Subgroups — C₂₃×#18776 #19166402

Order 5888.

1192 conjugacy classes of subgroups.

The subgroup lattice has 520 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×#3226	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3100	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#3037	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1649	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1627	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×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₂₃	—	2	1	no	—
11	C₂×C₂₃×#137	—	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×#500	C₂²	1	2	yes	complemented
25	C₂₃×#486	C₂²	1	2	yes	complemented
26	C₂₃×#480	—	2	2	no	—
27	C₂₃×#480	—	2	1	no	—
28	C₂₃×#474	—	2	2	no	—
29	C₂₃×#474	C₂²	1	2	yes	—
30	C₂₃×#464	—	2	2	no	—
31	C₂₃×#462	C₂²	1	2	yes	complemented
32	C₂₃×#460	C₂²	1	2	yes	complemented
33	C₂₃×#459	C₂²	1	2	yes	—
34	C₂₃×#447	—	2	1	no	—
35	C₄×D₈×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	4	no	—
49	C₂²×Q₈×C₂₃	—	4	4	no	—
50	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
51	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₂₃	—	2	2	no	—
53	C₂³×C₄×C₂₃	—	2	1	no	—
54	C₂₃×#137	—	4	4	no	—
55	C₂×Q₁₆×C₂₃	—	2	2	no	—
56	C₂×SD₁₆×C₂₃	—	2	2	no	—
57	C₂×D₈×C₂₃	—	2	2	no	—
58	C₂×M₁₆×C₂₃	—	2	1	no	—
59	C₂₃×C₄⋊Q₈	—	2	4	no	—
60	C₂₃×C₄⋊Q₈	—	2	4	no	—
61	C₂₃×C₄⋊Q₈	—	2	1	no	—
62	C₂₃×C₄⋊Q₈	—	2	1	no	—
63	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
64	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₂₃×#127	C₂³	1	1	yes	characteristic
67	C₂₃×#124	—	2	1	no	—
68	C₂₃×C₂²⋊Q₈	—	2	4	no	—
69	C₂₃×C₂²⋊Q₈	—	2	8	no	—
70	C₂₃×C₂²⋊Q₈	—	2	4	no	—
71	C₂₃×C₂²⋊Q₈	—	2	8	no	—
72	C₂₃×#121	—	2	1	no	—
73	C₂₃×#121	—	2	1	no	—
74	C₄×Q₈×C₂₃	—	2	1	no	—
75	C₄×Q₈×C₂₃	—	2	1	no	—
76	C₄×Q₈×C₂₃	D₄	1	2	yes	—
77	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
78	C₄×D₄×C₂₃	—	2	4	no	—
79	C₄×D₄×C₂₃	—	2	4	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	—	2	1	no	—
84	C₄×D₄×C₂₃	D₄	1	2	yes	—
85	C₄×D₄×C₂₃	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
89	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₄²×C₂₃	—	2	1	no	—
95	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
96	C₂₃×#107	—	2	2	no	—
97	C₂₃×#107	—	2	4	no	—
98	C₂₃×C₄⋊C₈	—	2	1	no	—
99	C₂₃×C₄⋊C₈	—	2	1	no	—
100	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
101	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
102	C₂₃×#103	—	2	2	no	—
103	C₂₃×#102	—	2	2	no	—
104	C₂₃×#102	—	2	2	no	—
105	C₂₃×#102	—	2	2	no	—
106	C₂₃×#102	—	2	4	no	—
107	C₂₃×C₂²⋊C₈	—	2	1	no	—
108	C₂₃×C₂²⋊C₈	—	2	1	no	—
109	C₄×C₈×C₂₃	—	2	1	no	—
110	C₄×C₈×C₂₃	C₂³	1	2	yes	—
111	C₂⁴×C₂₃	—	2	1	no	—
112	#41×C₂₃	—	4	2	no	—
113	#41×C₂₃	—	4	2	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	2	no	—
119	C₂×Q₈×C₂₃	—	2	1	no	—
120	C₂×Q₈×C₂₃	—	4	8	no	—
121	C₂×Q₈×C₂₃	—	4	8	no	—
122	C₂×Q₈×C₂₃	—	4	16	no	—
123	C₂×Q₈×C₂₃	—	4	8	no	—
124	C₂×Q₈×C₂₃	—	4	2	no	—
125	C₂×Q₈×C₂₃	—	4	2	no	—
126	C₂×Q₈×C₂₃	D₈	1	4	yes	complemented
127	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
128	C₂×D₄×C₂₃	—	2	4	no	—
129	C₂×D₄×C₂₃	—	2	1	no	—
130	C₂×D₄×C₂₃	—	2	1	no	—
131	C₂×D₄×C₂₃	—	2	1	no	—
132	C₂×D₄×C₂₃	—	2	1	no	—
133	C₂×D₄×C₂₃	—	4	1	no	—
134	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₂₃	—	2	1	no	—
136	C₂²×C₄×C₂₃	—	2	8	no	—
137	C₂²×C₄×C₂₃	—	2	1	no	—
138	C₂²×C₄×C₂₃	—	2	2	no	—
139	C₂²×C₄×C₂₃	—	2	1	no	—
140	C₂²×C₄×C₂₃	—	2	2	no	—
141	C₂²×C₄×C₂₃	—	2	2	no	—
142	C₂²×C₄×C₂₃	—	4	8	no	—
143	C₂²×C₄×C₂₃	—	4	2	no	—
144	C₂²×C₄×C₂₃	—	4	2	no	—
145	C₂²×C₄×C₂₃	—	4	1	no	—
146	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
149	Q₁₆×C₂₃	—	4	4	no	—
150	SD₁₆×C₂₃	—	4	4	no	—
151	D₈×C₂₃	—	4	4	no	—
152	M₁₆×C₂₃	—	4	2	no	—
153	C₂×C₈×C₂₃	—	2	2	no	—
154	C₂×C₈×C₂₃	—	2	2	no	—
155	C₂×C₈×C₂₃	—	2	2	no	—
156	C₄⋊C₄×C₂₃	—	2	8	no	—
157	C₄⋊C₄×C₂₃	—	2	2	no	—
158	C₄⋊C₄×C₂₃	—	2	2	no	—
159	C₄⋊C₄×C₂₃	—	2	1	no	—
160	C₄⋊C₄×C₂₃	—	2	1	no	—
161	C₄⋊C₄×C₂₃	—	2	2	no	—
162	C₄⋊C₄×C₂₃	—	2	2	no	—
163	C₄⋊C₄×C₂₃	—	2	2	no	—
164	C₄⋊C₄×C₂₃	—	2	1	no	—
165	C₄⋊C₄×C₂₃	—	2	1	no	—
166	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
167	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
168	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
169	C₂²⋊C₄×C₂₃	—	2	8	no	—
170	C₂²⋊C₄×C₂₃	—	2	2	no	—
171	C₂²⋊C₄×C₂₃	—	2	2	no	—
172	C₂²⋊C₄×C₂₃	—	4	1	no	—
173	C₂²⋊C₄×C₂₃	—	4	1	no	—
174	C₄²×C₂₃	—	2	2	no	—
175	C₄²×C₂₃	—	2	2	no	—
176	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
180	#18776	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂³×C₂₃	—	2	4	no	—
182	C₂³×C₂₃	—	2	2	no	—
183	C₂³×C₂₃	—	4	2	no	—
184	C₂³×C₂₃	—	4	2	no	—
185	C₂³×C₂₃	—	4	1	no	—
186	C₂³×C₂₃	#120	1	1	yes	characteristic
187	Q₈×C₂₃	—	2	2	no	—
188	Q₈×C₂₃	—	2	4	no	—
189	Q₈×C₂₃	—	4	16	no	—
190	Q₈×C₂₃	—	4	2	no	—
191	Q₈×C₂₃	—	4	2	no	—
192	Q₈×C₂₃	—	4	4	no	—
193	Q₈×C₂₃	—	4	1	no	—
194	Q₈×C₂₃	—	4	1	no	—
195	Q₈×C₂₃	C₂×D₈	1	4	yes	—
196	D₄×C₂₃	—	2	4	no	—
197	D₄×C₂₃	—	2	2	no	—
198	D₄×C₂₃	—	4	2	no	—
199	D₄×C₂₃	—	4	1	no	—
200	D₄×C₂₃	—	4	1	no	—
201	D₄×C₂₃	—	4	2	no	—
202	D₄×C₂₃	—	8	2	no	—
203	D₄×C₂₃	#137	1	4	yes	complemented
204	C₂×C₄×C₂₃	—	2	2	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	2	2	no	—
207	C₂×C₄×C₂₃	—	2	2	no	—
208	C₂×C₄×C₂₃	—	2	4	no	—
209	C₂×C₄×C₂₃	—	2	2	no	—
210	C₂×C₄×C₂₃	—	4	1	no	—
211	C₂×C₄×C₂₃	—	4	16	no	—
212	C₂×C₄×C₂₃	—	4	2	no	—
213	C₂×C₄×C₂₃	—	4	4	no	—
214	C₂×C₄×C₂₃	—	4	2	no	—
215	C₂×C₄×C₂₃	—	4	2	no	—
216	C₂×C₄×C₂₃	—	4	2	no	—
217	C₂×C₄×C₂₃	—	4	4	no	—
218	C₂×C₄×C₂₃	—	4	1	no	—
219	C₂×C₄×C₂₃	—	8	2	no	—
220	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
221	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
222	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
223	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
224	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
225	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
228	C₈×C₂₃	—	4	2	no	—
229	C₈×C₂₃	—	4	2	no	—
230	C₈×C₂₃	—	4	2	no	—
231	C₂×D₄×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
232	#3226	C₂×C₂₃	1	1	yes	characteristic, complemented
233	#3100	C₂×C₂₃	1	1	yes	characteristic
234	#3037	C₂×C₂₃	1	1	yes	characteristic, complemented
235	#1649	C₂×C₂₃	1	4	yes	complemented
236	#1627	C₂×C₂₃	1	4	yes	complemented
237	#1546	C₂×C₂₃	1	1	yes	characteristic, complemented
238	#1486	C₂×C₂₃	1	1	yes	characteristic, complemented
239	#1479	C₂×C₂₃	1	1	yes	characteristic, complemented
240	C₂²×C₂₃	—	2	1	no	—
241	C₂²×C₂₃	—	2	1	no	—
242	C₂²×C₂₃	—	2	4	no	—
243	C₂²×C₂₃	—	4	8	no	—
244	C₂²×C₂₃	—	4	2	no	—
245	C₂²×C₂₃	—	4	2	no	—
246	C₂²×C₂₃	—	8	1	no	—
247	C₂²×C₂₃	—	8	1	no	—
248	C₂²×C₂₃	—	8	1	no	—
249	C₂²×C₂₃	#447	1	2	yes	—
250	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
251	C₄×C₂₃	—	2	4	no	—
252	C₄×C₂₃	—	2	1	no	—
253	C₄×C₂₃	—	2	1	no	—
254	C₄×C₂₃	—	2	1	no	—
255	C₄×C₂₃	—	4	4	no	—
256	C₄×C₂₃	—	4	2	no	—
257	C₄×C₂₃	—	4	2	no	—
258	C₄×C₂₃	—	8	1	no	—
259	C₄×C₂₃	#447	1	2	yes	—
260	C₄×C₂₃	C₂²×D₈	1	1	yes	characteristic
261	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
262	C₄×C₂₃	C₂×#137	1	2	yes	—
263	C₂³×Q₈	—	2	1	no	—
264	C₂×#137	—	2	1	no	—
265	D₄×Q₈	—	2	2	no	—
266	D₄×Q₈	—	2	2	no	—
267	D₄×Q₈	—	2	2	no	—
268	D₄×Q₈	C₂²×C₂₃	1	4	yes	—
269	#532	C₂²×C₂₃	1	1	yes	characteristic
270	C₂×C₄⋊Q₈	—	2	1	no	—
271	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
272	C₂×C₂²⋊Q₈	—	2	1	no	—
273	C₂×C₂²⋊Q₈	—	2	2	no	—
274	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
275	C₂×C₄×D₄	—	2	1	no	—
276	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
277	#500	C₂²×C₂₃	1	2	yes	complemented
278	#486	C₂²×C₂₃	1	2	yes	complemented
279	#480	—	2	1	no	—
280	#480	—	2	2	no	—
281	#474	—	2	2	no	—
282	#474	C₂²×C₂₃	1	2	yes	—
283	#464	—	2	2	no	—
284	#462	C₂²×C₂₃	1	2	yes	complemented
285	#460	C₂²×C₂₃	1	2	yes	complemented
286	#459	C₂²×C₂₃	1	2	yes	—
287	#447	—	2	1	no	—
288	C₄×D₈	—	2	2	no	—
289	#431	C₂²×C₂₃	1	1	yes	characteristic
290	#431	C₂²×C₂₃	1	1	yes	characteristic
291	#423	C₂²×C₂₃	1	1	yes	characteristic
292	#417	—	2	1	no	—
293	#332	C₂²×C₂₃	1	4	yes	complemented
294	Q₈⋊C₈	C₂²×C₂₃	1	4	yes	complemented
295	D₄⋊C₈	C₂²×C₂₃	1	4	yes	—
296	C₂×C₂₃	—	2	2	no	—
297	C₂×C₂₃	—	4	4	no	—
298	C₂×C₂₃	—	16	1	no	—
299	C₂×C₂₃	C₂×#447	1	1	yes	characteristic
300	C₂×C₂₃	#3004	1	1	yes	characteristic
301	C₂×C₂₃	#3020	1	1	yes	characteristic
302	C₂×#41	—	2	1	no	—
303	C₂²×Q₈	—	2	2	no	—
304	C₂²×Q₈	—	2	4	no	—
305	C₂²×Q₈	—	2	4	no	—
306	C₂²×Q₈	—	2	1	no	—
307	C₂²×Q₈	—	4	4	no	—
308	C₂²×Q₈	—	4	4	no	—
309	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
310	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
311	C₂³×C₄	—	2	2	no	—
312	C₂³×C₄	—	2	1	no	—
313	#137	—	4	4	no	—
314	C₂×Q₁₆	—	2	2	no	—
315	C₂×SD₁₆	—	2	2	no	—
316	C₂×D₈	—	2	2	no	—
317	C₂×M₁₆	—	2	1	no	—
318	C₄⋊Q₈	—	2	4	no	—
319	C₄⋊Q₈	—	2	4	no	—
320	C₄⋊Q₈	—	2	1	no	—
321	C₄⋊Q₈	—	2	1	no	—
322	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
323	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
324	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
325	#127	C₂³×C₂₃	1	1	yes	characteristic
326	#124	—	2	1	no	—
327	C₂²⋊Q₈	—	2	4	no	—
328	C₂²⋊Q₈	—	2	8	no	—
329	C₂²⋊Q₈	—	2	4	no	—
330	C₂²⋊Q₈	—	2	8	no	—
331	#121	—	2	1	no	—
332	#121	—	2	1	no	—
333	C₄×Q₈	—	2	1	no	—
334	C₄×Q₈	—	2	1	no	—
335	C₄×Q₈	D₄×C₂₃	1	2	yes	—
336	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
337	C₄×D₄	—	2	4	no	—
338	C₄×D₄	—	2	4	no	—
339	C₄×D₄	—	2	1	no	—
340	C₄×D₄	—	2	1	no	—
341	C₄×D₄	—	2	1	no	—
342	C₄×D₄	—	2	1	no	—
343	C₄×D₄	D₄×C₂₃	1	2	yes	—
344	C₄×D₄	C₂³×C₂₃	1	2	yes	—
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
349	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
350	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₂²⋊C₄	—	2	1	no	—
353	C₂×C₄²	—	2	1	no	—
354	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
355	#107	—	2	2	no	—
356	#107	—	2	4	no	—
357	C₄⋊C₈	—	2	1	no	—
358	C₄⋊C₈	—	2	1	no	—
359	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
360	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
361	#103	—	2	2	no	—
362	#102	—	2	2	no	—
363	#102	—	2	2	no	—
364	#102	—	2	2	no	—
365	#102	—	2	4	no	—
366	C₂²⋊C₈	—	2	1	no	—
367	C₂²⋊C₈	—	2	1	no	—
368	C₄×C₈	—	2	1	no	—
369	C₄×C₈	C₂³×C₂₃	1	2	yes	—
370	C₂₃	#18776	1	1	yes	characteristic, direct product factor, Sylow
371	C₂⁴	—	2	1	no	—
372	#41	—	4	2	no	—
373	#41	—	4	2	no	—
374	C₂×Q₈	—	2	8	no	—
375	C₂×Q₈	—	2	1	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	—	2	2	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×Q₈	—	4	8	no	—
381	C₂×Q₈	—	4	8	no	—
382	C₂×Q₈	—	4	16	no	—
383	C₂×Q₈	—	4	8	no	—
384	C₂×Q₈	—	4	2	no	—
385	C₂×Q₈	—	4	2	no	—
386	C₂×Q₈	D₈×C₂₃	1	4	yes	complemented
387	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
388	C₂×D₄	—	2	4	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	2	1	no	—
393	C₂×D₄	—	4	1	no	—
394	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	8	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	4	8	no	—
403	C₂²×C₄	—	4	2	no	—
404	C₂²×C₄	—	4	2	no	—
405	C₂²×C₄	—	4	1	no	—
406	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
408	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
409	Q₁₆	—	4	4	no	—
410	SD₁₆	—	4	4	no	—
411	D₈	—	4	4	no	—
412	M₁₆	—	4	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₄⋊C₄	—	2	8	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	2	2	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
427	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
428	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
429	C₂²⋊C₄	—	2	8	no	—
430	C₂²⋊C₄	—	2	2	no	—
431	C₂²⋊C₄	—	2	2	no	—
432	C₂²⋊C₄	—	4	1	no	—
433	C₂²⋊C₄	—	4	1	no	—
434	C₄²	—	2	2	no	—
435	C₄²	—	2	2	no	—
436	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
437	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
438	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
439	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
440	C₂³	—	2	4	no	—
441	C₂³	—	2	2	no	—
442	C₂³	—	4	2	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	1	no	—
445	C₂³	C₂₃×#120	1	1	yes	characteristic
446	Q₈	—	2	2	no	—
447	Q₈	—	2	4	no	—
448	Q₈	—	4	16	no	—
449	Q₈	—	4	2	no	—
450	Q₈	—	4	2	no	—
451	Q₈	—	4	4	no	—
452	Q₈	—	4	1	no	—
453	Q₈	—	4	1	no	—
454	Q₈	C₂×D₈×C₂₃	1	4	yes	—
455	D₄	—	2	4	no	—
456	D₄	—	2	2	no	—
457	D₄	—	4	2	no	—
458	D₄	—	4	1	no	—
459	D₄	—	4	1	no	—
460	D₄	—	4	2	no	—
461	D₄	—	8	2	no	—
462	D₄	C₂₃×#137	1	4	yes	complemented
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	16	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	4	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	8	2	no	—
479	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
480	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
481	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
482	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
483	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
484	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
486	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
487	C₈	—	4	2	no	—
488	C₈	—	4	2	no	—
489	C₈	—	4	2	no	—
490	C₂²	—	2	1	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	2	4	no	—
493	C₂²	—	4	8	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	8	1	no	—
497	C₂²	—	8	1	no	—
498	C₂²	—	8	1	no	—
499	C₂²	C₂₃×#447	1	2	yes	—
500	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
501	C₄	—	2	4	no	—
502	C₄	—	2	1	no	—
503	C₄	—	2	1	no	—
504	C₄	—	2	1	no	—
505	C₄	—	4	4	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	—	8	1	no	—
509	C₄	C₂₃×#447	1	2	yes	—
510	C₄	C₂²×D₈×C₂₃	1	1	yes	characteristic
511	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
512	C₄	C₂×C₂₃×#137	1	2	yes	—
513	C₂	—	2	2	no	—
514	C₂	—	4	4	no	—
515	C₂	—	16	1	no	—
516	C₂	C₂×C₂₃×#447	1	1	yes	characteristic
517	C₂	C₂₃×#3004	1	1	yes	characteristic
518	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×#3100	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×#3226	C₂	semidirect
4. C₂₃×#3037	C₂	semidirect
5. C₂₃×#1649	C₂	semidirect
6. C₂₃×#1627	C₂	semidirect
7. C₂₃×#1546	C₂	semidirect
8. C₂₃×#1486	C₂	semidirect
9. C₂₃×#1479	C₂	semidirect
24. C₂₃×#500	C₂²	semidirect
25. C₂₃×#486	C₂²	semidirect
31. C₂₃×#462	C₂²	semidirect
32. C₂₃×#460	C₂²	semidirect
40. C₂₃×#332	C₂²	semidirect
41. C₂₃×Q₈⋊C₈	C₂²	semidirect
126. C₂×Q₈×C₂₃	D₈	semidirect
180. #18776	C₂₃	direct (N × H)
203. D₄×C₂₃	#137	semidirect
231. C₂×D₄×Q₈	C₂×C₂₃	semidirect
232. #3226	C₂×C₂₃	semidirect
234. #3037	C₂×C₂₃	semidirect
235. #1649	C₂×C₂₃	semidirect
236. #1627	C₂×C₂₃	semidirect
237. #1546	C₂×C₂₃	semidirect
238. #1486	C₂×C₂₃	semidirect
239. #1479	C₂×C₂₃	semidirect
277. #500	C₂²×C₂₃	semidirect
278. #486	C₂²×C₂₃	semidirect
284. #462	C₂²×C₂₃	semidirect
285. #460	C₂²×C₂₃	semidirect
293. #332	C₂²×C₂₃	semidirect
294. Q₈⋊C₈	C₂²×C₂₃	semidirect
370. C₂₃	#18776	direct (N × H)
386. C₂×Q₈	D₈×C₂₃	semidirect
462. D₄	C₂₃×#137	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×#3226	2
C₂₃×#3100	2
C₂₃×#3037	2
C₂₃×#1649	2
C₂₃×#1627	2
C₂₃×#1546	2
C₂₃×#1486	2
C₂₃×#1479	2
#18776	23

Subgroups — C23×#18776 #19166402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18776 #19166402