Subgroups — C₂₃×#18047 #19165673

Order 5888.

642 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2970	C₂	1	1	yes	maximal, characteristic
2	Q₈×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×#137	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1593	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1587	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1585	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1584	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1582	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1569	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1491	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1489	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#137	C₄	1	1	yes	characteristic
17	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#446	—	2	1	no	—
20	C₂₃×#446	—	2	1	no	—
21	C₂₃×#446	C₂²	1	1	yes	characteristic
22	C₂₃×#446	C₂²	1	1	yes	characteristic
23	C₂₃×#446	C₂²	1	1	yes	characteristic
24	C₂₃×#446	C₂²	1	1	yes	characteristic
25	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#441	C₄	1	1	yes	characteristic
28	C₂₃×#441	C₄	1	1	yes	characteristic
29	C₂₃×#440	C₄	1	1	yes	characteristic
30	C₂₃×#440	C₄	1	1	yes	characteristic
31	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
32	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
33	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#435	—	2	1	no	—
36	C₂₃×#435	—	2	1	no	—
37	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₈×D₄×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#432	—	2	1	no	—
40	C₂₃×#432	C₂²	1	1	yes	characteristic
41	C₂₃×#428	C₄	1	1	yes	characteristic
42	C₂₃×#423	—	2	1	no	—
43	C₂₃×#423	C₂²	1	1	yes	characteristic
44	C₂₃×#419	C₄	1	1	yes	characteristic
45	C₂₃×#417	C₄	1	1	yes	characteristic
46	C₄×M₁₆×C₂₃	—	2	1	no	—
47	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
48	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
49	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
50	C₂×C₂₃×#97	—	2	1	no	—
51	C₂₃×#335	C₂²	1	1	yes	characteristic
52	C₂₃×#335	C₂²	1	1	yes	characteristic
53	C₂₃×#334	C₂²	1	1	yes	characteristic
54	C₂₃×#334	C₂²	1	1	yes	characteristic
55	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
56	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
57	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
58	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
59	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
60	C₂₃×Q₈⋊C₈	C₂²	1	1	yes	characteristic
61	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
62	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
63	C₂₃×#322	C₂²	1	1	yes	characteristic
64	C₂₃×#322	C₂²	1	1	yes	characteristic
65	C₂₃×#322	C₂²	1	1	yes	characteristic
66	C₈²×C₂₃	C₂²	1	1	yes	characteristic
67	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
69	C₂₃×#137	—	2	4	no	—
70	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
72	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
73	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
74	C₂×M₁₆×C₂₃	—	2	1	no	—
75	C₂×M₁₆×C₂₃	—	2	1	no	—
76	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
77	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
78	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
79	C₂×M₁₆×C₂₃	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₂₃	—	2	1	no	—
81	C₂²×C₈×C₂₃	—	2	1	no	—
82	C₄×Q₈×C₂₃	—	2	1	no	—
83	C₄×Q₈×C₂₃	—	2	1	no	—
84	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
88	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
89	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
90	C₄×D₄×C₂₃	—	2	1	no	—
91	C₄×D₄×C₂₃	—	2	1	no	—
92	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
93	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₂₃×#117	—	2	1	no	—
95	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	—	2	1	no	—
99	C₂×C₄²×C₂₃	—	2	1	no	—
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
102	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
103	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
104	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
105	C₂₃×C₄⋊C₈	—	2	1	no	—
106	C₂₃×C₄⋊C₈	—	2	1	no	—
107	C₂₃×C₄⋊C₈	—	2	1	no	—
108	C₂₃×C₄⋊C₈	—	2	1	no	—
109	C₂₃×C₄⋊C₈	—	2	1	no	—
110	C₂₃×C₄⋊C₈	—	2	1	no	—
111	C₂₃×C₄⋊C₈	—	2	1	no	—
112	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
117	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
118	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
119	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
120	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
121	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
122	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
123	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
124	C₂₃×C₂²⋊C₈	—	2	1	no	—
125	C₂₃×C₂²⋊C₈	—	2	1	no	—
126	C₂₃×C₂²⋊C₈	—	2	1	no	—
127	C₂₃×#97	—	2	1	no	—
128	C₂₃×#97	—	2	1	no	—
129	C₂₃×#97	—	2	1	no	—
130	C₂₃×#97	—	2	1	no	—
131	C₂₃×#97	—	2	1	no	—
132	C₂₃×#97	—	2	1	no	—
133	C₂₃×#97	C₂×C₄	1	1	yes	characteristic
134	C₂₃×#97	C₂×C₄	1	1	yes	characteristic
135	C₂₃×#97	D₄	1	1	yes	characteristic
136	C₂₃×#97	D₄	1	1	yes	characteristic
137	C₄×C₈×C₂₃	—	2	1	no	—
138	C₄×C₈×C₂₃	—	2	1	no	—
139	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
145	#41×C₂₃	—	2	2	no	—
146	#41×C₂₃	—	4	1	no	—
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	2	2	no	—
149	C₂×Q₈×C₂₃	—	4	1	no	—
150	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
152	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
153	C₂×D₄×C₂₃	—	2	1	no	—
154	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	—	2	1	no	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	1	no	—
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
163	Q₁₆×C₂₃	—	2	2	no	—
164	Q₁₆×C₂₃	—	2	2	no	—
165	SD₁₆×C₂₃	—	2	2	no	—
166	SD₁₆×C₂₃	M₁₆	1	4	yes	complemented
167	M₁₆×C₂₃	—	2	2	no	—
168	M₁₆×C₂₃	—	2	2	no	—
169	M₁₆×C₂₃	—	4	1	no	—
170	M₁₆×C₂₃	—	4	1	no	—
171	M₁₆×C₂₃	—	4	2	no	—
172	M₁₆×C₂₃	—	4	2	no	—
173	C₂×C₈×C₂₃	—	2	1	no	—
174	C₂×C₈×C₂₃	—	2	1	no	—
175	C₂×C₈×C₂₃	—	2	1	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	4	2	no	—
180	C₂×C₈×C₂₃	—	4	2	no	—
181	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
182	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
183	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
184	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
185	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	—	2	1	no	—
192	C₄⋊C₄×C₂₃	—	2	1	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	—	2	1	no	—
195	C₄⋊C₄×C₂₃	—	2	1	no	—
196	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
197	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
200	C₂²⋊C₄×C₂₃	—	2	1	no	—
201	C₂²⋊C₄×C₂₃	—	2	1	no	—
202	C₂²⋊C₄×C₂₃	—	2	1	no	—
203	C₄²×C₂₃	—	2	1	no	—
204	C₄²×C₂₃	—	2	1	no	—
205	C₄²×C₂₃	—	2	1	no	—
206	C₄²×C₂₃	—	2	1	no	—
207	C₄²×C₂₃	—	2	1	no	—
208	C₄²×C₂₃	—	2	1	no	—
209	C₄²×C₂₃	—	2	1	no	—
210	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
211	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₂₃	#41	1	1	yes	characteristic
213	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
214	#18047	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂³×C₂₃	—	2	1	no	—
216	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
217	Q₈×C₂₃	—	2	1	no	—
218	Q₈×C₂₃	—	2	1	no	—
219	Q₈×C₂₃	—	2	1	no	—
220	Q₈×C₂₃	—	2	1	no	—
221	Q₈×C₂₃	—	2	1	no	—
222	Q₈×C₂₃	—	4	2	no	—
223	Q₈×C₂₃	C₂×M₁₆	1	2	yes	—
224	D₄×C₂₃	—	2	1	no	—
225	D₄×C₂₃	—	4	2	no	—
226	D₄×C₂₃	C₂×M₁₆	1	2	yes	—
227	C₂×C₄×C₂₃	—	2	1	no	—
228	C₂×C₄×C₂₃	—	2	1	no	—
229	C₂×C₄×C₂₃	—	2	1	no	—
230	C₂×C₄×C₂₃	—	2	1	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	1	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	2	no	—
240	C₂×C₄×C₂₃	—	4	2	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	8	1	no	—
244	C₂×C₄×C₂₃	—	8	1	no	—
245	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
246	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
247	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
248	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
249	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
251	C₈×C₂₃	—	2	2	no	—
252	C₈×C₂₃	—	2	1	no	—
253	C₈×C₂₃	—	2	1	no	—
254	C₈×C₂₃	—	2	1	no	—
255	C₈×C₂₃	—	2	1	no	—
256	C₈×C₂₃	—	2	1	no	—
257	C₈×C₂₃	—	2	1	no	—
258	C₈×C₂₃	—	4	1	no	—
259	C₈×C₂₃	—	4	1	no	—
260	C₈×C₂₃	—	4	1	no	—
261	C₈×C₂₃	—	4	1	no	—
262	C₈×C₂₃	—	4	1	no	—
263	C₈×C₂₃	C₂×M₁₆	1	2	yes	—
264	C₈×C₂₃	#137	1	2	yes	complemented
265	#2970	C₂×C₂₃	1	1	yes	characteristic
266	Q₈×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
267	C₄×#137	C₂×C₂₃	1	1	yes	characteristic
268	#1593	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1591	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1587	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1585	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1584	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1582	C₂×C₂₃	1	1	yes	characteristic, complemented
274	SD₁₆⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
275	C₈×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#1569	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#1491	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#1489	C₂×C₂₃	1	1	yes	characteristic
279	#1455	C₂×C₂₃	1	1	yes	characteristic, complemented
280	C₂²×C₂₃	—	2	1	no	—
281	C₂²×C₂₃	—	2	2	no	—
282	C₂²×C₂₃	—	2	1	no	—
283	C₂²×C₂₃	—	2	1	no	—
284	C₂²×C₂₃	—	4	1	no	—
285	C₂²×C₂₃	—	4	1	no	—
286	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
287	C₄×C₂₃	—	2	1	no	—
288	C₄×C₂₃	—	2	2	no	—
289	C₄×C₂₃	—	2	1	no	—
290	C₄×C₂₃	—	2	1	no	—
291	C₄×C₂₃	—	2	1	no	—
292	C₄×C₂₃	—	4	1	no	—
293	C₄×C₂₃	—	4	1	no	—
294	C₄×C₂₃	—	4	1	no	—
295	C₄×C₂₃	—	4	1	no	—
296	C₄×C₂₃	—	4	1	no	—
297	C₄×C₂₃	—	4	1	no	—
298	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
299	C₄×C₂₃	#568	1	1	yes	characteristic
300	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
301	C₄×C₂₃	#575	1	1	yes	characteristic
302	C₂×#137	C₄×C₂₃	1	1	yes	characteristic
303	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
304	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
305	#446	—	2	1	no	—
306	#446	—	2	1	no	—
307	#446	C₂²×C₂₃	1	1	yes	characteristic
308	#446	C₂²×C₂₃	1	1	yes	characteristic
309	#446	C₂²×C₂₃	1	1	yes	characteristic
310	#446	C₂²×C₂₃	1	1	yes	characteristic
311	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
312	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
313	#441	C₄×C₂₃	1	1	yes	characteristic
314	#441	C₄×C₂₃	1	1	yes	characteristic
315	#440	C₄×C₂₃	1	1	yes	characteristic
316	#440	C₄×C₂₃	1	1	yes	characteristic
317	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
318	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
319	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
320	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
321	#435	—	2	1	no	—
322	#435	—	2	1	no	—
323	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
324	C₈×D₄	C₂²×C₂₃	1	1	yes	characteristic
325	#432	—	2	1	no	—
326	#432	C₂²×C₂₃	1	1	yes	characteristic
327	#428	C₄×C₂₃	1	1	yes	characteristic
328	#423	—	2	1	no	—
329	#423	C₂²×C₂₃	1	1	yes	characteristic
330	#419	C₄×C₂₃	1	1	yes	characteristic
331	#417	C₄×C₂₃	1	1	yes	characteristic
332	C₄×M₁₆	—	2	1	no	—
333	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
334	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
335	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×#97	—	2	1	no	—
337	#335	C₂²×C₂₃	1	1	yes	characteristic
338	#335	C₂²×C₂₃	1	1	yes	characteristic
339	#334	C₂²×C₂₃	1	1	yes	characteristic
340	#334	C₂²×C₂₃	1	1	yes	characteristic
341	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
342	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
343	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
344	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
345	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
346	Q₈⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
347	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
348	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
349	#322	C₂²×C₂₃	1	1	yes	characteristic
350	#322	C₂²×C₂₃	1	1	yes	characteristic
351	#322	C₂²×C₂₃	1	1	yes	characteristic
352	C₈²	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×C₂₃	—	4	1	no	—
354	C₂×C₂₃	—	4	2	no	—
355	C₂×C₂₃	D₄×M₁₆	1	1	yes	characteristic
356	C₂×C₂₃	C₄×#137	1	1	yes	characteristic
357	C₂×C₂₃	#2956	1	1	yes	characteristic
358	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
359	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
360	#137	—	2	4	no	—
361	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
362	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
363	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
364	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
365	C₂×M₁₆	—	2	1	no	—
366	C₂×M₁₆	—	2	1	no	—
367	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
368	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
369	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
370	C₂×M₁₆	D₄×C₂₃	1	1	yes	characteristic
371	C₂²×C₈	—	2	1	no	—
372	C₂²×C₈	—	2	1	no	—
373	C₄×Q₈	—	2	1	no	—
374	C₄×Q₈	—	2	1	no	—
375	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
376	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
377	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
378	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
379	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
380	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
381	C₄×D₄	—	2	1	no	—
382	C₄×D₄	—	2	1	no	—
383	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
384	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
385	#117	—	2	1	no	—
386	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
389	C₂×C₄²	—	2	1	no	—
390	C₂×C₄²	—	2	1	no	—
391	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
392	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
393	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
394	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
395	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
396	C₄⋊C₈	—	2	1	no	—
397	C₄⋊C₈	—	2	1	no	—
398	C₄⋊C₈	—	2	1	no	—
399	C₄⋊C₈	—	2	1	no	—
400	C₄⋊C₈	—	2	1	no	—
401	C₄⋊C₈	—	2	1	no	—
402	C₄⋊C₈	—	2	1	no	—
403	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
404	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
405	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
406	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
407	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
408	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
409	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
410	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
411	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
412	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
413	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
414	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
415	C₂²⋊C₈	—	2	1	no	—
416	C₂²⋊C₈	—	2	1	no	—
417	C₂²⋊C₈	—	2	1	no	—
418	#97	—	2	1	no	—
419	#97	—	2	1	no	—
420	#97	—	2	1	no	—
421	#97	—	2	1	no	—
422	#97	—	2	1	no	—
423	#97	—	2	1	no	—
424	#97	C₂×C₄×C₂₃	1	1	yes	characteristic
425	#97	C₂×C₄×C₂₃	1	1	yes	characteristic
426	#97	D₄×C₂₃	1	1	yes	characteristic
427	#97	D₄×C₂₃	1	1	yes	characteristic
428	C₄×C₈	—	2	1	no	—
429	C₄×C₈	—	2	1	no	—
430	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
431	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
432	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
433	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
434	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
435	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
436	C₂₃	#18047	1	1	yes	characteristic, direct product factor, Sylow
437	#41	—	2	2	no	—
438	#41	—	4	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	2	no	—
441	C₂×Q₈	—	4	1	no	—
442	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
443	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
444	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
445	C₂×D₄	—	2	1	no	—
446	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
455	Q₁₆	—	2	2	no	—
456	Q₁₆	—	2	2	no	—
457	SD₁₆	—	2	2	no	—
458	SD₁₆	M₁₆×C₂₃	1	4	yes	complemented
459	M₁₆	—	2	2	no	—
460	M₁₆	—	2	2	no	—
461	M₁₆	—	4	1	no	—
462	M₁₆	—	4	1	no	—
463	M₁₆	—	4	2	no	—
464	M₁₆	—	4	2	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	—	2	1	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	4	2	no	—
472	C₂×C₈	—	4	2	no	—
473	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
474	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
475	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
476	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
477	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	1	no	—
485	C₄⋊C₄	—	2	1	no	—
486	C₄⋊C₄	—	2	1	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
489	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
490	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
491	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
492	C₂²⋊C₄	—	2	1	no	—
493	C₂²⋊C₄	—	2	1	no	—
494	C₂²⋊C₄	—	2	1	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	2	1	no	—
500	C₄²	—	2	1	no	—
501	C₄²	—	2	1	no	—
502	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
503	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
504	C₄²	#41×C₂₃	1	1	yes	characteristic
505	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
506	C₂³	—	2	1	no	—
507	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
508	Q₈	—	2	1	no	—
509	Q₈	—	2	1	no	—
510	Q₈	—	2	1	no	—
511	Q₈	—	2	1	no	—
512	Q₈	—	2	1	no	—
513	Q₈	—	4	2	no	—
514	Q₈	C₂×M₁₆×C₂₃	1	2	yes	—
515	D₄	—	2	1	no	—
516	D₄	—	4	2	no	—
517	D₄	C₂×M₁₆×C₂₃	1	2	yes	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	8	1	no	—
535	C₂×C₄	—	8	1	no	—
536	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
537	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
538	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
539	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
540	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
541	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
542	C₈	—	2	2	no	—
543	C₈	—	2	1	no	—
544	C₈	—	2	1	no	—
545	C₈	—	2	1	no	—
546	C₈	—	2	1	no	—
547	C₈	—	2	1	no	—
548	C₈	—	2	1	no	—
549	C₈	—	4	1	no	—
550	C₈	—	4	1	no	—
551	C₈	—	4	1	no	—
552	C₈	—	4	1	no	—
553	C₈	—	4	1	no	—
554	C₈	C₂×M₁₆×C₂₃	1	2	yes	—
555	C₈	C₂₃×#137	1	2	yes	complemented
556	C₂²	—	2	1	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	1	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	4	1	no	—
561	C₂²	—	4	1	no	—
562	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
563	C₄	—	2	1	no	—
564	C₄	—	2	2	no	—
565	C₄	—	2	1	no	—
566	C₄	—	2	1	no	—
567	C₄	—	2	1	no	—
568	C₄	—	4	1	no	—
569	C₄	—	4	1	no	—
570	C₄	—	4	1	no	—
571	C₄	—	4	1	no	—
572	C₄	—	4	1	no	—
573	C₄	—	4	1	no	—
574	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
575	C₄	C₂₃×#568	1	1	yes	characteristic
576	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
577	C₄	C₂₃×#575	1	1	yes	characteristic
578	C₂	—	4	1	no	—
579	C₂	—	4	2	no	—
580	C₂	D₄×M₁₆×C₂₃	1	1	yes	characteristic
581	C₂	C₄×C₂₃×#137	1	1	yes	characteristic
582	C₂	C₂₃×#2956	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₄×M₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆×C₂₃	C₂	semidirect
4. C₂₃×#1593	C₂	semidirect
5. C₂₃×#1591	C₂	semidirect
6. C₂₃×#1587	C₂	semidirect
7. C₂₃×#1585	C₂	semidirect
8. C₂₃×#1584	C₂	semidirect
9. C₂₃×#1582	C₂	semidirect
10. C₂₃×SD₁₆⋊C₈	C₂	semidirect
11. C₈×SD₁₆×C₂₃	C₂	semidirect
12. C₂₃×#1569	C₂	semidirect
13. C₂₃×#1491	C₂	semidirect
15. C₂₃×#1455	C₂	semidirect
166. SD₁₆×C₂₃	M₁₆	semidirect
214. #18047	C₂₃	direct (N × H)
264. C₈×C₂₃	#137	semidirect
266. Q₈×M₁₆	C₂×C₂₃	semidirect
268. #1593	C₂×C₂₃	semidirect
269. #1591	C₂×C₂₃	semidirect
270. #1587	C₂×C₂₃	semidirect
271. #1585	C₂×C₂₃	semidirect
272. #1584	C₂×C₂₃	semidirect
273. #1582	C₂×C₂₃	semidirect
274. SD₁₆⋊C₈	C₂×C₂₃	semidirect
275. C₈×SD₁₆	C₂×C₂₃	semidirect
276. #1569	C₂×C₂₃	semidirect
277. #1491	C₂×C₂₃	semidirect
279. #1455	C₂×C₂₃	semidirect
436. C₂₃	#18047	direct (N × H)
458. SD₁₆	M₁₆×C₂₃	semidirect
555. C₈	C₂₃×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2970	2
Q₈×M₁₆×C₂₃	2
C₄×C₂₃×#137	2
C₂₃×#1593	2
C₂₃×#1591	2
C₂₃×#1587	2
C₂₃×#1585	2
C₂₃×#1584	2
C₂₃×#1582	2
C₂₃×SD₁₆⋊C₈	2
C₈×SD₁₆×C₂₃	2
C₂₃×#1569	2
C₂₃×#1491	2
C₂₃×#1489	2
C₂₃×#1455	2
#18047	23

Subgroups — C23×#18047 #19165673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18047 #19165673