Subgroups — C₂₃×#24684 #19172310

Order 5888.

826 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#478	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2566	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2357	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2073	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2071	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2054	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#2054	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1991	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1949	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1939	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1926	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1925	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1804	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#478	—	2	4	no	—
21	C₄×D₈×C₂₃	—	2	4	no	—
22	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
33	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#396	—	2	1	no	—
36	C₂₃×#396	—	2	1	no	—
37	C₂₃×#390	C₂²	1	1	yes	characteristic
38	C₂₃×#390	C₂²	1	1	yes	characteristic
39	C₂₃×#390	C₂²	1	1	yes	characteristic
40	C₂₃×#390	C₂²	1	1	yes	characteristic
41	C₂₃×#388	—	2	1	no	—
42	C₂₃×#388	—	2	1	no	—
43	C₂₃×#385	—	2	1	no	—
44	C₂₃×#385	—	2	1	no	—
45	C₂₃×#384	C₂²	1	1	yes	characteristic
46	C₂₃×#382	C₂²	1	1	yes	characteristic
47	C₂₃×#382	C₂²	1	1	yes	characteristic
48	C₂₃×#381	—	2	1	no	—
49	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
50	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
51	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
52	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#340	C₂²	1	1	yes	characteristic
58	C₂₃×#340	C₂²	1	1	yes	characteristic
59	C₂₃×#336	C₂²	1	1	yes	characteristic
60	C₂₃×#336	C₂²	1	1	yes	characteristic
61	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₂₃	—	2	1	no	—
64	C₂³×C₄×C₂₃	—	2	1	no	—
65	C₂×D₈×C₂₃	—	2	2	no	—
66	C₂×D₈×C₂₃	—	2	2	no	—
67	C₂×D₈×C₂₃	—	2	2	no	—
68	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
72	C₂₃×#125	—	2	2	no	—
73	C₂₃×#125	—	2	2	no	—
74	C₄×D₄×C₂₃	—	2	2	no	—
75	C₄×D₄×C₂₃	—	2	2	no	—
76	C₄×D₄×C₂₃	—	2	2	no	—
77	C₄×D₄×C₂₃	D₄	1	4	yes	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
90	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
98	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄²×C₂₃	—	2	1	no	—
101	C₂×C₄²×C₂₃	—	2	1	no	—
102	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂₃×#107	—	2	2	no	—
106	C₂₃×#107	—	2	2	no	—
107	C₂₃×#106	—	2	2	no	—
108	C₂₃×C₄⋊C₈	—	2	2	no	—
109	C₂₃×C₄⋊C₈	—	2	2	no	—
110	C₂₃×C₄⋊C₈	—	2	2	no	—
111	C₂₃×#102	—	2	2	no	—
112	C₂₃×#102	—	2	2	no	—
113	C₂₃×#102	—	2	2	no	—
114	C₂₃×#102	—	2	2	no	—
115	C₂₃×#102	—	2	2	no	—
116	C₂₃×#102	—	2	2	no	—
117	C₄×C₈×C₂₃	—	2	2	no	—
118	C₂₃×#95	—	2	1	no	—
119	C₂₃×#95	—	2	1	no	—
120	C₂₃×#95	—	2	1	no	—
121	C₂₃×#95	—	2	1	no	—
122	C₂₃×#95	—	2	1	no	—
123	C₂₃×#95	—	2	1	no	—
124	C₂₃×#95	D₄	1	1	yes	characteristic
125	C₂₃×#95	D₄	1	1	yes	characteristic
126	C₂⁴×C₂₃	—	2	1	no	—
127	C₂⁴×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	—	2	1	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	—	2	1	no	—
134	C₂×D₄×C₂₃	—	2	1	no	—
135	C₂×D₄×C₂₃	—	2	1	no	—
136	C₂×D₄×C₂₃	—	2	1	no	—
137	C₂×D₄×C₂₃	—	2	1	no	—
138	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
139	C₂×D₄×C₂₃	#41	1	2	yes	—
140	C₂²×C₄×C₂₃	—	2	1	no	—
141	C₂²×C₄×C₂₃	—	2	4	no	—
142	C₂²×C₄×C₂₃	—	2	1	no	—
143	C₂²×C₄×C₂₃	—	2	1	no	—
144	C₂²×C₄×C₂₃	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	2	1	no	—
147	C₂²×C₄×C₂₃	—	4	2	no	—
148	C₂²×C₄×C₂₃	—	8	1	no	—
149	C₂²×C₄×C₂₃	—	8	1	no	—
150	C₂²×C₄×C₂₃	—	8	1	no	—
151	C₂²×C₄×C₂₃	—	8	1	no	—
152	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
159	D₈×C₂₃	—	2	8	no	—
160	C₂×C₈×C₂₃	—	2	1	no	—
161	C₂×C₈×C₂₃	—	2	1	no	—
162	C₂×C₈×C₂₃	—	2	1	no	—
163	C₂×C₈×C₂₃	—	2	1	no	—
164	C₂×C₈×C₂₃	—	2	1	no	—
165	C₂×C₈×C₂₃	—	2	1	no	—
166	C₂×C₈×C₂₃	—	2	1	no	—
167	C₂×C₈×C₂₃	—	2	1	no	—
168	C₂×C₈×C₂₃	—	2	1	no	—
169	C₂×C₈×C₂₃	—	2	1	no	—
170	C₂×C₈×C₂₃	—	2	1	no	—
171	C₂×C₈×C₂₃	#41	1	2	yes	—
172	C₄⋊C₄×C₂₃	—	2	2	no	—
173	C₄⋊C₄×C₂₃	—	2	1	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	—	2	1	no	—
178	C₄⋊C₄×C₂₃	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	2	1	no	—
183	C₄⋊C₄×C₂₃	—	2	1	no	—
184	C₄⋊C₄×C₂₃	—	4	1	no	—
185	C₄⋊C₄×C₂₃	—	4	2	no	—
186	C₄⋊C₄×C₂₃	—	4	2	no	—
187	C₄⋊C₄×C₂₃	—	8	1	no	—
188	C₄⋊C₄×C₂₃	—	8	1	no	—
189	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
190	C₂²⋊C₄×C₂₃	—	2	4	no	—
191	C₂²⋊C₄×C₂₃	—	4	2	no	—
192	C₂²⋊C₄×C₂₃	—	4	2	no	—
193	C₂²⋊C₄×C₂₃	—	4	2	no	—
194	C₂²⋊C₄×C₂₃	—	4	2	no	—
195	C₂²⋊C₄×C₂₃	—	4	2	no	—
196	C₂²⋊C₄×C₂₃	—	8	1	no	—
197	C₂²⋊C₄×C₂₃	—	8	1	no	—
198	C₂²⋊C₄×C₂₃	—	8	1	no	—
199	C₂²⋊C₄×C₂₃	—	8	1	no	—
200	C₄²×C₂₃	—	2	1	no	—
201	C₄²×C₂₃	—	2	1	no	—
202	C₄²×C₂₃	—	2	1	no	—
203	C₄²×C₂₃	—	2	1	no	—
204	C₄²×C₂₃	—	8	1	no	—
205	C₄²×C₂₃	—	8	1	no	—
206	C₄²×C₂₃	C₂×D₄	1	2	yes	—
207	C₄²×C₂₃	#41	1	2	yes	—
208	#24684	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂³×C₂₃	—	2	2	no	—
210	C₂³×C₂₃	—	2	2	no	—
211	C₂³×C₂₃	—	4	1	no	—
212	C₂³×C₂₃	—	4	1	no	—
213	C₂³×C₂₃	—	4	1	no	—
214	C₂³×C₂₃	—	4	1	no	—
215	C₂³×C₂₃	—	4	1	no	—
216	C₂³×C₂₃	—	4	1	no	—
217	C₂³×C₂₃	—	4	1	no	—
218	C₂³×C₂₃	—	4	1	no	—
219	C₂³×C₂₃	—	4	1	no	—
220	C₂³×C₂₃	—	4	1	no	—
221	C₂³×C₂₃	—	4	1	no	—
222	C₂³×C₂₃	—	4	1	no	—
223	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
224	D₄×C₂₃	—	2	2	no	—
225	D₄×C₂₃	—	2	2	no	—
226	D₄×C₂₃	—	2	2	no	—
227	D₄×C₂₃	—	2	2	no	—
228	D₄×C₂₃	—	4	1	no	—
229	D₄×C₂₃	—	4	1	no	—
230	D₄×C₂₃	—	4	1	no	—
231	D₄×C₂₃	—	4	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	—	2	2	no	—
239	C₂×C₄×C₂₃	—	2	1	no	—
240	C₂×C₄×C₂₃	—	2	1	no	—
241	C₂×C₄×C₂₃	—	2	1	no	—
242	C₂×C₄×C₂₃	—	2	1	no	—
243	C₂×C₄×C₂₃	—	2	1	no	—
244	C₂×C₄×C₂₃	—	2	1	no	—
245	C₂×C₄×C₂₃	—	2	1	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	4	1	no	—
250	C₂×C₄×C₂₃	—	4	1	no	—
251	C₂×C₄×C₂₃	—	4	1	no	—
252	C₂×C₄×C₂₃	—	4	1	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	4	1	no	—
258	C₂×C₄×C₂₃	—	4	1	no	—
259	C₂×C₄×C₂₃	—	4	1	no	—
260	C₂×C₄×C₂₃	—	4	1	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	8	4	no	—
266	C₂×C₄×C₂₃	—	8	4	no	—
267	C₂×C₄×C₂₃	#121	1	2	yes	—
268	C₂×C₄×C₂₃	#121	1	2	yes	—
269	C₂×C₄×C₂₃	#135	1	2	yes	—
270	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
272	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
275	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
276	C₈×C₂₃	—	2	2	no	—
277	C₈×C₂₃	—	4	1	no	—
278	C₈×C₂₃	—	4	1	no	—
279	C₈×C₂₃	—	4	1	no	—
280	C₂×#478	C₂×C₂₃	1	1	yes	characteristic, complemented
281	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic
282	#2566	C₂×C₂₃	1	1	yes	characteristic, complemented
283	#2357	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#2073	C₂×C₂₃	1	1	yes	characteristic, complemented
285	#2071	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#2054	C₂×C₂₃	1	1	yes	characteristic
287	#2054	C₂×C₂₃	1	1	yes	characteristic
288	#1991	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#1949	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#1947	C₂×C₂₃	1	1	yes	characteristic, complemented
291	#1939	C₂×C₂₃	1	1	yes	characteristic, complemented
292	#1926	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#1925	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#1804	C₂×C₂₃	1	1	yes	characteristic, complemented
295	C₂²×C₂₃	—	4	2	no	—
296	C₂²×C₂₃	—	4	2	no	—
297	C₂²×C₂₃	—	4	2	no	—
298	C₂²×C₂₃	—	4	2	no	—
299	C₂²×C₂₃	—	4	2	no	—
300	C₂²×C₂₃	—	4	2	no	—
301	C₂²×C₂₃	—	8	1	no	—
302	C₂²×C₂₃	—	8	1	no	—
303	C₂²×C₂₃	—	8	1	no	—
304	C₂²×C₂₃	—	8	1	no	—
305	C₂²×C₂₃	—	8	1	no	—
306	C₂²×C₂₃	—	8	1	no	—
307	C₂²×C₂₃	—	8	1	no	—
308	C₂²×C₂₃	—	8	1	no	—
309	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
310	C₂²×C₂₃	#547	1	1	yes	characteristic
311	C₂²×C₂₃	#548	1	1	yes	characteristic
312	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
313	C₂²×C₂₃	#575	1	1	yes	characteristic
314	C₂²×C₂₃	#576	1	1	yes	characteristic
315	C₂²×C₂₃	#576	1	1	yes	characteristic
316	C₄×C₂₃	—	2	2	no	—
317	C₄×C₂₃	—	2	2	no	—
318	C₄×C₂₃	—	2	2	no	—
319	C₄×C₂₃	—	4	1	no	—
320	C₄×C₂₃	—	4	1	no	—
321	C₄×C₂₃	—	4	1	no	—
322	C₄×C₂₃	—	4	2	no	—
323	C₄×C₂₃	—	8	1	no	—
324	C₄×C₂₃	—	8	1	no	—
325	C₄×C₂₃	—	8	1	no	—
326	C₄×C₂₃	—	8	1	no	—
327	C₄×C₂₃	—	8	1	no	—
328	C₄×C₂₃	#525	1	1	yes	characteristic
329	C₄×C₂₃	#536	1	1	yes	characteristic
330	C₄×C₂₃	#552	1	1	yes	characteristic
331	C₄×C₂₃	#556	1	1	yes	characteristic
332	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
333	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
334	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
335	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
336	#478	—	2	4	no	—
337	C₄×D₈	—	2	4	no	—
338	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
351	#396	—	2	1	no	—
352	#396	—	2	1	no	—
353	#390	C₂²×C₂₃	1	1	yes	characteristic
354	#390	C₂²×C₂₃	1	1	yes	characteristic
355	#390	C₂²×C₂₃	1	1	yes	characteristic
356	#390	C₂²×C₂₃	1	1	yes	characteristic
357	#388	—	2	1	no	—
358	#388	—	2	1	no	—
359	#385	—	2	1	no	—
360	#385	—	2	1	no	—
361	#384	C₂²×C₂₃	1	1	yes	characteristic
362	#382	C₂²×C₂₃	1	1	yes	characteristic
363	#382	C₂²×C₂₃	1	1	yes	characteristic
364	#381	—	2	1	no	—
365	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
366	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
367	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
368	C₄×C₂²⋊C₄	—	2	1	no	—
369	#340	C₂²×C₂₃	1	1	yes	characteristic
370	#340	C₂²×C₂₃	1	1	yes	characteristic
371	#340	C₂²×C₂₃	1	1	yes	characteristic
372	#340	C₂²×C₂₃	1	1	yes	characteristic
373	#340	C₂²×C₂₃	1	1	yes	characteristic
374	#340	C₂²×C₂₃	1	1	yes	characteristic
375	#336	C₂²×C₂₃	1	1	yes	characteristic
376	#336	C₂²×C₂₃	1	1	yes	characteristic
377	C₂×C₂₃	—	8	2	no	—
378	C₂×C₂₃	—	8	2	no	—
379	C₂×C₂₃	#2415	1	1	yes	characteristic
380	C₂×C₂₃	#3035	1	1	yes	characteristic
381	C₂×C₂₃	#3041	1	1	yes	characteristic
382	C₂×C₂₃	#3339	1	1	yes	characteristic
383	C₂×C₂₃	#3355	1	1	yes	characteristic
384	C₂×C₂₃	#3361	1	1	yes	characteristic
385	C₂×C₂₃	#3370	1	1	yes	characteristic
386	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
387	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
388	C₂³×C₄	—	2	1	no	—
389	C₂³×C₄	—	2	1	no	—
390	C₂×D₈	—	2	2	no	—
391	C₂×D₈	—	2	2	no	—
392	C₂×D₈	—	2	2	no	—
393	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
394	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
395	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
396	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
397	#125	—	2	2	no	—
398	#125	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	D₄×C₂₃	1	4	yes	—
403	C₂×C₄⋊C₄	—	2	1	no	—
404	C₂×C₄⋊C₄	—	2	1	no	—
405	C₂×C₄⋊C₄	—	2	1	no	—
406	C₂×C₄⋊C₄	—	2	1	no	—
407	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
408	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
411	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
412	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
413	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
414	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
415	C₂×C₂²⋊C₄	—	2	1	no	—
416	C₂×C₂²⋊C₄	—	2	1	no	—
417	C₂×C₂²⋊C₄	—	2	1	no	—
418	C₂×C₂²⋊C₄	—	2	1	no	—
419	C₂×C₂²⋊C₄	—	2	1	no	—
420	C₂×C₂²⋊C₄	—	2	1	no	—
421	C₂×C₂²⋊C₄	—	2	1	no	—
422	C₂×C₂²⋊C₄	—	2	1	no	—
423	C₂×C₂²⋊C₄	—	2	1	no	—
424	C₂×C₂²⋊C₄	—	2	1	no	—
425	C₂×C₄²	—	2	1	no	—
426	C₂×C₄²	—	2	1	no	—
427	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
428	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
429	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
430	#107	—	2	2	no	—
431	#107	—	2	2	no	—
432	#106	—	2	2	no	—
433	C₄⋊C₈	—	2	2	no	—
434	C₄⋊C₈	—	2	2	no	—
435	C₄⋊C₈	—	2	2	no	—
436	#102	—	2	2	no	—
437	#102	—	2	2	no	—
438	#102	—	2	2	no	—
439	#102	—	2	2	no	—
440	#102	—	2	2	no	—
441	#102	—	2	2	no	—
442	C₄×C₈	—	2	2	no	—
443	#95	—	2	1	no	—
444	#95	—	2	1	no	—
445	#95	—	2	1	no	—
446	#95	—	2	1	no	—
447	#95	—	2	1	no	—
448	#95	—	2	1	no	—
449	#95	D₄×C₂₃	1	1	yes	characteristic
450	#95	D₄×C₂₃	1	1	yes	characteristic
451	C₂₃	#24684	1	1	yes	characteristic, direct product factor, Sylow
452	C₂⁴	—	2	1	no	—
453	C₂⁴	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	—	2	1	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	1	no	—
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	—	2	1	no	—
464	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
465	C₂×D₄	#41×C₂₃	1	2	yes	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	4	2	no	—
474	C₂²×C₄	—	8	1	no	—
475	C₂²×C₄	—	8	1	no	—
476	C₂²×C₄	—	8	1	no	—
477	C₂²×C₄	—	8	1	no	—
478	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
485	D₈	—	2	8	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	1	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	#41×C₂₃	1	2	yes	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	4	1	no	—
511	C₄⋊C₄	—	4	2	no	—
512	C₄⋊C₄	—	4	2	no	—
513	C₄⋊C₄	—	8	1	no	—
514	C₄⋊C₄	—	8	1	no	—
515	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	4	2	no	—
518	C₂²⋊C₄	—	4	2	no	—
519	C₂²⋊C₄	—	4	2	no	—
520	C₂²⋊C₄	—	4	2	no	—
521	C₂²⋊C₄	—	4	2	no	—
522	C₂²⋊C₄	—	8	1	no	—
523	C₂²⋊C₄	—	8	1	no	—
524	C₂²⋊C₄	—	8	1	no	—
525	C₂²⋊C₄	—	8	1	no	—
526	C₄²	—	2	1	no	—
527	C₄²	—	2	1	no	—
528	C₄²	—	2	1	no	—
529	C₄²	—	2	1	no	—
530	C₄²	—	8	1	no	—
531	C₄²	—	8	1	no	—
532	C₄²	C₂×D₄×C₂₃	1	2	yes	—
533	C₄²	#41×C₂₃	1	2	yes	—
534	C₂³	—	2	2	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	4	1	no	—
537	C₂³	—	4	1	no	—
538	C₂³	—	4	1	no	—
539	C₂³	—	4	1	no	—
540	C₂³	—	4	1	no	—
541	C₂³	—	4	1	no	—
542	C₂³	—	4	1	no	—
543	C₂³	—	4	1	no	—
544	C₂³	—	4	1	no	—
545	C₂³	—	4	1	no	—
546	C₂³	—	4	1	no	—
547	C₂³	—	4	1	no	—
548	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
549	D₄	—	2	2	no	—
550	D₄	—	2	2	no	—
551	D₄	—	2	2	no	—
552	D₄	—	2	2	no	—
553	D₄	—	4	1	no	—
554	D₄	—	4	1	no	—
555	D₄	—	4	1	no	—
556	D₄	—	4	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	8	4	no	—
591	C₂×C₄	—	8	4	no	—
592	C₂×C₄	C₂₃×#121	1	2	yes	—
593	C₂×C₄	C₂₃×#121	1	2	yes	—
594	C₂×C₄	C₂₃×#135	1	2	yes	—
595	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
596	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
597	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
598	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
599	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
600	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
601	C₈	—	2	2	no	—
602	C₈	—	4	1	no	—
603	C₈	—	4	1	no	—
604	C₈	—	4	1	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	8	1	no	—
612	C₂²	—	8	1	no	—
613	C₂²	—	8	1	no	—
614	C₂²	—	8	1	no	—
615	C₂²	—	8	1	no	—
616	C₂²	—	8	1	no	—
617	C₂²	—	8	1	no	—
618	C₂²	—	8	1	no	—
619	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
620	C₂²	C₂₃×#547	1	1	yes	characteristic
621	C₂²	C₂₃×#548	1	1	yes	characteristic
622	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
623	C₂²	C₂₃×#575	1	1	yes	characteristic
624	C₂²	C₂₃×#576	1	1	yes	characteristic
625	C₂²	C₂₃×#576	1	1	yes	characteristic
626	C₄	—	2	2	no	—
627	C₄	—	2	2	no	—
628	C₄	—	2	2	no	—
629	C₄	—	4	2	no	—
630	C₄	—	4	1	no	—
631	C₄	—	4	1	no	—
632	C₄	—	4	1	no	—
633	C₄	—	8	1	no	—
634	C₄	—	8	1	no	—
635	C₄	—	8	1	no	—
636	C₄	—	8	1	no	—
637	C₄	—	8	1	no	—
638	C₄	C₂₃×#525	1	1	yes	characteristic
639	C₄	C₂₃×#536	1	1	yes	characteristic
640	C₄	C₂₃×#552	1	1	yes	characteristic
641	C₄	C₂₃×#556	1	1	yes	characteristic
642	C₂	—	8	2	no	—
643	C₂	—	8	2	no	—
644	C₂	C₂₃×#2415	1	1	yes	characteristic
645	C₂	C₂₃×#3035	1	1	yes	characteristic
646	C₂	C₂₃×#3041	1	1	yes	characteristic
647	C₂	C₂₃×#3339	1	1	yes	characteristic
648	C₂	C₂₃×#3355	1	1	yes	characteristic
649	C₂	C₂₃×#3361	1	1	yes	characteristic
650	C₂	C₂₃×#3370	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₄⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#478	C₂	semidirect
3. C₂₃×#2566	C₂	semidirect
4. C₂₃×#2357	C₂	semidirect
5. C₂₃×#2073	C₂	semidirect
6. C₂₃×#2071	C₂	semidirect
9. C₂₃×#1991	C₂	semidirect
10. C₂₃×#1949	C₂	semidirect
11. C₂₃×#1947	C₂	semidirect
12. C₂₃×#1939	C₂	semidirect
13. C₂₃×#1926	C₂	semidirect
14. C₂₃×#1925	C₂	semidirect
15. C₂₃×#1804	C₂	semidirect
208. #24684	C₂₃	direct (N × H)
280. C₂×#478	C₂×C₂₃	semidirect
282. #2566	C₂×C₂₃	semidirect
283. #2357	C₂×C₂₃	semidirect
284. #2073	C₂×C₂₃	semidirect
285. #2071	C₂×C₂₃	semidirect
288. #1991	C₂×C₂₃	semidirect
289. #1949	C₂×C₂₃	semidirect
290. #1947	C₂×C₂₃	semidirect
291. #1939	C₂×C₂₃	semidirect
292. #1926	C₂×C₂₃	semidirect
293. #1925	C₂×C₂₃	semidirect
294. #1804	C₂×C₂₃	semidirect
451. C₂₃	#24684	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#478	2
C₂×C₄×D₈×C₂₃	2
C₂₃×#2566	2
C₂₃×#2357	2
C₂₃×#2073	2
C₂₃×#2071	2
C₂₃×#2054	2
C₂₃×#1991	2
C₂₃×#1949	2
C₂₃×#1947	2
C₂₃×#1939	2
C₂₃×#1926	2
C₂₃×#1925	2
C₂₃×#1804	2
#24684	23

Subgroups — C23×#24684 #19172310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#24684 #19172310