Subgroups — C₂₃×#21517 #19169143

Order 5888.

922 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3035	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#443	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1980	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1929	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1876	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1806	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1764	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×#135	C₂²	1	2	yes	—
13	C₂×C₂₃×#131	C₂²	1	2	yes	—
14	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
16	C₂₃×#532	C₂²	1	1	yes	characteristic
17	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#464	C₄	1	2	yes	complemented
19	C₂₃×#463	C₄	1	2	yes	complemented
20	C₂₃×#462	C₄	1	1	yes	characteristic, complemented
21	C₂₃×#461	C₄	1	1	yes	characteristic, complemented
22	C₂₃×#460	C₄	1	1	yes	characteristic, complemented
23	C₂₃×#459	C₄	1	1	yes	characteristic, complemented
24	C₂₃×#443	—	2	8	no	—
25	C₂₃×#436	—	2	1	no	—
26	C₂₃×#436	—	2	1	no	—
27	C₈×D₄×C₂₃	—	2	1	no	—
28	C₈×D₄×C₂₃	—	2	1	no	—
29	C₂×C₂₃×#108	C₂²	1	2	yes	—
30	C₂₃×#423	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#104	C₂²	1	2	yes	—
33	C₂×C₂₃×#104	C₂²	1	2	yes	—
34	C₂×C₂₃×#104	C₂²	1	2	yes	—
35	C₂₃×#416	C₂²	1	2	yes	—
36	C₂₃×#408	C₂²	1	2	yes	—
37	C₂₃×#405	—	2	1	no	—
38	C₂₃×#405	—	2	1	no	—
39	C₂₃×#405	C₂²	1	1	yes	characteristic
40	C₂₃×#405	C₂²	1	1	yes	characteristic
41	C₄×M₁₆×C₂₃	—	2	1	no	—
42	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₂₃	—	2	1	no	—
44	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
45	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
47	C₂₃×#341	C₂²	1	2	yes	—
48	C₂₃×#339	C₂²	1	2	yes	—
49	C₂₃×#336	C₂²	1	2	yes	—
50	C₂×#41×C₂₃	C₂³	1	2	yes	—
51	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×#135	—	2	4	no	—
53	C₂₃×#135	—	2	4	no	—
54	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
55	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
56	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	—
57	C₂×D₈×C₂₃	C₂×C₄	1	2	yes	—
58	C₂₃×#131	—	2	2	no	—
59	C₂₃×#131	—	2	2	no	—
60	C₂₃×#131	—	2	2	no	—
61	C₂₃×#131	—	2	4	no	—
62	C₂₃×#131	—	2	4	no	—
63	C₂₃×#131	D₄	1	4	yes	complemented
64	C₂×M₁₆×C₂₃	—	2	1	no	—
65	C₂×M₁₆×C₂₃	—	2	1	no	—
66	C₂×M₁₆×C₂₃	—	2	2	no	—
67	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
68	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₂₃	—	2	1	no	—
71	C₂²×C₈×C₂₃	—	2	1	no	—
72	C₂²×C₈×C₂₃	—	2	2	no	—
73	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
74	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
77	C₂₃×#127	C₂×C₄	1	1	yes	characteristic
78	C₂₃×#124	C₂×C₄	1	2	yes	—
79	C₂₃×#121	—	2	1	no	—
80	C₂₃×#121	—	2	1	no	—
81	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
82	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
83	C₄×D₄×C₂₃	—	2	1	no	—
84	C₄×D₄×C₂₃	—	2	1	no	—
85	C₄×D₄×C₂₃	—	2	1	no	—
86	C₄×D₄×C₂₃	—	2	1	no	—
87	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
88	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
89	C₂₃×#117	—	2	1	no	—
90	C₂₃×#117	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄²×C₂₃	—	2	1	no	—
93	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
94	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
95	C₂₃×#108	—	2	4	no	—
96	C₂₃×C₄⋊C₈	—	2	1	no	—
97	C₂₃×C₄⋊C₈	—	2	1	no	—
98	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
99	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
100	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
101	C₂₃×#104	—	2	4	no	—
102	C₂₃×#104	—	2	4	no	—
103	C₂₃×#104	—	2	4	no	—
104	C₂₃×#103	C₂×C₄	1	2	yes	—
105	C₂₃×#103	C₂×C₄	1	2	yes	—
106	C₂₃×#102	C₂×C₄	1	2	yes	—
107	C₂₃×#102	C₂×C₄	1	2	yes	—
108	C₂₃×C₂²⋊C₈	—	2	2	no	—
109	C₂₃×C₂²⋊C₈	—	2	2	no	—
110	C₂₃×#97	—	2	1	no	—
111	C₂₃×#97	—	2	1	no	—
112	C₂₃×#97	—	2	1	no	—
113	C₄×C₈×C₂₃	—	2	1	no	—
114	C₄×C₈×C₂₃	—	2	4	no	—
115	C₄×C₈×C₂₃	—	2	1	no	—
116	C₄×C₈×C₂₃	—	2	1	no	—
117	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
118	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
119	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
120	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
121	C₂₃×#95	—	2	1	no	—
122	#41×C₂₃	—	2	1	no	—
123	#41×C₂₃	—	2	1	no	—
124	#41×C₂₃	—	2	2	no	—
125	#41×C₂₃	—	2	2	no	—
126	#41×C₂₃	—	2	2	no	—
127	#41×C₂₃	—	2	2	no	—
128	#41×C₂₃	C₂×D₄	1	2	yes	—
129	#41×C₂₃	#41	1	2	yes	—
130	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
131	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×D₄×C₂₃	—	2	1	no	—
133	C₂×D₄×C₂₃	—	2	2	no	—
134	C₂×D₄×C₂₃	—	2	1	no	—
135	C₂×D₄×C₂₃	—	2	1	no	—
136	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
137	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	—	2	1	no	—
139	C₂²×C₄×C₂₃	—	2	2	no	—
140	C₂²×C₄×C₂₃	—	2	1	no	—
141	C₂²×C₄×C₂₃	—	2	1	no	—
142	C₂²×C₄×C₂₃	—	2	2	no	—
143	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
146	Q₁₆×C₂₃	—	2	4	no	—
147	SD₁₆×C₂₃	—	2	4	no	—
148	SD₁₆×C₂₃	—	2	4	no	—
149	D₈×C₂₃	—	2	4	no	—
150	M₁₆×C₂₃	—	2	4	no	—
151	M₁₆×C₂₃	—	2	1	no	—
152	M₁₆×C₂₃	—	2	1	no	—
153	M₁₆×C₂₃	—	2	4	no	—
154	M₁₆×C₂₃	—	2	2	no	—
155	M₁₆×C₂₃	—	2	2	no	—
156	M₁₆×C₂₃	—	4	4	no	—
157	M₁₆×C₂₃	C₂×D₄	1	2	yes	—
158	M₁₆×C₂₃	#41	1	2	yes	—
159	C₂×C₈×C₂₃	—	2	4	no	—
160	C₂×C₈×C₂₃	—	2	2	no	—
161	C₂×C₈×C₂₃	—	2	1	no	—
162	C₂×C₈×C₂₃	—	2	1	no	—
163	C₂×C₈×C₂₃	—	2	1	no	—
164	C₂×C₈×C₂₃	—	2	1	no	—
165	C₂×C₈×C₂₃	—	2	2	no	—
166	C₂×C₈×C₂₃	—	2	4	no	—
167	C₂×C₈×C₂₃	—	2	2	no	—
168	C₂×C₈×C₂₃	—	4	4	no	—
169	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
170	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
171	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
175	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₂₃	#41	1	2	yes	—
177	C₄⋊C₄×C₂₃	—	2	4	no	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
183	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
184	C₂²⋊C₄×C₂₃	—	2	1	no	—
185	C₂²⋊C₄×C₂₃	—	2	1	no	—
186	C₂²⋊C₄×C₂₃	—	2	1	no	—
187	C₂²⋊C₄×C₂₃	—	2	2	no	—
188	C₂²⋊C₄×C₂₃	—	2	2	no	—
189	C₄²×C₂₃	—	2	1	no	—
190	C₄²×C₂₃	—	2	1	no	—
191	C₄²×C₂₃	—	2	1	no	—
192	C₄²×C₂₃	—	2	2	no	—
193	C₄²×C₂₃	—	2	2	no	—
194	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
195	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
196	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
198	#21517	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂³×C₂₃	—	2	1	no	—
200	C₂³×C₂₃	—	2	2	no	—
201	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
202	Q₈×C₂₃	—	2	1	no	—
203	Q₈×C₂₃	—	2	2	no	—
204	Q₈×C₂₃	—	2	2	no	—
205	Q₈×C₂₃	C₄×D₄	1	2	yes	—
206	D₄×C₂₃	—	2	2	no	—
207	D₄×C₂₃	—	2	2	no	—
208	D₄×C₂₃	—	2	2	no	—
209	D₄×C₂₃	—	2	1	no	—
210	D₄×C₂₃	—	2	2	no	—
211	D₄×C₂₃	—	4	2	no	—
212	D₄×C₂₃	—	4	2	no	—
213	D₄×C₂₃	—	4	2	no	—
214	D₄×C₂₃	—	4	2	no	—
215	D₄×C₂₃	C₄×D₄	1	2	yes	—
216	C₂×C₄×C₂₃	—	2	1	no	—
217	C₂×C₄×C₂₃	—	2	2	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	4	no	—
221	C₂×C₄×C₂₃	—	2	1	no	—
222	C₂×C₄×C₂₃	—	2	1	no	—
223	C₂×C₄×C₂₃	—	2	2	no	—
224	C₂×C₄×C₂₃	—	4	1	no	—
225	C₂×C₄×C₂₃	—	4	2	no	—
226	C₂×C₄×C₂₃	—	4	2	no	—
227	C₂×C₄×C₂₃	—	4	2	no	—
228	C₂×C₄×C₂₃	—	4	2	no	—
229	C₂×C₄×C₂₃	—	4	2	no	—
230	C₂×C₄×C₂₃	—	4	2	no	—
231	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
232	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
233	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
234	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
235	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
236	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
240	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
241	C₈×C₂₃	—	2	2	no	—
242	C₈×C₂₃	—	2	2	no	—
243	C₈×C₂₃	—	2	2	no	—
244	C₈×C₂₃	—	2	1	no	—
245	C₈×C₂₃	—	2	2	no	—
246	C₈×C₂₃	—	2	1	no	—
247	C₈×C₂₃	—	4	2	no	—
248	C₈×C₂₃	#121	1	2	yes	—
249	C₈×C₂₃	#121	1	2	yes	—
250	#3035	C₂×C₂₃	1	1	yes	characteristic
251	C₂×#443	C₂×C₂₃	1	2	yes	complemented
252	#2931	C₂×C₂₃	1	1	yes	characteristic, complemented
253	C₄×#131	C₂×C₂₃	1	1	yes	characteristic, complemented
254	#1980	C₂×C₂₃	1	1	yes	characteristic, complemented
255	#1950	C₂×C₂₃	1	1	yes	characteristic, complemented
256	#1929	C₂×C₂₃	1	2	yes	complemented
257	#1876	C₂×C₂₃	1	2	yes	complemented
258	#1806	C₂×C₂₃	1	1	yes	characteristic, complemented
259	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#1764	C₂×C₂₃	1	2	yes	complemented
261	C₂²×C₂₃	—	2	1	no	—
262	C₂²×C₂₃	—	2	1	no	—
263	C₂²×C₂₃	—	2	2	no	—
264	C₂²×C₂₃	—	4	1	no	—
265	C₂²×C₂₃	—	4	1	no	—
266	C₂²×C₂₃	—	4	2	no	—
267	C₂²×C₂₃	—	4	2	no	—
268	C₂²×C₂₃	—	4	2	no	—
269	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
270	C₂²×C₂₃	#529	1	1	yes	characteristic
271	C₂²×C₂₃	#532	1	1	yes	characteristic
272	C₄×C₂₃	—	2	1	no	—
273	C₄×C₂₃	—	2	1	no	—
274	C₄×C₂₃	—	2	1	no	—
275	C₄×C₂₃	—	2	2	no	—
276	C₄×C₂₃	—	4	4	no	—
277	C₄×C₂₃	—	4	1	no	—
278	C₄×C₂₃	—	4	1	no	—
279	C₄×C₂₃	—	4	2	no	—
280	C₄×C₂₃	#443	1	2	yes	complemented
281	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
282	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
283	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
284	C₄×C₂₃	#525	1	1	yes	characteristic
285	C₂×#135	C₂²×C₂₃	1	2	yes	—
286	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
287	C₂×#131	C₂²×C₂₃	1	2	yes	—
288	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
289	#532	C₂²×C₂₃	1	1	yes	characteristic
290	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
291	#464	C₄×C₂₃	1	2	yes	complemented
292	#463	C₄×C₂₃	1	2	yes	complemented
293	#462	C₄×C₂₃	1	1	yes	characteristic, complemented
294	#461	C₄×C₂₃	1	1	yes	characteristic, complemented
295	#460	C₄×C₂₃	1	1	yes	characteristic, complemented
296	#459	C₄×C₂₃	1	1	yes	characteristic, complemented
297	#443	—	2	8	no	—
298	#436	—	2	1	no	—
299	#436	—	2	1	no	—
300	C₈×D₄	—	2	1	no	—
301	C₈×D₄	—	2	1	no	—
302	C₂×#108	C₂²×C₂₃	1	2	yes	—
303	#423	C₂²×C₂₃	1	1	yes	characteristic
304	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
305	C₂×#104	C₂²×C₂₃	1	2	yes	—
306	C₂×#104	C₂²×C₂₃	1	2	yes	—
307	C₂×#104	C₂²×C₂₃	1	2	yes	—
308	#416	C₂²×C₂₃	1	2	yes	—
309	#408	C₂²×C₂₃	1	2	yes	—
310	#405	—	2	1	no	—
311	#405	—	2	1	no	—
312	#405	C₂²×C₂₃	1	1	yes	characteristic
313	#405	C₂²×C₂₃	1	1	yes	characteristic
314	C₄×M₁₆	—	2	1	no	—
315	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×C₄×C₈	—	2	1	no	—
317	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
318	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
319	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
320	#341	C₂²×C₂₃	1	2	yes	—
321	#339	C₂²×C₂₃	1	2	yes	—
322	#336	C₂²×C₂₃	1	2	yes	—
323	C₂×C₂₃	—	2	1	no	—
324	C₂×C₂₃	—	2	1	no	—
325	C₂×C₂₃	—	4	2	no	—
326	C₂×C₂₃	—	8	2	no	—
327	C₂×C₂₃	C₄×#121	1	1	yes	characteristic
328	C₂×C₂₃	C₂×#443	1	1	yes	characteristic
329	C₂×C₂₃	#2955	1	1	yes	characteristic
330	C₂×#41	C₂³×C₂₃	1	2	yes	—
331	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
332	#135	—	2	4	no	—
333	#135	—	2	4	no	—
334	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
335	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
336	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	—
337	C₂×D₈	C₂×C₄×C₂₃	1	2	yes	—
338	#131	—	2	2	no	—
339	#131	—	2	2	no	—
340	#131	—	2	2	no	—
341	#131	—	2	4	no	—
342	#131	—	2	4	no	—
343	#131	D₄×C₂₃	1	4	yes	complemented
344	C₂×M₁₆	—	2	1	no	—
345	C₂×M₁₆	—	2	1	no	—
346	C₂×M₁₆	—	2	2	no	—
347	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
348	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
349	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
350	C₂²×C₈	—	2	1	no	—
351	C₂²×C₈	—	2	1	no	—
352	C₂²×C₈	—	2	2	no	—
353	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
354	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
355	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
356	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
357	#127	C₂×C₄×C₂₃	1	1	yes	characteristic
358	#124	C₂×C₄×C₂₃	1	2	yes	—
359	#121	—	2	1	no	—
360	#121	—	2	1	no	—
361	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
362	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
363	C₄×D₄	—	2	1	no	—
364	C₄×D₄	—	2	1	no	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
368	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
369	#117	—	2	1	no	—
370	#117	C₂³×C₂₃	1	1	yes	characteristic
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄²	—	2	1	no	—
373	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
374	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
375	#108	—	2	4	no	—
376	C₄⋊C₈	—	2	1	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
379	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
380	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
381	#104	—	2	4	no	—
382	#104	—	2	4	no	—
383	#104	—	2	4	no	—
384	#103	C₂×C₄×C₂₃	1	2	yes	—
385	#103	C₂×C₄×C₂₃	1	2	yes	—
386	#102	C₂×C₄×C₂₃	1	2	yes	—
387	#102	C₂×C₄×C₂₃	1	2	yes	—
388	C₂²⋊C₈	—	2	2	no	—
389	C₂²⋊C₈	—	2	2	no	—
390	#97	—	2	1	no	—
391	#97	—	2	1	no	—
392	#97	—	2	1	no	—
393	C₄×C₈	—	2	1	no	—
394	C₄×C₈	—	2	1	no	—
395	C₄×C₈	—	2	4	no	—
396	C₄×C₈	—	2	1	no	—
397	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
398	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
399	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
400	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
401	#95	—	2	1	no	—
402	C₂₃	#21517	1	1	yes	characteristic, direct product factor, Sylow
403	#41	—	2	1	no	—
404	#41	—	2	1	no	—
405	#41	—	2	2	no	—
406	#41	—	2	2	no	—
407	#41	—	2	2	no	—
408	#41	—	2	2	no	—
409	#41	C₂×D₄×C₂₃	1	2	yes	—
410	#41	#41×C₂₃	1	2	yes	—
411	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
412	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
418	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
425	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
426	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
427	Q₁₆	—	2	4	no	—
428	SD₁₆	—	2	4	no	—
429	SD₁₆	—	2	4	no	—
430	D₈	—	2	4	no	—
431	M₁₆	—	2	4	no	—
432	M₁₆	—	2	1	no	—
433	M₁₆	—	2	1	no	—
434	M₁₆	—	2	4	no	—
435	M₁₆	—	2	2	no	—
436	M₁₆	—	2	2	no	—
437	M₁₆	—	4	4	no	—
438	M₁₆	C₂×D₄×C₂₃	1	2	yes	—
439	M₁₆	#41×C₂₃	1	2	yes	—
440	C₂×C₈	—	2	4	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	1	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	1	no	—
446	C₂×C₈	—	2	2	no	—
447	C₂×C₈	—	2	4	no	—
448	C₂×C₈	—	2	2	no	—
449	C₂×C₈	—	4	4	no	—
450	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
451	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
452	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
453	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
455	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
456	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
457	C₂×C₈	#41×C₂₃	1	2	yes	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	1	no	—
462	C₄⋊C₄	—	2	1	no	—
463	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
464	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
465	C₂²⋊C₄	—	2	1	no	—
466	C₂²⋊C₄	—	2	1	no	—
467	C₂²⋊C₄	—	2	1	no	—
468	C₂²⋊C₄	—	2	2	no	—
469	C₂²⋊C₄	—	2	2	no	—
470	C₄²	—	2	1	no	—
471	C₄²	—	2	1	no	—
472	C₄²	—	2	1	no	—
473	C₄²	—	2	2	no	—
474	C₄²	—	2	2	no	—
475	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
476	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
477	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₂³	—	2	1	no	—
480	C₂³	—	2	2	no	—
481	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
482	Q₈	—	2	1	no	—
483	Q₈	—	2	2	no	—
484	Q₈	—	2	2	no	—
485	Q₈	C₄×D₄×C₂₃	1	2	yes	—
486	D₄	—	2	2	no	—
487	D₄	—	2	2	no	—
488	D₄	—	2	2	no	—
489	D₄	—	2	1	no	—
490	D₄	—	2	2	no	—
491	D₄	—	4	2	no	—
492	D₄	—	4	2	no	—
493	D₄	—	4	2	no	—
494	D₄	—	4	2	no	—
495	D₄	C₄×D₄×C₂₃	1	2	yes	—
496	C₂×C₄	—	2	1	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
512	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
513	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
514	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
515	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
516	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
517	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
518	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
519	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
520	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
521	C₈	—	2	2	no	—
522	C₈	—	2	2	no	—
523	C₈	—	2	2	no	—
524	C₈	—	2	1	no	—
525	C₈	—	2	2	no	—
526	C₈	—	2	1	no	—
527	C₈	—	4	2	no	—
528	C₈	C₂₃×#121	1	2	yes	—
529	C₈	C₂₃×#121	1	2	yes	—
530	C₂²	—	2	1	no	—
531	C₂²	—	2	1	no	—
532	C₂²	—	2	2	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	1	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
539	C₂²	C₂₃×#529	1	1	yes	characteristic
540	C₂²	C₂₃×#532	1	1	yes	characteristic
541	C₄	—	2	1	no	—
542	C₄	—	2	1	no	—
543	C₄	—	2	1	no	—
544	C₄	—	2	2	no	—
545	C₄	—	4	4	no	—
546	C₄	—	4	1	no	—
547	C₄	—	4	1	no	—
548	C₄	—	4	2	no	—
549	C₄	C₂₃×#443	1	2	yes	complemented
550	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
551	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
552	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
553	C₄	C₂₃×#525	1	1	yes	characteristic
554	C₂	—	2	1	no	—
555	C₂	—	2	1	no	—
556	C₂	—	4	2	no	—
557	C₂	—	8	2	no	—
558	C₂	C₄×C₂₃×#121	1	1	yes	characteristic
559	C₂	C₂×C₂₃×#443	1	1	yes	characteristic
560	C₂	C₂₃×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#3035	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#443	C₂	semidirect
3. C₂₃×#2931	C₂	semidirect
4. C₄×C₂₃×#131	C₂	semidirect
5. C₂₃×#1980	C₂	semidirect
6. C₂₃×#1950	C₂	semidirect
7. C₂₃×#1929	C₂	semidirect
8. C₂₃×#1876	C₂	semidirect
9. C₂₃×#1806	C₂	semidirect
10. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
11. C₂₃×#1764	C₂	semidirect
18. C₂₃×#464	C₄	semidirect
19. C₂₃×#463	C₄	semidirect
20. C₂₃×#462	C₄	semidirect
21. C₂₃×#461	C₄	semidirect
22. C₂₃×#460	C₄	semidirect
23. C₂₃×#459	C₄	semidirect
63. C₂₃×#131	D₄	semidirect
198. #21517	C₂₃	direct (N × H)
251. C₂×#443	C₂×C₂₃	semidirect
252. #2931	C₂×C₂₃	semidirect
253. C₄×#131	C₂×C₂₃	semidirect
254. #1980	C₂×C₂₃	semidirect
255. #1950	C₂×C₂₃	semidirect
256. #1929	C₂×C₂₃	semidirect
257. #1876	C₂×C₂₃	semidirect
258. #1806	C₂×C₂₃	semidirect
259. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
260. #1764	C₂×C₂₃	semidirect
280. C₄×C₂₃	#443	semidirect
291. #464	C₄×C₂₃	semidirect
292. #463	C₄×C₂₃	semidirect
293. #462	C₄×C₂₃	semidirect
294. #461	C₄×C₂₃	semidirect
295. #460	C₄×C₂₃	semidirect
296. #459	C₄×C₂₃	semidirect
343. #131	D₄×C₂₃	semidirect
402. C₂₃	#21517	direct (N × H)
549. C₄	C₂₃×#443	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3035	2
C₂×C₂₃×#443	2
C₂₃×#2931	2
C₄×C₂₃×#131	2
C₂₃×#1980	2
C₂₃×#1950	2
C₂₃×#1929	2
C₂₃×#1876	2
C₂₃×#1806	2
C₈×C₄⋊C₄×C₂₃	2
C₂₃×#1764	2
#21517	23

Subgroups — C23×#21517 #19169143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21517 #19169143