Subgroups — C₂₃×#18831 #19166457

Order 5888.

838 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3509	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3230	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3116	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1650	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1648	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1638	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1636	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1556	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1497	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1488	C₂	1	1	yes	maximal, characteristic
12	C₂₃×#557	—	2	1	no	—
13	C₂₃×#557	C₂²	1	2	yes	—
14	C₂₃×#556	—	2	1	no	—
15	C₂₃×#556	C₂²	1	2	yes	—
16	C₂₃×#552	—	2	2	no	—
17	C₂₃×#552	—	2	2	no	—
18	C₂₃×#541	—	2	2	no	—
19	C₂₃×#536	—	2	1	no	—
20	C₂₃×#533	—	2	1	no	—
21	C₂₃×#533	—	2	1	no	—
22	C₂₃×#533	C₂²	1	1	yes	characteristic
23	C₂₃×#533	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂₃×#519	—	2	1	no	—
26	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#501	—	2	1	no	—
28	C₂₃×#500	C₂²	1	2	yes	—
29	C₂₃×#491	—	2	1	no	—
30	C₂₃×#487	C₂²	1	2	yes	—
31	C₂₃×#485	—	2	2	no	—
32	C₂₃×#485	—	2	1	no	—
33	C₂₃×#484	—	2	2	no	—
34	C₂₃×#477	—	2	2	no	—
35	C₂₃×#477	C₂²	1	1	yes	characteristic
36	C₂₃×#477	C₂²	1	2	yes	—
37	C₂₃×#476	C₂²	1	1	yes	characteristic
38	C₂₃×#475	C₂²	1	2	yes	—
39	C₂₃×#462	C₂²	1	1	yes	characteristic
40	C₂₃×#461	C₂²	1	1	yes	characteristic
41	C₂₃×#452	—	2	1	no	—
42	C₂₃×#441	—	2	1	no	—
43	C₂₃×#440	—	2	1	no	—
44	C₂₃×#432	C₂²	1	1	yes	characteristic
45	C₂₃×#432	C₂²	1	1	yes	characteristic
46	C₂₃×#431	C₂²	1	1	yes	characteristic
47	C₂₃×#332	C₂²	1	2	yes	—
48	C₂₃×#332	C₂²	1	2	yes	—
49	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
51	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
52	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
54	C₂²×Q₈×C₂₃	—	2	1	no	—
55	C₂×Q₁₆×C₂₃	—	2	1	no	—
56	C₂×SD₁₆×C₂₃	—	2	1	no	—
57	C₂₃×C₄⋊Q₈	—	2	2	no	—
58	C₂₃×C₄⋊Q₈	—	2	1	no	—
59	C₂₃×C₄⋊Q₈	—	2	1	no	—
60	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
61	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
62	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
63	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₂₃×#126	—	2	4	no	—
67	C₂₃×#126	—	2	4	no	—
68	C₂₃×#125	—	2	2	no	—
69	C₂₃×#125	—	2	4	no	—
70	C₂₃×#125	—	2	2	no	—
71	C₂₃×#125	—	2	4	no	—
72	C₂₃×#125	—	2	1	no	—
73	C₂₃×#125	—	2	1	no	—
74	C₂₃×#124	—	2	2	no	—
75	C₂₃×#123	—	2	4	no	—
76	C₂₃×C₂²⋊Q₈	—	2	2	no	—
77	C₂₃×C₂²⋊Q₈	—	2	2	no	—
78	C₂₃×C₂²⋊Q₈	—	2	2	no	—
79	C₂₃×C₂²⋊Q₈	—	2	2	no	—
80	C₂₃×C₂²⋊Q₈	—	2	1	no	—
81	C₂₃×C₂²⋊Q₈	—	2	1	no	—
82	C₂₃×C₂²⋊Q₈	—	2	1	no	—
83	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
84	C₄×Q₈×C₂₃	—	2	2	no	—
85	C₄×Q₈×C₂₃	—	2	2	no	—
86	C₄×Q₈×C₂₃	—	2	2	no	—
87	C₄×Q₈×C₂₃	—	2	2	no	—
88	C₄×Q₈×C₂₃	—	2	2	no	—
89	C₄×Q₈×C₂₃	—	2	2	no	—
90	C₄×Q₈×C₂₃	—	2	1	no	—
91	C₄×Q₈×C₂₃	—	2	1	no	—
92	C₄×Q₈×C₂₃	—	2	1	no	—
93	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
94	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
95	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
96	C₄×D₄×C₂₃	—	2	1	no	—
97	C₄×D₄×C₂₃	—	2	1	no	—
98	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
99	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂₃×#117	—	2	4	no	—
101	C₂₃×#117	—	2	2	no	—
102	C₂₃×#117	—	2	1	no	—
103	C₂₃×#117	—	2	2	no	—
104	C₂₃×#117	—	2	1	no	—
105	C₂₃×#117	—	2	1	no	—
106	C₂₃×#117	D₄	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
108	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
109	C₂×C₄²×C₂₃	—	2	1	no	—
110	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
111	C₂₃×#107	—	2	1	no	—
112	C₂₃×#107	—	2	4	no	—
113	C₂₃×#107	—	2	2	no	—
114	C₂₃×#106	—	2	1	no	—
115	C₂₃×#106	—	2	2	no	—
116	C₂₃×C₄⋊C₈	—	2	1	no	—
117	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
118	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
119	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₂₃×#103	—	2	1	no	—
121	C₂₃×#103	—	2	2	no	—
122	C₂₃×#103	—	2	2	no	—
123	C₂₃×#103	—	2	1	no	—
124	C₂₃×#103	—	2	1	no	—
125	C₂₃×#103	—	2	4	no	—
126	C₂₃×#102	—	2	1	no	—
127	C₂₃×C₂²⋊C₈	—	2	1	no	—
128	C₂₃×C₂²⋊C₈	—	2	1	no	—
129	C₂₃×C₂²⋊C₈	—	2	1	no	—
130	C₂₃×#97	—	2	1	no	—
131	C₂₃×#97	—	2	1	no	—
132	C₄×C₈×C₂₃	C₂³	1	2	yes	—
133	#41×C₂₃	—	4	1	no	—
134	#41×C₂₃	—	4	1	no	—
135	C₂×Q₈×C₂₃	—	2	1	no	—
136	C₂×Q₈×C₂₃	—	2	2	no	—
137	C₂×Q₈×C₂₃	—	2	2	no	—
138	C₂×Q₈×C₂₃	—	2	1	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	4	2	no	—
141	C₂×Q₈×C₂₃	—	4	2	no	—
142	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
143	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂×D₄×C₂₃	—	2	1	no	—
145	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	—	2	2	no	—
147	C₂²×C₄×C₂₃	—	2	1	no	—
148	C₂²×C₄×C₂₃	—	2	1	no	—
149	C₂²×C₄×C₂₃	—	2	1	no	—
150	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	Q₁₆×C₂₃	—	4	2	no	—
154	SD₁₆×C₂₃	—	4	2	no	—
155	C₂×C₈×C₂₃	—	2	1	no	—
156	C₂×C₈×C₂₃	—	2	2	no	—
157	C₂×C₈×C₂₃	—	2	2	no	—
158	C₂×C₈×C₂₃	—	2	1	no	—
159	C₄⋊C₄×C₂₃	—	2	4	no	—
160	C₄⋊C₄×C₂₃	—	2	4	no	—
161	C₄⋊C₄×C₂₃	—	2	4	no	—
162	C₄⋊C₄×C₂₃	—	2	4	no	—
163	C₄⋊C₄×C₂₃	—	2	2	no	—
164	C₄⋊C₄×C₂₃	—	2	2	no	—
165	C₄⋊C₄×C₂₃	—	2	1	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	2	no	—
170	C₄⋊C₄×C₂₃	—	2	2	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	2	no	—
173	C₄⋊C₄×C₂₃	—	2	1	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	—	4	1	no	—
178	C₄⋊C₄×C₂₃	—	4	1	no	—
179	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
181	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
183	C₂²⋊C₄×C₂₃	—	2	4	no	—
184	C₂²⋊C₄×C₂₃	—	2	4	no	—
185	C₂²⋊C₄×C₂₃	—	2	1	no	—
186	C₂²⋊C₄×C₂₃	—	2	2	no	—
187	C₂²⋊C₄×C₂₃	—	2	1	no	—
188	C₂²⋊C₄×C₂₃	—	2	1	no	—
189	C₂²⋊C₄×C₂₃	—	4	1	no	—
190	C₄²×C₂₃	—	2	4	no	—
191	C₄²×C₂₃	—	2	4	no	—
192	C₄²×C₂₃	—	2	1	no	—
193	C₄²×C₂₃	—	2	1	no	—
194	C₄²×C₂₃	—	2	1	no	—
195	C₄²×C₂₃	—	2	2	no	—
196	C₄²×C₂₃	—	4	1	no	—
197	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
201	#18831	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂³×C₂₃	—	2	1	no	—
203	C₂³×C₂₃	#120	1	1	yes	characteristic
204	Q₈×C₂₃	—	2	1	no	—
205	Q₈×C₂₃	—	2	1	no	—
206	Q₈×C₂₃	—	2	4	no	—
207	Q₈×C₂₃	—	4	2	no	—
208	Q₈×C₂₃	—	4	4	no	—
209	Q₈×C₂₃	—	4	2	no	—
210	Q₈×C₂₃	—	4	2	no	—
211	Q₈×C₂₃	—	4	2	no	—
212	Q₈×C₂₃	—	4	1	no	—
213	Q₈×C₂₃	—	4	1	no	—
214	D₄×C₂₃	—	2	1	no	—
215	D₄×C₂₃	—	2	1	no	—
216	D₄×C₂₃	—	4	2	no	—
217	C₂×C₄×C₂₃	—	2	1	no	—
218	C₂×C₄×C₂₃	—	2	1	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	1	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	2	2	no	—
224	C₂×C₄×C₂₃	—	2	4	no	—
225	C₂×C₄×C₂₃	—	2	1	no	—
226	C₂×C₄×C₂₃	—	4	1	no	—
227	C₂×C₄×C₂₃	—	4	2	no	—
228	C₂×C₄×C₂₃	—	4	2	no	—
229	C₂×C₄×C₂₃	—	4	4	no	—
230	C₂×C₄×C₂₃	—	4	2	no	—
231	C₂×C₄×C₂₃	—	4	1	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
234	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
235	C₂×C₄×C₂₃	#120	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	C₂²×D₄	1	1	yes	characteristic
239	C₈×C₂₃	—	4	2	no	—
240	C₈×C₂₃	—	4	1	no	—
241	C₈×C₂₃	—	4	1	no	—
242	C₈×C₂₃	—	4	2	no	—
243	#3509	C₂×C₂₃	1	1	yes	characteristic
244	#3237	C₂×C₂₃	1	1	yes	characteristic, complemented
245	#3230	C₂×C₂₃	1	1	yes	characteristic, complemented
246	#3116	C₂×C₂₃	1	1	yes	characteristic
247	#1650	C₂×C₂₃	1	2	yes	complemented
248	#1648	C₂×C₂₃	1	2	yes	complemented
249	#1638	C₂×C₂₃	1	2	yes	complemented
250	#1636	C₂×C₂₃	1	2	yes	complemented
251	#1556	C₂×C₂₃	1	1	yes	characteristic, complemented
252	#1497	C₂×C₂₃	1	1	yes	characteristic, complemented
253	#1488	C₂×C₂₃	1	1	yes	characteristic
254	C₂²×C₂₃	—	2	1	no	—
255	C₂²×C₂₃	—	2	1	no	—
256	C₂²×C₂₃	—	2	1	no	—
257	C₂²×C₂₃	—	4	1	no	—
258	C₂²×C₂₃	—	4	1	no	—
259	C₂²×C₂₃	—	4	1	no	—
260	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
261	C₄×C₂₃	—	2	1	no	—
262	C₄×C₂₃	—	2	1	no	—
263	C₄×C₂₃	—	2	1	no	—
264	C₄×C₂₃	—	2	1	no	—
265	C₄×C₂₃	—	4	1	no	—
266	C₄×C₂₃	—	4	4	no	—
267	C₄×C₂₃	—	4	1	no	—
268	C₄×C₂₃	—	4	2	no	—
269	C₄×C₂₃	—	4	1	no	—
270	C₄×C₂₃	—	4	2	no	—
271	C₄×C₂₃	—	8	1	no	—
272	C₄×C₂₃	—	8	1	no	—
273	C₄×C₂₃	#575	1	1	yes	characteristic
274	C₄×C₂₃	#575	1	1	yes	characteristic
275	C₄×C₂₃	#578	1	2	yes	—
276	#557	—	2	1	no	—
277	#557	C₂²×C₂₃	1	2	yes	—
278	#556	—	2	1	no	—
279	#556	C₂²×C₂₃	1	2	yes	—
280	#552	—	2	2	no	—
281	#552	—	2	2	no	—
282	#541	—	2	2	no	—
283	#536	—	2	1	no	—
284	#533	—	2	1	no	—
285	#533	—	2	1	no	—
286	#533	C₂²×C₂₃	1	1	yes	characteristic
287	#533	C₂²×C₂₃	1	1	yes	characteristic
288	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
289	#519	—	2	1	no	—
290	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
291	#501	—	2	1	no	—
292	#500	C₂²×C₂₃	1	2	yes	—
293	#491	—	2	1	no	—
294	#487	C₂²×C₂₃	1	2	yes	—
295	#485	—	2	2	no	—
296	#485	—	2	1	no	—
297	#484	—	2	2	no	—
298	#477	—	2	2	no	—
299	#477	C₂²×C₂₃	1	1	yes	characteristic
300	#477	C₂²×C₂₃	1	2	yes	—
301	#476	C₂²×C₂₃	1	1	yes	characteristic
302	#475	C₂²×C₂₃	1	2	yes	—
303	#462	C₂²×C₂₃	1	1	yes	characteristic
304	#461	C₂²×C₂₃	1	1	yes	characteristic
305	#452	—	2	1	no	—
306	#441	—	2	1	no	—
307	#440	—	2	1	no	—
308	#432	C₂²×C₂₃	1	1	yes	characteristic
309	#432	C₂²×C₂₃	1	1	yes	characteristic
310	#431	C₂²×C₂₃	1	1	yes	characteristic
311	#332	C₂²×C₂₃	1	2	yes	—
312	#332	C₂²×C₂₃	1	2	yes	—
313	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
314	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
315	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
316	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
317	C₂×C₂₃	—	4	1	no	—
318	C₂×C₂₃	—	8	1	no	—
319	C₂×C₂₃	#3005	1	1	yes	characteristic
320	C₂×C₂₃	#3011	1	1	yes	characteristic
321	C₂×C₂₃	#3020	1	1	yes	characteristic
322	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
323	C₂²×Q₈	—	2	1	no	—
324	C₂×Q₁₆	—	2	1	no	—
325	C₂×SD₁₆	—	2	1	no	—
326	C₄⋊Q₈	—	2	2	no	—
327	C₄⋊Q₈	—	2	1	no	—
328	C₄⋊Q₈	—	2	1	no	—
329	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
330	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
331	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
332	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
333	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
334	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
335	#126	—	2	4	no	—
336	#126	—	2	4	no	—
337	#125	—	2	2	no	—
338	#125	—	2	4	no	—
339	#125	—	2	2	no	—
340	#125	—	2	4	no	—
341	#125	—	2	1	no	—
342	#125	—	2	1	no	—
343	#124	—	2	2	no	—
344	#123	—	2	4	no	—
345	C₂²⋊Q₈	—	2	2	no	—
346	C₂²⋊Q₈	—	2	2	no	—
347	C₂²⋊Q₈	—	2	2	no	—
348	C₂²⋊Q₈	—	2	2	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	C₂²⋊Q₈	—	2	1	no	—
351	C₂²⋊Q₈	—	2	1	no	—
352	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
353	C₄×Q₈	—	2	2	no	—
354	C₄×Q₈	—	2	2	no	—
355	C₄×Q₈	—	2	2	no	—
356	C₄×Q₈	—	2	2	no	—
357	C₄×Q₈	—	2	2	no	—
358	C₄×Q₈	—	2	2	no	—
359	C₄×Q₈	—	2	1	no	—
360	C₄×Q₈	—	2	1	no	—
361	C₄×Q₈	—	2	1	no	—
362	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
363	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
364	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
365	C₄×D₄	—	2	1	no	—
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
368	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
369	#117	—	2	4	no	—
370	#117	—	2	2	no	—
371	#117	—	2	1	no	—
372	#117	—	2	2	no	—
373	#117	—	2	1	no	—
374	#117	—	2	1	no	—
375	#117	D₄×C₂₃	1	1	yes	characteristic
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
378	C₂×C₄²	—	2	1	no	—
379	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
380	#107	—	2	1	no	—
381	#107	—	2	4	no	—
382	#107	—	2	2	no	—
383	#106	—	2	1	no	—
384	#106	—	2	2	no	—
385	C₄⋊C₈	—	2	1	no	—
386	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
387	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
388	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
389	#103	—	2	1	no	—
390	#103	—	2	2	no	—
391	#103	—	2	2	no	—
392	#103	—	2	1	no	—
393	#103	—	2	1	no	—
394	#103	—	2	4	no	—
395	#102	—	2	1	no	—
396	C₂²⋊C₈	—	2	1	no	—
397	C₂²⋊C₈	—	2	1	no	—
398	C₂²⋊C₈	—	2	1	no	—
399	#97	—	2	1	no	—
400	#97	—	2	1	no	—
401	C₄×C₈	C₂³×C₂₃	1	2	yes	—
402	C₂₃	#18831	1	1	yes	characteristic, direct product factor, Sylow
403	#41	—	4	1	no	—
404	#41	—	4	1	no	—
405	C₂×Q₈	—	2	1	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	1	no	—
410	C₂×Q₈	—	4	2	no	—
411	C₂×Q₈	—	4	2	no	—
412	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
413	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
421	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
422	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
423	Q₁₆	—	4	2	no	—
424	SD₁₆	—	4	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	—	2	1	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	1	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	4	no	—
431	C₄⋊C₄	—	2	4	no	—
432	C₄⋊C₄	—	2	4	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	—	2	1	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	1	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	2	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	4	1	no	—
448	C₄⋊C₄	—	4	1	no	—
449	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
450	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
451	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
452	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	1	no	—
456	C₂²⋊C₄	—	2	2	no	—
457	C₂²⋊C₄	—	2	1	no	—
458	C₂²⋊C₄	—	2	1	no	—
459	C₂²⋊C₄	—	4	1	no	—
460	C₄²	—	2	4	no	—
461	C₄²	—	2	4	no	—
462	C₄²	—	2	1	no	—
463	C₄²	—	2	1	no	—
464	C₄²	—	2	1	no	—
465	C₄²	—	2	2	no	—
466	C₄²	—	4	1	no	—
467	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
468	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
469	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
470	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
471	C₂³	—	2	1	no	—
472	C₂³	C₂₃×#120	1	1	yes	characteristic
473	Q₈	—	2	1	no	—
474	Q₈	—	2	1	no	—
475	Q₈	—	2	4	no	—
476	Q₈	—	4	4	no	—
477	Q₈	—	4	2	no	—
478	Q₈	—	4	2	no	—
479	Q₈	—	4	2	no	—
480	Q₈	—	4	2	no	—
481	Q₈	—	4	1	no	—
482	Q₈	—	4	1	no	—
483	D₄	—	2	1	no	—
484	D₄	—	2	1	no	—
485	D₄	—	4	2	no	—
486	C₂×C₄	—	2	1	no	—
487	C₂×C₄	—	2	1	no	—
488	C₂×C₄	—	2	2	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	4	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	4	1	no	—
496	C₂×C₄	—	4	2	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
503	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
504	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
505	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
506	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
508	C₈	—	4	2	no	—
509	C₈	—	4	1	no	—
510	C₈	—	4	1	no	—
511	C₈	—	4	2	no	—
512	C₂²	—	2	1	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	4	1	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	1	no	—
518	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
519	C₄	—	2	1	no	—
520	C₄	—	2	1	no	—
521	C₄	—	2	1	no	—
522	C₄	—	2	1	no	—
523	C₄	—	4	1	no	—
524	C₄	—	4	4	no	—
525	C₄	—	4	1	no	—
526	C₄	—	4	2	no	—
527	C₄	—	4	1	no	—
528	C₄	—	4	2	no	—
529	C₄	—	8	1	no	—
530	C₄	—	8	1	no	—
531	C₄	C₂₃×#575	1	1	yes	characteristic
532	C₄	C₂₃×#575	1	1	yes	characteristic
533	C₄	C₂₃×#578	1	2	yes	—
534	C₂	—	4	1	no	—
535	C₂	—	8	1	no	—
536	C₂	C₂₃×#3005	1	1	yes	characteristic
537	C₂	C₂₃×#3011	1	1	yes	characteristic
538	C₂	C₂₃×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3237	C₂	semidirect
3. C₂₃×#3230	C₂	semidirect
5. C₂₃×#1650	C₂	semidirect
6. C₂₃×#1648	C₂	semidirect
7. C₂₃×#1638	C₂	semidirect
8. C₂₃×#1636	C₂	semidirect
9. C₂₃×#1556	C₂	semidirect
10. C₂₃×#1497	C₂	semidirect
201. #18831	C₂₃	direct (N × H)
244. #3237	C₂×C₂₃	semidirect
245. #3230	C₂×C₂₃	semidirect
247. #1650	C₂×C₂₃	semidirect
248. #1648	C₂×C₂₃	semidirect
249. #1638	C₂×C₂₃	semidirect
250. #1636	C₂×C₂₃	semidirect
251. #1556	C₂×C₂₃	semidirect
252. #1497	C₂×C₂₃	semidirect
402. C₂₃	#18831	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×#3509	2
C₂₃×#3237	2
C₂₃×#3230	2
C₂₃×#3116	2
C₂₃×#1650	2
C₂₃×#1648	2
C₂₃×#1638	2
C₂₃×#1636	2
C₂₃×#1556	2
C₂₃×#1497	2
C₂₃×#1488	2
#18831	23

Subgroups — C23×#18831 #19166457

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18831 #19166457