂×C₄×C₂₃	—	2	1	no	—
250	C₂×C₄×C₂₃	—	2	2	no	—
251	C₂×C₄×C₂₃	—	4	2	no	—
252	C₂×C₄×C₂₃	—	4	1	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	2	no	—
257	C₂×C₄×C₂₃	—	4	1	no	—
258	C₂×C₄×C₂₃	—	4	2	no	—
259	C₂×C₄×C₂₃	—	4	1	no	—
260	C₂×C₄×C₂₃	—	4	2	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	2	no	—
263	C₂×C₄×C₂₃	—	4	2	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	8	2	no	—
266	C₂×C₄×C₂₃	—	8	1	no	—
267	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
268	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
269	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
270	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
271	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
274	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
275	C₈×C₂₃	—	4	1	no	—
276	C₈×C₂₃	—	4	1	no	—
277	C₈×C₂₃	—	4	2	no	—
278	#3495	C₂×C₂₃	1	1	yes	characteristic
279	#2924	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#2346	C₂×C₂₃	1	1	yes	characteristic, complemented
281	#1520	C₂×C₂₃	1	1	yes	characteristic, complemented
282	#1518	C₂×C₂₃	1	2	yes	complemented
283	#1513	C₂×C₂₃	1	1	yes	characteristic
284	#1512	C₂×C₂₃	1	2	yes	complemented
285	#1508	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#1506	C₂×C₂₃	1	2	yes	complemented
287	#1501	C₂×C₂₃	1	1	yes	characteristic
288	#1500	C₂×C₂₃	1	2	yes	complemented
289	C₂²×C₂₃	—	2	1	no	—
290	C₂²×C₂₃	—	2	1	no	—
291	C₂²×C₂₃	—	2	1	no	—
292	C₂²×C₂₃	—	2	1	no	—
293	C₂²×C₂₃	—	2	1	no	—
294	C₂²×C₂₃	—	2	1	no	—
295	C₂²×C₂₃	—	2	1	no	—
296	C₂²×C₂₃	—	2	1	no	—
297	C₂²×C₂₃	—	4	1	no	—
298	C₂²×C₂₃	—	4	1	no	—
299	C₂²×C₂₃	—	4	1	no	—
300	C₂²×C₂₃	—	8	1	no	—
301	C₂²×C₂₃	—	8	1	no	—
302	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
303	C₂²×C₂₃	#577	1	1	yes	characteristic
304	C₂²×C₂₃	#578	1	1	yes	characteristic
305	C₄×C₂₃	—	2	1	no	—
306	C₄×C₂₃	—	2	1	no	—
307	C₄×C₂₃	—	2	1	no	—
308	C₄×C₂₃	—	2	2	no	—
309	C₄×C₂₃	—	2	2	no	—
310	C₄×C₂₃	—	2	2	no	—
311	C₄×C₂₃	—	2	2	no	—
312	C₄×C₂₃	—	4	2	no	—
313	C₄×C₂₃	—	4	1	no	—
314	C₄×C₂₃	—	4	2	no	—
315	C₄×C₂₃	—	4	1	no	—
316	C₄×C₂₃	—	4	2	no	—
317	C₄×C₂₃	—	4	1	no	—
318	C₄×C₂₃	—	8	1	no	—
319	C₄×C₂₃	—	8	2	no	—
320	C₄×C₂₃	#410	1	2	yes	—
321	#556	—	2	1	no	—
322	#554	—	2	1	no	—
323	#552	—	2	2	no	—
324	#552	—	2	1	no	—
325	#552	—	2	1	no	—
326	#551	—	2	2	no	—
327	#544	C₄×C₂₃	1	1	yes	characteristic, complemented
328	#542	C₄×C₂₃	1	2	yes	complemented
329	#541	C₄×C₂₃	1	2	yes	complemented
330	#536	C₄×C₂₃	1	1	yes	characteristic, complemented
331	#533	C₂²×C₂₃	1	2	yes	—
332	#529	C₂²×C₂₃	1	2	yes	—
333	#525	C₂²×C₂₃	1	1	yes	characteristic
334	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
335	#520	C₄×C₂₃	1	1	yes	characteristic, complemented
336	#519	C₄×C₂₃	1	1	yes	characteristic, complemented
337	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
338	#433	—	2	2	no	—
339	#432	C₂²×C₂₃	1	2	yes	—
340	#431	C₂²×C₂₃	1	2	yes	—
341	#424	—	2	1	no	—
342	#407	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
344	#405	—	2	1	no	—
345	#389	—	2	1	no	—
346	#388	—	2	2	no	—
347	#382	C₂²×C₂₃	1	2	yes	—
348	#380	C₂²×C₂₃	1	1	yes	characteristic
349	C₄×C₄⋊C₄	—	2	1	no	—
350	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
351	#376	C₂²×C₂₃	1	2	yes	—
352	#328	C₂²×C₂₃	1	2	yes	—
353	#328	C₂²×C₂₃	1	2	yes	—
354	#328	C₂²×C₂₃	1	1	yes	characteristic
355	#328	C₂²×C₂₃	1	1	yes	characteristic
356	#328	C₂²×C₂₃	1	2	yes	—
357	#328	C₂²×C₂₃	1	2	yes	—
358	#328	C₂²×C₂₃	1	2	yes	—
359	#327	C₂²×C₂₃	1	1	yes	characteristic
360	#327	C₂²×C₂₃	1	2	yes	—
361	#327	C₂²×C₂₃	1	1	yes	characteristic
362	C₂×C₂₃	—	2	1	no	—
363	C₂×C₂₃	—	2	1	no	—
364	C₂×C₂₃	—	4	1	no	—
365	C₂×C₂₃	—	4	1	no	—
366	C₂×C₂₃	—	8	1	no	—
367	C₂×C₂₃	C₂×#410	1	1	yes	characteristic
368	C₂×C₂₃	#2898	1	1	yes	characteristic
369	C₂×C₂₃	#2901	1	1	yes	characteristic
370	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
371	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
372	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
373	C₂×M₁₆	—	2	1	no	—
374	C₂²×C₈	—	2	1	no	—
375	C₄⋊Q₈	—	2	1	no	—
376	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
377	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
378	#126	—	2	2	no	—
379	#126	—	2	2	no	—
380	#126	—	2	2	no	—
381	#126	—	2	2	no	—
382	#125	—	2	2	no	—
383	#125	—	2	2	no	—
384	#125	—	2	1	no	—
385	#125	—	2	2	no	—
386	#125	C₂×C₄×C₂₃	1	2	yes	—
387	#124	—	2	2	no	—
388	#124	C₂×C₄×C₂₃	1	2	yes	—
389	#123	—	2	1	no	—
390	#123	—	2	2	no	—
391	#123	—	2	2	no	—
392	#123	—	2	2	no	—
393	C₂²⋊Q₈	—	2	1	no	—
394	C₂²⋊Q₈	—	2	1	no	—
395	C₂²⋊Q₈	—	2	2	no	—
396	C₂²⋊Q₈	—	2	1	no	—
397	C₂²⋊Q₈	—	2	1	no	—
398	C₂²⋊Q₈	—	2	2	no	—
399	C₂²⋊Q₈	—	2	2	no	—
400	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
401	C₂²⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
402	C₂²⋊Q₈	C₂³×C₂₃	1	2	yes	—
403	C₂²⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
404	#121	C₂×C₄×C₂₃	1	1	yes	characteristic
405	#121	C₂³×C₂₃	1	1	yes	characteristic
406	#120	—	2	1	no	—
407	C₄×Q₈	—	2	1	no	—
408	C₄×Q₈	—	2	1	no	—
409	C₄×Q₈	—	2	2	no	—
410	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
411	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
412	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	2	no	—
416	C₄×D₄	—	2	2	no	—
417	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
418	#117	—	2	2	no	—
419	#117	—	2	1	no	—
420	#117	—	2	2	no	—
421	#117	—	2	2	no	—
422	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
423	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
424	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
425	#117	C₂×C₄×C₂₃	1	2	yes	—
426	#117	D₄×C₂₃	1	1	yes	characteristic
427	#117	D₄×C₂₃	1	2	yes	—
428	#117	D₄×C₂₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	—	2	1	no	—
430	C₂×C₄⋊C₄	—	2	1	no	—
431	C₂×C₄⋊C₄	—	2	1	no	—
432	C₂×C₄⋊C₄	—	2	2	no	—
433	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
434	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
435	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
436	C₂×C₂²⋊C₄	—	2	1	no	—
437	C₂×C₂²⋊C₄	—	2	1	no	—
438	C₂×C₂²⋊C₄	—	2	1	no	—
439	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
440	C₂×C₄²	—	2	2	no	—
441	C₂×C₄²	—	2	1	no	—
442	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
443	C₄⋊C₈	—	2	2	no	—
444	C₄⋊C₈	—	2	2	no	—
445	C₂²⋊C₈	—	2	1	no	—
446	C₂²⋊C₈	—	2	1	no	—
447	C₂²⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
448	C₂²⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
449	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
450	#97	—	2	2	no	—
451	C₄×C₈	—	2	2	no	—
452	#95	—	2	2	no	—
453	#95	—	2	2	no	—
454	#95	C₂³×C₂₃	1	1	yes	characteristic
455	#95	C₂³×C₂₃	1	2	yes	—
456	#95	C₂³×C₂₃	1	1	yes	characteristic
457	C₂₃	#17605	1	1	yes	characteristic, direct product factor, Sylow
458	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
459	#41	—	4	2	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	—	2	2	no	—
462	C₂×Q₈	—	4	2	no	—
463	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
464	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	2	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	4	2	no	—
477	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
478	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
480	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
483	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
485	M₁₆	—	4	2	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	2	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	4	2	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	4	2	no	—
504	C₄⋊C₄	—	8	2	no	—
505	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
506	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
507	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
508	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
509	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
510	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
511	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
512	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
513	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	1	no	—
516	C₂²⋊C₄	—	2	1	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	2	1	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	1	no	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₂²⋊C₄	—	2	1	no	—
524	C₂²⋊C₄	—	4	2	no	—
525	C₂²⋊C₄	—	8	1	no	—
526	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
527	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
528	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
529	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
530	C₄²	—	2	2	no	—
531	C₄²	—	2	2	no	—
532	C₄²	—	2	2	no	—
533	C₄²	—	2	1	no	—
534	C₄²	—	4	2	no	—
535	C₄²	—	4	4	no	—
536	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
537	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
538	C₄²	C₂×D₄×C₂₃	1	2	yes	—
539	C₄²	C₂×D₄×C₂₃	1	2	yes	—
540	C₂³	—	2	1	no	—
541	C₂³	—	2	1	no	—
542	C₂³	—	2	1	no	—
543	C₂³	—	4	1	no	—
544	C₂³	—	4	1	no	—
545	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
546	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
547	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
548	Q₈	—	4	2	no	—
549	Q₈	—	4	1	no	—
550	Q₈	—	4	2	no	—
551	Q₈	—	8	2	no	—
552	D₄	—	4	1	no	—
553	D₄	—	4	2	no	—
554	D₄	—	4	1	no	—
555	D₄	—	4	1	no	—
556	D₄	—	8	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	4	2	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	2	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	2	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	2	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	8	2	no	—
583	C₂×C₄	—	8	1	no	—
584	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
585	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
586	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
587	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
588	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
589	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
590	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
591	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
592	C₈	—	4	1	no	—
593	C₈	—	4	1	no	—
594	C₈	—	4	2	no	—
595	C₂²	—	2	1	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	2	1	no	—
600	C₂²	—	2	1	no	—
601	C₂²	—	2	1	no	—
602	C₂²	—	2	1	no	—
603	C₂²	—	4	1	no	—
604	C₂²	—	4	1	no	—
605	C₂²	—	4	1	no	—
606	C₂²	—	8	1	no	—
607	C₂²	—	8	1	no	—
608	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
609	C₂²	C₂₃×#577	1	1	yes	characteristic
610	C₂²	C₂₃×#578	1	1	yes	characteristic
611	C₄	—	2	1	no	—
612	C₄	—	2	1	no	—
613	C₄	—	2	1	no	—
614	C₄	—	2	2	no	—
615	C₄	—	2	2	no	—
616	C₄	—	2	2	no	—
617	C₄	—	2	2	no	—
618	C₄	—	4	2	no	—
619	C₄	—	4	1	no	—
620	C₄	—	4	2	no	—
621	C₄	—	4	1	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	1	no	—
624	C₄	—	8	1	no	—
625	C₄	—	8	2	no	—
626	C₄	C₂₃×#410	1	2	yes	—
627	C₂	—	2	1	no	—
628	C₂	—	2	1	no	—
629	C₂	—	4	1	no	—
630	C₂	—	4	1	no	—
631	C₂	—	8	1	no	—
632	C₂	C₂×C₂₃×#410	1	1	yes	characteristic
633	C₂	C₂₃×#2898	1	1	yes	characteristic
634	C₂	C₂₃×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2924	C₂	semidirect
3. C₂₃×#2346	C₂	semidirect
4. C₂₃×#1520	C₂	semidirect
5. C₂₃×#1518	C₂	semidirect
7. C₂₃×#1512	C₂	semidirect
8. C₂₃×#1508	C₂	semidirect
9. C₂₃×#1506	C₂	semidirect
11. C₂₃×#1500	C₂	semidirect
18. C₂₃×#544	C₄	semidirect
19. C₂₃×#542	C₄	semidirect
20. C₂₃×#541	C₄	semidirect
21. C₂₃×#536	C₄	semidirect
26. C₂₃×#520	C₄	semidirect
27. C₂₃×#519	C₄	semidirect
222. #17605	C₂₃	direct (N × H)
279. #2924	C₂×C₂₃	semidirect
280. #2346	C₂×C₂₃	semidirect
281. #1520	C₂×C₂₃	semidirect
282. #1518	C₂×C₂₃	semidirect
284. #1512	C₂×C₂₃	semidirect
285. #1508	C₂×C₂₃	semidirect
286. #1506	C₂×C₂₃	semidirect
288. #1500	C₂×C₂₃	semidirect
327. #544	C₄×C₂₃	semidirect
328. #542	C₄×C₂₃	semidirect
329. #541	C₄×C₂₃	semidirect
330. #536	C₄×C₂₃	semidirect
335. #520	C₄×C₂₃	semidirect
336. #519	C₄×C₂₃	semidirect
457. C₂₃	#17605	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×#3495	2
C₂₃×#2924	2
C₂₃×#2346	2
C₂₃×#1520	2
C₂₃×#1518	2
C₂₃×#1513	2
C₂₃×#1512	2
C₂₃×#1508	2
C₂₃×#1506	2
C₂₃×#1501	2
C₂₃×#1500	2
#17605	23

Subgroups — C23×#17605 #19165231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17605 #19165231