Subgroups — C₂₃×#33594 #19181220

Order 5888.

696 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3414	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#443	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2214	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2213	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2212	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2148	C₂	1	2	yes	maximal, complemented
9	C₂×C₂₃×#135	—	2	1	no	—
10	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#135	C₂²	1	2	yes	—
12	C₂×C₂₃×#131	—	2	1	no	—
13	C₂₃×#532	—	2	1	no	—
14	C₂₃×#510	—	2	2	no	—
15	C₂₃×#510	C₂²	1	4	yes	—
16	C₂₃×#509	—	2	2	no	—
17	C₂₃×#509	C₂²	1	4	yes	—
18	C₂₃×#508	—	2	4	no	—
19	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
20	C₂₃×#495	—	2	1	no	—
21	C₂₃×#495	C₂²	1	2	yes	complemented
22	C₂₃×#494	C₂²	1	1	yes	characteristic, complemented
23	C₂₃×#493	C₂²	1	1	yes	characteristic, complemented
24	C₂₃×#492	—	2	1	no	—
25	C₂₃×#443	—	2	1	no	—
26	C₂₃×#443	—	2	1	no	—
27	C₂₃×#443	C₂²	1	2	yes	—
28	C₂₃×#443	C₂²	1	2	yes	—
29	C₂×C₂₃×#108	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#104	—	2	1	no	—
31	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#364	C₂²	1	2	yes	complemented
33	C₂₃×#364	C₂²	1	2	yes	complemented
34	C₂₃×#362	C₂²	1	4	yes	complemented
35	C₂₃×#361	C₂²	1	4	yes	complemented
36	C₂×#41×C₂₃	—	2	1	no	—
37	C₂×#41×C₂₃	—	2	2	no	—
38	C₂₃×#135	—	2	1	no	—
39	C₂₃×#135	—	2	2	no	—
40	C₂₃×#135	—	2	2	no	—
41	C₂₃×#135	—	2	1	no	—
42	C₂₃×#135	—	2	2	no	—
43	C₂₃×#135	—	2	2	no	—
44	C₂₃×#135	—	4	1	no	—
45	C₂₃×#135	—	4	1	no	—
46	C₂₃×#135	—	4	1	no	—
47	C₂₃×#135	—	4	1	no	—
48	C₂₃×#135	D₄	1	2	yes	complemented
49	C₂₃×#135	C₂³	1	2	yes	—
50	C₂×Q₁₆×C₂₃	—	2	1	no	—
51	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic, complemented
52	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
53	C₂×SD₁₆×C₂₃	—	2	2	no	—
54	C₂×SD₁₆×C₂₃	—	2	2	no	—
55	C₂×SD₁₆×C₂₃	—	2	1	no	—
56	C₂×D₈×C₂₃	—	2	1	no	—
57	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic, complemented
58	C₂×D₈×C₂₃	C₂³	1	2	yes	—
59	C₂₃×#131	—	2	2	no	—
60	C₂₃×#131	—	2	2	no	—
61	C₂₃×#131	—	4	1	no	—
62	C₂₃×#131	—	4	1	no	—
63	C₂×M₁₆×C₂₃	—	2	1	no	—
64	C₂×M₁₆×C₂₃	—	4	1	no	—
65	C₂²×C₈×C₂₃	—	4	1	no	—
66	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂₃×C₄⋊Q₈	—	2	1	no	—
69	C₂₃×#127	—	2	1	no	—
70	C₂₃×#124	—	2	2	no	—
71	C₂₃×#121	—	4	2	no	—
72	C₄×D₄×C₂₃	—	4	2	no	—
73	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
74	C₂₃×Q₃₂	—	2	4	no	—
75	C₂₃×SD₃₂	—	2	4	no	—
76	C₂₃×SD₃₂	—	2	4	no	—
77	C₂₃×D₁₆	—	2	4	no	—
78	C₂₃×M₃₂	—	2	2	no	—
79	C₂₃×M₃₂	C₂³	1	4	yes	—
80	C₂×C₁₆×C₂₃	—	2	2	no	—
81	C₂₃×#108	—	2	1	no	—
82	C₂₃×#108	C₂³	1	2	yes	—
83	C₂₃×#104	—	2	2	no	—
84	C₂₃×#104	—	2	2	no	—
85	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
86	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
87	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
89	#41×C₂₃	—	2	2	no	—
90	#41×C₂₃	—	2	2	no	—
91	#41×C₂₃	—	4	1	no	—
92	#41×C₂₃	—	4	1	no	—
93	#41×C₂₃	—	4	2	no	—
94	#41×C₂₃	—	4	2	no	—
95	#41×C₂₃	—	4	2	no	—
96	#41×C₂₃	—	4	2	no	—
97	C₂×Q₈×C₂₃	—	2	1	no	—
98	C₂×Q₈×C₂₃	—	2	2	no	—
99	C₂×D₄×C₂₃	—	2	1	no	—
100	C₂×D₄×C₂₃	—	2	2	no	—
101	C₂×D₄×C₂₃	—	4	2	no	—
102	C₂×D₄×C₂₃	—	4	1	no	—
103	C₂×D₄×C₂₃	—	4	2	no	—
104	C₂²×C₄×C₂₃	—	2	1	no	—
105	C₂²×C₄×C₂₃	—	4	1	no	—
106	C₂²×C₄×C₂₃	—	4	2	no	—
107	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
108	Q₁₆×C₂₃	—	2	1	no	—
109	Q₁₆×C₂₃	—	2	2	no	—
110	Q₁₆×C₂₃	—	2	2	no	—
111	Q₁₆×C₂₃	—	4	1	no	—
112	Q₁₆×C₂₃	—	4	1	no	—
113	Q₁₆×C₂₃	C₂×D₄	1	2	yes	—
114	SD₁₆×C₂₃	—	2	2	no	—
115	SD₁₆×C₂₃	—	2	2	no	—
116	SD₁₆×C₂₃	—	4	2	no	—
117	SD₁₆×C₂₃	—	4	2	no	—
118	SD₁₆×C₂₃	—	4	2	no	—
119	SD₁₆×C₂₃	—	4	2	no	—
120	D₈×C₂₃	—	2	1	no	—
121	D₈×C₂₃	—	2	2	no	—
122	D₈×C₂₃	—	2	2	no	—
123	D₈×C₂₃	—	4	1	no	—
124	D₈×C₂₃	—	4	1	no	—
125	D₈×C₂₃	C₂×D₄	1	2	yes	—
126	M₁₆×C₂₃	—	2	2	no	—
127	M₁₆×C₂₃	—	4	1	no	—
128	M₁₆×C₂₃	—	4	2	no	—
129	M₁₆×C₂₃	—	4	2	no	—
130	C₂×C₈×C₂₃	—	4	1	no	—
131	C₂×C₈×C₂₃	—	4	2	no	—
132	C₂×C₈×C₂₃	—	4	2	no	—
133	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
134	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
135	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
136	C₂×C₈×C₂₃	D₈	1	1	yes	characteristic
137	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
143	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
144	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
145	C₄⋊C₄×C₂₃	—	4	2	no	—
146	C₂²⋊C₄×C₂₃	—	4	4	no	—
147	C₄²×C₂₃	—	2	1	no	—
148	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₁₆×C₂₃	—	2	4	no	—
151	#33594	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂³×C₂₃	—	4	1	no	—
153	C₂³×C₂₃	—	4	2	no	—
154	C₂³×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
155	Q₈×C₂₃	—	2	2	no	—
156	Q₈×C₂₃	—	4	1	no	—
157	Q₈×C₂₃	—	4	2	no	—
158	Q₈×C₂₃	—	4	2	no	—
159	D₄×C₂₃	—	2	2	no	—
160	D₄×C₂₃	—	4	2	no	—
161	D₄×C₂₃	—	4	2	no	—
162	D₄×C₂₃	—	4	2	no	—
163	D₄×C₂₃	—	4	2	no	—
164	D₄×C₂₃	—	4	1	no	—
165	D₄×C₂₃	—	8	4	no	—
166	D₄×C₂₃	—	8	2	no	—
167	D₄×C₂₃	—	8	2	no	—
168	C₂×C₄×C₂₃	—	2	1	no	—
169	C₂×C₄×C₂₃	—	2	1	no	—
170	C₂×C₄×C₂₃	—	4	2	no	—
171	C₂×C₄×C₂₃	—	4	1	no	—
172	C₂×C₄×C₂₃	—	4	2	no	—
173	C₂×C₄×C₂₃	—	4	2	no	—
174	C₂×C₄×C₂₃	—	4	2	no	—
175	C₂×C₄×C₂₃	—	8	2	no	—
176	C₂×C₄×C₂₃	—	8	2	no	—
177	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
178	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
179	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
180	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
181	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
183	C₈×C₂₃	—	4	2	no	—
184	C₈×C₂₃	#121	1	1	yes	characteristic
185	C₈×C₂₃	#121	1	1	yes	characteristic
186	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
187	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
188	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
189	C₈×C₂₃	C₂×D₈	1	1	yes	characteristic
190	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
191	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
192	#3414	C₂×C₂₃	1	2	yes	complemented
193	#3147	C₂×C₂₃	1	1	yes	characteristic, complemented
194	C₂×#443	C₂×C₂₃	1	1	yes	characteristic, complemented
195	#2214	C₂×C₂₃	1	2	yes	complemented
196	#2213	C₂×C₂₃	1	2	yes	complemented
197	#2212	C₂×C₂₃	1	4	yes	complemented
198	C₂×#364	C₂×C₂₃	1	1	yes	characteristic, complemented
199	#2148	C₂×C₂₃	1	2	yes	complemented
200	C₂²×C₂₃	—	4	1	no	—
201	C₂²×C₂₃	—	4	2	no	—
202	C₂²×C₂₃	—	8	2	no	—
203	C₂²×C₂₃	—	8	4	no	—
204	C₂²×C₂₃	—	8	2	no	—
205	C₂²×C₂₃	#459	1	1	yes	characteristic
206	C₂²×C₂₃	#459	1	1	yes	characteristic
207	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₄×C₂₃	—	2	2	no	—
209	C₄×C₂₃	—	4	1	no	—
210	C₄×C₂₃	—	4	2	no	—
211	C₄×C₂₃	—	8	2	no	—
212	C₄×C₂₃	#459	1	1	yes	characteristic
213	C₄×C₂₃	#459	1	1	yes	characteristic
214	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
215	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
216	C₂×#135	—	2	1	no	—
217	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
218	C₂×#135	C₂²×C₂₃	1	2	yes	—
219	C₂×#131	—	2	1	no	—
220	#532	—	2	1	no	—
221	#510	—	2	2	no	—
222	#510	C₂²×C₂₃	1	4	yes	—
223	#509	—	2	2	no	—
224	#509	C₂²×C₂₃	1	4	yes	—
225	#508	—	2	4	no	—
226	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
227	#495	—	2	1	no	—
228	#495	C₂²×C₂₃	1	2	yes	complemented
229	#494	C₂²×C₂₃	1	1	yes	characteristic, complemented
230	#493	C₂²×C₂₃	1	1	yes	characteristic, complemented
231	#492	—	2	1	no	—
232	#443	—	2	1	no	—
233	#443	—	2	1	no	—
234	#443	C₂²×C₂₃	1	2	yes	—
235	#443	C₂²×C₂₃	1	2	yes	—
236	C₂×#108	C₂²×C₂₃	1	1	yes	characteristic
237	C₂×#104	—	2	1	no	—
238	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
239	#364	C₂²×C₂₃	1	2	yes	complemented
240	#364	C₂²×C₂₃	1	2	yes	complemented
241	#362	C₂²×C₂₃	1	4	yes	complemented
242	#361	C₂²×C₂₃	1	4	yes	complemented
243	C₂×C₂₃	—	2	1	no	—
244	C₂×C₂₃	—	2	1	no	—
245	C₂×C₂₃	—	2	1	no	—
246	C₂×C₂₃	—	8	2	no	—
247	C₂×C₂₃	—	16	2	no	—
248	C₂×C₂₃	C₂×#459	1	1	yes	characteristic
249	C₂×#41	—	2	1	no	—
250	C₂×#41	—	2	2	no	—
251	#135	—	2	1	no	—
252	#135	—	2	2	no	—
253	#135	—	2	2	no	—
254	#135	—	2	1	no	—
255	#135	—	2	2	no	—
256	#135	—	2	2	no	—
257	#135	—	4	1	no	—
258	#135	—	4	1	no	—
259	#135	—	4	1	no	—
260	#135	—	4	1	no	—
261	#135	D₄×C₂₃	1	2	yes	complemented
262	#135	C₂³×C₂₃	1	2	yes	—
263	C₂×Q₁₆	—	2	1	no	—
264	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic, complemented
265	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
266	C₂×SD₁₆	—	2	2	no	—
267	C₂×SD₁₆	—	2	2	no	—
268	C₂×SD₁₆	—	2	1	no	—
269	C₂×D₈	—	2	1	no	—
270	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic, complemented
271	C₂×D₈	C₂³×C₂₃	1	2	yes	—
272	#131	—	2	2	no	—
273	#131	—	2	2	no	—
274	#131	—	4	1	no	—
275	#131	—	4	1	no	—
276	C₂×M₁₆	—	2	1	no	—
277	C₂×M₁₆	—	4	1	no	—
278	C₂²×C₈	—	4	1	no	—
279	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
281	C₄⋊Q₈	—	2	1	no	—
282	#127	—	2	1	no	—
283	#124	—	2	2	no	—
284	#121	—	4	2	no	—
285	C₄×D₄	—	4	2	no	—
286	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
287	Q₃₂	—	2	4	no	—
288	SD₃₂	—	2	4	no	—
289	SD₃₂	—	2	4	no	—
290	D₁₆	—	2	4	no	—
291	M₃₂	—	2	2	no	—
292	M₃₂	C₂³×C₂₃	1	4	yes	—
293	C₂×C₁₆	—	2	2	no	—
294	#108	—	2	1	no	—
295	#108	C₂³×C₂₃	1	2	yes	—
296	#104	—	2	2	no	—
297	#104	—	2	2	no	—
298	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
299	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
300	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
301	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
302	C₂₃	#33594	1	1	yes	characteristic, direct product factor, Sylow
303	#41	—	2	2	no	—
304	#41	—	2	2	no	—
305	#41	—	4	1	no	—
306	#41	—	4	1	no	—
307	#41	—	4	2	no	—
308	#41	—	4	2	no	—
309	#41	—	4	2	no	—
310	#41	—	4	2	no	—
311	C₂×Q₈	—	2	1	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×D₄	—	2	1	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	4	2	no	—
316	C₂×D₄	—	4	1	no	—
317	C₂×D₄	—	4	2	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	4	1	no	—
320	C₂²×C₄	—	4	2	no	—
321	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
322	Q₁₆	—	2	1	no	—
323	Q₁₆	—	2	2	no	—
324	Q₁₆	—	2	2	no	—
325	Q₁₆	—	4	1	no	—
326	Q₁₆	—	4	1	no	—
327	Q₁₆	C₂×D₄×C₂₃	1	2	yes	—
328	SD₁₆	—	2	2	no	—
329	SD₁₆	—	2	2	no	—
330	SD₁₆	—	4	2	no	—
331	SD₁₆	—	4	2	no	—
332	SD₁₆	—	4	2	no	—
333	SD₁₆	—	4	2	no	—
334	D₈	—	2	1	no	—
335	D₈	—	2	2	no	—
336	D₈	—	2	2	no	—
337	D₈	—	4	1	no	—
338	D₈	—	4	1	no	—
339	D₈	C₂×D₄×C₂₃	1	2	yes	—
340	M₁₆	—	2	2	no	—
341	M₁₆	—	4	1	no	—
342	M₁₆	—	4	2	no	—
343	M₁₆	—	4	2	no	—
344	C₂×C₈	—	4	1	no	—
345	C₂×C₈	—	4	2	no	—
346	C₂×C₈	—	4	2	no	—
347	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
348	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
349	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
350	C₂×C₈	D₈×C₂₃	1	1	yes	characteristic
351	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
352	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
353	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
354	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
355	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
356	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
357	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
358	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
359	C₄⋊C₄	—	4	2	no	—
360	C₂²⋊C₄	—	4	4	no	—
361	C₄²	—	2	1	no	—
362	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
363	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
364	C₁₆	—	2	4	no	—
365	C₂³	—	4	1	no	—
366	C₂³	—	4	2	no	—
367	C₂³	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
368	Q₈	—	2	2	no	—
369	Q₈	—	4	1	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	4	2	no	—
372	D₄	—	2	2	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	2	no	—
375	D₄	—	4	2	no	—
376	D₄	—	4	1	no	—
377	D₄	—	4	2	no	—
378	D₄	—	8	4	no	—
379	D₄	—	8	2	no	—
380	D₄	—	8	2	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	8	2	no	—
389	C₂×C₄	—	8	2	no	—
390	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
391	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
392	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
393	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
394	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
395	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
396	C₈	—	4	2	no	—
397	C₈	C₂₃×#121	1	1	yes	characteristic
398	C₈	C₂₃×#121	1	1	yes	characteristic
399	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
400	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
401	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
402	C₈	C₂×D₈×C₂₃	1	1	yes	characteristic
403	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
404	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
405	C₂²	—	4	1	no	—
406	C₂²	—	4	2	no	—
407	C₂²	—	8	2	no	—
408	C₂²	—	8	4	no	—
409	C₂²	—	8	2	no	—
410	C₂²	C₂₃×#459	1	1	yes	characteristic
411	C₂²	C₂₃×#459	1	1	yes	characteristic
412	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
413	C₄	—	2	2	no	—
414	C₄	—	4	1	no	—
415	C₄	—	4	2	no	—
416	C₄	—	8	2	no	—
417	C₄	C₂₃×#459	1	1	yes	characteristic
418	C₄	C₂₃×#459	1	1	yes	characteristic
419	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
420	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
421	C₂	—	2	1	no	—
422	C₂	—	2	1	no	—
423	C₂	—	2	1	no	—
424	C₂	—	8	2	no	—
425	C₂	—	16	2	no	—
426	C₂	C₂×C₂₃×#459	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#459
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂×C₂₃	C₂×#459
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3414	C₂	semidirect
2. C₂₃×#3147	C₂	semidirect
3. C₂×C₂₃×#443	C₂	semidirect
4. C₂₃×#2214	C₂	semidirect
5. C₂₃×#2213	C₂	semidirect
6. C₂₃×#2212	C₂	semidirect
7. C₂×C₂₃×#364	C₂	semidirect
8. C₂₃×#2148	C₂	semidirect
21. C₂₃×#495	C₂²	semidirect
22. C₂₃×#494	C₂²	semidirect
23. C₂₃×#493	C₂²	semidirect
32. C₂₃×#364	C₂²	semidirect
33. C₂₃×#364	C₂²	semidirect
34. C₂₃×#362	C₂²	semidirect
35. C₂₃×#361	C₂²	semidirect
48. C₂₃×#135	D₄	semidirect
51. C₂×Q₁₆×C₂₃	D₄	semidirect
57. C₂×D₈×C₂₃	D₄	semidirect
151. #33594	C₂₃	direct (N × H)
192. #3414	C₂×C₂₃	semidirect
193. #3147	C₂×C₂₃	semidirect
194. C₂×#443	C₂×C₂₃	semidirect
195. #2214	C₂×C₂₃	semidirect
196. #2213	C₂×C₂₃	semidirect
197. #2212	C₂×C₂₃	semidirect
198. C₂×#364	C₂×C₂₃	semidirect
199. #2148	C₂×C₂₃	semidirect
228. #495	C₂²×C₂₃	semidirect
229. #494	C₂²×C₂₃	semidirect
230. #493	C₂²×C₂₃	semidirect
239. #364	C₂²×C₂₃	semidirect
240. #364	C₂²×C₂₃	semidirect
241. #362	C₂²×C₂₃	semidirect
242. #361	C₂²×C₂₃	semidirect
261. #135	D₄×C₂₃	semidirect
264. C₂×Q₁₆	D₄×C₂₃	semidirect
270. C₂×D₈	D₄×C₂₃	semidirect
302. C₂₃	#33594	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×#3414	2
C₂₃×#3147	2
C₂×C₂₃×#443	2
C₂₃×#2214	2
C₂₃×#2213	2
C₂₃×#2212	2
C₂×C₂₃×#364	2
C₂₃×#2148	2
#33594	23

Subgroups — C23×#33594 #19181220

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33594 #19181220