Subgroups — C₂₃×#18757 #19166383

Order 5888.

1170 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#547	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3101	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3077	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1654	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1635	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1486	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1478	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	—	2	1	no	—
11	C₂₃×#547	—	2	2	no	—
12	C₂₃×#547	—	2	2	no	—
13	C₂₃×#547	—	2	2	no	—
14	C₂₃×#547	C₂²	1	4	yes	—
15	C₂₃×#533	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#123	—	2	2	no	—
18	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₂₃×#121	—	2	1	no	—
20	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
23	C₂₃×#498	C₂²	1	2	yes	complemented
24	C₂₃×#487	C₂²	1	2	yes	complemented
25	C₂₃×#483	—	2	2	no	—
26	C₂₃×#483	—	2	1	no	—
27	C₂₃×#478	—	2	2	no	—
28	C₂₃×#476	C₂²	1	2	yes	complemented
29	C₂₃×#476	C₂²	1	2	yes	complemented
30	C₂₃×#475	—	2	2	no	—
31	C₂₃×#474	C₂²	1	2	yes	complemented
32	C₂₃×#461	C₂²	1	2	yes	complemented
33	C₂₃×#450	—	2	1	no	—
34	C₄×SD₁₆×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×#418	—	2	1	no	—
40	C₂₃×#332	C₂²	1	4	yes	complemented
41	C₂₃×D₄⋊C₈	C₂²	1	4	yes	complemented
42	C₂₃×D₄⋊C₈	C₂²	1	4	yes	complemented
43	C₂×#41×C₂₃	—	2	4	no	—
44	C₂×#41×C₂₃	—	2	4	no	—
45	C₂×#41×C₂₃	—	4	4	no	—
46	C₂×#41×C₂₃	—	4	4	no	—
47	C₂²×Q₈×C₂₃	—	2	1	no	—
48	C₂²×D₄×C₂₃	—	2	2	no	—
49	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
51	C₂³×C₄×C₂₃	—	2	2	no	—
52	C₂³×C₄×C₂₃	—	2	1	no	—
53	C₂³×C₄×C₂₃	—	2	1	no	—
54	C₂×SD₁₆×C₂₃	—	2	2	no	—
55	C₂×M₁₆×C₂₃	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	—	2	1	no	—
57	C₂₃×C₄⋊Q₈	—	2	1	no	—
58	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
59	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
62	C₂₃×#125	—	2	1	no	—
63	C₂₃×#123	—	2	8	no	—
64	C₂₃×#123	—	2	8	no	—
65	C₂₃×C₂²⋊Q₈	—	2	4	no	—
66	C₂₃×C₂²⋊Q₈	—	2	4	no	—
67	C₂₃×C₂²⋊Q₈	—	2	1	no	—
68	C₂₃×C₂²⋊Q₈	—	2	1	no	—
69	C₂₃×#121	—	2	4	no	—
70	C₂₃×#121	—	2	4	no	—
71	C₄×Q₈×C₂₃	—	2	1	no	—
72	C₄×Q₈×C₂₃	—	2	1	no	—
73	C₄×D₄×C₂₃	—	2	1	no	—
74	C₄×D₄×C₂₃	—	2	1	no	—
75	C₄×D₄×C₂₃	—	2	1	no	—
76	C₄×D₄×C₂₃	—	2	1	no	—
77	C₄×D₄×C₂₃	D₄	1	2	yes	—
78	C₄×D₄×C₂₃	D₄	1	2	yes	—
79	C₄×D₄×C₂₃	C₂³	1	2	yes	—
80	C₄×D₄×C₂₃	C₂³	1	2	yes	—
81	C₂₃×#117	—	2	1	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
86	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
89	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
93	C₂₃×#107	—	2	2	no	—
94	C₂₃×#106	—	2	2	no	—
95	C₂₃×#106	—	2	2	no	—
96	C₂₃×#106	—	2	4	no	—
97	C₂₃×C₄⋊C₈	—	2	1	no	—
98	C₂₃×C₄⋊C₈	—	2	1	no	—
99	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
100	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
101	C₂₃×#103	—	2	2	no	—
102	C₂₃×#103	—	2	4	no	—
103	C₂₃×#102	—	2	2	no	—
104	C₂₃×#102	—	2	2	no	—
105	C₂₃×#102	—	2	2	no	—
106	C₂₃×C₂²⋊C₈	—	2	1	no	—
107	C₂₃×C₂²⋊C₈	—	2	1	no	—
108	C₄×C₈×C₂₃	—	2	1	no	—
109	C₄×C₈×C₂₃	C₂³	1	2	yes	—
110	C₂⁴×C₂₃	—	2	1	no	—
111	C₂⁴×C₂₃	—	2	1	no	—
112	#41×C₂₃	—	4	8	no	—
113	#41×C₂₃	—	4	16	no	—
114	#41×C₂₃	—	4	8	no	—
115	C₂×Q₈×C₂₃	—	2	2	no	—
116	C₂×Q₈×C₂₃	—	2	2	no	—
117	C₂×Q₈×C₂₃	—	2	1	no	—
118	C₂×Q₈×C₂₃	—	2	1	no	—
119	C₂×Q₈×C₂₃	—	4	2	no	—
120	C₂×Q₈×C₂₃	—	4	2	no	—
121	C₂×D₄×C₂₃	—	2	8	no	—
122	C₂×D₄×C₂₃	—	2	1	no	—
123	C₂×D₄×C₂₃	—	2	1	no	—
124	C₂×D₄×C₂₃	—	2	2	no	—
125	C₂×D₄×C₂₃	—	4	2	no	—
126	C₂×D₄×C₂₃	—	4	8	no	—
127	C₂×D₄×C₂₃	SD₁₆	1	4	yes	complemented
128	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
129	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₂₃	—	2	8	no	—
131	C₂²×C₄×C₂₃	—	2	1	no	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	2	no	—
134	C₂²×C₄×C₂₃	—	2	2	no	—
135	C₂²×C₄×C₂₃	—	2	2	no	—
136	C₂²×C₄×C₂₃	—	2	2	no	—
137	C₂²×C₄×C₂₃	—	2	2	no	—
138	C₂²×C₄×C₂₃	—	4	4	no	—
139	C₂²×C₄×C₂₃	—	4	8	no	—
140	C₂²×C₄×C₂₃	—	4	2	no	—
141	C₂²×C₄×C₂₃	—	4	2	no	—
142	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	SD₁₆×C₂₃	—	4	4	no	—
146	M₁₆×C₂₃	—	4	2	no	—
147	C₂×C₈×C₂₃	—	2	2	no	—
148	C₂×C₈×C₂₃	—	2	2	no	—
149	C₂×C₈×C₂₃	—	2	2	no	—
150	C₄⋊C₄×C₂₃	—	2	8	no	—
151	C₄⋊C₄×C₂₃	—	2	1	no	—
152	C₄⋊C₄×C₂₃	—	2	2	no	—
153	C₄⋊C₄×C₂₃	—	2	1	no	—
154	C₄⋊C₄×C₂₃	—	2	2	no	—
155	C₄⋊C₄×C₂₃	—	2	2	no	—
156	C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₄⋊C₄×C₂₃	—	2	1	no	—
158	C₄⋊C₄×C₂₃	—	2	1	no	—
159	C₄⋊C₄×C₂₃	—	2	1	no	—
160	C₄⋊C₄×C₂₃	—	4	1	no	—
161	C₄⋊C₄×C₂₃	—	4	1	no	—
162	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
163	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
165	C₂²⋊C₄×C₂₃	—	2	8	no	—
166	C₂²⋊C₄×C₂₃	—	2	2	no	—
167	C₂²⋊C₄×C₂₃	—	2	2	no	—
168	C₂²⋊C₄×C₂₃	—	2	2	no	—
169	C₂²⋊C₄×C₂₃	—	2	2	no	—
170	C₂²⋊C₄×C₂₃	—	4	1	no	—
171	C₄²×C₂₃	—	4	1	no	—
172	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
176	#18757	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₂₃	—	2	2	no	—
178	C₂³×C₂₃	—	2	4	no	—
179	C₂³×C₂₃	—	2	2	no	—
180	C₂³×C₂₃	—	4	2	no	—
181	C₂³×C₂₃	—	4	2	no	—
182	C₂³×C₂₃	—	4	2	no	—
183	C₂³×C₂₃	—	4	2	no	—
184	C₂³×C₂₃	—	4	2	no	—
185	C₂³×C₂₃	#120	1	1	yes	characteristic
186	Q₈×C₂₃	—	4	2	no	—
187	Q₈×C₂₃	—	4	2	no	—
188	Q₈×C₂₃	—	4	2	no	—
189	Q₈×C₂₃	—	4	4	no	—
190	Q₈×C₂₃	—	4	2	no	—
191	D₄×C₂₃	—	2	2	no	—
192	D₄×C₂₃	—	2	2	no	—
193	D₄×C₂₃	—	2	2	no	—
194	D₄×C₂₃	—	2	4	no	—
195	D₄×C₂₃	—	4	2	no	—
196	D₄×C₂₃	—	4	16	no	—
197	D₄×C₂₃	C₂×SD₁₆	1	4	yes	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	2	2	no	—
201	C₂×C₄×C₂₃	—	2	4	no	—
202	C₂×C₄×C₂₃	—	4	1	no	—
203	C₂×C₄×C₂₃	—	4	4	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	4	16	no	—
206	C₂×C₄×C₂₃	—	4	2	no	—
207	C₂×C₄×C₂₃	—	4	4	no	—
208	C₂×C₄×C₂₃	—	4	2	no	—
209	C₂×C₄×C₂₃	—	4	2	no	—
210	C₂×C₄×C₂₃	—	4	4	no	—
211	C₂×C₄×C₂₃	—	4	4	no	—
212	C₂×C₄×C₂₃	—	4	1	no	—
213	C₂×C₄×C₂₃	—	4	1	no	—
214	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
215	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
216	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
217	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
218	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
219	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
222	C₈×C₂₃	—	4	2	no	—
223	C₈×C₂₃	—	4	2	no	—
224	C₈×C₂₃	—	4	2	no	—
225	C₂×#547	C₂×C₂₃	1	1	yes	characteristic
226	#3229	C₂×C₂₃	1	1	yes	characteristic, complemented
227	#3101	C₂×C₂₃	1	1	yes	characteristic, complemented
228	#3077	C₂×C₂₃	1	1	yes	characteristic, complemented
229	#1654	C₂×C₂₃	1	4	yes	complemented
230	#1635	C₂×C₂₃	1	4	yes	complemented
231	#1546	C₂×C₂₃	1	1	yes	characteristic, complemented
232	#1486	C₂×C₂₃	1	1	yes	characteristic, complemented
233	#1478	C₂×C₂₃	1	1	yes	characteristic, complemented
234	C₂²×C₂₃	—	2	1	no	—
235	C₂²×C₂₃	—	2	1	no	—
236	C₂²×C₂₃	—	2	4	no	—
237	C₂²×C₂₃	—	4	8	no	—
238	C₂²×C₂₃	—	4	2	no	—
239	C₂²×C₂₃	—	4	2	no	—
240	C₂²×C₂₃	—	4	2	no	—
241	C₂²×C₂₃	—	4	2	no	—
242	C₂²×C₂₃	—	4	2	no	—
243	C₂²×C₂₃	—	8	4	no	—
244	C₂²×C₂₃	#450	1	2	yes	—
245	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
246	C₄×C₂₃	—	2	1	no	—
247	C₄×C₂₃	—	2	1	no	—
248	C₄×C₂₃	—	2	1	no	—
249	C₄×C₂₃	—	4	2	no	—
250	C₄×C₂₃	—	4	4	no	—
251	C₄×C₂₃	—	4	2	no	—
252	C₄×C₂₃	—	8	1	no	—
253	C₄×C₂₃	—	8	1	no	—
254	C₄×C₂₃	#450	1	2	yes	—
255	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
256	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
257	C₄×C₂₃	#575	1	2	yes	—
258	C₂²×#41	—	2	1	no	—
259	#547	—	2	2	no	—
260	#547	—	2	2	no	—
261	#547	—	2	2	no	—
262	#547	C₂²×C₂₃	1	4	yes	—
263	#533	C₂²×C₂₃	1	1	yes	characteristic
264	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
265	C₂×#123	—	2	2	no	—
266	C₂×C₂²⋊Q₈	—	2	1	no	—
267	C₂×#121	—	2	1	no	—
268	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
269	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
270	C₂²×C₄⋊C₄	—	2	1	no	—
271	#498	C₂²×C₂₃	1	2	yes	complemented
272	#487	C₂²×C₂₃	1	2	yes	complemented
273	#483	—	2	1	no	—
274	#483	—	2	2	no	—
275	#478	—	2	2	no	—
276	#476	C₂²×C₂₃	1	2	yes	complemented
277	#476	C₂²×C₂₃	1	2	yes	complemented
278	#475	—	2	2	no	—
279	#474	C₂²×C₂₃	1	2	yes	complemented
280	#461	C₂²×C₂₃	1	2	yes	complemented
281	#450	—	2	1	no	—
282	C₄×SD₁₆	—	2	2	no	—
283	#431	C₂²×C₂₃	1	1	yes	characteristic
284	#431	C₂²×C₂₃	1	1	yes	characteristic
285	#428	—	2	1	no	—
286	#423	C₂²×C₂₃	1	1	yes	characteristic
287	#418	—	2	1	no	—
288	#332	C₂²×C₂₃	1	4	yes	complemented
289	D₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
290	D₄⋊C₈	C₂²×C₂₃	1	4	yes	complemented
291	C₂×C₂₃	—	2	2	no	—
292	C₂×C₂₃	—	4	4	no	—
293	C₂×C₂₃	—	8	2	no	—
294	C₂×C₂₃	C₂×#450	1	1	yes	characteristic
295	C₂×C₂₃	#3005	1	1	yes	characteristic
296	C₂×C₂₃	#3020	1	1	yes	characteristic
297	C₂×#41	—	2	4	no	—
298	C₂×#41	—	2	4	no	—
299	C₂×#41	—	4	4	no	—
300	C₂×#41	—	4	4	no	—
301	C₂²×Q₈	—	2	1	no	—
302	C₂²×D₄	—	2	2	no	—
303	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
304	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
305	C₂³×C₄	—	2	2	no	—
306	C₂³×C₄	—	2	1	no	—
307	C₂³×C₄	—	2	1	no	—
308	C₂×SD₁₆	—	2	2	no	—
309	C₂×M₁₆	—	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	C₂³×C₂₃	1	1	yes	characteristic
314	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
315	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
316	#125	—	2	1	no	—
317	#123	—	2	8	no	—
318	#123	—	2	8	no	—
319	C₂²⋊Q₈	—	2	4	no	—
320	C₂²⋊Q₈	—	2	4	no	—
321	C₂²⋊Q₈	—	2	1	no	—
322	C₂²⋊Q₈	—	2	1	no	—
323	#121	—	2	4	no	—
324	#121	—	2	4	no	—
325	C₄×Q₈	—	2	1	no	—
326	C₄×Q₈	—	2	1	no	—
327	C₄×D₄	—	2	1	no	—
328	C₄×D₄	—	2	1	no	—
329	C₄×D₄	—	2	1	no	—
330	C₄×D₄	—	2	1	no	—
331	C₄×D₄	D₄×C₂₃	1	2	yes	—
332	C₄×D₄	D₄×C₂₃	1	2	yes	—
333	C₄×D₄	C₂³×C₂₃	1	2	yes	—
334	C₄×D₄	C₂³×C₂₃	1	2	yes	—
335	#117	—	2	1	no	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	—	2	4	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	—	2	4	no	—
340	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
341	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
342	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₂²⋊C₄	—	2	1	no	—
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
347	#107	—	2	2	no	—
348	#106	—	2	2	no	—
349	#106	—	2	2	no	—
350	#106	—	2	4	no	—
351	C₄⋊C₈	—	2	1	no	—
352	C₄⋊C₈	—	2	1	no	—
353	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
354	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
355	#103	—	2	2	no	—
356	#103	—	2	4	no	—
357	#102	—	2	2	no	—
358	#102	—	2	2	no	—
359	#102	—	2	2	no	—
360	C₂²⋊C₈	—	2	1	no	—
361	C₂²⋊C₈	—	2	1	no	—
362	C₄×C₈	—	2	1	no	—
363	C₄×C₈	C₂³×C₂₃	1	2	yes	—
364	C₂₃	#18757	1	1	yes	characteristic, direct product factor, Sylow
365	C₂⁴	—	2	1	no	—
366	C₂⁴	—	2	1	no	—
367	#41	—	4	8	no	—
368	#41	—	4	16	no	—
369	#41	—	4	8	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	—	4	2	no	—
375	C₂×Q₈	—	4	2	no	—
376	C₂×D₄	—	2	8	no	—
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	4	2	no	—
381	C₂×D₄	—	4	8	no	—
382	C₂×D₄	SD₁₆×C₂₃	1	4	yes	complemented
383	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
384	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
385	C₂²×C₄	—	2	8	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	2	1	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	4	4	no	—
394	C₂²×C₄	—	4	8	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	4	2	no	—
397	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
400	SD₁₆	—	4	4	no	—
401	M₁₆	—	4	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	2	no	—
404	C₂×C₈	—	2	2	no	—
405	C₄⋊C₄	—	2	8	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	4	1	no	—
416	C₄⋊C₄	—	4	1	no	—
417	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
418	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
419	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	2	no	—
422	C₂²⋊C₄	—	2	2	no	—
423	C₂²⋊C₄	—	2	2	no	—
424	C₂²⋊C₄	—	2	2	no	—
425	C₂²⋊C₄	—	4	1	no	—
426	C₄²	—	4	1	no	—
427	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
428	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
429	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
430	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
431	C₂³	—	2	4	no	—
432	C₂³	—	2	2	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	4	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	C₂₃×#120	1	1	yes	characteristic
440	Q₈	—	4	2	no	—
441	Q₈	—	4	2	no	—
442	Q₈	—	4	2	no	—
443	Q₈	—	4	4	no	—
444	Q₈	—	4	2	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	2	no	—
447	D₄	—	2	2	no	—
448	D₄	—	2	4	no	—
449	D₄	—	4	2	no	—
450	D₄	—	4	16	no	—
451	D₄	C₂×SD₁₆×C₂₃	1	4	yes	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	—	4	16	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	4	no	—
465	C₂×C₄	—	4	4	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
469	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
470	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
471	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
472	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
473	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
474	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
476	C₈	—	4	2	no	—
477	C₈	—	4	2	no	—
478	C₈	—	4	2	no	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	4	8	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	2	no	—
487	C₂²	—	4	2	no	—
488	C₂²	—	8	4	no	—
489	C₂²	C₂₃×#450	1	2	yes	—
490	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
491	C₄	—	2	1	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	4	no	—
496	C₄	—	4	2	no	—
497	C₄	—	8	1	no	—
498	C₄	—	8	1	no	—
499	C₄	C₂₃×#450	1	2	yes	—
500	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
501	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
502	C₄	C₂₃×#575	1	2	yes	—
503	C₂	—	2	2	no	—
504	C₂	—	4	4	no	—
505	C₂	—	8	2	no	—
506	C₂	C₂×C₂₃×#450	1	1	yes	characteristic
507	C₂	C₂₃×#3005	1	1	yes	characteristic
508	C₂	C₂₃×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#547	C₂
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
2. C₂₃×#3229	C₂	semidirect
3. C₂₃×#3101	C₂	semidirect
4. C₂₃×#3077	C₂	semidirect
5. C₂₃×#1654	C₂	semidirect
6. C₂₃×#1635	C₂	semidirect
7. C₂₃×#1546	C₂	semidirect
8. C₂₃×#1486	C₂	semidirect
9. C₂₃×#1478	C₂	semidirect
23. C₂₃×#498	C₂²	semidirect
24. C₂₃×#487	C₂²	semidirect
28. C₂₃×#476	C₂²	semidirect
29. C₂₃×#476	C₂²	semidirect
31. C₂₃×#474	C₂²	semidirect
32. C₂₃×#461	C₂²	semidirect
40. C₂₃×#332	C₂²	semidirect
41. C₂₃×D₄⋊C₈	C₂²	semidirect
42. C₂₃×D₄⋊C₈	C₂²	semidirect
127. C₂×D₄×C₂₃	SD₁₆	semidirect
176. #18757	C₂₃	direct (N × H)
226. #3229	C₂×C₂₃	semidirect
227. #3101	C₂×C₂₃	semidirect
228. #3077	C₂×C₂₃	semidirect
229. #1654	C₂×C₂₃	semidirect
230. #1635	C₂×C₂₃	semidirect
231. #1546	C₂×C₂₃	semidirect
232. #1486	C₂×C₂₃	semidirect
233. #1478	C₂×C₂₃	semidirect
271. #498	C₂²×C₂₃	semidirect
272. #487	C₂²×C₂₃	semidirect
276. #476	C₂²×C₂₃	semidirect
277. #476	C₂²×C₂₃	semidirect
279. #474	C₂²×C₂₃	semidirect
280. #461	C₂²×C₂₃	semidirect
288. #332	C₂²×C₂₃	semidirect
289. D₄⋊C₈	C₂²×C₂₃	semidirect
290. D₄⋊C₈	C₂²×C₂₃	semidirect
364. C₂₃	#18757	direct (N × H)
382. C₂×D₄	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×#547	2
C₂₃×#3229	2
C₂₃×#3101	2
C₂₃×#3077	2
C₂₃×#1654	2
C₂₃×#1635	2
C₂₃×#1546	2
C₂₃×#1486	2
C₂₃×#1478	2
#18757	23

Subgroups — C23×#18757 #19166383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18757 #19166383