Subgroups — C₂₃×#60944 #19208570

Order 5888.

1846 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3444	C₂	1	2	yes	maximal, complemented
2	C₂₃×#3250	C₂	1	4	yes	maximal, complemented
3	C₂₃×#3231	C₂	1	2	yes	maximal, complemented
4	C₂₃×#3228	C₂	1	2	yes	maximal, complemented
5	C₂₃×#3210	C₂	1	2	yes	maximal, complemented
6	C₂₃×#3207	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3191	C₂	1	4	yes	maximal, complemented
8	C₂₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3167	C₂	1	2	yes	maximal, complemented
11	C₂₃×#3102	C₂	1	8	yes	maximal, complemented
12	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#533	C₂²	1	2	yes	—
14	C₂₃×#533	C₂²	1	2	yes	—
15	C₂₃×#533	C₂²	1	2	yes	—
16	C₂₃×#532	C₂²	1	2	yes	—
17	C₂₃×#532	C₂²	1	2	yes	—
18	C₂₃×#532	C₂²	1	2	yes	—
19	C₂₃×#529	—	2	2	no	—
20	C₂₃×#529	—	2	2	no	—
21	C₂₃×#529	—	2	2	no	—
22	C₂₃×#529	—	2	2	no	—
23	C₂₃×#529	C₂²	1	4	yes	—
24	C₂₃×#529	C₂²	1	4	yes	—
25	C₂₃×#525	—	2	2	no	—
26	C₂₃×#525	—	2	2	no	—
27	C₂₃×#525	C₂²	1	4	yes	—
28	C₄×#41×C₂₃	—	2	2	no	—
29	C₄×#41×C₂₃	—	2	2	no	—
30	C₄×#41×C₂₃	C₂²	1	4	yes	—
31	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#500	C₂²	1	1	yes	characteristic, complemented
33	C₂₃×#500	C₂²	1	1	yes	characteristic, complemented
34	C₂₃×#498	C₂²	1	2	yes	complemented
35	C₂₃×#494	C₂²	1	1	yes	characteristic, complemented
36	C₂₃×#493	C₂²	1	1	yes	characteristic, complemented
37	C₂₃×#492	C₂²	1	2	yes	complemented
38	C₂₃×#487	C₂²	1	2	yes	complemented
39	C₂₃×#487	C₂²	1	2	yes	complemented
40	C₂₃×#486	C₂²	1	2	yes	complemented
41	C₂₃×#486	C₂²	1	2	yes	complemented
42	C₂₃×#485	C₂²	1	8	yes	—
43	C₂₃×#482	C₂²	1	8	yes	—
44	C₂₃×#479	C₂²	1	8	yes	complemented
45	C₂₃×#478	C₂²	1	8	yes	complemented
46	C₂₃×#467	C₂²	1	4	yes	—
47	C₂₃×#466	C₂²	1	4	yes	—
48	C₂₃×#465	C₂²	1	8	yes	—
49	C₂₃×#464	C₂²	1	8	yes	complemented
50	C₂₃×#463	C₂²	1	8	yes	complemented
51	C₂₃×#452	C₂²	1	8	yes	—
52	C₂₃×#449	C₂²	1	8	yes	—
53	C₄×Q₁₆×C₂₃	C₂²	1	4	yes	complemented
54	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
55	C₄×D₈×C₂₃	C₂²	1	4	yes	complemented
56	C₂₃×#431	C₂²	1	2	yes	—
57	C₂₃×#431	C₂²	1	2	yes	—
58	C₂₃×#431	C₂²	1	4	yes	—
59	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
60	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
61	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
62	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
63	C₂×#41×C₂₃	—	2	2	no	—
64	C₂×#41×C₂₃	—	2	2	no	—
65	C₂×#41×C₂₃	C₂³	1	4	yes	—
66	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
67	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
68	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
70	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
71	C₂×D₈×C₂₃	C₂³	1	4	yes	—
72	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
73	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
74	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
75	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
76	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
77	C₂₃×#127	C₂³	1	2	yes	—
78	C₂₃×#126	—	2	4	no	—
79	C₂₃×#126	—	2	4	no	—
80	C₂₃×#125	—	2	2	no	—
81	C₂₃×#125	—	2	2	no	—
82	C₂₃×#125	C₂³	1	4	yes	—
83	C₂₃×#124	—	2	2	no	—
84	C₂₃×#124	—	2	2	no	—
85	C₂₃×#124	C₂³	1	4	yes	—
86	C₂₃×#123	—	2	4	no	—
87	C₂₃×#123	—	2	4	no	—
88	C₂₃×#123	—	2	4	no	—
89	C₂₃×C₂²⋊Q₈	—	2	2	no	—
90	C₂₃×C₂²⋊Q₈	—	2	2	no	—
91	C₂₃×C₂²⋊Q₈	—	2	2	no	—
92	C₂₃×C₂²⋊Q₈	—	2	2	no	—
93	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
94	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
95	C₂₃×#121	—	2	2	no	—
96	C₂₃×#121	—	2	2	no	—
97	C₂₃×#121	—	2	2	no	—
98	C₂₃×#121	—	2	2	no	—
99	C₂₃×#121	C₂³	1	4	yes	—
100	C₂₃×#121	C₂³	1	4	yes	—
101	C₂₃×#120	—	2	4	no	—
102	C₄×Q₈×C₂₃	—	2	2	no	—
103	C₄×Q₈×C₂₃	—	2	2	no	—
104	C₄×Q₈×C₂₃	—	2	2	no	—
105	C₄×Q₈×C₂₃	—	2	2	no	—
106	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
107	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
108	C₄×D₄×C₂₃	—	2	4	no	—
109	C₄×D₄×C₂₃	—	2	4	no	—
110	C₄×D₄×C₂₃	—	2	4	no	—
111	C₄×D₄×C₂₃	—	2	4	no	—
112	C₄×D₄×C₂₃	—	2	4	no	—
113	C₄×D₄×C₂₃	—	2	4	no	—
114	C₄×D₄×C₂₃	—	2	2	no	—
115	C₄×D₄×C₂₃	—	2	2	no	—
116	C₄×D₄×C₂₃	—	2	2	no	—
117	C₄×D₄×C₂₃	—	2	2	no	—
118	C₄×D₄×C₂₃	C₂³	1	4	yes	—
119	C₄×D₄×C₂₃	C₂³	1	4	yes	—
120	C₂₃×#117	—	2	2	no	—
121	C₂₃×#117	—	2	2	no	—
122	C₂₃×#117	—	2	4	no	—
123	C₂₃×#117	—	2	4	no	—
124	C₂₃×#117	—	2	4	no	—
125	C₂₃×#117	C₂³	1	4	yes	—
126	C₂×C₄²×C₂₃	—	2	4	no	—
127	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₂₃	D₄	1	2	yes	—
129	C₂×C₄²×C₂₃	D₄	1	2	yes	—
130	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
131	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
134	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
135	C₂₃×#107	C₂³	1	4	yes	—
136	C₂₃×#107	C₂³	1	4	yes	—
137	C₂₃×#106	C₂³	1	8	yes	—
138	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
139	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
140	C₂₃×#103	C₂³	1	8	yes	—
141	C₂₃×#103	C₂³	1	8	yes	—
142	C₂₃×#102	C₂³	1	8	yes	—
143	C₂₃×#102	C₂³	1	8	yes	—
144	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
145	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
146	C₄×C₈×C₂₃	C₂³	1	2	yes	—
147	C₄×C₈×C₂₃	C₂³	1	2	yes	—
148	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
149	C₄×C₈×C₂₃	C₂³	1	2	yes	—
150	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
151	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
152	#41×C₂₃	—	4	8	no	—
153	#41×C₂₃	—	4	8	no	—
154	#41×C₂₃	—	4	8	no	—
155	C₂×Q₈×C₂₃	—	2	2	no	—
156	C₂×Q₈×C₂₃	—	2	2	no	—
157	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
158	C₂×D₄×C₂₃	—	2	4	no	—
159	C₂×D₄×C₂₃	—	2	4	no	—
160	C₂×D₄×C₂₃	—	2	4	no	—
161	C₂×D₄×C₂₃	—	2	2	no	—
162	C₂×D₄×C₂₃	—	2	2	no	—
163	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₂₃	—	2	4	no	—
165	C₂²×C₄×C₂₃	—	2	4	no	—
166	C₂²×C₄×C₂₃	—	2	4	no	—
167	C₂²×C₄×C₂₃	—	2	4	no	—
168	C₂²×C₄×C₂₃	—	4	2	no	—
169	C₂²×C₄×C₂₃	—	4	2	no	—
170	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
182	Q₁₆×C₂₃	—	4	4	no	—
183	SD₁₆×C₂₃	—	4	8	no	—
184	D₈×C₂₃	—	4	4	no	—
185	C₂×C₈×C₂₃	—	4	2	no	—
186	C₂×C₈×C₂₃	—	4	2	no	—
187	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
188	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
189	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
190	C₄⋊C₄×C₂₃	—	2	2	no	—
191	C₄⋊C₄×C₂₃	—	2	2	no	—
192	C₄⋊C₄×C₂₃	—	2	2	no	—
193	C₄⋊C₄×C₂₃	—	2	2	no	—
194	C₄⋊C₄×C₂₃	—	2	4	no	—
195	C₄⋊C₄×C₂₃	—	2	4	no	—
196	C₄⋊C₄×C₂₃	—	2	4	no	—
197	C₄⋊C₄×C₂₃	—	2	4	no	—
198	C₄⋊C₄×C₂₃	—	2	4	no	—
199	C₄⋊C₄×C₂₃	—	2	4	no	—
200	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
201	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
202	C₂²⋊C₄×C₂₃	—	2	4	no	—
203	C₂²⋊C₄×C₂₃	—	2	4	no	—
204	C₂²⋊C₄×C₂₃	—	2	4	no	—
205	C₂²⋊C₄×C₂₃	—	2	4	no	—
206	C₂²⋊C₄×C₂₃	—	2	4	no	—
207	C₂²⋊C₄×C₂₃	—	2	4	no	—
208	C₂²⋊C₄×C₂₃	—	2	4	no	—
209	C₂²⋊C₄×C₂₃	—	2	4	no	—
210	C₂²⋊C₄×C₂₃	—	2	4	no	—
211	C₄²×C₂₃	—	2	4	no	—
212	C₄²×C₂₃	—	2	4	no	—
213	C₄²×C₂₃	—	2	4	no	—
214	C₄²×C₂₃	C₂×D₄	1	2	yes	—
215	C₄²×C₂₃	C₂×D₄	1	2	yes	—
216	C₄²×C₂₃	C₂×D₄	1	2	yes	—
217	C₄²×C₂₃	C₂×D₄	1	2	yes	—
218	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₂₃	C₂×D₄	1	2	yes	—
220	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
221	C₄²×C₂₃	C₂⁴	1	2	yes	—
222	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
223	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
224	#60944	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂³×C₂₃	—	2	2	no	—
226	C₂³×C₂₃	—	2	4	no	—
227	C₂³×C₂₃	—	4	1	no	—
228	C₂³×C₂₃	—	4	1	no	—
229	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
230	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
231	Q₈×C₂₃	—	2	8	no	—
232	Q₈×C₂₃	—	4	4	no	—
233	Q₈×C₂₃	—	4	4	no	—
234	D₄×C₂₃	—	2	8	no	—
235	D₄×C₂₃	—	4	8	no	—
236	D₄×C₂₃	—	4	8	no	—
237	D₄×C₂₃	—	4	8	no	—
238	D₄×C₂₃	—	4	4	no	—
239	D₄×C₂₃	—	4	4	no	—
240	C₂×C₄×C₂₃	—	2	2	no	—
241	C₂×C₄×C₂₃	—	2	4	no	—
242	C₂×C₄×C₂₃	—	2	4	no	—
243	C₂×C₄×C₂₃	—	2	2	no	—
244	C₂×C₄×C₂₃	—	2	4	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	4	no	—
247	C₂×C₄×C₂₃	—	2	4	no	—
248	C₂×C₄×C₂₃	—	2	4	no	—
249	C₂×C₄×C₂₃	—	2	4	no	—
250	C₂×C₄×C₂₃	—	4	1	no	—
251	C₂×C₄×C₂₃	—	4	2	no	—
252	C₂×C₄×C₂₃	—	4	1	no	—
253	C₂×C₄×C₂₃	—	4	2	no	—
254	C₂×C₄×C₂₃	—	4	8	no	—
255	C₂×C₄×C₂₃	—	4	8	no	—
256	C₂×C₄×C₂₃	—	4	8	no	—
257	C₂×C₄×C₂₃	#135	1	4	yes	—
258	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
262	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
267	C₈×C₂₃	—	2	2	no	—
268	C₈×C₂₃	—	2	4	no	—
269	C₈×C₂₃	—	2	2	no	—
270	#3444	C₂×C₂₃	1	2	yes	complemented
271	#3250	C₂×C₂₃	1	4	yes	complemented
272	#3231	C₂×C₂₃	1	2	yes	complemented
273	#3228	C₂×C₂₃	1	2	yes	complemented
274	#3210	C₂×C₂₃	1	2	yes	complemented
275	#3207	C₂×C₂₃	1	2	yes	complemented
276	#3191	C₂×C₂₃	1	4	yes	complemented
277	#3170	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#3169	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#3167	C₂×C₂₃	1	2	yes	complemented
280	#3102	C₂×C₂₃	1	8	yes	complemented
281	C₂×#431	C₂×C₂₃	1	1	yes	characteristic, complemented
282	C₂²×C₂₃	—	2	1	no	—
283	C₂²×C₂₃	—	2	1	no	—
284	C₂²×C₂₃	—	2	2	no	—
285	C₂²×C₂₃	—	2	2	no	—
286	C₂²×C₂₃	—	2	2	no	—
287	C₂²×C₂₃	—	4	4	no	—
288	C₂²×C₂₃	—	4	4	no	—
289	C₂²×C₂₃	—	4	4	no	—
290	C₂²×C₂₃	—	4	4	no	—
291	C₂²×C₂₃	C₂×#135	1	2	yes	—
292	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
293	C₄×C₂₃	—	2	2	no	—
294	C₄×C₂₃	—	2	1	no	—
295	C₄×C₂₃	—	2	2	no	—
296	C₄×C₂₃	—	2	2	no	—
297	C₄×C₂₃	—	2	1	no	—
298	C₄×C₂₃	—	4	4	no	—
299	C₄×C₂₃	—	4	4	no	—
300	C₄×C₂₃	—	4	4	no	—
301	C₄×C₂₃	C₂×#135	1	2	yes	—
302	C₄×C₂₃	C₂×#135	1	2	yes	—
303	C₄×C₂₃	#575	1	2	yes	—
304	C₄×C₂₃	#585	1	2	yes	—
305	#533	C₂²×C₂₃	1	2	yes	—
306	#533	C₂²×C₂₃	1	2	yes	—
307	#533	C₂²×C₂₃	1	2	yes	—
308	#532	C₂²×C₂₃	1	2	yes	—
309	#532	C₂²×C₂₃	1	2	yes	—
310	#532	C₂²×C₂₃	1	2	yes	—
311	#529	—	2	2	no	—
312	#529	—	2	2	no	—
313	#529	—	2	2	no	—
314	#529	—	2	2	no	—
315	#529	C₂²×C₂₃	1	4	yes	—
316	#529	C₂²×C₂₃	1	4	yes	—
317	#525	—	2	2	no	—
318	#525	—	2	2	no	—
319	#525	C₂²×C₂₃	1	4	yes	—
320	C₄×#41	—	2	2	no	—
321	C₄×#41	—	2	2	no	—
322	C₄×#41	C₂²×C₂₃	1	4	yes	—
323	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
324	#500	C₂²×C₂₃	1	1	yes	characteristic, complemented
325	#500	C₂²×C₂₃	1	1	yes	characteristic, complemented
326	#498	C₂²×C₂₃	1	2	yes	complemented
327	#494	C₂²×C₂₃	1	1	yes	characteristic, complemented
328	#493	C₂²×C₂₃	1	1	yes	characteristic, complemented
329	#492	C₂²×C₂₃	1	2	yes	complemented
330	#487	C₂²×C₂₃	1	2	yes	complemented
331	#487	C₂²×C₂₃	1	2	yes	complemented
332	#486	C₂²×C₂₃	1	2	yes	complemented
333	#486	C₂²×C₂₃	1	2	yes	complemented
334	#485	C₂²×C₂₃	1	8	yes	—
335	#482	C₂²×C₂₃	1	8	yes	—
336	#479	C₂²×C₂₃	1	8	yes	complemented
337	#478	C₂²×C₂₃	1	8	yes	complemented
338	#467	C₂²×C₂₃	1	4	yes	—
339	#466	C₂²×C₂₃	1	4	yes	—
340	#465	C₂²×C₂₃	1	8	yes	—
341	#464	C₂²×C₂₃	1	8	yes	complemented
342	#463	C₂²×C₂₃	1	8	yes	complemented
343	#452	C₂²×C₂₃	1	8	yes	—
344	#449	C₂²×C₂₃	1	8	yes	—
345	C₄×Q₁₆	C₂²×C₂₃	1	4	yes	complemented
346	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
347	C₄×D₈	C₂²×C₂₃	1	4	yes	complemented
348	#431	C₂²×C₂₃	1	4	yes	—
349	#431	C₂²×C₂₃	1	2	yes	—
350	#431	C₂²×C₂₃	1	2	yes	—
351	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
352	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
353	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×C₂₃	—	2	2	no	—
356	C₂×C₂₃	—	4	2	no	—
357	C₂×C₂₃	—	8	4	no	—
358	C₂×C₂₃	#3449	1	1	yes	characteristic
359	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
360	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
361	C₂×#41	—	2	2	no	—
362	C₂×#41	—	2	2	no	—
363	C₂×#41	C₂³×C₂₃	1	4	yes	—
364	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
365	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
366	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
367	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
368	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
369	C₂×D₈	C₂³×C₂₃	1	4	yes	—
370	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
371	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
372	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
373	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
374	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
375	#127	C₂³×C₂₃	1	2	yes	—
376	#126	—	2	4	no	—
377	#126	—	2	4	no	—
378	#125	—	2	2	no	—
379	#125	—	2	2	no	—
380	#125	C₂³×C₂₃	1	4	yes	—
381	#124	—	2	2	no	—
382	#124	—	2	2	no	—
383	#124	C₂³×C₂₃	1	4	yes	—
384	#123	—	2	4	no	—
385	#123	—	2	4	no	—
386	#123	—	2	4	no	—
387	C₂²⋊Q₈	—	2	2	no	—
388	C₂²⋊Q₈	—	2	2	no	—
389	C₂²⋊Q₈	—	2	2	no	—
390	C₂²⋊Q₈	—	2	2	no	—
391	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
392	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
393	#121	—	2	2	no	—
394	#121	—	2	2	no	—
395	#121	—	2	2	no	—
396	#121	—	2	2	no	—
397	#121	C₂³×C₂₃	1	4	yes	—
398	#121	C₂³×C₂₃	1	4	yes	—
399	#120	—	2	4	no	—
400	C₄×Q₈	—	2	2	no	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	—	2	2	no	—
403	C₄×Q₈	—	2	2	no	—
404	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
405	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
406	C₄×D₄	—	2	4	no	—
407	C₄×D₄	—	2	4	no	—
408	C₄×D₄	—	2	4	no	—
409	C₄×D₄	—	2	4	no	—
410	C₄×D₄	—	2	4	no	—
411	C₄×D₄	—	2	4	no	—
412	C₄×D₄	—	2	2	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	—	2	2	no	—
416	C₄×D₄	C₂³×C₂₃	1	4	yes	—
417	C₄×D₄	C₂³×C₂₃	1	4	yes	—
418	#117	—	2	2	no	—
419	#117	—	2	2	no	—
420	#117	—	2	4	no	—
421	#117	—	2	4	no	—
422	#117	—	2	4	no	—
423	#117	C₂³×C₂₃	1	4	yes	—
424	C₂×C₄²	—	2	4	no	—
425	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
426	C₂×C₄²	D₄×C₂₃	1	2	yes	—
427	C₂×C₄²	D₄×C₂₃	1	2	yes	—
428	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
429	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
430	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
433	#107	C₂³×C₂₃	1	4	yes	—
434	#107	C₂³×C₂₃	1	4	yes	—
435	#106	C₂³×C₂₃	1	8	yes	—
436	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
437	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
438	#103	C₂³×C₂₃	1	8	yes	—
439	#103	C₂³×C₂₃	1	8	yes	—
440	#102	C₂³×C₂₃	1	8	yes	—
441	#102	C₂³×C₂₃	1	8	yes	—
442	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
443	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
444	C₄×C₈	C₂³×C₂₃	1	2	yes	—
445	C₄×C₈	C₂³×C₂₃	1	2	yes	—
446	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
447	C₄×C₈	C₂³×C₂₃	1	2	yes	—
448	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
449	C₂₃	#60944	1	1	yes	characteristic, direct product factor, Sylow
450	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
451	#41	—	4	8	no	—
452	#41	—	4	8	no	—
453	#41	—	4	8	no	—
454	C₂×Q₈	—	2	2	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	4	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	2	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	4	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	2	no	—
469	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
470	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
471	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
472	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
473	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
476	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
478	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
480	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
481	Q₁₆	—	4	4	no	—
482	SD₁₆	—	4	8	no	—
483	D₈	—	4	4	no	—
484	C₂×C₈	—	4	2	no	—
485	C₂×C₈	—	4	2	no	—
486	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
487	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
488	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	4	no	—
499	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
500	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	4	no	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	2	4	no	—
509	C₂²⋊C₄	—	2	4	no	—
510	C₄²	—	2	4	no	—
511	C₄²	—	2	4	no	—
512	C₄²	—	2	4	no	—
513	C₄²	C₂×D₄×C₂₃	1	2	yes	—
514	C₄²	C₂×D₄×C₂₃	1	2	yes	—
515	C₄²	C₂×D₄×C₂₃	1	2	yes	—
516	C₄²	C₂×D₄×C₂₃	1	2	yes	—
517	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
518	C₄²	C₂×D₄×C₂₃	1	2	yes	—
519	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
520	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
521	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
522	C₄²	C₂⁴×C₂₃	1	2	yes	—
523	C₂³	—	2	2	no	—
524	C₂³	—	2	4	no	—
525	C₂³	—	4	1	no	—
526	C₂³	—	4	1	no	—
527	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
528	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
529	Q₈	—	2	8	no	—
530	Q₈	—	4	4	no	—
531	Q₈	—	4	4	no	—
532	D₄	—	2	8	no	—
533	D₄	—	4	8	no	—
534	D₄	—	4	8	no	—
535	D₄	—	4	8	no	—
536	D₄	—	4	4	no	—
537	D₄	—	4	4	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	2	no	—
552	C₂×C₄	—	4	8	no	—
553	C₂×C₄	—	4	8	no	—
554	C₂×C₄	—	4	8	no	—
555	C₂×C₄	C₂₃×#135	1	4	yes	—
556	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
557	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
558	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
559	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
560	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
561	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
562	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
563	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
564	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
565	C₈	—	2	2	no	—
566	C₈	—	2	4	no	—
567	C₈	—	2	2	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	2	1	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	4	4	no	—
574	C₂²	—	4	4	no	—
575	C₂²	—	4	4	no	—
576	C₂²	—	4	4	no	—
577	C₂²	C₂×C₂₃×#135	1	2	yes	—
578	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
579	C₄	—	2	2	no	—
580	C₄	—	2	1	no	—
581	C₄	—	2	2	no	—
582	C₄	—	2	2	no	—
583	C₄	—	2	1	no	—
584	C₄	—	4	4	no	—
585	C₄	—	4	4	no	—
586	C₄	—	4	4	no	—
587	C₄	C₂×C₂₃×#135	1	2	yes	—
588	C₄	C₂×C₂₃×#135	1	2	yes	—
589	C₄	C₂₃×#575	1	2	yes	—
590	C₄	C₂₃×#585	1	2	yes	—
591	C₂	—	2	2	no	—
592	C₂	—	4	2	no	—
593	C₂	—	8	4	no	—
594	C₂	C₂₃×#3449	1	1	yes	characteristic
595	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
596	C₂	C₂×C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3444	C₂	semidirect
2. C₂₃×#3250	C₂	semidirect
3. C₂₃×#3231	C₂	semidirect
4. C₂₃×#3228	C₂	semidirect
5. C₂₃×#3210	C₂	semidirect
6. C₂₃×#3207	C₂	semidirect
7. C₂₃×#3191	C₂	semidirect
8. C₂₃×#3170	C₂	semidirect
9. C₂₃×#3169	C₂	semidirect
10. C₂₃×#3167	C₂	semidirect
11. C₂₃×#3102	C₂	semidirect
12. C₂×C₂₃×#431	C₂	semidirect
32. C₂₃×#500	C₂²	semidirect
33. C₂₃×#500	C₂²	semidirect
34. C₂₃×#498	C₂²	semidirect
35. C₂₃×#494	C₂²	semidirect
36. C₂₃×#493	C₂²	semidirect
37. C₂₃×#492	C₂²	semidirect
38. C₂₃×#487	C₂²	semidirect
39. C₂₃×#487	C₂²	semidirect
40. C₂₃×#486	C₂²	semidirect
41. C₂₃×#486	C₂²	semidirect
44. C₂₃×#479	C₂²	semidirect
45. C₂₃×#478	C₂²	semidirect
49. C₂₃×#464	C₂²	semidirect
50. C₂₃×#463	C₂²	semidirect
53. C₄×Q₁₆×C₂₃	C₂²	semidirect
54. C₄×SD₁₆×C₂₃	C₂²	semidirect
55. C₄×D₈×C₂₃	C₂²	semidirect
224. #60944	C₂₃	direct (N × H)
270. #3444	C₂×C₂₃	semidirect
271. #3250	C₂×C₂₃	semidirect
272. #3231	C₂×C₂₃	semidirect
273. #3228	C₂×C₂₃	semidirect
274. #3210	C₂×C₂₃	semidirect
275. #3207	C₂×C₂₃	semidirect
276. #3191	C₂×C₂₃	semidirect
277. #3170	C₂×C₂₃	semidirect
278. #3169	C₂×C₂₃	semidirect
279. #3167	C₂×C₂₃	semidirect
280. #3102	C₂×C₂₃	semidirect
281. C₂×#431	C₂×C₂₃	semidirect
324. #500	C₂²×C₂₃	semidirect
325. #500	C₂²×C₂₃	semidirect
326. #498	C₂²×C₂₃	semidirect
327. #494	C₂²×C₂₃	semidirect
328. #493	C₂²×C₂₃	semidirect
329. #492	C₂²×C₂₃	semidirect
330. #487	C₂²×C₂₃	semidirect
331. #487	C₂²×C₂₃	semidirect
332. #486	C₂²×C₂₃	semidirect
333. #486	C₂²×C₂₃	semidirect
336. #479	C₂²×C₂₃	semidirect
337. #478	C₂²×C₂₃	semidirect
341. #464	C₂²×C₂₃	semidirect
342. #463	C₂²×C₂₃	semidirect
345. C₄×Q₁₆	C₂²×C₂₃	semidirect
346. C₄×SD₁₆	C₂²×C₂₃	semidirect
347. C₄×D₈	C₂²×C₂₃	semidirect
449. C₂₃	#60944	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×#3444	2
C₂₃×#3250	2
C₂₃×#3231	2
C₂₃×#3228	2
C₂₃×#3210	2
C₂₃×#3207	2
C₂₃×#3191	2
C₂₃×#3170	2
C₂₃×#3169	2
C₂₃×#3167	2
C₂₃×#3102	2
C₂×C₂₃×#431	2
#60944	23

Subgroups — C23×#60944 #19208570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60944 #19208570