Subgroups — C₂₃×#18758 #19166384

Order 5888.

1192 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×D₄×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3227	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3036	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1651	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1627	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×#1478	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₂₃	—	2	1	no	—
11	C₂×C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
12	D₄×Q₈×C₂₃	—	2	2	no	—
13	D₄×Q₈×C₂₃	—	2	2	no	—
14	D₄×Q₈×C₂₃	—	2	2	no	—
15	D₄×Q₈×C₂₃	C₂²	1	4	yes	—
16	C₂₃×#532	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
18	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
20	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
21	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₂₃	—	2	1	no	—
23	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#498	C₂²	1	2	yes	complemented
25	C₂₃×#486	C₂²	1	2	yes	complemented
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	complemented
30	C₂₃×#463	—	2	2	no	—
31	C₂₃×#461	C₂²	1	2	yes	—
32	C₂₃×#460	C₂²	1	2	yes	complemented
33	C₂₃×#459	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	complemented
41	C₂₃×Q₈⋊C₈	C₂²	1	4	yes	complemented
42	C₂₃×D₄⋊C₈	C₂²	1	4	yes	—
43	C₂×#41×C₂₃	—	2	1	no	—
44	C₂²×Q₈×C₂₃	—	2	4	no	—
45	C₂²×Q₈×C₂₃	—	2	2	no	—
46	C₂²×Q₈×C₂₃	—	2	4	no	—
47	C₂²×Q₈×C₂₃	—	2	1	no	—
48	C₂²×Q₈×C₂₃	—	4	8	no	—
49	C₂²×Q₈×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×AGL(1,ℤ/8)	—	4	4	no	—
54	C₂×SD₁₆×C₂₃	—	2	2	no	—
55	C₂×SD₁₆×C₂₃	—	2	2	no	—
56	C₂×D₈×C₂₃	—	2	2	no	—
57	C₂×M₁₆×C₂₃	—	2	1	no	—
58	C₂₃×C₄⋊Q₈	—	2	4	no	—
59	C₂₃×C₄⋊Q₈	—	2	4	no	—
60	C₂₃×C₄⋊Q₈	—	2	1	no	—
61	C₂₃×C₄⋊Q₈	—	2	1	no	—
62	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
63	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₂₃×#127	C₂³	1	1	yes	characteristic
66	C₂₃×#124	—	2	1	no	—
67	C₂₃×C₂²⋊Q₈	—	2	4	no	—
68	C₂₃×C₂²⋊Q₈	—	2	8	no	—
69	C₂₃×C₂²⋊Q₈	—	2	4	no	—
70	C₂₃×C₂²⋊Q₈	—	2	8	no	—
71	C₂₃×#121	—	2	1	no	—
72	C₂₃×#121	—	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	C₂³	1	2	yes	—
77	C₄×D₄×C₂₃	—	2	4	no	—
78	C₄×D₄×C₂₃	—	2	4	no	—
79	C₄×D₄×C₂₃	—	2	1	no	—
80	C₄×D₄×C₂₃	—	2	1	no	—
81	C₄×D₄×C₂₃	—	2	1	no	—
82	C₄×D₄×C₂₃	—	2	1	no	—
83	C₄×D₄×C₂₃	D₄	1	2	yes	—
84	C₄×D₄×C₂₃	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄²×C₂₃	—	2	1	no	—
94	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
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	2	no	—
103	C₂₃×#102	—	2	2	no	—
104	C₂₃×#102	—	2	2	no	—
105	C₂₃×#102	—	2	4	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	#41×C₂₃	—	4	2	no	—
112	#41×C₂₃	—	4	2	no	—
113	C₂×Q₈×C₂₃	—	2	4	no	—
114	C₂×Q₈×C₂₃	—	2	8	no	—
115	C₂×Q₈×C₂₃	—	2	1	no	—
116	C₂×Q₈×C₂₃	—	2	2	no	—
117	C₂×Q₈×C₂₃	—	2	2	no	—
118	C₂×Q₈×C₂₃	—	2	1	no	—
119	C₂×Q₈×C₂₃	—	4	8	no	—
120	C₂×Q₈×C₂₃	—	4	4	no	—
121	C₂×Q₈×C₂₃	—	4	16	no	—
122	C₂×Q₈×C₂₃	—	4	8	no	—
123	C₂×Q₈×C₂₃	—	4	8	no	—
124	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
125	C₂×D₄×C₂₃	—	2	1	no	—
126	C₂×D₄×C₂₃	—	2	2	no	—
127	C₂×D₄×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	—	2	1	no	—
129	C₂×D₄×C₂₃	—	2	1	no	—
130	C₂×D₄×C₂₃	—	4	1	no	—
131	C₂×D₄×C₂₃	SD₁₆	1	4	yes	complemented
132	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	8	no	—
135	C₂²×C₄×C₂₃	—	2	1	no	—
136	C₂²×C₄×C₂₃	—	2	2	no	—
137	C₂²×C₄×C₂₃	—	2	1	no	—
138	C₂²×C₄×C₂₃	—	2	2	no	—
139	C₂²×C₄×C₂₃	—	2	2	no	—
140	C₂²×C₄×C₂₃	—	4	8	no	—
141	C₂²×C₄×C₂₃	—	4	4	no	—
142	C₂²×C₄×C₂₃	—	4	1	no	—
143	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
146	SD₁₆×C₂₃	—	4	4	no	—
147	SD₁₆×C₂₃	—	4	4	no	—
148	D₈×C₂₃	—	4	4	no	—
149	M₁₆×C₂₃	—	4	2	no	—
150	C₂×C₈×C₂₃	—	2	2	no	—
151	C₂×C₈×C₂₃	—	2	2	no	—
152	C₂×C₈×C₂₃	—	2	2	no	—
153	C₄⋊C₄×C₂₃	—	2	8	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	2	no	—
159	C₄⋊C₄×C₂₃	—	2	2	no	—
160	C₄⋊C₄×C₂₃	—	2	2	no	—
161	C₄⋊C₄×C₂₃	—	2	1	no	—
162	C₄⋊C₄×C₂₃	—	2	1	no	—
163	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
164	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
165	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
166	C₂²⋊C₄×C₂₃	—	2	8	no	—
167	C₂²⋊C₄×C₂₃	—	2	2	no	—
168	C₂²⋊C₄×C₂₃	—	2	2	no	—
169	C₂²⋊C₄×C₂₃	—	4	1	no	—
170	C₂²⋊C₄×C₂₃	—	4	1	no	—
171	C₄²×C₂₃	—	2	2	no	—
172	C₄²×C₂₃	—	2	2	no	—
173	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
177	#18758	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂³×C₂₃	—	2	2	no	—
179	C₂³×C₂₃	—	2	4	no	—
180	C₂³×C₂₃	—	4	2	no	—
181	C₂³×C₂₃	—	4	2	no	—
182	C₂³×C₂₃	—	4	1	no	—
183	C₂³×C₂₃	#120	1	1	yes	characteristic
184	Q₈×C₂₃	—	2	4	no	—
185	Q₈×C₂₃	—	2	2	no	—
186	Q₈×C₂₃	—	4	16	no	—
187	Q₈×C₂₃	—	4	4	no	—
188	Q₈×C₂₃	—	4	4	no	—
189	Q₈×C₂₃	—	4	1	no	—
190	Q₈×C₂₃	—	4	1	no	—
191	Q₈×C₂₃	AGL(1,ℤ/8)	1	4	yes	complemented
192	D₄×C₂₃	—	2	2	no	—
193	D₄×C₂₃	—	2	4	no	—
194	D₄×C₂₃	—	4	2	no	—
195	D₄×C₂₃	—	4	1	no	—
196	D₄×C₂₃	—	4	2	no	—
197	D₄×C₂₃	—	4	1	no	—
198	D₄×C₂₃	—	8	2	no	—
199	D₄×C₂₃	C₂×SD₁₆	1	4	yes	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	2	2	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	2	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	2	no	—
206	C₂×C₄×C₂₃	—	4	1	no	—
207	C₂×C₄×C₂₃	—	4	16	no	—
208	C₂×C₄×C₂₃	—	4	4	no	—
209	C₂×C₄×C₂₃	—	4	2	no	—
210	C₂×C₄×C₂₃	—	4	4	no	—
211	C₂×C₄×C₂₃	—	4	2	no	—
212	C₂×C₄×C₂₃	—	4	4	no	—
213	C₂×C₄×C₂₃	—	4	1	no	—
214	C₂×C₄×C₂₃	—	8	2	no	—
215	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
216	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
217	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
218	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
219	C₂×C₄×C₂₃	#137	1	2	yes	—
220	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
223	C₈×C₂₃	—	4	2	no	—
224	C₈×C₂₃	—	4	2	no	—
225	C₈×C₂₃	—	4	2	no	—
226	C₂×D₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
227	#3227	C₂×C₂₃	1	1	yes	characteristic, complemented
228	#3103	C₂×C₂₃	1	1	yes	characteristic, complemented
229	#3036	C₂×C₂₃	1	1	yes	characteristic
230	#1651	C₂×C₂₃	1	4	yes	complemented
231	#1627	C₂×C₂₃	1	4	yes	complemented
232	#1549	C₂×C₂₃	1	1	yes	characteristic, complemented
233	#1487	C₂×C₂₃	1	1	yes	characteristic, complemented
234	#1478	C₂×C₂₃	1	1	yes	characteristic, complemented
235	C₂²×C₂₃	—	2	1	no	—
236	C₂²×C₂₃	—	2	1	no	—
237	C₂²×C₂₃	—	2	4	no	—
238	C₂²×C₂₃	—	4	8	no	—
239	C₂²×C₂₃	—	4	2	no	—
240	C₂²×C₂₃	—	4	2	no	—
241	C₂²×C₂₃	—	8	1	no	—
242	C₂²×C₂₃	—	8	1	no	—
243	C₂²×C₂₃	—	8	1	no	—
244	C₂²×C₂₃	#448	1	2	yes	—
245	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
246	C₄×C₂₃	—	2	4	no	—
247	C₄×C₂₃	—	2	1	no	—
248	C₄×C₂₃	—	2	1	no	—
249	C₄×C₂₃	—	2	1	no	—
250	C₄×C₂₃	—	4	4	no	—
251	C₄×C₂₃	—	4	2	no	—
252	C₄×C₂₃	—	4	2	no	—
253	C₄×C₂₃	—	8	1	no	—
254	C₄×C₂₃	#448	1	2	yes	—
255	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
256	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
257	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
258	C₂³×Q₈	—	2	1	no	—
259	C₂×AGL(1,ℤ/8)	—	2	1	no	—
260	D₄×Q₈	—	2	2	no	—
261	D₄×Q₈	—	2	2	no	—
262	D₄×Q₈	—	2	2	no	—
263	D₄×Q₈	C₂²×C₂₃	1	4	yes	—
264	#532	C₂²×C₂₃	1	1	yes	characteristic
265	C₂×C₄⋊Q₈	—	2	1	no	—
266	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
267	C₂×C₂²⋊Q₈	—	2	1	no	—
268	C₂×C₂²⋊Q₈	—	2	2	no	—
269	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
270	C₂×C₄×D₄	—	2	1	no	—
271	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
272	#498	C₂²×C₂₃	1	2	yes	complemented
273	#486	C₂²×C₂₃	1	2	yes	complemented
274	#481	—	2	1	no	—
275	#481	—	2	2	no	—
276	#476	—	2	2	no	—
277	#475	C₂²×C₂₃	1	2	yes	complemented
278	#463	—	2	2	no	—
279	#461	C₂²×C₂₃	1	2	yes	—
280	#460	C₂²×C₂₃	1	2	yes	complemented
281	#459	C₂²×C₂₃	1	2	yes	—
282	#448	—	2	1	no	—
283	C₄×SD₁₆	—	2	2	no	—
284	#431	C₂²×C₂₃	1	1	yes	characteristic
285	#431	C₂²×C₂₃	1	1	yes	characteristic
286	#423	C₂²×C₂₃	1	1	yes	characteristic
287	#417	—	2	1	no	—
288	#332	C₂²×C₂₃	1	4	yes	complemented
289	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	complemented
290	D₄⋊C₈	C₂²×C₂₃	1	4	yes	—
291	C₂×C₂₃	—	2	2	no	—
292	C₂×C₂₃	—	4	4	no	—
293	C₂×C₂₃	—	16	1	no	—
294	C₂×C₂₃	C₂×#448	1	1	yes	characteristic
295	C₂×C₂₃	#3003	1	1	yes	characteristic
296	C₂×C₂₃	#3020	1	1	yes	characteristic
297	C₂×#41	—	2	1	no	—
298	C₂²×Q₈	—	2	2	no	—
299	C₂²×Q₈	—	2	4	no	—
300	C₂²×Q₈	—	2	4	no	—
301	C₂²×Q₈	—	2	1	no	—
302	C₂²×Q₈	—	4	8	no	—
303	C₂²×Q₈	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	AGL(1,ℤ/8)	—	4	4	no	—
308	C₂×SD₁₆	—	2	2	no	—
309	C₂×SD₁₆	—	2	2	no	—
310	C₂×D₈	—	2	2	no	—
311	C₂×M₁₆	—	2	1	no	—
312	C₄⋊Q₈	—	2	4	no	—
313	C₄⋊Q₈	—	2	4	no	—
314	C₄⋊Q₈	—	2	1	no	—
315	C₄⋊Q₈	—	2	1	no	—
316	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
317	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
318	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
319	#127	C₂³×C₂₃	1	1	yes	characteristic
320	#124	—	2	1	no	—
321	C₂²⋊Q₈	—	2	4	no	—
322	C₂²⋊Q₈	—	2	8	no	—
323	C₂²⋊Q₈	—	2	4	no	—
324	C₂²⋊Q₈	—	2	8	no	—
325	#121	—	2	1	no	—
326	#121	—	2	1	no	—
327	C₄×Q₈	—	2	1	no	—
328	C₄×Q₈	—	2	1	no	—
329	C₄×Q₈	D₄×C₂₃	1	2	yes	—
330	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
331	C₄×D₄	—	2	4	no	—
332	C₄×D₄	—	2	4	no	—
333	C₄×D₄	—	2	1	no	—
334	C₄×D₄	—	2	1	no	—
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	D₄×C₂₃	1	2	yes	—
338	C₄×D₄	C₂³×C₂₃	1	2	yes	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
343	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
344	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	—	2	1	no	—
347	C₂×C₄²	—	2	1	no	—
348	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
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	2	no	—
357	#102	—	2	2	no	—
358	#102	—	2	2	no	—
359	#102	—	2	4	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	#18758	1	1	yes	characteristic, direct product factor, Sylow
365	C₂⁴	—	2	1	no	—
366	#41	—	4	2	no	—
367	#41	—	4	2	no	—
368	C₂×Q₈	—	2	4	no	—
369	C₂×Q₈	—	2	8	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	2	2	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	—	4	8	no	—
375	C₂×Q₈	—	4	4	no	—
376	C₂×Q₈	—	4	16	no	—
377	C₂×Q₈	—	4	8	no	—
378	C₂×Q₈	—	4	8	no	—
379	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	4	1	no	—
386	C₂×D₄	SD₁₆×C₂₃	1	4	yes	complemented
387	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
388	C₂²×C₄	—	2	1	no	—
389	C₂²×C₄	—	2	8	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	4	8	no	—
396	C₂²×C₄	—	4	4	no	—
397	C₂²×C₄	—	4	1	no	—
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	SD₁₆	—	4	4	no	—
402	SD₁₆	—	4	4	no	—
403	D₈	—	4	4	no	—
404	M₁₆	—	4	2	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₄⋊C₄	—	2	8	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	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
419	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
420	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
421	C₂²⋊C₄	—	2	8	no	—
422	C₂²⋊C₄	—	2	2	no	—
423	C₂²⋊C₄	—	2	2	no	—
424	C₂²⋊C₄	—	4	1	no	—
425	C₂²⋊C₄	—	4	1	no	—
426	C₄²	—	2	2	no	—
427	C₄²	—	2	2	no	—
428	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
429	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
430	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
431	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
432	C₂³	—	2	4	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	4	1	no	—
437	C₂³	C₂₃×#120	1	1	yes	characteristic
438	Q₈	—	2	4	no	—
439	Q₈	—	2	2	no	—
440	Q₈	—	4	16	no	—
441	Q₈	—	4	4	no	—
442	Q₈	—	4	4	no	—
443	Q₈	—	4	1	no	—
444	Q₈	—	4	1	no	—
445	Q₈	C₂₃×AGL(1,ℤ/8)	1	4	yes	complemented
446	D₄	—	2	2	no	—
447	D₄	—	2	4	no	—
448	D₄	—	4	2	no	—
449	D₄	—	4	1	no	—
450	D₄	—	4	2	no	—
451	D₄	—	4	1	no	—
452	D₄	—	8	2	no	—
453	D₄	C₂×SD₁₆×C₂₃	1	4	yes	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	16	no	—
462	C₂×C₄	—	4	4	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	4	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	8	2	no	—
469	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
470	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
471	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
472	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
473	C₂×C₄	C₂₃×#137	1	2	yes	—
474	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
475	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
476	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
477	C₈	—	4	2	no	—
478	C₈	—	4	2	no	—
479	C₈	—	4	2	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	4	8	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	8	1	no	—
487	C₂²	—	8	1	no	—
488	C₂²	—	8	1	no	—
489	C₂²	C₂₃×#448	1	2	yes	—
490	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
491	C₄	—	2	4	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	2	1	no	—
495	C₄	—	4	4	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	8	1	no	—
499	C₄	C₂₃×#448	1	2	yes	—
500	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
501	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
502	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
503	C₂	—	2	2	no	—
504	C₂	—	4	4	no	—
505	C₂	—	16	1	no	—
506	C₂	C₂×C₂₃×#448	1	1	yes	characteristic
507	C₂	C₂₃×#3003	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	C₂₃×#3036	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
1. C₂×D₄×Q₈×C₂₃	C₂	semidirect
2. C₂₃×#3227	C₂	semidirect
3. C₂₃×#3103	C₂	semidirect
5. C₂₃×#1651	C₂	semidirect
6. C₂₃×#1627	C₂	semidirect
7. C₂₃×#1549	C₂	semidirect
8. C₂₃×#1487	C₂	semidirect
9. C₂₃×#1478	C₂	semidirect
24. C₂₃×#498	C₂²	semidirect
25. C₂₃×#486	C₂²	semidirect
29. C₂₃×#475	C₂²	semidirect
32. C₂₃×#460	C₂²	semidirect
40. C₂₃×#332	C₂²	semidirect
41. C₂₃×Q₈⋊C₈	C₂²	semidirect
131. C₂×D₄×C₂₃	SD₁₆	semidirect
177. #18758	C₂₃	direct (N × H)
191. Q₈×C₂₃	AGL(1,ℤ/8)	semidirect
226. C₂×D₄×Q₈	C₂×C₂₃	semidirect
227. #3227	C₂×C₂₃	semidirect
228. #3103	C₂×C₂₃	semidirect
230. #1651	C₂×C₂₃	semidirect
231. #1627	C₂×C₂₃	semidirect
232. #1549	C₂×C₂₃	semidirect
233. #1487	C₂×C₂₃	semidirect
234. #1478	C₂×C₂₃	semidirect
272. #498	C₂²×C₂₃	semidirect
273. #486	C₂²×C₂₃	semidirect
277. #475	C₂²×C₂₃	semidirect
280. #460	C₂²×C₂₃	semidirect
288. #332	C₂²×C₂₃	semidirect
289. Q₈⋊C₈	C₂²×C₂₃	semidirect
364. C₂₃	#18758	direct (N × H)
386. C₂×D₄	SD₁₆×C₂₃	semidirect
445. Q₈	C₂₃×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₂₃	2
C₂₃×#3227	2
C₂₃×#3103	2
C₂₃×#3036	2
C₂₃×#1651	2
C₂₃×#1627	2
C₂₃×#1549	2
C₂₃×#1487	2
C₂₃×#1478	2
#18758	23

Subgroups — C23×#18758 #19166384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18758 #19166384