Subgroups — C₂₃×#19125 #19166751

Order 5888.

726 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3035	C₂	1	2	yes	maximal, complemented
3	C₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1682	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1681	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1674	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1673	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1668	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1667	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1540	C₂	1	2	yes	maximal, complemented
12	C₂×C₂₃×#135	—	2	2	no	—
13	C₂₃×#532	C₂²	1	2	yes	—
14	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#500	C₂²	1	1	yes	characteristic
16	C₂₃×#493	C₂²	1	1	yes	characteristic
17	C₂₃×#486	C₂²	1	2	yes	—
18	C₂₃×#466	—	2	1	no	—
19	C₂₃×#466	—	2	1	no	—
20	C₂₃×#464	—	2	2	no	—
21	C₂₃×#463	—	2	2	no	—
22	C₂₃×#462	C₂²	1	2	yes	—
23	C₂₃×#461	C₂²	1	2	yes	—
24	C₂₃×#460	C₂²	1	2	yes	—
25	C₂₃×#459	C₂²	1	2	yes	—
26	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×D₈×C₂₃	—	2	1	no	—
29	C₄×D₈×C₂₃	—	2	1	no	—
30	C₈×D₄×C₂₃	—	2	1	no	—
31	C₈×D₄×C₂₃	—	2	1	no	—
32	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
33	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₂₃×#431	—	2	1	no	—
35	C₂₃×#431	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
37	C₂₃×#416	—	2	2	no	—
38	C₂×C₄×C₈×C₂₃	—	2	1	no	—
39	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
40	C₂₃×#335	C₂²	1	1	yes	characteristic
41	C₂₃×#335	C₂²	1	1	yes	characteristic
42	C₂₃×#335	C₂²	1	2	yes	—
43	C₂₃×#332	C₂²	1	2	yes	—
44	C₂₃×#332	C₂²	1	2	yes	—
45	C₂₃×#331	C₂²	1	2	yes	—
46	C₂₃×#331	C₂²	1	2	yes	—
47	C₂×#41×C₂₃	—	2	2	no	—
48	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
49	C₂₃×#135	—	4	8	no	—
50	C₂×Q₁₆×C₂₃	—	2	2	no	—
51	C₂×SD₁₆×C₂₃	—	2	2	no	—
52	C₂×SD₁₆×C₂₃	—	2	2	no	—
53	C₂×D₈×C₂₃	—	2	2	no	—
54	C₂×D₈×C₂₃	—	2	1	no	—
55	C₂×D₈×C₂₃	—	2	1	no	—
56	C₂²×C₈×C₂₃	—	2	2	no	—
57	C₂²×C₈×C₂₃	—	2	1	no	—
58	C₂²×C₈×C₂₃	—	2	1	no	—
59	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
61	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
62	C₂₃×#127	C₂³	1	2	yes	—
63	C₂₃×#124	—	2	2	no	—
64	C₂₃×#121	—	2	2	no	—
65	C₂₃×#121	—	2	2	no	—
66	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
67	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₂₃	—	2	1	no	—
69	C₄×D₄×C₂₃	—	2	1	no	—
70	C₄×D₄×C₂₃	—	2	2	no	—
71	C₄×D₄×C₂₃	—	2	2	no	—
72	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
73	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂₃×#117	—	2	1	no	—
75	C₂₃×#117	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₂₃	—	2	1	no	—
77	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
78	C₂₃×#107	—	2	1	no	—
79	C₂₃×#107	—	2	1	no	—
80	C₂₃×C₄⋊C₈	—	2	1	no	—
81	C₂₃×C₄⋊C₈	—	2	1	no	—
82	C₂₃×C₄⋊C₈	—	2	2	no	—
83	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
84	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
86	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
87	C₂₃×#103	—	2	2	no	—
88	C₂₃×#103	—	2	2	no	—
89	C₂₃×#102	—	2	2	no	—
90	C₂₃×#102	—	2	2	no	—
91	C₂₃×#102	—	2	2	no	—
92	C₂₃×#102	—	2	2	no	—
93	C₂₃×C₂²⋊C₈	—	2	1	no	—
94	C₂₃×C₂²⋊C₈	—	2	1	no	—
95	C₂₃×C₂²⋊C₈	—	2	1	no	—
96	C₄×C₈×C₂₃	—	2	1	no	—
97	C₄×C₈×C₂₃	—	2	1	no	—
98	C₄×C₈×C₂₃	—	2	1	no	—
99	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
100	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
101	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
102	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
103	#41×C₂₃	—	2	2	no	—
104	#41×C₂₃	—	4	4	no	—
105	#41×C₂₃	—	4	4	no	—
106	#41×C₂₃	D₈	1	4	yes	complemented
107	C₂×Q₈×C₂₃	—	2	2	no	—
108	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂×D₄×C₂₃	—	2	1	no	—
110	C₂×D₄×C₂₃	—	2	2	no	—
111	C₂×D₄×C₂₃	—	2	2	no	—
112	C₂×D₄×C₂₃	—	2	2	no	—
113	C₂×D₄×C₂₃	—	4	2	no	—
114	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₂₃	—	2	1	no	—
116	C₂²×C₄×C₂₃	—	2	1	no	—
117	C₂²×C₄×C₂₃	—	2	1	no	—
118	C₂²×C₄×C₂₃	—	4	2	no	—
119	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
122	Q₁₆×C₂₃	—	4	4	no	—
123	SD₁₆×C₂₃	—	4	4	no	—
124	SD₁₆×C₂₃	—	4	4	no	—
125	D₈×C₂₃	—	4	2	no	—
126	D₈×C₂₃	—	4	4	no	—
127	D₈×C₂₃	—	4	2	no	—
128	C₂×C₈×C₂₃	—	2	2	no	—
129	C₂×C₈×C₂₃	—	2	2	no	—
130	C₂×C₈×C₂₃	—	2	2	no	—
131	C₂×C₈×C₂₃	—	2	1	no	—
132	C₂×C₈×C₂₃	—	2	1	no	—
133	C₂×C₈×C₂₃	—	4	1	no	—
134	C₂×C₈×C₂₃	—	4	2	no	—
135	C₂×C₈×C₂₃	—	4	4	no	—
136	C₂×C₈×C₂₃	—	4	2	no	—
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	C₄⋊C₄×C₂₃	—	2	1	no	—
142	C₄⋊C₄×C₂₃	—	2	1	no	—
143	C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₄⋊C₄×C₂₃	—	2	2	no	—
145	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
146	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	—	4	2	no	—
151	C₂²⋊C₄×C₂₃	—	4	2	no	—
152	C₄²×C₂₃	—	2	1	no	—
153	C₄²×C₂₃	—	2	1	no	—
154	C₄²×C₂₃	—	2	1	no	—
155	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₂₃	#41	1	1	yes	characteristic
158	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
159	#19125	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₂₃	—	2	1	no	—
161	C₂³×C₂₃	—	4	2	no	—
162	C₂³×C₂₃	#121	1	1	yes	characteristic
163	Q₈×C₂₃	—	2	1	no	—
164	Q₈×C₂₃	—	4	2	no	—
165	Q₈×C₂₃	—	4	2	no	—
166	Q₈×C₂₃	C₂×D₈	1	2	yes	—
167	D₄×C₂₃	—	2	4	no	—
168	D₄×C₂₃	—	2	1	no	—
169	D₄×C₂₃	—	4	2	no	—
170	D₄×C₂₃	—	4	2	no	—
171	D₄×C₂₃	—	4	2	no	—
172	D₄×C₂₃	—	4	2	no	—
173	D₄×C₂₃	—	8	2	no	—
174	D₄×C₂₃	—	8	4	no	—
175	D₄×C₂₃	C₂×D₈	1	2	yes	—
176	C₂×C₄×C₂₃	—	2	1	no	—
177	C₂×C₄×C₂₃	—	2	1	no	—
178	C₂×C₄×C₂₃	—	2	1	no	—
179	C₂×C₄×C₂₃	—	2	4	no	—
180	C₂×C₄×C₂₃	—	2	1	no	—
181	C₂×C₄×C₂₃	—	2	1	no	—
182	C₂×C₄×C₂₃	—	4	1	no	—
183	C₂×C₄×C₂₃	—	4	2	no	—
184	C₂×C₄×C₂₃	—	4	1	no	—
185	C₂×C₄×C₂₃	—	4	1	no	—
186	C₂×C₄×C₂₃	—	4	2	no	—
187	C₂×C₄×C₂₃	—	8	4	no	—
188	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
189	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
190	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
191	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
192	C₂×C₄×C₂₃	#135	1	2	yes	—
193	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
196	C₈×C₂₃	—	2	1	no	—
197	C₈×C₂₃	—	2	1	no	—
198	C₈×C₂₃	—	2	1	no	—
199	C₈×C₂₃	—	4	2	no	—
200	C₈×C₂₃	—	4	2	no	—
201	C₈×C₂₃	—	4	1	no	—
202	C₈×C₂₃	—	4	1	no	—
203	C₈×C₂₃	#135	1	2	yes	complemented
204	#3169	C₂×C₂₃	1	1	yes	characteristic, complemented
205	#3035	C₂×C₂₃	1	2	yes	complemented
206	C₈×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
207	#1682	C₂×C₂₃	1	2	yes	complemented
208	#1681	C₂×C₂₃	1	2	yes	complemented
209	#1674	C₂×C₂₃	1	1	yes	characteristic, complemented
210	#1673	C₂×C₂₃	1	1	yes	characteristic, complemented
211	#1668	C₂×C₂₃	1	1	yes	characteristic, complemented
212	#1667	C₂×C₂₃	1	1	yes	characteristic, complemented
213	C₂×#335	C₂×C₂₃	1	1	yes	characteristic, complemented
214	#1540	C₂×C₂₃	1	2	yes	complemented
215	C₂²×C₂₃	—	2	1	no	—
216	C₂²×C₂₃	—	2	1	no	—
217	C₂²×C₂₃	—	2	2	no	—
218	C₂²×C₂₃	—	4	1	no	—
219	C₂²×C₂₃	—	4	1	no	—
220	C₂²×C₂₃	—	8	2	no	—
221	C₂²×C₂₃	—	8	2	no	—
222	C₂²×C₂₃	—	8	2	no	—
223	C₂²×C₂₃	#466	1	2	yes	—
224	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
225	C₄×C₂₃	—	2	1	no	—
226	C₄×C₂₃	—	2	2	no	—
227	C₄×C₂₃	—	2	1	no	—
228	C₄×C₂₃	—	2	1	no	—
229	C₄×C₂₃	—	4	1	no	—
230	C₄×C₂₃	—	4	1	no	—
231	C₄×C₂₃	—	8	2	no	—
232	C₄×C₂₃	#466	1	2	yes	—
233	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
234	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
235	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
236	C₄×C₂₃	#575	1	1	yes	characteristic
237	C₂×#135	—	2	2	no	—
238	#532	C₂²×C₂₃	1	2	yes	—
239	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
240	#500	C₂²×C₂₃	1	1	yes	characteristic
241	#493	C₂²×C₂₃	1	1	yes	characteristic
242	#486	C₂²×C₂₃	1	2	yes	—
243	#466	—	2	1	no	—
244	#466	—	2	1	no	—
245	#464	—	2	2	no	—
246	#463	—	2	2	no	—
247	#462	C₂²×C₂₃	1	2	yes	—
248	#461	C₂²×C₂₃	1	2	yes	—
249	#460	C₂²×C₂₃	1	2	yes	—
250	#459	C₂²×C₂₃	1	2	yes	—
251	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
252	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
253	C₄×D₈	—	2	1	no	—
254	C₄×D₈	—	2	1	no	—
255	C₈×D₄	—	2	1	no	—
256	C₈×D₄	—	2	1	no	—
257	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
258	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
259	#431	—	2	1	no	—
260	#431	C₂²×C₂₃	1	1	yes	characteristic
261	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
262	#416	—	2	2	no	—
263	C₂×C₄×C₈	—	2	1	no	—
264	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
265	#335	C₂²×C₂₃	1	1	yes	characteristic
266	#335	C₂²×C₂₃	1	1	yes	characteristic
267	#335	C₂²×C₂₃	1	2	yes	—
268	#332	C₂²×C₂₃	1	2	yes	—
269	#332	C₂²×C₂₃	1	2	yes	—
270	#331	C₂²×C₂₃	1	2	yes	—
271	#331	C₂²×C₂₃	1	2	yes	—
272	C₂×C₂₃	—	2	2	no	—
273	C₂×C₂₃	—	4	2	no	—
274	C₂×C₂₃	—	16	2	no	—
275	C₂×C₂₃	#3035	1	1	yes	characteristic
276	C₂×C₂₃	C₂×#466	1	1	yes	characteristic
277	C₂×C₂₃	#3067	1	1	yes	characteristic
278	C₂×#41	—	2	2	no	—
279	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
280	#135	—	4	8	no	—
281	C₂×Q₁₆	—	2	2	no	—
282	C₂×SD₁₆	—	2	2	no	—
283	C₂×SD₁₆	—	2	2	no	—
284	C₂×D₈	—	2	2	no	—
285	C₂×D₈	—	2	1	no	—
286	C₂×D₈	—	2	1	no	—
287	C₂²×C₈	—	2	2	no	—
288	C₂²×C₈	—	2	1	no	—
289	C₂²×C₈	—	2	1	no	—
290	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
291	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
292	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
293	#127	C₂³×C₂₃	1	2	yes	—
294	#124	—	2	2	no	—
295	#121	—	2	2	no	—
296	#121	—	2	2	no	—
297	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
298	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
299	C₄×D₄	—	2	1	no	—
300	C₄×D₄	—	2	1	no	—
301	C₄×D₄	—	2	2	no	—
302	C₄×D₄	—	2	2	no	—
303	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
304	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
305	#117	—	2	1	no	—
306	#117	D₄×C₂₃	1	1	yes	characteristic
307	C₂×C₄²	—	2	1	no	—
308	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
309	#107	—	2	1	no	—
310	#107	—	2	1	no	—
311	C₄⋊C₈	—	2	1	no	—
312	C₄⋊C₈	—	2	1	no	—
313	C₄⋊C₈	—	2	2	no	—
314	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
315	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
316	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
317	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
318	#103	—	2	2	no	—
319	#103	—	2	2	no	—
320	#102	—	2	2	no	—
321	#102	—	2	2	no	—
322	#102	—	2	2	no	—
323	#102	—	2	2	no	—
324	C₂²⋊C₈	—	2	1	no	—
325	C₂²⋊C₈	—	2	1	no	—
326	C₂²⋊C₈	—	2	1	no	—
327	C₄×C₈	—	2	1	no	—
328	C₄×C₈	—	2	1	no	—
329	C₄×C₈	—	2	1	no	—
330	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
331	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
332	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
333	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
334	C₂₃	#19125	1	1	yes	characteristic, direct product factor, Sylow
335	#41	—	2	2	no	—
336	#41	—	4	4	no	—
337	#41	—	4	4	no	—
338	#41	D₈×C₂₃	1	4	yes	complemented
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
341	C₂×D₄	—	2	1	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	4	2	no	—
346	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	4	2	no	—
351	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
353	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
354	Q₁₆	—	4	4	no	—
355	SD₁₆	—	4	4	no	—
356	SD₁₆	—	4	4	no	—
357	D₈	—	4	2	no	—
358	D₈	—	4	4	no	—
359	D₈	—	4	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	1	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	4	1	no	—
366	C₂×C₈	—	4	2	no	—
367	C₂×C₈	—	4	4	no	—
368	C₂×C₈	—	4	2	no	—
369	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
370	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
371	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
372	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
378	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
379	C₂²⋊C₄	—	2	1	no	—
380	C₂²⋊C₄	—	2	1	no	—
381	C₂²⋊C₄	—	2	1	no	—
382	C₂²⋊C₄	—	4	2	no	—
383	C₂²⋊C₄	—	4	2	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	1	no	—
387	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
388	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
389	C₄²	#41×C₂₃	1	1	yes	characteristic
390	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
391	C₂³	—	2	1	no	—
392	C₂³	—	4	2	no	—
393	C₂³	C₂₃×#121	1	1	yes	characteristic
394	Q₈	—	2	1	no	—
395	Q₈	—	4	2	no	—
396	Q₈	—	4	2	no	—
397	Q₈	C₂×D₈×C₂₃	1	2	yes	—
398	D₄	—	2	4	no	—
399	D₄	—	2	1	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	2	no	—
402	D₄	—	4	2	no	—
403	D₄	—	4	2	no	—
404	D₄	—	8	2	no	—
405	D₄	—	8	4	no	—
406	D₄	C₂×D₈×C₂₃	1	2	yes	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	8	4	no	—
419	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
420	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
421	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
422	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
423	C₂×C₄	C₂₃×#135	1	2	yes	—
424	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
425	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
426	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
427	C₈	—	2	1	no	—
428	C₈	—	2	1	no	—
429	C₈	—	2	1	no	—
430	C₈	—	4	2	no	—
431	C₈	—	4	2	no	—
432	C₈	—	4	1	no	—
433	C₈	—	4	1	no	—
434	C₈	C₂₃×#135	1	2	yes	complemented
435	C₂²	—	2	1	no	—
436	C₂²	—	2	1	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	4	1	no	—
439	C₂²	—	4	1	no	—
440	C₂²	—	8	2	no	—
441	C₂²	—	8	2	no	—
442	C₂²	—	8	2	no	—
443	C₂²	C₂₃×#466	1	2	yes	—
444	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
445	C₄	—	2	1	no	—
446	C₄	—	2	2	no	—
447	C₄	—	2	1	no	—
448	C₄	—	2	1	no	—
449	C₄	—	4	1	no	—
450	C₄	—	4	1	no	—
451	C₄	—	8	2	no	—
452	C₄	C₂₃×#466	1	2	yes	—
453	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
454	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
455	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
456	C₄	C₂₃×#575	1	1	yes	characteristic
457	C₂	—	2	2	no	—
458	C₂	—	4	2	no	—
459	C₂	—	16	2	no	—
460	C₂	C₂₃×#3035	1	1	yes	characteristic
461	C₂	C₂×C₂₃×#466	1	1	yes	characteristic
462	C₂	C₂₃×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3169	C₂	semidirect
2. C₂₃×#3035	C₂	semidirect
3. C₈×#41×C₂₃	C₂	semidirect
4. C₂₃×#1682	C₂	semidirect
5. C₂₃×#1681	C₂	semidirect
6. C₂₃×#1674	C₂	semidirect
7. C₂₃×#1673	C₂	semidirect
8. C₂₃×#1668	C₂	semidirect
9. C₂₃×#1667	C₂	semidirect
10. C₂×C₂₃×#335	C₂	semidirect
11. C₂₃×#1540	C₂	semidirect
106. #41×C₂₃	D₈	semidirect
159. #19125	C₂₃	direct (N × H)
203. C₈×C₂₃	#135	semidirect
204. #3169	C₂×C₂₃	semidirect
205. #3035	C₂×C₂₃	semidirect
206. C₈×#41	C₂×C₂₃	semidirect
207. #1682	C₂×C₂₃	semidirect
208. #1681	C₂×C₂₃	semidirect
209. #1674	C₂×C₂₃	semidirect
210. #1673	C₂×C₂₃	semidirect
211. #1668	C₂×C₂₃	semidirect
212. #1667	C₂×C₂₃	semidirect
213. C₂×#335	C₂×C₂₃	semidirect
214. #1540	C₂×C₂₃	semidirect
334. C₂₃	#19125	direct (N × H)
338. #41	D₈×C₂₃	semidirect
434. C₈	C₂₃×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3169	2
C₂₃×#3035	2
C₈×#41×C₂₃	2
C₂₃×#1682	2
C₂₃×#1681	2
C₂₃×#1674	2
C₂₃×#1673	2
C₂₃×#1668	2
C₂₃×#1667	2
C₂×C₂₃×#335	2
C₂₃×#1540	2
#19125	23

Subgroups — C23×#19125 #19166751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19125 #19166751