Subgroups — C₂₃×#33645 #19181271

Order 5888.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3050	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2237	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2236	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2235	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2234	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2233	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2232	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#2158	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2138	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×#525	—	2	1	no	—
15	C₂×C₂₃×#117	—	2	1	no	—
16	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂₃×#467	C₂²	1	1	yes	characteristic
18	C₂₃×#467	C₂²	1	1	yes	characteristic
19	C₂₃×#466	C₂²	1	1	yes	characteristic
20	C₂₃×#466	C₂²	1	1	yes	characteristic
21	C₂₃×#465	—	2	1	no	—
22	C₂₃×#465	—	2	1	no	—
23	C₂₃×#427	—	2	1	no	—
24	C₂₃×#427	—	2	1	no	—
25	C₂₃×#427	C₂²	1	2	yes	—
26	C₂₃×#427	C₂²	1	2	yes	—
27	C₂₃×#427	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#106	—	2	1	no	—
31	C₂₃×#416	—	2	1	no	—
32	C₂₃×#367	C₂²	1	2	yes	complemented
33	C₂₃×#367	C₂²	1	2	yes	complemented
34	C₂₃×#366	C₂²	1	2	yes	complemented
35	C₂₃×#366	C₂²	1	2	yes	complemented
36	C₂₃×#358	C₂²	1	2	yes	complemented
37	C₂₃×#358	C₂²	1	2	yes	complemented
38	C₂₃×#357	C₂²	1	2	yes	complemented
39	C₂₃×#357	C₂²	1	2	yes	complemented
40	C₂₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₂₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₂₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
43	C₂×#41×C₂₃	—	2	1	no	—
44	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
45	C₂₃×#135	—	2	4	no	—
46	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×SD₁₆×C₂₃	—	2	1	no	—
48	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₂₃	—	2	4	no	—
50	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂₃×#123	—	4	1	no	—
57	C₂₃×C₂²⋊Q₈	—	2	1	no	—
58	C₂₃×C₂²⋊Q₈	—	2	1	no	—
59	C₂₃×#121	—	2	1	no	—
60	C₂₃×#121	—	2	1	no	—
61	C₂₃×#120	—	4	1	no	—
62	C₄×D₄×C₂₃	—	4	1	no	—
63	C₄×D₄×C₂₃	—	4	1	no	—
64	C₂₃×#117	—	2	1	no	—
65	C₂₃×#117	—	2	2	no	—
66	C₂₃×#117	—	2	2	no	—
67	C₂₃×#117	—	4	1	no	—
68	C₂₃×#117	—	4	1	no	—
69	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
70	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
72	C₂×C₄²×C₂₃	—	4	1	no	—
73	C₂×C₁₆×C₂₃	—	2	4	no	—
74	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
75	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
76	C₂₃×#107	—	2	1	no	—
77	C₂₃×#107	—	2	1	no	—
78	C₂₃×#107	C₂³	1	2	yes	—
79	C₂₃×#107	C₂³	1	2	yes	—
80	C₂₃×#107	C₂³	1	1	yes	characteristic
81	C₂₃×#107	C₂³	1	1	yes	characteristic
82	C₂₃×#106	—	2	2	no	—
83	C₂₃×#106	—	2	2	no	—
84	C₂₃×#106	—	2	1	no	—
85	C₂₃×#103	—	2	1	no	—
86	C₂₃×#103	—	2	1	no	—
87	C₂₃×#102	—	2	1	no	—
88	C₂₃×#102	—	2	1	no	—
89	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
90	#41×C₂₃	—	4	2	no	—
91	#41×C₂₃	—	4	2	no	—
92	C₂×Q₈×C₂₃	—	2	1	no	—
93	C₂×D₄×C₂₃	—	2	1	no	—
94	C₂×D₄×C₂₃	—	4	1	no	—
95	C₂×D₄×C₂₃	—	4	1	no	—
96	C₂×D₄×C₂₃	—	4	1	no	—
97	C₂²×C₄×C₂₃	—	2	2	no	—
98	C₂²×C₄×C₂₃	—	2	2	no	—
99	C₂²×C₄×C₂₃	—	4	1	no	—
100	C₂²×C₄×C₂₃	—	4	1	no	—
101	C₂²×C₄×C₂₃	—	4	1	no	—
102	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
103	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
104	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
105	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
108	Q₁₆×C₂₃	—	2	2	no	—
109	SD₁₆×C₂₃	—	4	2	no	—
110	D₈×C₂₃	—	2	2	no	—
111	C₂×C₈×C₂₃	—	2	2	no	—
112	C₂×C₈×C₂₃	—	2	2	no	—
113	C₂×C₈×C₂₃	—	2	2	no	—
114	C₂×C₈×C₂₃	—	2	2	no	—
115	C₂×C₈×C₂₃	—	2	2	no	—
116	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₂₃	#41	1	4	yes	—
120	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
121	C₄⋊C₄×C₂₃	—	2	1	no	—
122	C₄⋊C₄×C₂₃	—	2	1	no	—
123	C₄⋊C₄×C₂₃	—	2	2	no	—
124	C₄⋊C₄×C₂₃	—	2	2	no	—
125	C₄⋊C₄×C₂₃	—	4	1	no	—
126	C₄⋊C₄×C₂₃	—	4	1	no	—
127	C₄⋊C₄×C₂₃	—	4	1	no	—
128	C₄⋊C₄×C₂₃	—	4	1	no	—
129	C₂²⋊C₄×C₂₃	—	4	2	no	—
130	C₂²⋊C₄×C₂₃	—	4	2	no	—
131	C₂²⋊C₄×C₂₃	—	4	1	no	—
132	C₂²⋊C₄×C₂₃	—	4	1	no	—
133	C₂²⋊C₄×C₂₃	—	4	1	no	—
134	C₂²⋊C₄×C₂₃	—	4	1	no	—
135	C₂²⋊C₄×C₂₃	—	4	1	no	—
136	C₄²×C₂₃	—	4	2	no	—
137	C₄²×C₂₃	—	4	2	no	—
138	C₄²×C₂₃	—	4	1	no	—
139	C₁₆×C₂₃	—	2	2	no	—
140	C₁₆×C₂₃	—	2	2	no	—
141	#33645	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂³×C₂₃	—	2	2	no	—
143	C₂³×C₂₃	—	2	2	no	—
144	C₂³×C₂₃	—	4	1	no	—
145	C₂³×C₂₃	—	4	1	no	—
146	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
147	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
148	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
149	Q₈×C₂₃	—	4	1	no	—
150	Q₈×C₂₃	—	4	1	no	—
151	D₄×C₂₃	—	4	1	no	—
152	D₄×C₂₃	—	4	1	no	—
153	D₄×C₂₃	—	8	2	no	—
154	D₄×C₂₃	—	8	2	no	—
155	D₄×C₂₃	—	8	2	no	—
156	C₂×C₄×C₂₃	—	2	2	no	—
157	C₂×C₄×C₂₃	—	2	1	no	—
158	C₂×C₄×C₂₃	—	2	1	no	—
159	C₂×C₄×C₂₃	—	2	1	no	—
160	C₂×C₄×C₂₃	—	2	1	no	—
161	C₂×C₄×C₂₃	—	2	2	no	—
162	C₂×C₄×C₂₃	—	2	2	no	—
163	C₂×C₄×C₂₃	—	4	2	no	—
164	C₂×C₄×C₂₃	—	4	2	no	—
165	C₂×C₄×C₂₃	—	4	1	no	—
166	C₂×C₄×C₂₃	—	4	1	no	—
167	C₂×C₄×C₂₃	—	4	1	no	—
168	C₂×C₄×C₂₃	—	8	2	no	—
169	C₂×C₄×C₂₃	—	8	2	no	—
170	C₂×C₄×C₂₃	—	8	2	no	—
171	C₂×C₄×C₂₃	#123	1	2	yes	—
172	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
175	C₂×C₄×C₂₃	#137	1	2	yes	—
176	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
177	C₈×C₂₃	—	2	1	no	—
178	C₈×C₂₃	—	2	1	no	—
179	C₈×C₂₃	#123	1	2	yes	—
180	C₈×C₂₃	C₂×#41	1	2	yes	—
181	#3050	C₂×C₂₃	1	1	yes	characteristic
182	C₂×#427	C₂×C₂₃	1	1	yes	characteristic, complemented
183	#2237	C₂×C₂₃	1	2	yes	complemented
184	#2236	C₂×C₂₃	1	1	yes	characteristic, complemented
185	#2235	C₂×C₂₃	1	1	yes	characteristic, complemented
186	#2234	C₂×C₂₃	1	2	yes	complemented
187	#2233	C₂×C₂₃	1	1	yes	characteristic, complemented
188	#2232	C₂×C₂₃	1	1	yes	characteristic, complemented
189	#2158	C₂×C₂₃	1	2	yes	complemented
190	#2138	C₂×C₂₃	1	2	yes	complemented
191	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
192	C₂²×C₂₃	—	2	1	no	—
193	C₂²×C₂₃	—	2	1	no	—
194	C₂²×C₂₃	—	2	1	no	—
195	C₂²×C₂₃	—	2	1	no	—
196	C₂²×C₂₃	—	2	1	no	—
197	C₂²×C₂₃	—	2	1	no	—
198	C₂²×C₂₃	—	2	1	no	—
199	C₂²×C₂₃	—	4	4	no	—
200	C₂²×C₂₃	—	8	1	no	—
201	C₂²×C₂₃	—	8	1	no	—
202	C₂²×C₂₃	—	8	1	no	—
203	C₂²×C₂₃	#480	1	2	yes	—
204	C₂²×C₂₃	#508	1	2	yes	—
205	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
206	C₄×C₂₃	—	2	1	no	—
207	C₄×C₂₃	—	2	1	no	—
208	C₄×C₂₃	—	8	1	no	—
209	C₄×C₂₃	—	8	2	no	—
210	C₄×C₂₃	—	8	1	no	—
211	C₄×C₂₃	—	8	2	no	—
212	C₄×C₂₃	—	8	1	no	—
213	C₄×C₂₃	#480	1	2	yes	—
214	C₄×C₂₃	C₂×#123	1	1	yes	characteristic
215	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
216	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
217	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
218	#525	—	2	1	no	—
219	C₂×#117	—	2	1	no	—
220	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
221	#467	C₂²×C₂₃	1	1	yes	characteristic
222	#467	C₂²×C₂₃	1	1	yes	characteristic
223	#466	C₂²×C₂₃	1	1	yes	characteristic
224	#466	C₂²×C₂₃	1	1	yes	characteristic
225	#465	—	2	1	no	—
226	#465	—	2	1	no	—
227	#427	—	2	1	no	—
228	#427	—	2	1	no	—
229	#427	C₂²×C₂₃	1	2	yes	—
230	#427	C₂²×C₂₃	1	2	yes	—
231	#427	C₂²×C₂₃	1	1	yes	characteristic
232	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
233	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
234	C₂×#106	—	2	1	no	—
235	#416	—	2	1	no	—
236	#367	C₂²×C₂₃	1	2	yes	complemented
237	#367	C₂²×C₂₃	1	2	yes	complemented
238	#366	C₂²×C₂₃	1	2	yes	complemented
239	#366	C₂²×C₂₃	1	2	yes	complemented
240	#358	C₂²×C₂₃	1	2	yes	complemented
241	#358	C₂²×C₂₃	1	2	yes	complemented
242	#357	C₂²×C₂₃	1	2	yes	complemented
243	#357	C₂²×C₂₃	1	2	yes	complemented
244	C₂²⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
245	C₂²⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
246	C₂²⋊C₁₆	C₂²×C₂₃	1	2	yes	—
247	C₂×C₂₃	—	2	2	no	—
248	C₂×C₂₃	—	2	2	no	—
249	C₂×C₂₃	—	4	1	no	—
250	C₂×C₂₃	—	16	1	no	—
251	C₂×C₂₃	C₂×#480	1	1	yes	characteristic
252	C₂×C₂₃	C₂×#508	1	1	yes	characteristic
253	C₂×C₂₃	#3414	1	1	yes	characteristic
254	C₂×#41	—	2	1	no	—
255	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
256	#135	—	2	4	no	—
257	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
258	C₂×SD₁₆	—	2	1	no	—
259	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
260	C₂²×C₈	—	2	4	no	—
261	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
262	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
263	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
265	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
266	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
267	#123	—	4	1	no	—
268	C₂²⋊Q₈	—	2	1	no	—
269	C₂²⋊Q₈	—	2	1	no	—
270	#121	—	2	1	no	—
271	#121	—	2	1	no	—
272	#120	—	4	1	no	—
273	C₄×D₄	—	4	1	no	—
274	C₄×D₄	—	4	1	no	—
275	#117	—	2	2	no	—
276	#117	—	2	2	no	—
277	#117	—	2	1	no	—
278	#117	—	4	1	no	—
279	#117	—	4	1	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₂²⋊C₄	—	4	1	no	—
283	C₂×C₄²	—	4	1	no	—
284	C₂×C₁₆	—	2	4	no	—
285	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
286	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
287	#107	—	2	1	no	—
288	#107	—	2	1	no	—
289	#107	C₂³×C₂₃	1	2	yes	—
290	#107	C₂³×C₂₃	1	2	yes	—
291	#107	C₂³×C₂₃	1	1	yes	characteristic
292	#107	C₂³×C₂₃	1	1	yes	characteristic
293	#106	—	2	2	no	—
294	#106	—	2	2	no	—
295	#106	—	2	1	no	—
296	#103	—	2	1	no	—
297	#103	—	2	1	no	—
298	#102	—	2	1	no	—
299	#102	—	2	1	no	—
300	C₂₃	#33645	1	1	yes	characteristic, direct product factor, Sylow
301	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
302	#41	—	4	2	no	—
303	#41	—	4	2	no	—
304	C₂×Q₈	—	2	1	no	—
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	—	4	1	no	—
307	C₂×D₄	—	4	1	no	—
308	C₂×D₄	—	4	1	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	4	1	no	—
312	C₂²×C₄	—	4	1	no	—
313	C₂²×C₄	—	4	1	no	—
314	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
315	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
316	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
317	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
320	Q₁₆	—	2	2	no	—
321	SD₁₆	—	4	2	no	—
322	D₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
329	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
330	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
331	C₂×C₈	#41×C₂₃	1	4	yes	—
332	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	4	1	no	—
338	C₄⋊C₄	—	4	1	no	—
339	C₄⋊C₄	—	4	1	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	2	no	—
343	C₂²⋊C₄	—	4	1	no	—
344	C₂²⋊C₄	—	4	1	no	—
345	C₂²⋊C₄	—	4	1	no	—
346	C₂²⋊C₄	—	4	1	no	—
347	C₂²⋊C₄	—	4	1	no	—
348	C₄²	—	4	2	no	—
349	C₄²	—	4	2	no	—
350	C₄²	—	4	1	no	—
351	C₁₆	—	2	2	no	—
352	C₁₆	—	2	2	no	—
353	C₂³	—	2	2	no	—
354	C₂³	—	2	2	no	—
355	C₂³	—	4	1	no	—
356	C₂³	—	4	1	no	—
357	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
358	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
359	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
360	Q₈	—	4	1	no	—
361	Q₈	—	4	1	no	—
362	D₄	—	4	1	no	—
363	D₄	—	4	1	no	—
364	D₄	—	8	2	no	—
365	D₄	—	8	2	no	—
366	D₄	—	8	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	8	2	no	—
380	C₂×C₄	—	8	2	no	—
381	C₂×C₄	—	8	2	no	—
382	C₂×C₄	C₂₃×#123	1	2	yes	—
383	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
384	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
385	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
386	C₂×C₄	C₂₃×#137	1	2	yes	—
387	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
388	C₈	—	2	1	no	—
389	C₈	—	2	1	no	—
390	C₈	C₂₃×#123	1	2	yes	—
391	C₈	C₂×#41×C₂₃	1	2	yes	—
392	C₂²	—	2	1	no	—
393	C₂²	—	2	1	no	—
394	C₂²	—	2	1	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	2	1	no	—
398	C₂²	—	2	1	no	—
399	C₂²	—	4	4	no	—
400	C₂²	—	8	1	no	—
401	C₂²	—	8	1	no	—
402	C₂²	—	8	1	no	—
403	C₂²	C₂₃×#480	1	2	yes	—
404	C₂²	C₂₃×#508	1	2	yes	—
405	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
406	C₄	—	2	1	no	—
407	C₄	—	2	1	no	—
408	C₄	—	8	1	no	—
409	C₄	—	8	2	no	—
410	C₄	—	8	1	no	—
411	C₄	—	8	2	no	—
412	C₄	—	8	1	no	—
413	C₄	C₂₃×#480	1	2	yes	—
414	C₄	C₂×C₂₃×#123	1	1	yes	characteristic
415	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
416	C₂	—	2	2	no	—
417	C₂	—	2	2	no	—
418	C₂	—	4	1	no	—
419	C₂	—	16	1	no	—
420	C₂	C₂×C₂₃×#480	1	1	yes	characteristic
421	C₂	C₂×C₂₃×#508	1	1	yes	characteristic
422	C₂	C₂₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×D₈
prime radical	C₂₃×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#427	C₂	semidirect
3. C₂₃×#2237	C₂	semidirect
4. C₂₃×#2236	C₂	semidirect
5. C₂₃×#2235	C₂	semidirect
6. C₂₃×#2234	C₂	semidirect
7. C₂₃×#2233	C₂	semidirect
8. C₂₃×#2232	C₂	semidirect
9. C₂₃×#2158	C₂	semidirect
10. C₂₃×#2138	C₂	semidirect
11. C₂×C₂₃×C₂²⋊C₁₆	C₂	semidirect
32. C₂₃×#367	C₂²	semidirect
33. C₂₃×#367	C₂²	semidirect
34. C₂₃×#366	C₂²	semidirect
35. C₂₃×#366	C₂²	semidirect
36. C₂₃×#358	C₂²	semidirect
37. C₂₃×#358	C₂²	semidirect
38. C₂₃×#357	C₂²	semidirect
39. C₂₃×#357	C₂²	semidirect
141. #33645	C₂₃	direct (N × H)
182. C₂×#427	C₂×C₂₃	semidirect
183. #2237	C₂×C₂₃	semidirect
184. #2236	C₂×C₂₃	semidirect
185. #2235	C₂×C₂₃	semidirect
186. #2234	C₂×C₂₃	semidirect
187. #2233	C₂×C₂₃	semidirect
188. #2232	C₂×C₂₃	semidirect
189. #2158	C₂×C₂₃	semidirect
190. #2138	C₂×C₂₃	semidirect
191. C₂×C₂²⋊C₁₆	C₂×C₂₃	semidirect
236. #367	C₂²×C₂₃	semidirect
237. #367	C₂²×C₂₃	semidirect
238. #366	C₂²×C₂₃	semidirect
239. #366	C₂²×C₂₃	semidirect
240. #358	C₂²×C₂₃	semidirect
241. #358	C₂²×C₂₃	semidirect
242. #357	C₂²×C₂₃	semidirect
243. #357	C₂²×C₂₃	semidirect
300. C₂₃	#33645	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×#3050	2
C₂×C₂₃×#427	2
C₂₃×#2237	2
C₂₃×#2236	2
C₂₃×#2235	2
C₂₃×#2234	2
C₂₃×#2233	2
C₂₃×#2232	2
C₂₃×#2158	2
C₂₃×#2138	2
C₂×C₂₃×C₂²⋊C₁₆	2
#33645	23

Subgroups — C23×#33645 #19181271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33645 #19181271