Subgroups — C₂₃×#10537 #19158163

Order 5888.

554 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1982	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×#332	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#331	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₂₃	—	2	1	no	—
9	C₂²×SD₁₆×C₂₃	—	2	1	no	—
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
12	C₂₃×#492	—	2	2	no	—
13	C₂₃×#492	—	2	2	no	—
14	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#102	—	2	1	no	—
19	C₂×C₂₃×#102	—	2	1	no	—
20	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#340	—	2	1	no	—
23	C₂₃×#340	—	2	1	no	—
24	C₂₃×#336	—	2	1	no	—
25	C₂₃×#332	—	2	2	no	—
26	C₂₃×#331	—	2	2	no	—
27	C₂²×Q₈×C₂₃	—	2	1	no	—
28	C₂²×D₄×C₂₃	—	2	1	no	—
29	C₂²×D₄×C₂₃	—	2	1	no	—
30	C₂²×D₄×C₂₃	—	2	1	no	—
31	C₂×SD₁₆×C₂₃	—	2	4	no	—
32	C₂×SD₁₆×C₂₃	—	2	4	no	—
33	C₂×SD₁₆×C₂₃	—	4	2	no	—
34	C₂×SD₁₆×C₂₃	—	4	2	no	—
35	C₂×SD₁₆×C₂₃	—	4	2	no	—
36	C₂×SD₁₆×C₂₃	—	4	2	no	—
37	C₂²×C₈×C₂₃	—	2	1	no	—
38	C₂²×C₈×C₂₃	—	2	1	no	—
39	C₂²×C₈×C₂₃	—	2	1	no	—
40	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
42	C₂₃×C₄⋊Q₈	—	2	1	no	—
43	C₂₃×C₄⋊Q₈	—	2	1	no	—
44	C₂₃×C₄⋊Q₈	—	2	1	no	—
45	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
46	C₂₃×#127	—	2	1	no	—
47	C₂₃×#127	—	2	1	no	—
48	C₂₃×#127	—	2	1	no	—
49	C₂₃×#127	D₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₂₃	—	2	1	no	—
55	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
56	C₂₃×C₄⋊C₈	—	2	1	no	—
57	C₂₃×C₄⋊C₈	—	2	1	no	—
58	C₂₃×C₄⋊C₈	—	2	1	no	—
59	C₂₃×C₄⋊C₈	—	2	1	no	—
60	C₂₃×C₄⋊C₈	—	2	1	no	—
61	C₂₃×C₄⋊C₈	—	2	1	no	—
62	C₂₃×C₄⋊C₈	—	4	2	no	—
63	C₂₃×#102	—	4	2	no	—
64	C₂₃×#102	—	4	2	no	—
65	C₄×C₈×C₂₃	—	2	1	no	—
66	C₄×C₈×C₂₃	—	2	1	no	—
67	C₂₃×#95	—	2	1	no	—
68	C₂⁴×C₂₃	—	4	1	no	—
69	C₂×Q₈×C₂₃	—	2	2	no	—
70	C₂×Q₈×C₂₃	—	4	1	no	—
71	C₂×Q₈×C₂₃	—	4	1	no	—
72	C₂×Q₈×C₂₃	—	4	1	no	—
73	C₂×Q₈×C₂₃	—	4	1	no	—
74	C₂×Q₈×C₂₃	—	4	1	no	—
75	C₂×D₄×C₂₃	—	2	2	no	—
76	C₂×D₄×C₂₃	—	2	2	no	—
77	C₂×D₄×C₂₃	—	2	2	no	—
78	C₂×D₄×C₂₃	—	4	1	no	—
79	C₂×D₄×C₂₃	—	4	1	no	—
80	C₂×D₄×C₂₃	—	4	1	no	—
81	C₂×D₄×C₂₃	—	4	1	no	—
82	C₂×D₄×C₂₃	—	4	1	no	—
83	C₂×D₄×C₂₃	—	4	1	no	—
84	C₂×D₄×C₂₃	—	4	1	no	—
85	C₂×D₄×C₂₃	—	4	1	no	—
86	C₂×D₄×C₂₃	—	4	1	no	—
87	C₂×D₄×C₂₃	—	4	1	no	—
88	C₂×D₄×C₂₃	—	4	1	no	—
89	C₂×D₄×C₂₃	—	8	1	no	—
90	C₂×D₄×C₂₃	—	8	1	no	—
91	C₂²×C₄×C₂₃	—	2	1	no	—
92	C₂²×C₄×C₂₃	—	2	1	no	—
93	C₂²×C₄×C₂₃	—	4	1	no	—
94	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
97	SD₁₆×C₂₃	—	4	8	no	—
98	SD₁₆×C₂₃	—	4	8	no	—
99	C₂×C₈×C₂₃	—	2	1	no	—
100	C₂×C₈×C₂₃	—	2	1	no	—
101	C₂×C₈×C₂₃	—	2	1	no	—
102	C₂×C₈×C₂₃	—	2	1	no	—
103	C₂×C₈×C₂₃	—	4	1	no	—
104	C₂×C₈×C₂₃	—	4	1	no	—
105	C₂×C₈×C₂₃	—	4	1	no	—
106	C₂×C₈×C₂₃	—	4	1	no	—
107	C₂×C₈×C₂₃	—	4	1	no	—
108	C₂×C₈×C₂₃	—	4	1	no	—
109	C₂×C₈×C₂₃	—	4	1	no	—
110	C₂×C₈×C₂₃	—	4	1	no	—
111	C₂×C₈×C₂₃	—	4	1	no	—
112	C₂×C₈×C₂₃	D₈	1	2	yes	—
113	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
114	C₄⋊C₄×C₂₃	—	2	1	no	—
115	C₄⋊C₄×C₂₃	—	2	2	no	—
116	C₄⋊C₄×C₂₃	—	2	2	no	—
117	C₄⋊C₄×C₂₃	—	2	1	no	—
118	C₄⋊C₄×C₂₃	—	2	1	no	—
119	C₄⋊C₄×C₂₃	—	4	1	no	—
120	C₄⋊C₄×C₂₃	—	4	1	no	—
121	C₄⋊C₄×C₂₃	SD₁₆	1	2	yes	complemented
122	C₄²×C₂₃	—	4	1	no	—
123	C₄²×C₂₃	—	4	1	no	—
124	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₂₃	#41	1	1	yes	characteristic
127	C₄²×C₂₃	#41	1	1	yes	characteristic
128	#10537	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₂₃	—	4	2	no	—
130	C₂³×C₂₃	—	8	1	no	—
131	C₂³×C₂₃	—	8	1	no	—
132	C₂³×C₂₃	—	8	1	no	—
133	C₂³×C₂₃	—	8	1	no	—
134	C₂³×C₂₃	—	8	1	no	—
135	C₂³×C₂₃	—	8	1	no	—
136	C₂³×C₂₃	#121	1	1	yes	characteristic
137	Q₈×C₂₃	—	4	2	no	—
138	Q₈×C₂₃	—	4	2	no	—
139	Q₈×C₂₃	—	8	1	no	—
140	Q₈×C₂₃	—	8	1	no	—
141	D₄×C₂₃	—	4	2	no	—
142	D₄×C₂₃	—	4	2	no	—
143	D₄×C₂₃	—	4	2	no	—
144	D₄×C₂₃	—	4	2	no	—
145	D₄×C₂₃	—	8	2	no	—
146	D₄×C₂₃	—	8	2	no	—
147	D₄×C₂₃	—	8	1	no	—
148	D₄×C₂₃	—	8	1	no	—
149	D₄×C₂₃	—	8	1	no	—
150	D₄×C₂₃	—	8	1	no	—
151	C₂×C₄×C₂₃	—	2	1	no	—
152	C₂×C₄×C₂₃	—	2	1	no	—
153	C₂×C₄×C₂₃	—	2	2	no	—
154	C₂×C₄×C₂₃	—	4	1	no	—
155	C₂×C₄×C₂₃	—	4	1	no	—
156	C₂×C₄×C₂₃	—	4	2	no	—
157	C₂×C₄×C₂₃	—	4	1	no	—
158	C₂×C₄×C₂₃	—	4	1	no	—
159	C₂×C₄×C₂₃	—	4	1	no	—
160	C₂×C₄×C₂₃	—	8	1	no	—
161	C₂×C₄×C₂₃	—	8	1	no	—
162	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
163	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
164	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
165	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
166	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
167	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
168	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
169	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
170	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
171	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
172	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
175	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
176	C₈×C₂₃	—	2	2	no	—
177	C₈×C₂₃	—	2	2	no	—
178	C₈×C₂₃	—	8	1	no	—
179	C₈×C₂₃	—	8	1	no	—
180	C₈×C₂₃	—	8	1	no	—
181	C₂×#492	C₂×C₂₃	1	1	yes	characteristic
182	#1988	C₂×C₂₃	1	1	yes	characteristic, complemented
183	#1982	C₂×C₂₃	1	1	yes	characteristic
184	#1939	C₂×C₂₃	1	1	yes	characteristic, complemented
185	C₂×#332	C₂×C₂₃	1	1	yes	characteristic, complemented
186	C₂×#331	C₂×C₂₃	1	1	yes	characteristic
187	#1277	C₂×C₂₃	1	1	yes	characteristic, complemented
188	C₂²×C₂₃	—	8	2	no	—
189	C₂²×C₂₃	—	8	2	no	—
190	C₂²×C₂₃	—	8	2	no	—
191	C₂²×C₂₃	—	16	1	no	—
192	C₂²×C₂₃	—	16	1	no	—
193	C₂²×C₂₃	—	16	1	no	—
194	C₂²×C₂₃	—	16	1	no	—
195	C₂²×C₂₃	#394	1	1	yes	characteristic
196	C₂²×C₂₃	#459	1	1	yes	characteristic
197	C₂²×C₂₃	#460	1	1	yes	characteristic
198	C₂²×C₂₃	#465	1	1	yes	characteristic
199	C₂²×C₂₃	#468	1	1	yes	characteristic
200	C₂²×C₂₃	#471	1	1	yes	characteristic
201	C₂²×C₂₃	#471	1	1	yes	characteristic
202	C₄×C₂₃	—	2	1	no	—
203	C₄×C₂₃	—	2	1	no	—
204	C₄×C₂₃	—	4	2	no	—
205	C₄×C₂₃	—	8	1	no	—
206	C₄×C₂₃	—	8	2	no	—
207	C₄×C₂₃	#447	1	1	yes	characteristic
208	C₄×C₂₃	#450	1	1	yes	characteristic
209	C₄×C₂₃	#460	1	1	yes	characteristic
210	C₄×C₂₃	#461	1	1	yes	characteristic
211	C₄×C₂₃	#463	1	1	yes	characteristic
212	C₄×C₂₃	#464	1	1	yes	characteristic
213	C₄×C₂₃	#486	1	1	yes	characteristic
214	C₄×C₂₃	#490	1	1	yes	characteristic
215	C₂²×SD₁₆	—	2	1	no	—
216	C₂²×SD₁₆	—	2	1	no	—
217	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
218	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
219	#492	—	2	2	no	—
220	#492	—	2	2	no	—
221	C₂×C₄⋊C₈	—	2	1	no	—
222	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
223	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
224	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
225	C₂×#102	—	2	1	no	—
226	C₂×#102	—	2	1	no	—
227	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
228	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
229	#340	—	2	1	no	—
230	#340	—	2	1	no	—
231	#336	—	2	1	no	—
232	#332	—	2	2	no	—
233	#331	—	2	2	no	—
234	C₂×C₂₃	—	16	2	no	—
235	C₂×C₂₃	#1665	1	1	yes	characteristic
236	C₂×C₂₃	#1666	1	1	yes	characteristic
237	C₂×C₂₃	#1685	1	1	yes	characteristic
238	C₂×C₂₃	#1686	1	1	yes	characteristic
239	C₂×C₂₃	#2041	1	1	yes	characteristic
240	C₂×C₂₃	#2046	1	1	yes	characteristic
241	C₂×C₂₃	#2055	1	1	yes	characteristic
242	C₂²×Q₈	—	2	1	no	—
243	C₂²×D₄	—	2	1	no	—
244	C₂²×D₄	—	2	1	no	—
245	C₂²×D₄	—	2	1	no	—
246	C₂×SD₁₆	—	2	4	no	—
247	C₂×SD₁₆	—	2	4	no	—
248	C₂×SD₁₆	—	4	2	no	—
249	C₂×SD₁₆	—	4	2	no	—
250	C₂×SD₁₆	—	4	2	no	—
251	C₂×SD₁₆	—	4	2	no	—
252	C₂²×C₈	—	2	1	no	—
253	C₂²×C₈	—	2	1	no	—
254	C₂²×C₈	—	2	1	no	—
255	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
256	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
257	C₄⋊Q₈	—	2	1	no	—
258	C₄⋊Q₈	—	2	1	no	—
259	C₄⋊Q₈	—	2	1	no	—
260	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
261	#127	—	2	1	no	—
262	#127	—	2	1	no	—
263	#127	—	2	1	no	—
264	#127	D₄×C₂₃	1	2	yes	—
265	C₂×C₄⋊C₄	—	2	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
268	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
269	C₂×C₄²	—	2	1	no	—
270	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	2	1	no	—
277	C₄⋊C₈	—	4	2	no	—
278	#102	—	4	2	no	—
279	#102	—	4	2	no	—
280	C₄×C₈	—	2	1	no	—
281	C₄×C₈	—	2	1	no	—
282	#95	—	2	1	no	—
283	C₂₃	#10537	1	1	yes	characteristic, direct product factor, Sylow
284	C₂⁴	—	4	1	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	—	4	1	no	—
287	C₂×Q₈	—	4	1	no	—
288	C₂×Q₈	—	4	1	no	—
289	C₂×Q₈	—	4	1	no	—
290	C₂×Q₈	—	4	1	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	4	1	no	—
295	C₂×D₄	—	4	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	4	1	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	—	4	1	no	—
301	C₂×D₄	—	4	1	no	—
302	C₂×D₄	—	4	1	no	—
303	C₂×D₄	—	4	1	no	—
304	C₂×D₄	—	4	1	no	—
305	C₂×D₄	—	8	1	no	—
306	C₂×D₄	—	8	1	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	4	1	no	—
310	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
312	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
313	SD₁₆	—	4	8	no	—
314	SD₁₆	—	4	8	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	4	1	no	—
320	C₂×C₈	—	4	1	no	—
321	C₂×C₈	—	4	1	no	—
322	C₂×C₈	—	4	1	no	—
323	C₂×C₈	—	4	1	no	—
324	C₂×C₈	—	4	1	no	—
325	C₂×C₈	—	4	1	no	—
326	C₂×C₈	—	4	1	no	—
327	C₂×C₈	—	4	1	no	—
328	C₂×C₈	D₈×C₂₃	1	2	yes	—
329	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	4	1	no	—
336	C₄⋊C₄	—	4	1	no	—
337	C₄⋊C₄	SD₁₆×C₂₃	1	2	yes	complemented
338	C₄²	—	4	1	no	—
339	C₄²	—	4	1	no	—
340	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
341	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
342	C₄²	#41×C₂₃	1	1	yes	characteristic
343	C₄²	#41×C₂₃	1	1	yes	characteristic
344	C₂³	—	4	2	no	—
345	C₂³	—	8	1	no	—
346	C₂³	—	8	1	no	—
347	C₂³	—	8	1	no	—
348	C₂³	—	8	1	no	—
349	C₂³	—	8	1	no	—
350	C₂³	—	8	1	no	—
351	C₂³	C₂₃×#121	1	1	yes	characteristic
352	Q₈	—	4	2	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	8	1	no	—
355	Q₈	—	8	1	no	—
356	D₄	—	4	2	no	—
357	D₄	—	4	2	no	—
358	D₄	—	4	2	no	—
359	D₄	—	4	2	no	—
360	D₄	—	8	2	no	—
361	D₄	—	8	2	no	—
362	D₄	—	8	1	no	—
363	D₄	—	8	1	no	—
364	D₄	—	8	1	no	—
365	D₄	—	8	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	8	1	no	—
376	C₂×C₄	—	8	1	no	—
377	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
378	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
379	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
380	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
381	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
382	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
383	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
384	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
385	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
386	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
387	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
388	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
389	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
390	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
391	C₈	—	2	2	no	—
392	C₈	—	2	2	no	—
393	C₈	—	8	1	no	—
394	C₈	—	8	1	no	—
395	C₈	—	8	1	no	—
396	C₂²	—	8	2	no	—
397	C₂²	—	8	2	no	—
398	C₂²	—	8	2	no	—
399	C₂²	—	16	1	no	—
400	C₂²	—	16	1	no	—
401	C₂²	—	16	1	no	—
402	C₂²	—	16	1	no	—
403	C₂²	C₂₃×#394	1	1	yes	characteristic
404	C₂²	C₂₃×#459	1	1	yes	characteristic
405	C₂²	C₂₃×#460	1	1	yes	characteristic
406	C₂²	C₂₃×#465	1	1	yes	characteristic
407	C₂²	C₂₃×#468	1	1	yes	characteristic
408	C₂²	C₂₃×#471	1	1	yes	characteristic
409	C₂²	C₂₃×#471	1	1	yes	characteristic
410	C₄	—	2	1	no	—
411	C₄	—	2	1	no	—
412	C₄	—	4	2	no	—
413	C₄	—	8	1	no	—
414	C₄	—	8	2	no	—
415	C₄	C₂₃×#447	1	1	yes	characteristic
416	C₄	C₂₃×#450	1	1	yes	characteristic
417	C₄	C₂₃×#460	1	1	yes	characteristic
418	C₄	C₂₃×#461	1	1	yes	characteristic
419	C₄	C₂₃×#463	1	1	yes	characteristic
420	C₄	C₂₃×#464	1	1	yes	characteristic
421	C₄	C₂₃×#486	1	1	yes	characteristic
422	C₄	C₂₃×#490	1	1	yes	characteristic
423	C₂	—	16	2	no	—
424	C₂	C₂₃×#1665	1	1	yes	characteristic
425	C₂	C₂₃×#1666	1	1	yes	characteristic
426	C₂	C₂₃×#1685	1	1	yes	characteristic
427	C₂	C₂₃×#1686	1	1	yes	characteristic
428	C₂	C₂₃×#2041	1	1	yes	characteristic
429	C₂	C₂₃×#2046	1	1	yes	characteristic
430	C₂	C₂₃×#2055	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1988	C₂	semidirect
4. C₂₃×#1939	C₂	semidirect
5. C₂×C₂₃×#332	C₂	semidirect
7. C₂₃×#1277	C₂	semidirect
121. C₄⋊C₄×C₂₃	SD₁₆	semidirect
128. #10537	C₂₃	direct (N × H)
182. #1988	C₂×C₂₃	semidirect
184. #1939	C₂×C₂₃	semidirect
185. C₂×#332	C₂×C₂₃	semidirect
187. #1277	C₂×C₂₃	semidirect
283. C₂₃	#10537	direct (N × H)
337. C₄⋊C₄	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#492	2
C₂₃×#1988	2
C₂₃×#1982	2
C₂₃×#1939	2
C₂×C₂₃×#332	2
C₂×C₂₃×#331	2
C₂₃×#1277	2
#10537	23

Subgroups — C23×#10537 #19158163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10537 #19158163