Subgroups — C₂₃×#18751 #19166377

Order 5888.

986 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#559	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3208	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3104	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3078	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1640	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1630	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1545	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#559	—	2	2	no	—
11	C₂₃×#559	—	2	2	no	—
12	C₂₃×#559	—	2	2	no	—
13	C₂₃×#559	C₂²	1	4	yes	—
14	C₂₃×#533	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#124	—	2	1	no	—
17	C₂×C₂₃×#124	—	2	1	no	—
18	C₂×C₂₃×#123	—	2	2	no	—
19	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	—	2	1	no	—
22	C₂×C₄×D₄×C₂₃	—	2	1	no	—
23	C₂₃×#492	C₂²	1	2	yes	—
24	C₂₃×#486	C₂²	1	2	yes	—
25	C₂₃×#484	—	2	1	no	—
26	C₂₃×#479	—	2	2	no	—
27	C₂₃×#477	C₂²	1	2	yes	complemented
28	C₂₃×#477	C₂²	1	2	yes	complemented
29	C₂₃×#475	C₂²	1	2	yes	complemented
30	C₂₃×#462	C₂²	1	2	yes	complemented
31	C₂₃×#460	—	2	2	no	—
32	C₂₃×#451	—	2	1	no	—
33	C₂₃×#450	—	2	2	no	—
34	C₄×Q₁₆×C₂₃	—	2	2	no	—
35	C₂₃×#431	C₂²	1	1	yes	characteristic
36	C₂₃×#431	C₂²	1	1	yes	characteristic
37	C₂₃×#428	—	2	1	no	—
38	C₂₃×#423	C₂²	1	1	yes	characteristic
39	C₂₃×#419	—	2	1	no	—
40	C₂₃×#331	C₂²	1	4	yes	—
41	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
43	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
45	C₂²×D₄×C₂₃	—	2	1	no	—
46	C₂²×D₄×C₂₃	—	2	1	no	—
47	C₂²×D₄×C₂₃	D₄	1	2	yes	—
48	C₂³×C₄×C₂₃	—	2	2	no	—
49	C₂×Q₁₆×C₂₃	—	2	2	no	—
50	C₂×SD₁₆×C₂₃	—	2	4	no	—
51	C₂×M₁₆×C₂₃	—	2	1	no	—
52	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
53	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
54	C₂₃×#127	—	2	1	no	—
55	C₂₃×#127	—	2	1	no	—
56	C₂₃×#127	D₄	1	2	yes	—
57	C₂₃×#127	C₂³	1	2	yes	—
58	C₂₃×#125	—	2	1	no	—
59	C₂₃×#124	—	2	4	no	—
60	C₂₃×#124	—	2	4	no	—
61	C₂₃×#124	—	2	4	no	—
62	C₂₃×#124	—	2	4	no	—
63	C₂₃×#123	—	2	8	no	—
64	C₂₃×#123	—	2	8	no	—
65	C₂₃×C₂²⋊Q₈	—	2	1	no	—
66	C₂₃×C₂²⋊Q₈	—	2	1	no	—
67	C₄×Q₈×C₂₃	—	2	1	no	—
68	C₄×Q₈×C₂₃	—	2	1	no	—
69	C₄×Q₈×C₂₃	—	2	1	no	—
70	C₄×Q₈×C₂₃	—	2	1	no	—
71	C₄×Q₈×C₂₃	—	2	1	no	—
72	C₄×Q₈×C₂₃	—	2	1	no	—
73	C₄×Q₈×C₂₃	D₄	1	2	yes	—
74	C₄×Q₈×C₂₃	D₄	1	2	yes	—
75	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
76	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
77	C₄×D₄×C₂₃	—	2	4	no	—
78	C₄×D₄×C₂₃	—	2	4	no	—
79	C₄×D₄×C₂₃	—	2	4	no	—
80	C₄×D₄×C₂₃	—	2	4	no	—
81	C₂₃×#117	—	2	1	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
86	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
89	C₂×C₄²×C₂₃	—	2	1	no	—
90	C₂×C₄²×C₂₃	—	2	1	no	—
91	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
92	C₂₃×#107	—	2	2	no	—
93	C₂₃×#107	—	2	2	no	—
94	C₂₃×#106	—	2	2	no	—
95	C₂₃×C₄⋊C₈	—	2	1	no	—
96	C₂₃×C₄⋊C₈	—	2	1	no	—
97	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
98	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
99	C₂₃×#103	—	2	2	no	—
100	C₂₃×#103	—	2	2	no	—
101	C₂₃×#103	—	2	2	no	—
102	C₂₃×#103	—	2	2	no	—
103	C₂₃×#102	—	2	4	no	—
104	C₂₃×C₂²⋊C₈	—	2	1	no	—
105	C₂₃×C₂²⋊C₈	—	2	1	no	—
106	C₄×C₈×C₂₃	—	2	1	no	—
107	C₄×C₈×C₂₃	C₂³	1	2	yes	—
108	C₂⁴×C₂₃	—	2	2	no	—
109	C₂×Q₈×C₂₃	—	2	1	no	—
110	C₂×Q₈×C₂₃	—	2	1	no	—
111	C₂×Q₈×C₂₃	—	2	2	no	—
112	C₂×Q₈×C₂₃	—	2	2	no	—
113	C₂×Q₈×C₂₃	—	2	1	no	—
114	C₂×Q₈×C₂₃	—	2	1	no	—
115	C₂×Q₈×C₂₃	—	4	2	no	—
116	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
117	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
118	C₂×D₄×C₂₃	—	2	2	no	—
119	C₂×D₄×C₂₃	—	2	2	no	—
120	C₂×D₄×C₂₃	—	2	2	no	—
121	C₂×D₄×C₂₃	—	2	2	no	—
122	C₂×D₄×C₂₃	—	2	2	no	—
123	C₂×D₄×C₂₃	—	4	4	no	—
124	C₂×D₄×C₂₃	—	4	2	no	—
125	C₂×D₄×C₂₃	—	4	2	no	—
126	C₂×D₄×C₂₃	—	8	2	no	—
127	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
128	C₂²×C₄×C₂₃	—	2	1	no	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	—	2	1	no	—
131	C₂²×C₄×C₂₃	—	2	1	no	—
132	C₂²×C₄×C₂₃	—	2	8	no	—
133	C₂²×C₄×C₂₃	—	8	4	no	—
134	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
137	Q₁₆×C₂₃	—	4	4	no	—
138	SD₁₆×C₂₃	—	8	4	no	—
139	M₁₆×C₂₃	—	4	2	no	—
140	C₂×C₈×C₂₃	—	2	2	no	—
141	C₂×C₈×C₂₃	—	2	2	no	—
142	C₂×C₈×C₂₃	—	2	2	no	—
143	C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₄⋊C₄×C₂₃	—	2	2	no	—
145	C₄⋊C₄×C₂₃	—	2	1	no	—
146	C₄⋊C₄×C₂₃	—	2	1	no	—
147	C₄⋊C₄×C₂₃	—	2	2	no	—
148	C₄⋊C₄×C₂₃	—	2	2	no	—
149	C₄⋊C₄×C₂₃	—	2	1	no	—
150	C₄⋊C₄×C₂₃	—	2	1	no	—
151	C₄⋊C₄×C₂₃	—	2	1	no	—
152	C₄⋊C₄×C₂₃	—	2	1	no	—
153	C₄⋊C₄×C₂₃	—	4	1	no	—
154	C₄⋊C₄×C₂₃	—	4	1	no	—
155	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
157	C₂²⋊C₄×C₂₃	—	2	8	no	—
158	C₂²⋊C₄×C₂₃	—	2	8	no	—
159	C₂²⋊C₄×C₂₃	—	2	8	no	—
160	C₂²⋊C₄×C₂₃	—	4	1	no	—
161	C₄²×C₂₃	—	2	2	no	—
162	C₄²×C₂₃	—	2	2	no	—
163	C₄²×C₂₃	—	2	2	no	—
164	C₄²×C₂₃	—	2	2	no	—
165	C₄²×C₂₃	—	4	1	no	—
166	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
170	#18751	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂³×C₂₃	—	2	4	no	—
172	C₂³×C₂₃	—	4	4	no	—
173	C₂³×C₂₃	—	4	4	no	—
174	C₂³×C₂₃	—	4	4	no	—
175	C₂³×C₂₃	#120	1	1	yes	characteristic
176	Q₈×C₂₃	—	2	2	no	—
177	Q₈×C₂₃	—	2	2	no	—
178	Q₈×C₂₃	—	2	2	no	—
179	Q₈×C₂₃	—	2	2	no	—
180	Q₈×C₂₃	—	4	2	no	—
181	Q₈×C₂₃	—	4	2	no	—
182	Q₈×C₂₃	—	4	2	no	—
183	Q₈×C₂₃	—	4	2	no	—
184	D₄×C₂₃	—	4	4	no	—
185	D₄×C₂₃	—	4	4	no	—
186	D₄×C₂₃	—	4	2	no	—
187	D₄×C₂₃	—	4	2	no	—
188	D₄×C₂₃	—	4	4	no	—
189	D₄×C₂₃	—	4	4	no	—
190	D₄×C₂₃	—	8	2	no	—
191	C₂×C₄×C₂₃	—	2	2	no	—
192	C₂×C₄×C₂₃	—	2	2	no	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	4	1	no	—
197	C₂×C₄×C₂₃	—	4	2	no	—
198	C₂×C₄×C₂₃	—	4	2	no	—
199	C₂×C₄×C₂₃	—	4	2	no	—
200	C₂×C₄×C₂₃	—	4	2	no	—
201	C₂×C₄×C₂₃	—	4	1	no	—
202	C₂×C₄×C₂₃	—	4	1	no	—
203	C₂×C₄×C₂₃	—	8	8	no	—
204	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
205	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
206	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
207	C₂×C₄×C₂₃	#135	1	2	yes	—
208	C₂×C₄×C₂₃	#137	1	2	yes	—
209	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
212	C₈×C₂₃	—	4	2	no	—
213	C₈×C₂₃	—	4	2	no	—
214	C₈×C₂₃	—	4	2	no	—
215	C₂×#559	C₂×C₂₃	1	1	yes	characteristic
216	#3208	C₂×C₂₃	1	1	yes	characteristic
217	#3104	C₂×C₂₃	1	1	yes	characteristic, complemented
218	#3078	C₂×C₂₃	1	1	yes	characteristic, complemented
219	#1640	C₂×C₂₃	1	4	yes	complemented
220	#1630	C₂×C₂₃	1	4	yes	complemented
221	#1545	C₂×C₂₃	1	1	yes	characteristic
222	#1487	C₂×C₂₃	1	1	yes	characteristic, complemented
223	#1479	C₂×C₂₃	1	1	yes	characteristic, complemented
224	C₂²×C₂₃	—	2	1	no	—
225	C₂²×C₂₃	—	2	1	no	—
226	C₂²×C₂₃	—	4	4	no	—
227	C₂²×C₂₃	—	4	4	no	—
228	C₂²×C₂₃	—	4	4	no	—
229	C₂²×C₂₃	—	8	8	no	—
230	C₂²×C₂₃	#452	1	2	yes	—
231	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
232	C₄×C₂₃	—	2	1	no	—
233	C₄×C₂₃	—	2	1	no	—
234	C₄×C₂₃	—	2	1	no	—
235	C₄×C₂₃	—	4	2	no	—
236	C₄×C₂₃	—	4	2	no	—
237	C₄×C₂₃	—	4	2	no	—
238	C₄×C₂₃	—	4	2	no	—
239	C₄×C₂₃	—	8	1	no	—
240	C₄×C₂₃	—	8	1	no	—
241	C₄×C₂₃	#452	1	2	yes	—
242	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
243	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
244	C₄×C₂₃	#575	1	2	yes	—
245	#559	—	2	2	no	—
246	#559	—	2	2	no	—
247	#559	—	2	2	no	—
248	#559	C₂²×C₂₃	1	4	yes	—
249	#533	C₂²×C₂₃	1	1	yes	characteristic
250	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
251	C₂×#124	—	2	1	no	—
252	C₂×#124	—	2	1	no	—
253	C₂×#123	—	2	2	no	—
254	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
255	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
256	C₂×C₄×D₄	—	2	1	no	—
257	C₂×C₄×D₄	—	2	1	no	—
258	#492	C₂²×C₂₃	1	2	yes	—
259	#486	C₂²×C₂₃	1	2	yes	—
260	#484	—	2	1	no	—
261	#479	—	2	2	no	—
262	#477	C₂²×C₂₃	1	2	yes	complemented
263	#477	C₂²×C₂₃	1	2	yes	complemented
264	#475	C₂²×C₂₃	1	2	yes	complemented
265	#462	C₂²×C₂₃	1	2	yes	complemented
266	#460	—	2	2	no	—
267	#451	—	2	1	no	—
268	#450	—	2	2	no	—
269	C₄×Q₁₆	—	2	2	no	—
270	#431	C₂²×C₂₃	1	1	yes	characteristic
271	#431	C₂²×C₂₃	1	1	yes	characteristic
272	#428	—	2	1	no	—
273	#423	C₂²×C₂₃	1	1	yes	characteristic
274	#419	—	2	1	no	—
275	#331	C₂²×C₂₃	1	4	yes	—
276	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	complemented
277	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	complemented
278	C₂×C₂₃	—	2	2	no	—
279	C₂×C₂₃	—	8	4	no	—
280	C₂×C₂₃	C₂×#452	1	1	yes	characteristic
281	C₂×C₂₃	#3005	1	1	yes	characteristic
282	C₂×C₂₃	#3019	1	1	yes	characteristic
283	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
284	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
285	C₂²×D₄	—	2	1	no	—
286	C₂²×D₄	—	2	1	no	—
287	C₂²×D₄	D₄×C₂₃	1	2	yes	—
288	C₂³×C₄	—	2	2	no	—
289	C₂×Q₁₆	—	2	2	no	—
290	C₂×SD₁₆	—	2	4	no	—
291	C₂×M₁₆	—	2	1	no	—
292	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
293	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
294	#127	—	2	1	no	—
295	#127	—	2	1	no	—
296	#127	D₄×C₂₃	1	2	yes	—
297	#127	C₂³×C₂₃	1	2	yes	—
298	#125	—	2	1	no	—
299	#124	—	2	4	no	—
300	#124	—	2	4	no	—
301	#124	—	2	4	no	—
302	#124	—	2	4	no	—
303	#123	—	2	8	no	—
304	#123	—	2	8	no	—
305	C₂²⋊Q₈	—	2	1	no	—
306	C₂²⋊Q₈	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×Q₈	—	2	1	no	—
309	C₄×Q₈	—	2	1	no	—
310	C₄×Q₈	—	2	1	no	—
311	C₄×Q₈	—	2	1	no	—
312	C₄×Q₈	—	2	1	no	—
313	C₄×Q₈	D₄×C₂₃	1	2	yes	—
314	C₄×Q₈	D₄×C₂₃	1	2	yes	—
315	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
316	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
317	C₄×D₄	—	2	4	no	—
318	C₄×D₄	—	2	4	no	—
319	C₄×D₄	—	2	4	no	—
320	C₄×D₄	—	2	4	no	—
321	#117	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
325	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₄²	—	2	1	no	—
330	C₂×C₄²	—	2	1	no	—
331	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
332	#107	—	2	2	no	—
333	#107	—	2	2	no	—
334	#106	—	2	2	no	—
335	C₄⋊C₈	—	2	1	no	—
336	C₄⋊C₈	—	2	1	no	—
337	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
338	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
339	#103	—	2	2	no	—
340	#103	—	2	2	no	—
341	#103	—	2	2	no	—
342	#103	—	2	2	no	—
343	#102	—	2	4	no	—
344	C₂²⋊C₈	—	2	1	no	—
345	C₂²⋊C₈	—	2	1	no	—
346	C₄×C₈	—	2	1	no	—
347	C₄×C₈	C₂³×C₂₃	1	2	yes	—
348	C₂₃	#18751	1	1	yes	characteristic, direct product factor, Sylow
349	C₂⁴	—	2	2	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×Q₈	—	2	1	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	2	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	—	2	1	no	—
356	C₂×Q₈	—	4	2	no	—
357	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
358	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	4	4	no	—
365	C₂×D₄	—	4	2	no	—
366	C₂×D₄	—	4	2	no	—
367	C₂×D₄	—	8	2	no	—
368	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	8	no	—
374	C₂²×C₄	—	8	4	no	—
375	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
378	Q₁₆	—	4	4	no	—
379	SD₁₆	—	8	4	no	—
380	M₁₆	—	4	2	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	4	1	no	—
395	C₄⋊C₄	—	4	1	no	—
396	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
397	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
398	C₂²⋊C₄	—	2	8	no	—
399	C₂²⋊C₄	—	2	8	no	—
400	C₂²⋊C₄	—	2	8	no	—
401	C₂²⋊C₄	—	4	1	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	2	2	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	2	2	no	—
406	C₄²	—	4	1	no	—
407	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
408	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
409	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
410	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
411	C₂³	—	2	4	no	—
412	C₂³	—	4	4	no	—
413	C₂³	—	4	4	no	—
414	C₂³	—	4	4	no	—
415	C₂³	C₂₃×#120	1	1	yes	characteristic
416	Q₈	—	2	2	no	—
417	Q₈	—	2	2	no	—
418	Q₈	—	2	2	no	—
419	Q₈	—	2	2	no	—
420	Q₈	—	4	2	no	—
421	Q₈	—	4	2	no	—
422	Q₈	—	4	2	no	—
423	Q₈	—	4	2	no	—
424	D₄	—	4	4	no	—
425	D₄	—	4	4	no	—
426	D₄	—	4	2	no	—
427	D₄	—	4	2	no	—
428	D₄	—	4	4	no	—
429	D₄	—	4	4	no	—
430	D₄	—	8	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	8	8	no	—
444	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
445	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
446	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
447	C₂×C₄	C₂₃×#135	1	2	yes	—
448	C₂×C₄	C₂₃×#137	1	2	yes	—
449	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
450	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
451	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
452	C₈	—	4	2	no	—
453	C₈	—	4	2	no	—
454	C₈	—	4	2	no	—
455	C₂²	—	2	1	no	—
456	C₂²	—	2	1	no	—
457	C₂²	—	4	4	no	—
458	C₂²	—	4	4	no	—
459	C₂²	—	4	4	no	—
460	C₂²	—	8	8	no	—
461	C₂²	C₂₃×#452	1	2	yes	—
462	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
463	C₄	—	2	1	no	—
464	C₄	—	2	1	no	—
465	C₄	—	2	1	no	—
466	C₄	—	4	2	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	2	no	—
469	C₄	—	4	2	no	—
470	C₄	—	8	1	no	—
471	C₄	—	8	1	no	—
472	C₄	C₂₃×#452	1	2	yes	—
473	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
474	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
475	C₄	C₂₃×#575	1	2	yes	—
476	C₂	—	2	2	no	—
477	C₂	—	8	4	no	—
478	C₂	C₂×C₂₃×#452	1	1	yes	characteristic
479	C₂	C₂₃×#3005	1	1	yes	characteristic
480	C₂	C₂₃×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
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
3. C₂₃×#3104	C₂	semidirect
4. C₂₃×#3078	C₂	semidirect
5. C₂₃×#1640	C₂	semidirect
6. C₂₃×#1630	C₂	semidirect
8. C₂₃×#1487	C₂	semidirect
9. C₂₃×#1479	C₂	semidirect
27. C₂₃×#477	C₂²	semidirect
28. C₂₃×#477	C₂²	semidirect
29. C₂₃×#475	C₂²	semidirect
30. C₂₃×#462	C₂²	semidirect
41. C₂₃×Q₈⋊C₈	C₂²	semidirect
42. C₂₃×Q₈⋊C₈	C₂²	semidirect
170. #18751	C₂₃	direct (N × H)
217. #3104	C₂×C₂₃	semidirect
218. #3078	C₂×C₂₃	semidirect
219. #1640	C₂×C₂₃	semidirect
220. #1630	C₂×C₂₃	semidirect
222. #1487	C₂×C₂₃	semidirect
223. #1479	C₂×C₂₃	semidirect
262. #477	C₂²×C₂₃	semidirect
263. #477	C₂²×C₂₃	semidirect
264. #475	C₂²×C₂₃	semidirect
265. #462	C₂²×C₂₃	semidirect
276. Q₈⋊C₈	C₂²×C₂₃	semidirect
277. Q₈⋊C₈	C₂²×C₂₃	semidirect
348. C₂₃	#18751	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#559	2
C₂₃×#3208	2
C₂₃×#3104	2
C₂₃×#3078	2
C₂₃×#1640	2
C₂₃×#1630	2
C₂₃×#1545	2
C₂₃×#1487	2
C₂₃×#1479	2
#18751	23

Subgroups — C23×#18751 #19166377

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18751 #19166377