Subgroups — C₂₃×#17772 #19165398

Order 5888.

924 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3511	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2991	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2990	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1560	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1557	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1554	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1550	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1546	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1543	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1540	C₂	1	2	yes	maximal, complemented
12	C₂₃×#556	—	2	1	no	—
13	C₂₃×#555	—	2	1	no	—
14	C₂₃×#554	—	2	1	no	—
15	C₂₃×#553	—	2	2	no	—
16	C₂₃×#548	—	2	2	no	—
17	C₂₃×#547	—	2	1	no	—
18	C₂₃×#542	C₄	1	4	yes	—
19	C₂₃×#537	C₄	1	2	yes	—
20	C₂₃×#533	C₂²	1	2	yes	—
21	C₂₃×#532	C₂²	1	1	yes	characteristic
22	C₂₃×#532	C₂²	1	2	yes	—
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂₃×#520	C₄	1	2	yes	—
25	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#446	—	2	1	no	—
27	C₈×Q₈×C₂₃	—	2	1	no	—
28	C₂₃×#436	—	2	2	no	—
29	C₂₃×#436	—	2	1	no	—
30	C₂₃×#435	—	2	2	no	—
31	C₈×D₄×C₂₃	—	2	1	no	—
32	C₂₃×#432	C₂²	1	2	yes	—
33	C₂₃×#431	C₂²	1	1	yes	characteristic
34	C₂₃×#431	C₂²	1	2	yes	—
35	C₂₃×#431	C₂²	1	1	yes	characteristic
36	C₂₃×#423	C₂²	1	1	yes	characteristic
37	C₂₃×#423	C₂²	1	1	yes	characteristic
38	C₂₃×#423	C₂²	1	2	yes	—
39	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
40	C₂₃×#333	C₂²	1	1	yes	characteristic
41	C₂₃×#333	C₂²	1	1	yes	characteristic
42	C₂₃×#333	C₂²	1	2	yes	—
43	C₂₃×#332	C₂²	1	2	yes	—
44	C₂₃×#332	C₂²	1	2	yes	—
45	C₂₃×#332	C₂²	1	2	yes	—
46	C₂₃×#332	C₂²	1	2	yes	—
47	C₂₃×#331	C₂²	1	1	yes	characteristic
48	C₂₃×#331	C₂²	1	2	yes	—
49	C₂₃×#331	C₂²	1	1	yes	characteristic
50	C₂×#41×C₂₃	—	2	2	no	—
51	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
52	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×Q₈×C₂₃	—	2	1	no	—
54	C₂×M₁₆×C₂₃	—	2	1	no	—
55	C₂×M₁₆×C₂₃	—	2	1	no	—
56	C₂×M₁₆×C₂₃	—	2	2	no	—
57	C₂²×C₈×C₂₃	—	2	2	no	—
58	C₂₃×C₄⋊Q₈	—	2	1	no	—
59	C₂₃×C₄⋊Q₈	—	2	1	no	—
60	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
61	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
63	C₂₃×#127	C₂³	1	1	yes	characteristic
64	C₂₃×#126	—	2	4	no	—
65	C₂₃×#125	—	2	2	no	—
66	C₂₃×#124	—	2	4	no	—
67	C₂₃×#124	—	2	2	no	—
68	C₂₃×#124	C₂×C₄	1	2	yes	—
69	C₂₃×#123	—	2	4	no	—
70	C₂₃×C₂²⋊Q₈	—	2	2	no	—
71	C₂₃×C₂²⋊Q₈	—	2	2	no	—
72	C₂₃×C₂²⋊Q₈	—	2	4	no	—
73	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
74	C₂₃×#121	—	2	2	no	—
75	C₂₃×#121	—	2	4	no	—
76	C₂₃×#121	—	2	1	no	—
77	C₂₃×#121	—	2	1	no	—
78	C₂₃×#121	—	2	2	no	—
79	C₂₃×#121	C₂×C₄	1	4	yes	—
80	C₄×Q₈×C₂₃	—	2	2	no	—
81	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
82	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
83	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
84	C₄×D₄×C₂₃	—	2	4	no	—
85	C₄×D₄×C₂₃	—	2	1	no	—
86	C₄×D₄×C₂₃	—	2	2	no	—
87	C₄×D₄×C₂₃	—	2	1	no	—
88	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
89	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
90	C₄×D₄×C₂₃	D₄	1	2	yes	—
91	C₄×D₄×C₂₃	D₄	1	2	yes	—
92	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
93	C₂₃×#117	—	2	2	no	—
94	C₂₃×#117	—	2	2	no	—
95	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
96	C₂₃×#117	C₂×C₄	1	2	yes	—
97	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
100	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂₃×C₄⋊C₈	—	2	2	no	—
103	C₂₃×C₄⋊C₈	—	2	1	no	—
104	C₂₃×C₄⋊C₈	—	2	2	no	—
105	C₂₃×C₄⋊C₈	—	2	1	no	—
106	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
107	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
108	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
109	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
111	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₂₃×C₂²⋊C₈	—	2	1	no	—
113	C₂₃×C₂²⋊C₈	—	2	1	no	—
114	C₂₃×C₂²⋊C₈	—	2	2	no	—
115	C₂₃×C₂²⋊C₈	—	2	2	no	—
116	C₂₃×#97	—	2	2	no	—
117	C₄×C₈×C₂₃	—	2	1	no	—
118	C₄×C₈×C₂₃	—	2	2	no	—
119	C₄×C₈×C₂₃	—	2	1	no	—
120	#41×C₂₃	—	4	2	no	—
121	#41×C₂₃	—	4	4	no	—
122	#41×C₂₃	—	8	2	no	—
123	#41×C₂₃	—	8	2	no	—
124	C₂×Q₈×C₂₃	—	2	2	no	—
125	C₂×Q₈×C₂₃	—	2	2	no	—
126	C₂×Q₈×C₂₃	—	8	1	no	—
127	C₂×Q₈×C₂₃	—	8	1	no	—
128	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
129	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
130	C₂×D₄×C₂₃	—	2	2	no	—
131	C₂×D₄×C₂₃	—	2	4	no	—
132	C₂×D₄×C₂₃	—	2	1	no	—
133	C₂×D₄×C₂₃	—	2	1	no	—
134	C₂×D₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
135	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
136	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
137	C₂²×C₄×C₂₃	—	2	2	no	—
138	C₂²×C₄×C₂₃	—	2	2	no	—
139	C₂²×C₄×C₂₃	—	2	4	no	—
140	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
141	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
142	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
146	M₁₆×C₂₃	—	8	1	no	—
147	M₁₆×C₂₃	—	8	1	no	—
148	M₁₆×C₂₃	—	8	2	no	—
149	C₂×C₈×C₂₃	—	2	1	no	—
150	C₂×C₈×C₂₃	—	2	2	no	—
151	C₂×C₈×C₂₃	—	2	1	no	—
152	C₂×C₈×C₂₃	—	2	2	no	—
153	C₂×C₈×C₂₃	—	2	1	no	—
154	C₂×C₈×C₂₃	—	2	1	no	—
155	C₂×C₈×C₂₃	—	8	2	no	—
156	C₄⋊C₄×C₂₃	—	2	2	no	—
157	C₄⋊C₄×C₂₃	—	2	4	no	—
158	C₄⋊C₄×C₂₃	—	2	2	no	—
159	C₄⋊C₄×C₂₃	—	2	4	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	C₂²⋊C₄	1	1	yes	characteristic
164	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
165	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
166	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
167	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
168	C₂²⋊C₄×C₂₃	—	2	4	no	—
169	C₂²⋊C₄×C₂₃	—	2	4	no	—
170	C₂²⋊C₄×C₂₃	—	2	4	no	—
171	C₂²⋊C₄×C₂₃	—	2	2	no	—
172	C₂²⋊C₄×C₂₃	—	2	2	no	—
173	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
174	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
175	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
176	C₄²×C₂₃	—	2	4	no	—
177	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
178	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₂₃	C₂×D₄	1	2	yes	—
180	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
181	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₄²×C₂₃	C₂×D₄	1	2	yes	—
183	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
184	#17772	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂³×C₂₃	—	2	2	no	—
186	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
187	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	Q₈×C₂₃	—	2	2	no	—
189	Q₈×C₂₃	—	4	2	no	—
190	Q₈×C₂₃	—	4	2	no	—
191	Q₈×C₂₃	—	4	2	no	—
192	Q₈×C₂₃	—	8	2	no	—
193	D₄×C₂₃	—	2	2	no	—
194	D₄×C₂₃	—	4	2	no	—
195	D₄×C₂₃	—	4	2	no	—
196	D₄×C₂₃	—	4	4	no	—
197	D₄×C₂₃	—	4	1	no	—
198	D₄×C₂₃	—	4	1	no	—
199	D₄×C₂₃	—	4	1	no	—
200	D₄×C₂₃	—	4	1	no	—
201	D₄×C₂₃	—	8	4	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	1	no	—
204	C₂×C₄×C₂₃	—	2	1	no	—
205	C₂×C₄×C₂₃	—	2	2	no	—
206	C₂×C₄×C₂₃	—	2	1	no	—
207	C₂×C₄×C₂₃	—	2	2	no	—
208	C₂×C₄×C₂₃	—	2	1	no	—
209	C₂×C₄×C₂₃	—	2	2	no	—
210	C₂×C₄×C₂₃	—	2	2	no	—
211	C₂×C₄×C₂₃	—	2	2	no	—
212	C₂×C₄×C₂₃	—	2	4	no	—
213	C₂×C₄×C₂₃	—	4	2	no	—
214	C₂×C₄×C₂₃	—	4	4	no	—
215	C₂×C₄×C₂₃	—	8	4	no	—
216	C₂×C₄×C₂₃	—	8	2	no	—
217	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
218	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
221	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
222	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
224	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	C₈×C₂₃	—	4	1	no	—
227	C₈×C₂₃	—	4	2	no	—
228	C₈×C₂₃	—	4	1	no	—
229	C₈×C₂₃	—	4	1	no	—
230	C₈×C₂₃	—	4	2	no	—
231	C₈×C₂₃	—	4	1	no	—
232	#3511	C₂×C₂₃	1	1	yes	characteristic
233	#2991	C₂×C₂₃	1	1	yes	characteristic, complemented
234	#2990	C₂×C₂₃	1	1	yes	characteristic, complemented
235	#1560	C₂×C₂₃	1	1	yes	characteristic, complemented
236	#1557	C₂×C₂₃	1	2	yes	complemented
237	#1554	C₂×C₂₃	1	2	yes	complemented
238	#1550	C₂×C₂₃	1	2	yes	complemented
239	#1549	C₂×C₂₃	1	1	yes	characteristic, complemented
240	#1546	C₂×C₂₃	1	1	yes	characteristic, complemented
241	#1543	C₂×C₂₃	1	1	yes	characteristic, complemented
242	#1540	C₂×C₂₃	1	2	yes	complemented
243	C₂²×C₂₃	—	2	1	no	—
244	C₂²×C₂₃	—	2	2	no	—
245	C₂²×C₂₃	—	2	1	no	—
246	C₂²×C₂₃	—	2	2	no	—
247	C₂²×C₂₃	—	2	2	no	—
248	C₂²×C₂₃	—	2	1	no	—
249	C₂²×C₂₃	—	4	2	no	—
250	C₂²×C₂₃	—	4	2	no	—
251	C₂²×C₂₃	—	4	2	no	—
252	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
253	C₄×C₂₃	—	2	1	no	—
254	C₄×C₂₃	—	2	2	no	—
255	C₄×C₂₃	—	2	2	no	—
256	C₄×C₂₃	—	2	1	no	—
257	C₄×C₂₃	—	2	2	no	—
258	C₄×C₂₃	—	2	1	no	—
259	C₄×C₂₃	—	4	2	no	—
260	C₄×C₂₃	—	4	4	no	—
261	C₄×C₂₃	#413	1	2	yes	—
262	C₄×C₂₃	#576	1	1	yes	characteristic
263	C₄×C₂₃	#577	1	1	yes	characteristic
264	C₄×C₂₃	#577	1	1	yes	characteristic
265	C₄×C₂₃	#578	1	1	yes	characteristic
266	#556	—	2	1	no	—
267	#555	—	2	1	no	—
268	#554	—	2	1	no	—
269	#553	—	2	2	no	—
270	#548	—	2	2	no	—
271	#547	—	2	1	no	—
272	#542	C₄×C₂₃	1	4	yes	—
273	#537	C₄×C₂₃	1	2	yes	—
274	#533	C₂²×C₂₃	1	2	yes	—
275	#532	C₂²×C₂₃	1	1	yes	characteristic
276	#532	C₂²×C₂₃	1	2	yes	—
277	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
278	#520	C₄×C₂₃	1	2	yes	—
279	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
280	#446	—	2	1	no	—
281	C₈×Q₈	—	2	1	no	—
282	#436	—	2	2	no	—
283	#436	—	2	1	no	—
284	#435	—	2	2	no	—
285	C₈×D₄	—	2	1	no	—
286	#432	C₂²×C₂₃	1	2	yes	—
287	#431	C₂²×C₂₃	1	2	yes	—
288	#431	C₂²×C₂₃	1	1	yes	characteristic
289	#431	C₂²×C₂₃	1	1	yes	characteristic
290	#423	C₂²×C₂₃	1	1	yes	characteristic
291	#423	C₂²×C₂₃	1	1	yes	characteristic
292	#423	C₂²×C₂₃	1	2	yes	—
293	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
294	#333	C₂²×C₂₃	1	1	yes	characteristic
295	#333	C₂²×C₂₃	1	1	yes	characteristic
296	#333	C₂²×C₂₃	1	2	yes	—
297	#332	C₂²×C₂₃	1	2	yes	—
298	#332	C₂²×C₂₃	1	2	yes	—
299	#332	C₂²×C₂₃	1	2	yes	—
300	#332	C₂²×C₂₃	1	2	yes	—
301	#331	C₂²×C₂₃	1	1	yes	characteristic
302	#331	C₂²×C₂₃	1	2	yes	—
303	#331	C₂²×C₂₃	1	1	yes	characteristic
304	C₂×C₂₃	—	4	1	no	—
305	C₂×C₂₃	—	4	2	no	—
306	C₂×C₂₃	—	8	2	no	—
307	C₂×C₂₃	C₂×#413	1	1	yes	characteristic
308	C₂×C₂₃	#2897	1	1	yes	characteristic
309	C₂×C₂₃	#2898	1	1	yes	characteristic
310	C₂×#41	—	2	2	no	—
311	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
312	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
313	C₂²×Q₈	—	2	1	no	—
314	C₂×M₁₆	—	2	1	no	—
315	C₂×M₁₆	—	2	1	no	—
316	C₂×M₁₆	—	2	2	no	—
317	C₂²×C₈	—	2	2	no	—
318	C₄⋊Q₈	—	2	1	no	—
319	C₄⋊Q₈	—	2	1	no	—
320	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
321	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
322	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
323	#127	C₂³×C₂₃	1	1	yes	characteristic
324	#126	—	2	4	no	—
325	#125	—	2	2	no	—
326	#124	—	2	4	no	—
327	#124	—	2	2	no	—
328	#124	C₂×C₄×C₂₃	1	2	yes	—
329	#123	—	2	4	no	—
330	C₂²⋊Q₈	—	2	2	no	—
331	C₂²⋊Q₈	—	2	2	no	—
332	C₂²⋊Q₈	—	2	4	no	—
333	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
334	#121	—	2	2	no	—
335	#121	—	2	4	no	—
336	#121	—	2	1	no	—
337	#121	—	2	1	no	—
338	#121	—	2	2	no	—
339	#121	C₂×C₄×C₂₃	1	4	yes	—
340	C₄×Q₈	—	2	2	no	—
341	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
342	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
343	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
344	C₄×D₄	—	2	4	no	—
345	C₄×D₄	—	2	1	no	—
346	C₄×D₄	—	2	2	no	—
347	C₄×D₄	—	2	1	no	—
348	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
349	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
350	C₄×D₄	D₄×C₂₃	1	2	yes	—
351	C₄×D₄	D₄×C₂₃	1	2	yes	—
352	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
353	#117	—	2	2	no	—
354	#117	—	2	2	no	—
355	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
356	#117	C₂×C₄×C₂₃	1	2	yes	—
357	C₂×C₄⋊C₄	—	2	1	no	—
358	C₂×C₄⋊C₄	—	2	2	no	—
359	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
360	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
362	C₄⋊C₈	—	2	2	no	—
363	C₄⋊C₈	—	2	1	no	—
364	C₄⋊C₈	—	2	2	no	—
365	C₄⋊C₈	—	2	1	no	—
366	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
367	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
368	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
369	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
370	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
371	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
372	C₂²⋊C₈	—	2	1	no	—
373	C₂²⋊C₈	—	2	1	no	—
374	C₂²⋊C₈	—	2	2	no	—
375	C₂²⋊C₈	—	2	2	no	—
376	#97	—	2	2	no	—
377	C₄×C₈	—	2	1	no	—
378	C₄×C₈	—	2	2	no	—
379	C₄×C₈	—	2	1	no	—
380	C₂₃	#17772	1	1	yes	characteristic, direct product factor, Sylow
381	#41	—	4	2	no	—
382	#41	—	4	4	no	—
383	#41	—	8	2	no	—
384	#41	—	8	2	no	—
385	C₂×Q₈	—	2	2	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	8	1	no	—
388	C₂×Q₈	—	8	1	no	—
389	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
390	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	4	no	—
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
396	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
397	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
402	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
403	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
405	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
406	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
407	M₁₆	—	8	1	no	—
408	M₁₆	—	8	1	no	—
409	M₁₆	—	8	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	8	2	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
425	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
426	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
427	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
428	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
429	C₂²⋊C₄	—	2	4	no	—
430	C₂²⋊C₄	—	2	4	no	—
431	C₂²⋊C₄	—	2	4	no	—
432	C₂²⋊C₄	—	2	2	no	—
433	C₂²⋊C₄	—	2	2	no	—
434	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
435	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
436	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
437	C₄²	—	2	4	no	—
438	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
439	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
440	C₄²	C₂×D₄×C₂₃	1	2	yes	—
441	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
442	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
443	C₄²	C₂×D₄×C₂₃	1	2	yes	—
444	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
445	C₂³	—	2	2	no	—
446	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
447	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
448	Q₈	—	2	2	no	—
449	Q₈	—	4	2	no	—
450	Q₈	—	4	2	no	—
451	Q₈	—	4	2	no	—
452	Q₈	—	8	2	no	—
453	D₄	—	2	2	no	—
454	D₄	—	4	2	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	4	no	—
457	D₄	—	4	1	no	—
458	D₄	—	4	1	no	—
459	D₄	—	4	1	no	—
460	D₄	—	4	1	no	—
461	D₄	—	8	4	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	8	4	no	—
476	C₂×C₄	—	8	2	no	—
477	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
478	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
479	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
480	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
481	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
482	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
483	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
486	C₈	—	4	1	no	—
487	C₈	—	4	2	no	—
488	C₈	—	4	2	no	—
489	C₈	—	4	1	no	—
490	C₈	—	4	1	no	—
491	C₈	—	4	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	2	no	—
494	C₂²	—	2	1	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	4	2	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	2	no	—
501	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
502	C₄	—	2	1	no	—
503	C₄	—	2	2	no	—
504	C₄	—	2	2	no	—
505	C₄	—	2	1	no	—
506	C₄	—	2	2	no	—
507	C₄	—	2	1	no	—
508	C₄	—	4	2	no	—
509	C₄	—	4	4	no	—
510	C₄	C₂₃×#413	1	2	yes	—
511	C₄	C₂₃×#576	1	1	yes	characteristic
512	C₄	C₂₃×#577	1	1	yes	characteristic
513	C₄	C₂₃×#577	1	1	yes	characteristic
514	C₄	C₂₃×#578	1	1	yes	characteristic
515	C₂	—	4	1	no	—
516	C₂	—	4	2	no	—
517	C₂	—	8	2	no	—
518	C₂	C₂×C₂₃×#413	1	1	yes	characteristic
519	C₂	C₂₃×#2897	1	1	yes	characteristic
520	C₂	C₂₃×#2898	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#3511	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2991	C₂	semidirect
3. C₂₃×#2990	C₂	semidirect
4. C₂₃×#1560	C₂	semidirect
5. C₂₃×#1557	C₂	semidirect
6. C₂₃×#1554	C₂	semidirect
7. C₂₃×#1550	C₂	semidirect
8. C₂₃×#1549	C₂	semidirect
9. C₂₃×#1546	C₂	semidirect
10. C₂₃×#1543	C₂	semidirect
11. C₂₃×#1540	C₂	semidirect
184. #17772	C₂₃	direct (N × H)
233. #2991	C₂×C₂₃	semidirect
234. #2990	C₂×C₂₃	semidirect
235. #1560	C₂×C₂₃	semidirect
236. #1557	C₂×C₂₃	semidirect
237. #1554	C₂×C₂₃	semidirect
238. #1550	C₂×C₂₃	semidirect
239. #1549	C₂×C₂₃	semidirect
240. #1546	C₂×C₂₃	semidirect
241. #1543	C₂×C₂₃	semidirect
242. #1540	C₂×C₂₃	semidirect
380. C₂₃	#17772	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3511	2
C₂₃×#2991	2
C₂₃×#2990	2
C₂₃×#1560	2
C₂₃×#1557	2
C₂₃×#1554	2
C₂₃×#1550	2
C₂₃×#1549	2
C₂₃×#1546	2
C₂₃×#1543	2
C₂₃×#1540	2
#17772	23

Subgroups — C23×#17772 #19165398

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17772 #19165398