Subgroups — C₂₃×#25498 #19173124

Order 5888.

1132 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#467	C₂	1	2	yes	maximal
3	C₂₃×#2855	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2057	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2043	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2024	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2016	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1892	C₂	1	2	yes	maximal
9	C₂₃×#1878	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1863	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×#336	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#533	—	2	8	no	—
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#125	—	2	1	no	—
18	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
22	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂₃×#117	—	2	1	no	—
24	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#467	—	2	8	no	—
26	C₂×C₂₃×#107	C₂²	1	2	yes	—
27	C₂×C₂₃×#103	C₂²	1	2	yes	—
28	C₂×C₂₃×#103	C₂²	1	2	yes	—
29	C₂×C₂₃×#103	C₂²	1	2	yes	—
30	C₂×C₂₃×#103	C₂²	1	2	yes	—
31	C₂₃×#391	C₂²	1	1	yes	characteristic
32	C₂₃×#391	C₂²	1	1	yes	characteristic
33	C₂₃×#389	C₂²	1	1	yes	characteristic
34	C₂₃×#389	C₂²	1	1	yes	characteristic
35	C₂₃×#387	—	2	1	no	—
36	C₂₃×#387	—	2	1	no	—
37	C₂₃×#385	—	2	1	no	—
38	C₂₃×#381	—	2	1	no	—
39	C₂₃×#380	C₂²	1	1	yes	characteristic
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂₃×#340	C₂²	1	2	yes	—
42	C₂₃×#336	C₂²	1	1	yes	characteristic
43	C₂₃×#336	C₂²	1	1	yes	characteristic
44	C₂₃×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₂₃	—	2	1	no	—
46	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
48	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×Q₁₆×C₂₃	—	2	4	no	—
52	C₂×Q₁₆×C₂₃	—	2	4	no	—
53	C₂×Q₁₆×C₂₃	—	4	2	no	—
54	C₂²×C₈×C₂₃	—	2	4	no	—
55	C₂²×C₈×C₂₃	—	2	4	no	—
56	C₂²×C₈×C₂₃	—	4	2	no	—
57	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
59	C₂₃×C₄⋊Q₈	—	2	4	no	—
60	C₂₃×C₄⋊Q₈	—	2	4	no	—
61	C₂₃×#125	—	2	8	no	—
62	C₂₃×C₂²⋊Q₈	—	2	8	no	—
63	C₂₃×C₂²⋊Q₈	—	2	4	no	—
64	C₂₃×C₂²⋊Q₈	—	2	4	no	—
65	C₂₃×C₂²⋊Q₈	D₄	1	8	yes	—
66	C₄×Q₈×C₂₃	—	2	8	no	—
67	C₄×Q₈×C₂₃	—	2	4	no	—
68	C₄×Q₈×C₂₃	—	2	4	no	—
69	C₂₃×#117	—	2	8	no	—
70	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
72	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
82	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄²×C₂₃	—	2	4	no	—
86	C₂×C₄²×C₂₃	—	2	2	no	—
87	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
89	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
91	C₂₃×#107	—	2	4	no	—
92	C₂₃×#103	—	2	4	no	—
93	C₂₃×#103	—	2	4	no	—
94	C₂₃×#103	—	2	4	no	—
95	C₂₃×#103	—	4	2	no	—
96	C₂₃×#95	—	2	1	no	—
97	C₂₃×#95	—	2	1	no	—
98	C₂₃×#95	—	2	1	no	—
99	C₂₃×#95	—	2	1	no	—
100	C₂₃×#95	—	2	1	no	—
101	C₂₃×#95	—	2	1	no	—
102	C₂₃×#95	—	2	1	no	—
103	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×Q₈×C₂₃	—	2	4	no	—
105	C₂×Q₈×C₂₃	—	2	2	no	—
106	C₂×Q₈×C₂₃	—	2	2	no	—
107	C₂×Q₈×C₂₃	—	2	2	no	—
108	C₂×Q₈×C₂₃	—	2	2	no	—
109	C₂×Q₈×C₂₃	—	2	2	no	—
110	C₂×Q₈×C₂₃	—	2	2	no	—
111	C₂×Q₈×C₂₃	—	2	1	no	—
112	C₂×Q₈×C₂₃	—	2	2	no	—
113	C₂×Q₈×C₂₃	—	2	1	no	—
114	C₂×Q₈×C₂₃	—	4	2	no	—
115	C₂×Q₈×C₂₃	—	4	2	no	—
116	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
117	C₂²×C₄×C₂₃	—	2	2	no	—
118	C₂²×C₄×C₂₃	—	2	2	no	—
119	C₂²×C₄×C₂₃	—	2	2	no	—
120	C₂²×C₄×C₂₃	—	2	1	no	—
121	C₂²×C₄×C₂₃	—	2	1	no	—
122	C₂²×C₄×C₂₃	—	2	2	no	—
123	C₂²×C₄×C₂₃	—	2	2	no	—
124	C₂²×C₄×C₂₃	—	2	2	no	—
125	C₂²×C₄×C₂₃	—	2	2	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	—	4	1	no	—
131	C₂²×C₄×C₂₃	Q₁₆	1	4	yes	complemented
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	4	yes	—
135	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
139	Q₁₆×C₂₃	—	4	8	no	—
140	C₂×C₈×C₂₃	—	2	2	no	—
141	C₂×C₈×C₂₃	—	2	2	no	—
142	C₂×C₈×C₂₃	—	2	2	no	—
143	C₂×C₈×C₂₃	—	2	2	no	—
144	C₂×C₈×C₂₃	—	2	2	no	—
145	C₂×C₈×C₂₃	—	4	8	no	—
146	C₂×C₈×C₂₃	#41	1	4	yes	—
147	C₄⋊C₄×C₂₃	—	2	4	no	—
148	C₄⋊C₄×C₂₃	—	2	4	no	—
149	C₄⋊C₄×C₂₃	—	2	8	no	—
150	C₄⋊C₄×C₂₃	—	2	4	no	—
151	C₄⋊C₄×C₂₃	—	2	8	no	—
152	C₄⋊C₄×C₂₃	—	2	2	no	—
153	C₄⋊C₄×C₂₃	—	2	2	no	—
154	C₄⋊C₄×C₂₃	—	2	2	no	—
155	C₄⋊C₄×C₂₃	—	2	1	no	—
156	C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₄⋊C₄×C₂₃	—	4	2	no	—
158	C₄⋊C₄×C₂₃	—	4	2	no	—
159	C₄⋊C₄×C₂₃	—	4	2	no	—
160	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
161	C₂²⋊C₄×C₂₃	—	2	8	no	—
162	C₂²⋊C₄×C₂₃	—	4	2	no	—
163	C₂²⋊C₄×C₂₃	—	4	2	no	—
164	C₄²×C₂₃	—	2	8	no	—
165	C₄²×C₂₃	—	2	2	no	—
166	C₄²×C₂₃	—	2	2	no	—
167	C₄²×C₂₃	—	2	2	no	—
168	C₄²×C₂₃	—	2	2	no	—
169	#25498	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂³×C₂₃	—	2	1	no	—
171	C₂³×C₂₃	—	2	1	no	—
172	C₂³×C₂₃	—	2	1	no	—
173	C₂³×C₂₃	—	2	1	no	—
174	C₂³×C₂₃	#120	1	2	yes	—
175	C₂³×C₂₃	C₂×Q₁₆	1	2	yes	—
176	C₂³×C₂₃	#137	1	2	yes	—
177	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
178	Q₈×C₂₃	—	2	4	no	—
179	Q₈×C₂₃	—	2	4	no	—
180	Q₈×C₂₃	—	4	8	no	—
181	Q₈×C₂₃	—	4	2	no	—
182	Q₈×C₂₃	—	4	2	no	—
183	Q₈×C₂₃	—	4	2	no	—
184	Q₈×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	—	2	1	no	—
186	C₂×C₄×C₂₃	—	2	1	no	—
187	C₂×C₄×C₂₃	—	2	1	no	—
188	C₂×C₄×C₂₃	—	2	1	no	—
189	C₂×C₄×C₂₃	—	2	1	no	—
190	C₂×C₄×C₂₃	—	2	1	no	—
191	C₂×C₄×C₂₃	—	2	1	no	—
192	C₂×C₄×C₂₃	—	2	1	no	—
193	C₂×C₄×C₂₃	—	2	8	no	—
194	C₂×C₄×C₂₃	—	2	4	no	—
195	C₂×C₄×C₂₃	—	2	4	no	—
196	C₂×C₄×C₂₃	—	2	1	no	—
197	C₂×C₄×C₂₃	—	2	4	no	—
198	C₂×C₄×C₂₃	—	2	4	no	—
199	C₂×C₄×C₂₃	—	2	1	no	—
200	C₂×C₄×C₂₃	—	2	1	no	—
201	C₂×C₄×C₂₃	—	4	4	no	—
202	C₂×C₄×C₂₃	—	4	2	no	—
203	C₂×C₄×C₂₃	—	4	2	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	4	2	no	—
206	C₂×C₄×C₂₃	—	4	1	no	—
207	C₂×C₄×C₂₃	—	4	1	no	—
208	C₂×C₄×C₂₃	—	4	1	no	—
209	C₂×C₄×C₂₃	—	4	1	no	—
210	C₂×C₄×C₂₃	—	4	1	no	—
211	C₂×C₄×C₂₃	—	4	1	no	—
212	C₂×C₄×C₂₃	—	4	1	no	—
213	C₂×C₄×C₂₃	—	4	1	no	—
214	C₂×C₄×C₂₃	—	4	1	no	—
215	C₂×C₄×C₂₃	—	4	1	no	—
216	C₂×C₄×C₂₃	—	4	1	no	—
217	C₂×C₄×C₂₃	—	4	1	no	—
218	C₂×C₄×C₂₃	#120	1	2	yes	—
219	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
220	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
221	C₂×C₄×C₂₃	#135	1	2	yes	—
222	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
224	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
226	C₈×C₂₃	—	2	4	no	—
227	C₈×C₂₃	—	4	2	no	—
228	C₂×#533	C₂×C₂₃	1	1	yes	characteristic
229	C₂×#467	C₂×C₂₃	1	2	yes	—
230	#2855	C₂×C₂₃	1	1	yes	characteristic
231	#2057	C₂×C₂₃	1	1	yes	characteristic, complemented
232	#2043	C₂×C₂₃	1	1	yes	characteristic, complemented
233	#2024	C₂×C₂₃	1	1	yes	characteristic, complemented
234	#2016	C₂×C₂₃	1	1	yes	characteristic, complemented
235	#1892	C₂×C₂₃	1	2	yes	—
236	#1878	C₂×C₂₃	1	2	yes	complemented
237	#1863	C₂×C₂₃	1	2	yes	complemented
238	C₂×#336	C₂×C₂₃	1	1	yes	characteristic
239	C₂²×C₂₃	—	2	2	no	—
240	C₂²×C₂₃	—	2	2	no	—
241	C₂²×C₂₃	—	2	2	no	—
242	C₂²×C₂₃	—	2	2	no	—
243	C₂²×C₂₃	—	2	2	no	—
244	C₂²×C₂₃	—	2	2	no	—
245	C₂²×C₂₃	#451	1	4	yes	—
246	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
247	C₂²×C₂₃	#532	1	1	yes	characteristic
248	C₂²×C₂₃	#537	1	1	yes	characteristic
249	C₂²×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
250	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
251	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
252	C₂²×C₂₃	#575	1	1	yes	characteristic
253	C₄×C₂₃	—	2	2	no	—
254	C₄×C₂₃	—	2	2	no	—
255	C₄×C₂₃	—	2	2	no	—
256	C₄×C₂₃	—	2	2	no	—
257	C₄×C₂₃	—	4	4	no	—
258	C₄×C₂₃	—	4	1	no	—
259	C₄×C₂₃	—	4	4	no	—
260	C₄×C₂₃	—	8	1	no	—
261	C₄×C₂₃	—	8	1	no	—
262	C₄×C₂₃	—	8	1	no	—
263	C₄×C₂₃	—	8	1	no	—
264	C₄×C₂₃	#546	1	2	yes	—
265	C₄×C₂₃	#547	1	2	yes	—
266	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
267	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
268	#533	—	2	8	no	—
269	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
270	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
271	C₂×#125	—	2	1	no	—
272	C₂×C₂²⋊Q₈	—	2	1	no	—
273	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
274	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
275	C₂×C₄×Q₈	—	2	1	no	—
276	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
277	C₂×#117	—	2	1	no	—
278	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
279	#467	—	2	8	no	—
280	C₂×#107	C₂²×C₂₃	1	2	yes	—
281	C₂×#103	C₂²×C₂₃	1	2	yes	—
282	C₂×#103	C₂²×C₂₃	1	2	yes	—
283	C₂×#103	C₂²×C₂₃	1	2	yes	—
284	C₂×#103	C₂²×C₂₃	1	2	yes	—
285	#391	C₂²×C₂₃	1	1	yes	characteristic
286	#391	C₂²×C₂₃	1	1	yes	characteristic
287	#389	C₂²×C₂₃	1	1	yes	characteristic
288	#389	C₂²×C₂₃	1	1	yes	characteristic
289	#387	—	2	1	no	—
290	#387	—	2	1	no	—
291	#385	—	2	1	no	—
292	#381	—	2	1	no	—
293	#380	C₂²×C₂₃	1	1	yes	characteristic
294	#340	C₂²×C₂₃	1	2	yes	—
295	#340	C₂²×C₂₃	1	2	yes	—
296	#336	C₂²×C₂₃	1	1	yes	characteristic
297	#336	C₂²×C₂₃	1	2	yes	—
298	#336	C₂²×C₂₃	1	1	yes	characteristic
299	C₂×C₂₃	—	2	4	no	—
300	C₂×C₂₃	#2430	1	1	yes	characteristic
301	C₂×C₂₃	C₂×#451	1	1	yes	characteristic
302	C₂×C₂₃	#3002	1	1	yes	characteristic
303	C₂×C₂₃	#3170	1	1	yes	characteristic
304	C₂×C₂₃	#3181	1	1	yes	characteristic
305	C₂×C₂₃	#3248	1	1	yes	characteristic
306	C₂×C₂₃	#3249	1	1	yes	characteristic
307	C₂²×Q₈	—	2	1	no	—
308	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
309	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
310	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
311	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
312	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
313	C₂×Q₁₆	—	2	4	no	—
314	C₂×Q₁₆	—	2	4	no	—
315	C₂×Q₁₆	—	4	2	no	—
316	C₂²×C₈	—	2	4	no	—
317	C₂²×C₈	—	2	4	no	—
318	C₂²×C₈	—	4	2	no	—
319	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
320	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
321	C₄⋊Q₈	—	2	4	no	—
322	C₄⋊Q₈	—	2	4	no	—
323	#125	—	2	8	no	—
324	C₂²⋊Q₈	—	2	8	no	—
325	C₂²⋊Q₈	—	2	4	no	—
326	C₂²⋊Q₈	—	2	4	no	—
327	C₂²⋊Q₈	D₄×C₂₃	1	8	yes	—
328	C₄×Q₈	—	2	8	no	—
329	C₄×Q₈	—	2	4	no	—
330	C₄×Q₈	—	2	4	no	—
331	#117	—	2	8	no	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	1	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	1	no	—
340	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
342	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
344	C₂×C₂²⋊C₄	—	2	2	no	—
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₂²⋊C₄	—	2	1	no	—
347	C₂×C₄²	—	2	2	no	—
348	C₂×C₄²	—	2	4	no	—
349	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
350	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
351	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
352	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
353	#107	—	2	4	no	—
354	#103	—	2	4	no	—
355	#103	—	2	4	no	—
356	#103	—	2	4	no	—
357	#103	—	4	2	no	—
358	#95	—	2	1	no	—
359	#95	—	2	1	no	—
360	#95	—	2	1	no	—
361	#95	—	2	1	no	—
362	#95	—	2	1	no	—
363	#95	—	2	1	no	—
364	#95	—	2	1	no	—
365	C₂₃	#25498	1	1	yes	characteristic, direct product factor, Sylow
366	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
367	C₂×Q₈	—	2	4	no	—
368	C₂×Q₈	—	2	2	no	—
369	C₂×Q₈	—	2	2	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	1	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	1	no	—
377	C₂×Q₈	—	4	2	no	—
378	C₂×Q₈	—	4	2	no	—
379	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	4	1	no	—
394	C₂²×C₄	Q₁₆×C₂₃	1	4	yes	complemented
395	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
397	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
398	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
401	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
402	Q₁₆	—	4	8	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	4	8	no	—
409	C₂×C₈	#41×C₂₃	1	4	yes	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	4	no	—
412	C₄⋊C₄	—	2	8	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	4	2	no	—
421	C₄⋊C₄	—	4	2	no	—
422	C₄⋊C₄	—	4	2	no	—
423	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
424	C₂²⋊C₄	—	2	8	no	—
425	C₂²⋊C₄	—	4	2	no	—
426	C₂²⋊C₄	—	4	2	no	—
427	C₄²	—	2	8	no	—
428	C₄²	—	2	2	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	2	2	no	—
431	C₄²	—	2	2	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	2	1	no	—
434	C₂³	—	2	1	no	—
435	C₂³	—	2	1	no	—
436	C₂³	C₂₃×#120	1	2	yes	—
437	C₂³	C₂×Q₁₆×C₂₃	1	2	yes	—
438	C₂³	C₂₃×#137	1	2	yes	—
439	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
440	Q₈	—	2	4	no	—
441	Q₈	—	2	4	no	—
442	Q₈	—	4	8	no	—
443	Q₈	—	4	2	no	—
444	Q₈	—	4	2	no	—
445	Q₈	—	4	2	no	—
446	Q₈	—	4	2	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	8	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	C₂₃×#120	1	2	yes	—
481	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
482	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
483	C₂×C₄	C₂₃×#135	1	2	yes	—
484	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
485	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
486	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
487	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
488	C₈	—	2	4	no	—
489	C₈	—	4	2	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	2	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	2	no	—
496	C₂²	C₂₃×#451	1	4	yes	—
497	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
498	C₂²	C₂₃×#532	1	1	yes	characteristic
499	C₂²	C₂₃×#537	1	1	yes	characteristic
500	C₂²	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
501	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
502	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
503	C₂²	C₂₃×#575	1	1	yes	characteristic
504	C₄	—	2	2	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	2	no	—
507	C₄	—	2	2	no	—
508	C₄	—	4	4	no	—
509	C₄	—	4	1	no	—
510	C₄	—	4	4	no	—
511	C₄	—	8	1	no	—
512	C₄	—	8	1	no	—
513	C₄	—	8	1	no	—
514	C₄	—	8	1	no	—
515	C₄	C₂₃×#546	1	2	yes	—
516	C₄	C₂₃×#547	1	2	yes	—
517	C₂	—	2	4	no	—
518	C₂	C₂₃×#2430	1	1	yes	characteristic
519	C₂	C₂×C₂₃×#451	1	1	yes	characteristic
520	C₂	C₂₃×#3002	1	1	yes	characteristic
521	C₂	C₂₃×#3170	1	1	yes	characteristic
522	C₂	C₂₃×#3181	1	1	yes	characteristic
523	C₂	C₂₃×#3248	1	1	yes	characteristic
524	C₂	C₂₃×#3249	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#2057	C₂	semidirect
5. C₂₃×#2043	C₂	semidirect
6. C₂₃×#2024	C₂	semidirect
7. C₂₃×#2016	C₂	semidirect
9. C₂₃×#1878	C₂	semidirect
10. C₂₃×#1863	C₂	semidirect
131. C₂²×C₄×C₂₃	Q₁₆	semidirect
169. #25498	C₂₃	direct (N × H)
231. #2057	C₂×C₂₃	semidirect
232. #2043	C₂×C₂₃	semidirect
233. #2024	C₂×C₂₃	semidirect
234. #2016	C₂×C₂₃	semidirect
236. #1878	C₂×C₂₃	semidirect
237. #1863	C₂×C₂₃	semidirect
365. C₂₃	#25498	direct (N × H)
394. C₂²×C₄	Q₁₆×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×#533	2
C₂×C₂₃×#467	2
C₂₃×#2855	2
C₂₃×#2057	2
C₂₃×#2043	2
C₂₃×#2024	2
C₂₃×#2016	2
C₂₃×#1892	2
C₂₃×#1878	2
C₂₃×#1863	2
C₂×C₂₃×#336	2
#25498	23

Subgroups — C23×#25498 #19173124

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#25498 #19173124