Subgroups — C₂₃×#21549 #19169175

Order 5888.

938 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3054	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2930	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1975	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1946	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1937	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1896	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1811	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1765	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#131	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#525	C₂²	1	2	yes	—
17	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#470	C₄	1	2	yes	complemented
19	C₂₃×#469	C₄	1	2	yes	complemented
20	C₂₃×#468	C₄	1	4	yes	complemented
21	C₂₃×#443	—	2	4	no	—
22	C₂₃×#436	—	2	4	no	—
23	C₂₃×#435	—	2	4	no	—
24	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
26	C₂₃×#427	C₂²	1	1	yes	characteristic
27	C₂₃×#424	C₂²	1	2	yes	—
28	C₂×C₂₃×#104	C₂²	1	2	yes	—
29	C₂×C₂₃×#104	C₂²	1	2	yes	—
30	C₂₃×#416	C₂²	1	2	yes	—
31	C₂₃×#408	C₂²	1	2	yes	—
32	C₂₃×#407	C₂²	1	2	yes	—
33	C₂₃×#405	C₂²	1	2	yes	—
34	C₄×M₁₆×C₂₃	—	2	4	no	—
35	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#341	C₂²	1	2	yes	—
40	C₂₃×#339	C₂²	1	2	yes	—
41	C₂₃×#339	C₂²	1	2	yes	—
42	C₂×#41×C₂₃	C₂³	1	2	yes	—
43	C₂³×C₄×C₂₃	—	2	1	no	—
44	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₂₃×#135	—	2	2	no	—
46	C₂₃×#135	—	2	2	no	—
47	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
48	C₂×SD₁₆×C₂₃	C₂×C₄	1	2	yes	complemented
49	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic, complemented
50	C₂₃×#131	—	2	4	no	—
51	C₂₃×#131	—	2	4	no	—
52	C₂×M₁₆×C₂₃	—	2	2	no	—
53	C₂×M₁₆×C₂₃	—	2	2	no	—
54	C₂×M₁₆×C₂₃	—	2	2	no	—
55	C₂×M₁₆×C₂₃	—	2	2	no	—
56	C₂×M₁₆×C₂₃	—	2	2	no	—
57	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
58	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
59	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
60	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
61	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
62	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
63	C₂²×C₈×C₂₃	—	2	2	no	—
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂₃×#123	—	2	2	no	—
69	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
70	C₂₃×#121	C₂×C₄	1	4	yes	—
71	C₂₃×#120	—	2	2	no	—
72	C₄×D₄×C₂₃	—	2	4	no	—
73	C₂₃×#117	C₂³	1	2	yes	—
74	C₂×C₄²×C₂₃	—	2	1	no	—
75	C₂×C₄²×C₂₃	—	2	2	no	—
76	C₂×C₄²×C₂₃	—	2	2	no	—
77	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
79	C₂₃×#108	—	2	2	no	—
80	C₂₃×#108	—	2	2	no	—
81	C₂₃×#107	C₂×C₄	1	2	yes	complemented
82	C₂₃×#106	C₂×C₄	1	2	yes	complemented
83	C₂₃×C₄⋊C₈	—	2	4	no	—
84	C₂₃×#104	—	2	4	no	—
85	C₂₃×#104	—	2	4	no	—
86	C₂₃×#103	C₂×C₄	1	4	yes	complemented
87	C₂₃×#102	C₂×C₄	1	4	yes	complemented
88	C₂₃×C₂²⋊C₈	—	2	4	no	—
89	C₂₃×C₂²⋊C₈	—	2	4	no	—
90	C₂₃×#97	—	2	2	no	—
91	C₂₃×#97	—	2	2	no	—
92	C₂₃×#97	—	2	2	no	—
93	C₄×C₈×C₂₃	—	2	2	no	—
94	C₄×C₈×C₂₃	—	2	2	no	—
95	C₄×C₈×C₂₃	D₄	1	4	yes	complemented
96	C₂⁴×C₂₃	#41	1	1	yes	characteristic
97	#41×C₂₃	—	2	2	no	—
98	#41×C₂₃	—	2	2	no	—
99	#41×C₂₃	—	2	2	no	—
100	#41×C₂₃	—	2	2	no	—
101	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
102	C₂×D₄×C₂₃	—	2	2	no	—
103	C₂×D₄×C₂₃	—	2	4	no	—
104	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₂₃	—	2	1	no	—
106	C₂²×C₄×C₂₃	—	2	1	no	—
107	C₂²×C₄×C₂₃	—	2	1	no	—
108	C₂²×C₄×C₂₃	—	2	1	no	—
109	C₂²×C₄×C₂₃	—	2	1	no	—
110	C₂²×C₄×C₂₃	—	2	4	no	—
111	C₂²×C₄×C₂₃	—	2	1	no	—
112	C₂²×C₄×C₂₃	—	2	2	no	—
113	C₂²×C₄×C₂₃	—	4	4	no	—
114	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
117	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
118	Q₁₆×C₂₃	—	2	2	no	—
119	SD₁₆×C₂₃	—	2	4	no	—
120	D₈×C₂₃	—	2	2	no	—
121	M₁₆×C₂₃	—	2	4	no	—
122	M₁₆×C₂₃	—	2	4	no	—
123	M₁₆×C₂₃	—	2	4	no	—
124	M₁₆×C₂₃	—	2	4	no	—
125	M₁₆×C₂₃	—	2	2	no	—
126	M₁₆×C₂₃	—	2	2	no	—
127	M₁₆×C₂₃	—	2	2	no	—
128	M₁₆×C₂₃	—	2	2	no	—
129	M₁₆×C₂₃	—	4	4	no	—
130	C₂×C₈×C₂₃	—	2	1	no	—
131	C₂×C₈×C₂₃	—	2	1	no	—
132	C₂×C₈×C₂₃	—	2	1	no	—
133	C₂×C₈×C₂₃	—	2	1	no	—
134	C₂×C₈×C₂₃	—	2	2	no	—
135	C₂×C₈×C₂₃	—	2	2	no	—
136	C₂×C₈×C₂₃	—	4	4	no	—
137	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
138	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
141	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
143	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₂₃	#41	1	2	yes	—
146	C₂×C₈×C₂₃	#41	1	2	yes	—
147	C₄⋊C₄×C₂₃	—	2	4	no	—
148	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
149	C₂²⋊C₄×C₂₃	—	2	4	no	—
150	C₂²⋊C₄×C₂₃	—	2	2	no	—
151	C₂²⋊C₄×C₂₃	—	2	4	no	—
152	C₄²×C₂₃	—	2	4	no	—
153	C₄²×C₂₃	—	2	1	no	—
154	C₄²×C₂₃	—	2	2	no	—
155	C₄²×C₂₃	—	2	1	no	—
156	C₄²×C₂₃	C₂×D₄	1	2	yes	—
157	C₄²×C₂₃	#41	1	2	yes	—
158	#21549	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₂₃	—	2	1	no	—
160	C₂³×C₂₃	—	2	1	no	—
161	C₂³×C₂₃	—	2	2	no	—
162	C₂³×C₂₃	—	4	2	no	—
163	C₂³×C₂₃	C₄×D₄	1	2	yes	—
164	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
165	Q₈×C₂₃	—	2	2	no	—
166	Q₈×C₂₃	—	2	2	no	—
167	D₄×C₂₃	—	2	2	no	—
168	D₄×C₂₃	—	2	2	no	—
169	D₄×C₂₃	—	4	2	no	—
170	D₄×C₂₃	—	4	2	no	—
171	D₄×C₂₃	—	8	4	no	—
172	C₂×C₄×C₂₃	—	2	4	no	—
173	C₂×C₄×C₂₃	—	2	1	no	—
174	C₂×C₄×C₂₃	—	2	2	no	—
175	C₂×C₄×C₂₃	—	2	1	no	—
176	C₂×C₄×C₂₃	—	2	2	no	—
177	C₂×C₄×C₂₃	—	2	2	no	—
178	C₂×C₄×C₂₃	—	2	2	no	—
179	C₂×C₄×C₂₃	—	2	2	no	—
180	C₂×C₄×C₂₃	—	2	4	no	—
181	C₂×C₄×C₂₃	—	2	2	no	—
182	C₂×C₄×C₂₃	—	2	2	no	—
183	C₂×C₄×C₂₃	—	4	8	no	—
184	C₂×C₄×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	—	4	2	no	—
186	C₂×C₄×C₂₃	—	4	2	no	—
187	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
188	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
194	C₈×C₂₃	—	2	2	no	—
195	C₈×C₂₃	—	2	2	no	—
196	C₈×C₂₃	—	4	2	no	—
197	C₈×C₂₃	—	4	2	no	—
198	C₈×C₂₃	C₄×D₄	1	2	yes	—
199	C₈×C₂₃	C₄×D₄	1	2	yes	—
200	C₈×C₂₃	#121	1	2	yes	—
201	C₈×C₂₃	#121	1	2	yes	—
202	#3054	C₂×C₂₃	1	1	yes	characteristic
203	C₂×#443	C₂×C₂₃	1	1	yes	characteristic, complemented
204	#2930	C₂×C₂₃	1	2	yes	complemented
205	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
206	#1975	C₂×C₂₃	1	1	yes	characteristic, complemented
207	#1946	C₂×C₂₃	1	1	yes	characteristic, complemented
208	#1937	C₂×C₂₃	1	2	yes	complemented
209	#1896	C₂×C₂₃	1	2	yes	complemented
210	#1811	C₂×C₂₃	1	1	yes	characteristic, complemented
211	#1772	C₂×C₂₃	1	1	yes	characteristic, complemented
212	#1765	C₂×C₂₃	1	2	yes	complemented
213	C₂²×C₂₃	—	2	2	no	—
214	C₂²×C₂₃	—	2	4	no	—
215	C₂²×C₂₃	—	2	2	no	—
216	C₂²×C₂₃	—	4	2	no	—
217	C₂²×C₂₃	—	4	2	no	—
218	C₂²×C₂₃	—	4	4	no	—
219	C₂²×C₂₃	—	4	2	no	—
220	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
221	C₂²×C₂₃	#525	1	1	yes	characteristic
222	C₂²×C₂₃	#529	1	1	yes	characteristic
223	C₄×C₂₃	—	2	4	no	—
224	C₄×C₂₃	—	2	2	no	—
225	C₄×C₂₃	—	4	2	no	—
226	C₄×C₂₃	—	4	4	no	—
227	C₄×C₂₃	—	4	2	no	—
228	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
229	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
230	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
231	C₄×C₂₃	#532	1	1	yes	characteristic
232	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
233	C₂×#131	C₂²×C₂₃	1	2	yes	—
234	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
235	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
236	#525	C₂²×C₂₃	1	2	yes	—
237	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
238	#470	C₄×C₂₃	1	2	yes	complemented
239	#469	C₄×C₂₃	1	2	yes	complemented
240	#468	C₄×C₂₃	1	4	yes	complemented
241	#443	—	2	4	no	—
242	#436	—	2	4	no	—
243	#435	—	2	4	no	—
244	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
245	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
246	#427	C₂²×C₂₃	1	1	yes	characteristic
247	#424	C₂²×C₂₃	1	2	yes	—
248	C₂×#104	C₂²×C₂₃	1	2	yes	—
249	C₂×#104	C₂²×C₂₃	1	2	yes	—
250	#416	C₂²×C₂₃	1	2	yes	—
251	#408	C₂²×C₂₃	1	2	yes	—
252	#407	C₂²×C₂₃	1	2	yes	—
253	#405	C₂²×C₂₃	1	2	yes	—
254	C₄×M₁₆	—	2	4	no	—
255	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
256	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
257	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
259	#341	C₂²×C₂₃	1	2	yes	—
260	#339	C₂²×C₂₃	1	2	yes	—
261	#339	C₂²×C₂₃	1	2	yes	—
262	C₂×C₂₃	—	2	1	no	—
263	C₂×C₂₃	—	2	1	no	—
264	C₂×C₂₃	—	4	2	no	—
265	C₂×C₂₃	—	8	2	no	—
266	C₂×C₂₃	C₄×#121	1	1	yes	characteristic
267	C₂×C₂₃	#2956	1	2	yes	—
268	C₂×#41	C₂³×C₂₃	1	2	yes	—
269	C₂³×C₄	—	2	1	no	—
270	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
271	#135	—	2	2	no	—
272	#135	—	2	2	no	—
273	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
274	C₂×SD₁₆	C₂×C₄×C₂₃	1	2	yes	complemented
275	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic, complemented
276	#131	—	2	4	no	—
277	#131	—	2	4	no	—
278	C₂×M₁₆	—	2	2	no	—
279	C₂×M₁₆	—	2	2	no	—
280	C₂×M₁₆	—	2	2	no	—
281	C₂×M₁₆	—	2	2	no	—
282	C₂×M₁₆	—	2	2	no	—
283	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
284	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
285	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
286	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
287	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
288	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
289	C₂²×C₈	—	2	2	no	—
290	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
291	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
293	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
294	#123	—	2	2	no	—
295	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
296	#121	C₂×C₄×C₂₃	1	4	yes	—
297	#120	—	2	2	no	—
298	C₄×D₄	—	2	4	no	—
299	#117	C₂³×C₂₃	1	2	yes	—
300	C₂×C₄²	—	2	1	no	—
301	C₂×C₄²	—	2	2	no	—
302	C₂×C₄²	—	2	2	no	—
303	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
304	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
305	#108	—	2	2	no	—
306	#108	—	2	2	no	—
307	#107	C₂×C₄×C₂₃	1	2	yes	complemented
308	#106	C₂×C₄×C₂₃	1	2	yes	complemented
309	C₄⋊C₈	—	2	4	no	—
310	#104	—	2	4	no	—
311	#104	—	2	4	no	—
312	#103	C₂×C₄×C₂₃	1	4	yes	complemented
313	#102	C₂×C₄×C₂₃	1	4	yes	complemented
314	C₂²⋊C₈	—	2	4	no	—
315	C₂²⋊C₈	—	2	4	no	—
316	#97	—	2	2	no	—
317	#97	—	2	2	no	—
318	#97	—	2	2	no	—
319	C₄×C₈	—	2	2	no	—
320	C₄×C₈	—	2	2	no	—
321	C₄×C₈	D₄×C₂₃	1	4	yes	complemented
322	C₂₃	#21549	1	1	yes	characteristic, direct product factor, Sylow
323	C₂⁴	#41×C₂₃	1	1	yes	characteristic
324	#41	—	2	2	no	—
325	#41	—	2	2	no	—
326	#41	—	2	2	no	—
327	#41	—	2	2	no	—
328	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	4	no	—
331	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	4	4	no	—
341	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
342	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
343	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
344	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
345	Q₁₆	—	2	2	no	—
346	SD₁₆	—	2	4	no	—
347	D₈	—	2	2	no	—
348	M₁₆	—	2	4	no	—
349	M₁₆	—	2	4	no	—
350	M₁₆	—	2	4	no	—
351	M₁₆	—	2	4	no	—
352	M₁₆	—	2	2	no	—
353	M₁₆	—	2	2	no	—
354	M₁₆	—	2	2	no	—
355	M₁₆	—	2	2	no	—
356	M₁₆	—	4	4	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	4	4	no	—
364	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
365	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
366	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
367	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
368	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
369	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
370	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
371	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
372	C₂×C₈	#41×C₂₃	1	2	yes	—
373	C₂×C₈	#41×C₂₃	1	2	yes	—
374	C₄⋊C₄	—	2	4	no	—
375	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
376	C₂²⋊C₄	—	2	4	no	—
377	C₂²⋊C₄	—	2	2	no	—
378	C₂²⋊C₄	—	2	4	no	—
379	C₄²	—	2	4	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	2	1	no	—
383	C₄²	C₂×D₄×C₂₃	1	2	yes	—
384	C₄²	#41×C₂₃	1	2	yes	—
385	C₂³	—	2	1	no	—
386	C₂³	—	2	1	no	—
387	C₂³	—	2	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	C₄×D₄×C₂₃	1	2	yes	—
390	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
391	Q₈	—	2	2	no	—
392	Q₈	—	2	2	no	—
393	D₄	—	2	2	no	—
394	D₄	—	2	2	no	—
395	D₄	—	4	2	no	—
396	D₄	—	4	2	no	—
397	D₄	—	8	4	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	4	8	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
414	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
415	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
416	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
417	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
418	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
419	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
420	C₈	—	2	2	no	—
421	C₈	—	2	2	no	—
422	C₈	—	4	2	no	—
423	C₈	—	4	2	no	—
424	C₈	C₄×D₄×C₂₃	1	2	yes	—
425	C₈	C₄×D₄×C₂₃	1	2	yes	—
426	C₈	C₂₃×#121	1	2	yes	—
427	C₈	C₂₃×#121	1	2	yes	—
428	C₂²	—	2	2	no	—
429	C₂²	—	2	4	no	—
430	C₂²	—	2	2	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	4	4	no	—
434	C₂²	—	4	2	no	—
435	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
436	C₂²	C₂₃×#525	1	1	yes	characteristic
437	C₂²	C₂₃×#529	1	1	yes	characteristic
438	C₄	—	2	4	no	—
439	C₄	—	2	2	no	—
440	C₄	—	4	2	no	—
441	C₄	—	4	4	no	—
442	C₄	—	4	2	no	—
443	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
444	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
445	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
446	C₄	C₂₃×#532	1	1	yes	characteristic
447	C₂	—	2	1	no	—
448	C₂	—	2	1	no	—
449	C₂	—	4	2	no	—
450	C₂	—	8	2	no	—
451	C₂	C₄×C₂₃×#121	1	1	yes	characteristic
452	C₂	C₂₃×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂₃×#3054	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×M₁₆×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×#2930	C₂	semidirect
4. C₂×C₄×M₁₆×C₂₃	C₂	semidirect
5. C₂₃×#1975	C₂	semidirect
6. C₂₃×#1946	C₂	semidirect
7. C₂₃×#1937	C₂	semidirect
8. C₂₃×#1896	C₂	semidirect
9. C₂₃×#1811	C₂	semidirect
10. C₂₃×#1772	C₂	semidirect
11. C₂₃×#1765	C₂	semidirect
18. C₂₃×#470	C₄	semidirect
19. C₂₃×#469	C₄	semidirect
20. C₂₃×#468	C₄	semidirect
47. C₂×Q₁₆×C₂₃	C₂×C₄	semidirect
48. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
49. C₂×D₈×C₂₃	C₂×C₄	semidirect
81. C₂₃×#107	C₂×C₄	semidirect
82. C₂₃×#106	C₂×C₄	semidirect
86. C₂₃×#103	C₂×C₄	semidirect
87. C₂₃×#102	C₂×C₄	semidirect
95. C₄×C₈×C₂₃	D₄	semidirect
158. #21549	C₂₃	direct (N × H)
203. C₂×#443	C₂×C₂₃	semidirect
204. #2930	C₂×C₂₃	semidirect
205. C₂×C₄×M₁₆	C₂×C₂₃	semidirect
206. #1975	C₂×C₂₃	semidirect
207. #1946	C₂×C₂₃	semidirect
208. #1937	C₂×C₂₃	semidirect
209. #1896	C₂×C₂₃	semidirect
210. #1811	C₂×C₂₃	semidirect
211. #1772	C₂×C₂₃	semidirect
212. #1765	C₂×C₂₃	semidirect
238. #470	C₄×C₂₃	semidirect
239. #469	C₄×C₂₃	semidirect
240. #468	C₄×C₂₃	semidirect
273. C₂×Q₁₆	C₂×C₄×C₂₃	semidirect
274. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
275. C₂×D₈	C₂×C₄×C₂₃	semidirect
307. #107	C₂×C₄×C₂₃	semidirect
308. #106	C₂×C₄×C₂₃	semidirect
312. #103	C₂×C₄×C₂₃	semidirect
313. #102	C₂×C₄×C₂₃	semidirect
321. C₄×C₈	D₄×C₂₃	semidirect
322. C₂₃	#21549	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×#3054	2
C₂×C₂₃×#443	2
C₂₃×#2930	2
C₂×C₄×M₁₆×C₂₃	2
C₂₃×#1975	2
C₂₃×#1946	2
C₂₃×#1937	2
C₂₃×#1896	2
C₂₃×#1811	2
C₂₃×#1772	2
C₂₃×#1765	2
#21549	23

Subgroups — C23×#21549 #19169175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21549 #19169175