Subgroups — C₂₃×#18778 #19166404

Order 5888.

932 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3227	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3103	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3037	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1653	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1626	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#137	—	2	1	no	—
11	C₂₃×#557	—	2	2	no	—
12	C₂₃×#557	—	2	2	no	—
13	C₂₃×#557	—	2	2	no	—
14	C₂₃×#557	C₂²	1	4	yes	—
15	C₂₃×#532	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#125	—	2	2	no	—
19	C₂×C₂₃×#125	—	2	1	no	—
20	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
21	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
22	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#498	C₂²	1	2	yes	—
25	C₂₃×#486	C₂²	1	2	yes	—
26	C₂₃×#481	—	2	2	no	—
27	C₂₃×#481	—	2	1	no	—
28	C₂₃×#476	—	2	2	no	—
29	C₂₃×#475	C₂²	1	2	yes	—
30	C₂₃×#464	—	2	2	no	—
31	C₂₃×#462	C₂²	1	2	yes	—
32	C₂₃×#460	C₂²	1	2	yes	—
33	C₂₃×#460	C₂²	1	2	yes	—
34	C₂₃×#448	—	2	1	no	—
35	C₄×SD₁₆×C₂₃	—	2	2	no	—
36	C₂₃×#431	C₂²	1	1	yes	characteristic
37	C₂₃×#431	C₂²	1	1	yes	characteristic
38	C₂₃×#423	C₂²	1	1	yes	characteristic
39	C₂₃×#417	—	2	1	no	—
40	C₂₃×#332	C₂²	1	4	yes	—
41	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	—
42	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	—
43	C₂×#41×C₂₃	—	2	1	no	—
44	C₂²×Q₈×C₂₃	—	2	1	no	—
45	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
47	C₂₃×#137	—	4	4	no	—
48	C₂×Q₁₆×C₂₃	—	2	2	no	—
49	C₂×SD₁₆×C₂₃	—	2	2	no	—
50	C₂×SD₁₆×C₂₃	—	2	2	no	—
51	C₂×M₁₆×C₂₃	—	2	1	no	—
52	C₂₃×C₄⋊Q₈	—	2	4	no	—
53	C₂₃×C₄⋊Q₈	—	2	4	no	—
54	C₂₃×C₄⋊Q₈	—	2	1	no	—
55	C₂₃×C₄⋊Q₈	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
57	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
58	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
59	C₂₃×#127	C₂³	1	1	yes	characteristic
60	C₂₃×#125	—	2	8	no	—
61	C₂₃×#125	—	2	4	no	—
62	C₂₃×#125	—	2	8	no	—
63	C₂₃×#125	—	2	4	no	—
64	C₂₃×#124	—	2	1	no	—
65	C₂₃×#121	—	2	1	no	—
66	C₂₃×#121	—	2	1	no	—
67	C₄×Q₈×C₂₃	—	2	4	no	—
68	C₄×Q₈×C₂₃	—	2	4	no	—
69	C₄×Q₈×C₂₃	—	2	4	no	—
70	C₄×Q₈×C₂₃	—	2	4	no	—
71	C₄×Q₈×C₂₃	—	2	1	no	—
72	C₄×Q₈×C₂₃	—	2	1	no	—
73	C₄×Q₈×C₂₃	—	2	1	no	—
74	C₄×Q₈×C₂₃	—	2	1	no	—
75	C₄×Q₈×C₂₃	D₄	1	2	yes	—
76	C₄×Q₈×C₂₃	D₄	1	2	yes	—
77	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
78	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
79	C₄×D₄×C₂₃	—	2	1	no	—
80	C₄×D₄×C₂₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
90	C₂×C₄²×C₂₃	—	2	2	no	—
91	C₂×C₄²×C₂₃	—	2	1	no	—
92	C₂×C₄²×C₂₃	—	2	1	no	—
93	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
94	C₂₃×#106	—	2	2	no	—
95	C₂₃×#106	—	2	4	no	—
96	C₂₃×C₄⋊C₈	—	2	1	no	—
97	C₂₃×C₄⋊C₈	—	2	1	no	—
98	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
99	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
100	C₂₃×#103	—	2	2	no	—
101	C₂₃×#103	—	2	2	no	—
102	C₂₃×#102	—	2	2	no	—
103	C₂₃×#102	—	2	2	no	—
104	C₂₃×#102	—	2	4	no	—
105	C₂₃×C₂²⋊C₈	—	2	1	no	—
106	C₂₃×C₂²⋊C₈	—	2	1	no	—
107	C₄×C₈×C₂₃	—	2	1	no	—
108	C₄×C₈×C₂₃	C₂³	1	2	yes	—
109	#41×C₂₃	—	4	2	no	—
110	#41×C₂₃	—	4	2	no	—
111	C₂×Q₈×C₂₃	—	2	1	no	—
112	C₂×Q₈×C₂₃	—	2	1	no	—
113	C₂×Q₈×C₂₃	—	2	2	no	—
114	C₂×Q₈×C₂₃	—	2	2	no	—
115	C₂×Q₈×C₂₃	—	2	2	no	—
116	C₂×Q₈×C₂₃	—	2	1	no	—
117	C₂×Q₈×C₂₃	—	4	2	no	—
118	C₂×Q₈×C₂₃	—	4	4	no	—
119	C₂×Q₈×C₂₃	SD₁₆	1	4	yes	complemented
120	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
122	C₂×D₄×C₂₃	—	2	1	no	—
123	C₂×D₄×C₂₃	—	2	1	no	—
124	C₂×D₄×C₂₃	—	2	1	no	—
125	C₂×D₄×C₂₃	—	4	1	no	—
126	C₂²×C₄×C₂₃	—	2	1	no	—
127	C₂²×C₄×C₂₃	—	2	1	no	—
128	C₂²×C₄×C₂₃	—	2	1	no	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	—	2	2	no	—
131	C₂²×C₄×C₂₃	—	4	1	no	—
132	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
135	Q₁₆×C₂₃	—	4	4	no	—
136	SD₁₆×C₂₃	—	4	4	no	—
137	SD₁₆×C₂₃	—	4	4	no	—
138	M₁₆×C₂₃	—	4	2	no	—
139	C₂×C₈×C₂₃	—	2	2	no	—
140	C₂×C₈×C₂₃	—	2	2	no	—
141	C₂×C₈×C₂₃	—	2	2	no	—
142	C₄⋊C₄×C₂₃	—	2	8	no	—
143	C₄⋊C₄×C₂₃	—	2	8	no	—
144	C₄⋊C₄×C₂₃	—	2	8	no	—
145	C₄⋊C₄×C₂₃	—	2	2	no	—
146	C₄⋊C₄×C₂₃	—	2	2	no	—
147	C₄⋊C₄×C₂₃	—	2	2	no	—
148	C₄⋊C₄×C₂₃	—	2	1	no	—
149	C₄⋊C₄×C₂₃	—	2	1	no	—
150	C₄⋊C₄×C₂₃	—	2	2	no	—
151	C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₄⋊C₄×C₂₃	—	2	2	no	—
153	C₄⋊C₄×C₂₃	—	2	2	no	—
154	C₄⋊C₄×C₂₃	—	2	1	no	—
155	C₄⋊C₄×C₂₃	—	2	1	no	—
156	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
157	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
158	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
159	C₂²⋊C₄×C₂₃	—	4	1	no	—
160	C₂²⋊C₄×C₂₃	—	4	1	no	—
161	C₄²×C₂₃	—	2	8	no	—
162	C₄²×C₂₃	—	2	2	no	—
163	C₄²×C₂₃	—	2	2	no	—
164	C₄²×C₂₃	—	2	2	no	—
165	C₄²×C₂₃	—	2	2	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	#18778	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂³×C₂₃	—	4	1	no	—
172	C₂³×C₂₃	#120	1	1	yes	characteristic
173	Q₈×C₂₃	—	2	2	no	—
174	Q₈×C₂₃	—	2	4	no	—
175	Q₈×C₂₃	—	2	2	no	—
176	Q₈×C₂₃	—	2	2	no	—
177	Q₈×C₂₃	—	4	2	no	—
178	Q₈×C₂₃	—	4	4	no	—
179	Q₈×C₂₃	—	4	4	no	—
180	Q₈×C₂₃	—	4	1	no	—
181	Q₈×C₂₃	—	4	1	no	—
182	Q₈×C₂₃	C₂×SD₁₆	1	4	yes	—
183	D₄×C₂₃	—	4	2	no	—
184	D₄×C₂₃	—	4	1	no	—
185	D₄×C₂₃	—	4	1	no	—
186	D₄×C₂₃	—	4	2	no	—
187	D₄×C₂₃	—	8	2	no	—
188	C₂×C₄×C₂₃	—	2	2	no	—
189	C₂×C₄×C₂₃	—	2	4	no	—
190	C₂×C₄×C₂₃	—	2	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	4	no	—
195	C₂×C₄×C₂₃	—	4	1	no	—
196	C₂×C₄×C₂₃	—	4	2	no	—
197	C₂×C₄×C₂₃	—	4	2	no	—
198	C₂×C₄×C₂₃	—	4	2	no	—
199	C₂×C₄×C₂₃	—	4	4	no	—
200	C₂×C₄×C₂₃	—	4	1	no	—
201	C₂×C₄×C₂₃	—	8	2	no	—
202	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
203	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
204	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
205	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
206	C₂×C₄×C₂₃	#137	1	2	yes	—
207	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
210	C₈×C₂₃	—	4	2	no	—
211	C₈×C₂₃	—	4	2	no	—
212	C₈×C₂₃	—	4	2	no	—
213	C₂×#557	C₂×C₂₃	1	1	yes	characteristic, complemented
214	#3227	C₂×C₂₃	1	1	yes	characteristic
215	#3103	C₂×C₂₃	1	1	yes	characteristic
216	#3037	C₂×C₂₃	1	1	yes	characteristic
217	#1653	C₂×C₂₃	1	4	yes	complemented
218	#1626	C₂×C₂₃	1	4	yes	complemented
219	#1549	C₂×C₂₃	1	1	yes	characteristic, complemented
220	#1487	C₂×C₂₃	1	1	yes	characteristic, complemented
221	#1479	C₂×C₂₃	1	1	yes	characteristic, complemented
222	C₂²×C₂₃	—	2	1	no	—
223	C₂²×C₂₃	—	2	1	no	—
224	C₂²×C₂₃	—	8	1	no	—
225	C₂²×C₂₃	—	8	1	no	—
226	C₂²×C₂₃	—	8	1	no	—
227	C₂²×C₂₃	#448	1	2	yes	—
228	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
229	C₄×C₂₃	—	2	4	no	—
230	C₄×C₂₃	—	2	1	no	—
231	C₄×C₂₃	—	2	1	no	—
232	C₄×C₂₃	—	2	1	no	—
233	C₄×C₂₃	—	4	2	no	—
234	C₄×C₂₃	—	4	4	no	—
235	C₄×C₂₃	—	4	2	no	—
236	C₄×C₂₃	—	4	2	no	—
237	C₄×C₂₃	—	8	1	no	—
238	C₄×C₂₃	#448	1	2	yes	—
239	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
240	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
241	C₄×C₂₃	#575	1	2	yes	—
242	C₂×#137	—	2	1	no	—
243	#557	—	2	2	no	—
244	#557	—	2	2	no	—
245	#557	—	2	2	no	—
246	#557	C₂²×C₂₃	1	4	yes	—
247	#532	C₂²×C₂₃	1	1	yes	characteristic
248	C₂×C₄⋊Q₈	—	2	1	no	—
249	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
250	C₂×#125	—	2	2	no	—
251	C₂×#125	—	2	1	no	—
252	C₂×C₄×Q₈	—	2	1	no	—
253	C₂×C₄×Q₈	—	2	1	no	—
254	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
255	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
256	#498	C₂²×C₂₃	1	2	yes	—
257	#486	C₂²×C₂₃	1	2	yes	—
258	#481	—	2	1	no	—
259	#481	—	2	2	no	—
260	#476	—	2	2	no	—
261	#475	C₂²×C₂₃	1	2	yes	—
262	#464	—	2	2	no	—
263	#462	C₂²×C₂₃	1	2	yes	—
264	#460	C₂²×C₂₃	1	2	yes	—
265	#460	C₂²×C₂₃	1	2	yes	—
266	#448	—	2	1	no	—
267	C₄×SD₁₆	—	2	2	no	—
268	#431	C₂²×C₂₃	1	1	yes	characteristic
269	#431	C₂²×C₂₃	1	1	yes	characteristic
270	#423	C₂²×C₂₃	1	1	yes	characteristic
271	#417	—	2	1	no	—
272	#332	C₂²×C₂₃	1	4	yes	—
273	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	—
274	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	—
275	C₂×C₂₃	—	2	2	no	—
276	C₂×C₂₃	—	16	1	no	—
277	C₂×C₂₃	C₂×#448	1	1	yes	characteristic
278	C₂×C₂₃	#3005	1	1	yes	characteristic
279	C₂×C₂₃	#3020	1	1	yes	characteristic
280	C₂×#41	—	2	1	no	—
281	C₂²×Q₈	—	2	1	no	—
282	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
283	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
284	#137	—	4	4	no	—
285	C₂×Q₁₆	—	2	2	no	—
286	C₂×SD₁₆	—	2	2	no	—
287	C₂×SD₁₆	—	2	2	no	—
288	C₂×M₁₆	—	2	1	no	—
289	C₄⋊Q₈	—	2	4	no	—
290	C₄⋊Q₈	—	2	4	no	—
291	C₄⋊Q₈	—	2	1	no	—
292	C₄⋊Q₈	—	2	1	no	—
293	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
294	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
295	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
296	#127	C₂³×C₂₃	1	1	yes	characteristic
297	#125	—	2	8	no	—
298	#125	—	2	4	no	—
299	#125	—	2	8	no	—
300	#125	—	2	4	no	—
301	#124	—	2	1	no	—
302	#121	—	2	1	no	—
303	#121	—	2	1	no	—
304	C₄×Q₈	—	2	4	no	—
305	C₄×Q₈	—	2	4	no	—
306	C₄×Q₈	—	2	4	no	—
307	C₄×Q₈	—	2	4	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	D₄×C₂₃	1	2	yes	—
313	C₄×Q₈	D₄×C₂₃	1	2	yes	—
314	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
315	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	2	no	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	—	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×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
327	C₂×C₄²	—	2	2	no	—
328	C₂×C₄²	—	2	1	no	—
329	C₂×C₄²	—	2	1	no	—
330	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
331	#106	—	2	2	no	—
332	#106	—	2	4	no	—
333	C₄⋊C₈	—	2	1	no	—
334	C₄⋊C₈	—	2	1	no	—
335	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
336	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
337	#103	—	2	2	no	—
338	#103	—	2	2	no	—
339	#102	—	2	2	no	—
340	#102	—	2	2	no	—
341	#102	—	2	4	no	—
342	C₂²⋊C₈	—	2	1	no	—
343	C₂²⋊C₈	—	2	1	no	—
344	C₄×C₈	—	2	1	no	—
345	C₄×C₈	C₂³×C₂₃	1	2	yes	—
346	C₂₃	#18778	1	1	yes	characteristic, direct product factor, Sylow
347	#41	—	4	2	no	—
348	#41	—	4	2	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×Q₈	—	2	2	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	2	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	—	4	2	no	—
356	C₂×Q₈	—	4	4	no	—
357	C₂×Q₈	SD₁₆×C₂₃	1	4	yes	complemented
358	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
359	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
360	C₂×D₄	—	2	1	no	—
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	—	2	1	no	—
363	C₂×D₄	—	4	1	no	—
364	C₂²×C₄	—	2	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	4	1	no	—
370	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
373	Q₁₆	—	4	4	no	—
374	SD₁₆	—	4	4	no	—
375	SD₁₆	—	4	4	no	—
376	M₁₆	—	4	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₄⋊C₄	—	2	8	no	—
381	C₄⋊C₄	—	2	8	no	—
382	C₄⋊C₄	—	2	8	no	—
383	C₄⋊C₄	—	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	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
395	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
396	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
397	C₂²⋊C₄	—	4	1	no	—
398	C₂²⋊C₄	—	4	1	no	—
399	C₄²	—	2	8	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	2	2	no	—
403	C₄²	—	2	2	no	—
404	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
405	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
406	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
407	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
408	C₂³	—	4	1	no	—
409	C₂³	C₂₃×#120	1	1	yes	characteristic
410	Q₈	—	2	2	no	—
411	Q₈	—	2	4	no	—
412	Q₈	—	2	2	no	—
413	Q₈	—	2	2	no	—
414	Q₈	—	4	2	no	—
415	Q₈	—	4	4	no	—
416	Q₈	—	4	4	no	—
417	Q₈	—	4	1	no	—
418	Q₈	—	4	1	no	—
419	Q₈	C₂×SD₁₆×C₂₃	1	4	yes	—
420	D₄	—	4	2	no	—
421	D₄	—	4	1	no	—
422	D₄	—	4	1	no	—
423	D₄	—	4	2	no	—
424	D₄	—	8	2	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	4	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	8	2	no	—
439	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
440	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
441	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
442	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
443	C₂×C₄	C₂₃×#137	1	2	yes	—
444	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
445	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
446	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
447	C₈	—	4	2	no	—
448	C₈	—	4	2	no	—
449	C₈	—	4	2	no	—
450	C₂²	—	2	1	no	—
451	C₂²	—	2	1	no	—
452	C₂²	—	8	1	no	—
453	C₂²	—	8	1	no	—
454	C₂²	—	8	1	no	—
455	C₂²	C₂₃×#448	1	2	yes	—
456	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
457	C₄	—	2	4	no	—
458	C₄	—	2	1	no	—
459	C₄	—	2	1	no	—
460	C₄	—	2	1	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	4	no	—
463	C₄	—	4	2	no	—
464	C₄	—	4	2	no	—
465	C₄	—	8	1	no	—
466	C₄	C₂₃×#448	1	2	yes	—
467	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
468	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
469	C₄	C₂₃×#575	1	2	yes	—
470	C₂	—	2	2	no	—
471	C₂	—	16	1	no	—
472	C₂	C₂×C₂₃×#448	1	1	yes	characteristic
473	C₂	C₂₃×#3005	1	1	yes	characteristic
474	C₂	C₂₃×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#532	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
1. C₂×C₂₃×#557	C₂	semidirect
5. C₂₃×#1653	C₂	semidirect
6. C₂₃×#1626	C₂	semidirect
7. C₂₃×#1549	C₂	semidirect
8. C₂₃×#1487	C₂	semidirect
9. C₂₃×#1479	C₂	semidirect
119. C₂×Q₈×C₂₃	SD₁₆	semidirect
170. #18778	C₂₃	direct (N × H)
213. C₂×#557	C₂×C₂₃	semidirect
217. #1653	C₂×C₂₃	semidirect
218. #1626	C₂×C₂₃	semidirect
219. #1549	C₂×C₂₃	semidirect
220. #1487	C₂×C₂₃	semidirect
221. #1479	C₂×C₂₃	semidirect
346. C₂₃	#18778	direct (N × H)
357. C₂×Q₈	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×#557	2
C₂₃×#3227	2
C₂₃×#3103	2
C₂₃×#3037	2
C₂₃×#1653	2
C₂₃×#1626	2
C₂₃×#1549	2
C₂₃×#1487	2
C₂₃×#1479	2
#18778	23

Subgroups — C23×#18778 #19166404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18778 #19166404