Subgroups — C₂₃×#27037 #19174663

Order 5888.

700 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2494	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2360	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2074	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2066	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2036	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#2035	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1949	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1941	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1923	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1921	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1920	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1865	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1853	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1823	C₂	1	1	yes	maximal, characteristic
15	C₄×C₂₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#401	—	2	1	no	—
32	C₂₃×#397	—	2	1	no	—
33	C₂₃×#391	C₂²	1	1	yes	characteristic
34	C₂₃×#389	—	2	1	no	—
35	C₂₃×#389	C₂²	1	1	yes	characteristic
36	C₂₃×#389	C₂²	1	1	yes	characteristic
37	C₂₃×#389	C₂²	1	1	yes	characteristic
38	C₂₃×#388	—	2	1	no	—
39	C₂₃×#387	—	2	1	no	—
40	C₂₃×#387	—	2	1	no	—
41	C₂₃×#385	—	2	1	no	—
42	C₂₃×#382	C₂²	1	1	yes	characteristic
43	C₂₃×#380	C₂²	1	1	yes	characteristic
44	C₂₃×#380	C₂²	1	1	yes	characteristic
45	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
47	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
48	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
49	C₂₃×#340	C₂²	1	1	yes	characteristic
50	C₂₃×#340	C₂²	1	1	yes	characteristic
51	C₂₃×#340	C₂²	1	1	yes	characteristic
52	C₂₃×#340	C₂²	1	1	yes	characteristic
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×#336	C₂²	1	1	yes	characteristic
58	C₂₃×#336	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₂₃	—	2	1	no	—
61	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂₃×C₂²⋊Q₈	—	2	2	no	—
67	C₂₃×C₂²⋊Q₈	—	2	2	no	—
68	C₄×Q₈×C₂₃	—	2	2	no	—
69	C₄×Q₈×C₂₃	—	4	1	no	—
70	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
72	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
90	C₂×C₄²×C₂₃	—	2	1	no	—
91	C₂×C₄²×C₂₃	—	2	1	no	—
92	C₂×C₄²×C₂₃	—	2	1	no	—
93	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
95	C₂₃×#107	—	2	2	no	—
96	C₂₃×#106	—	2	2	no	—
97	C₂₃×#106	—	2	2	no	—
98	C₂₃×#106	—	4	1	no	—
99	C₂₃×C₄⋊C₈	—	2	2	no	—
100	C₂₃×#103	—	2	2	no	—
101	C₂₃×#103	—	2	2	no	—
102	C₂₃×#103	—	2	2	no	—
103	C₂₃×#103	—	4	1	no	—
104	C₂₃×C₂²⋊C₈	—	2	2	no	—
105	C₂₃×C₂²⋊C₈	—	4	1	no	—
106	C₄×C₈×C₂₃	—	4	1	no	—
107	C₂₃×#95	—	2	1	no	—
108	C₂₃×#95	—	2	1	no	—
109	C₂₃×#95	—	2	1	no	—
110	C₂₃×#95	—	2	1	no	—
111	C₂₃×#95	—	2	1	no	—
112	C₂₃×#95	—	2	1	no	—
113	C₂₃×#95	—	2	1	no	—
114	C₂₃×#95	—	2	1	no	—
115	C₂₃×#95	—	2	1	no	—
116	C₂₃×#95	—	2	1	no	—
117	C₂₃×#95	—	2	1	no	—
118	C₂₃×#95	D₄	1	1	yes	characteristic
119	C₂₃×#95	D₄	1	1	yes	characteristic
120	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
121	C₂×Q₈×C₂₃	—	2	1	no	—
122	C₂×Q₈×C₂₃	—	2	1	no	—
123	C₂×Q₈×C₂₃	—	2	1	no	—
124	C₂×Q₈×C₂₃	—	2	1	no	—
125	C₂×Q₈×C₂₃	—	2	1	no	—
126	C₂×Q₈×C₂₃	#41	1	2	yes	—
127	C₂²×C₄×C₂₃	—	2	1	no	—
128	C₂²×C₄×C₂₃	—	2	1	no	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	—	2	1	no	—
131	C₂²×C₄×C₂₃	—	2	1	no	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	2	1	no	—
136	C₂²×C₄×C₂₃	—	2	1	no	—
137	C₂²×C₄×C₂₃	—	2	1	no	—
138	C₂²×C₄×C₂₃	—	2	1	no	—
139	C₂²×C₄×C₂₃	—	2	1	no	—
140	C₂²×C₄×C₂₃	—	2	1	no	—
141	C₂²×C₄×C₂₃	—	4	2	no	—
142	C₂²×C₄×C₂₃	—	4	2	no	—
143	C₂²×C₄×C₂₃	—	8	1	no	—
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
150	C₂×C₈×C₂₃	—	2	1	no	—
151	C₂×C₈×C₂₃	—	2	1	no	—
152	C₂×C₈×C₂₃	—	2	1	no	—
153	C₂×C₈×C₂₃	—	2	1	no	—
154	C₂×C₈×C₂₃	—	2	1	no	—
155	C₂×C₈×C₂₃	—	2	1	no	—
156	C₂×C₈×C₂₃	—	2	1	no	—
157	C₂×C₈×C₂₃	—	2	1	no	—
158	C₂×C₈×C₂₃	—	2	1	no	—
159	C₂×C₈×C₂₃	—	2	1	no	—
160	C₂×C₈×C₂₃	—	2	1	no	—
161	C₂×C₈×C₂₃	#41	1	2	yes	—
162	C₄⋊C₄×C₂₃	—	2	4	no	—
163	C₄⋊C₄×C₂₃	—	2	1	no	—
164	C₄⋊C₄×C₂₃	—	2	1	no	—
165	C₄⋊C₄×C₂₃	—	2	1	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	1	no	—
170	C₄⋊C₄×C₂₃	—	2	1	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	1	no	—
173	C₄⋊C₄×C₂₃	—	2	1	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	4	2	no	—
176	C₄⋊C₄×C₂₃	—	4	2	no	—
177	C₄⋊C₄×C₂₃	—	4	2	no	—
178	C₄⋊C₄×C₂₃	—	8	1	no	—
179	C₄⋊C₄×C₂₃	—	8	1	no	—
180	C₄⋊C₄×C₂₃	#41	1	2	yes	—
181	C₄⋊C₄×C₂₃	#41	1	2	yes	—
182	C₂²⋊C₄×C₂₃	—	2	4	no	—
183	C₂²⋊C₄×C₂₃	—	2	2	no	—
184	C₂²⋊C₄×C₂₃	—	2	2	no	—
185	C₂²⋊C₄×C₂₃	—	4	2	no	—
186	C₂²⋊C₄×C₂₃	—	8	1	no	—
187	C₄²×C₂₃	—	2	1	no	—
188	C₄²×C₂₃	—	2	1	no	—
189	C₄²×C₂₃	—	2	1	no	—
190	C₄²×C₂₃	—	2	1	no	—
191	C₄²×C₂₃	—	4	2	no	—
192	C₄²×C₂₃	—	8	1	no	—
193	C₄²×C₂₃	—	8	1	no	—
194	#27037	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂³×C₂₃	—	2	1	no	—
196	C₂³×C₂₃	—	2	1	no	—
197	C₂³×C₂₃	—	2	1	no	—
198	C₂³×C₂₃	—	2	1	no	—
199	C₂³×C₂₃	—	2	1	no	—
200	C₂³×C₂₃	—	2	1	no	—
201	C₂³×C₂₃	#135	1	2	yes	—
202	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
203	Q₈×C₂₃	—	2	2	no	—
204	Q₈×C₂₃	—	2	2	no	—
205	Q₈×C₂₃	—	4	1	no	—
206	Q₈×C₂₃	—	4	1	no	—
207	C₂×C₄×C₂₃	—	2	1	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	1	no	—
212	C₂×C₄×C₂₃	—	2	1	no	—
213	C₂×C₄×C₂₃	—	2	2	no	—
214	C₂×C₄×C₂₃	—	2	1	no	—
215	C₂×C₄×C₂₃	—	2	1	no	—
216	C₂×C₄×C₂₃	—	2	1	no	—
217	C₂×C₄×C₂₃	—	2	1	no	—
218	C₂×C₄×C₂₃	—	2	1	no	—
219	C₂×C₄×C₂₃	—	2	1	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	1	no	—
222	C₂×C₄×C₂₃	—	2	1	no	—
223	C₂×C₄×C₂₃	—	2	1	no	—
224	C₂×C₄×C₂₃	—	4	1	no	—
225	C₂×C₄×C₂₃	—	4	1	no	—
226	C₂×C₄×C₂₃	—	4	1	no	—
227	C₂×C₄×C₂₃	—	4	1	no	—
228	C₂×C₄×C₂₃	—	4	1	no	—
229	C₂×C₄×C₂₃	—	4	1	no	—
230	C₂×C₄×C₂₃	—	4	1	no	—
231	C₂×C₄×C₂₃	—	4	1	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	—	4	1	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	4	1	no	—
247	C₂×C₄×C₂₃	—	4	1	no	—
248	C₂×C₄×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	8	4	no	—
250	C₂×C₄×C₂₃	#137	1	2	yes	—
251	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
252	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
255	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
256	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
257	C₈×C₂₃	—	2	2	no	—
258	C₈×C₂₃	—	4	1	no	—
259	C₈×C₂₃	—	4	1	no	—
260	C₈×C₂₃	—	4	1	no	—
261	#2494	C₂×C₂₃	1	1	yes	characteristic
262	#2360	C₂×C₂₃	1	1	yes	characteristic
263	#2074	C₂×C₂₃	1	1	yes	characteristic, complemented
264	#2066	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#2036	C₂×C₂₃	1	1	yes	characteristic
266	#2035	C₂×C₂₃	1	1	yes	characteristic
267	#1949	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1941	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1923	C₂×C₂₃	1	1	yes	characteristic
270	#1921	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#1920	C₂×C₂₃	1	1	yes	characteristic, complemented
272	#1865	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1853	C₂×C₂₃	1	1	yes	characteristic
274	#1823	C₂×C₂₃	1	1	yes	characteristic
275	C₄×#103	C₂×C₂₃	1	1	yes	characteristic, complemented
276	C₂²×C₂₃	—	2	2	no	—
277	C₂²×C₂₃	—	2	2	no	—
278	C₂²×C₂₃	—	2	2	no	—
279	C₂²×C₂₃	—	4	1	no	—
280	C₂²×C₂₃	—	4	1	no	—
281	C₂²×C₂₃	—	4	1	no	—
282	C₂²×C₂₃	—	4	1	no	—
283	C₂²×C₂₃	#525	1	1	yes	characteristic
284	C₂²×C₂₃	#546	1	1	yes	characteristic
285	C₂²×C₂₃	#547	1	1	yes	characteristic
286	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
287	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
288	C₂²×C₂₃	#577	1	1	yes	characteristic
289	C₂²×C₂₃	#577	1	1	yes	characteristic
290	C₄×C₂₃	—	2	2	no	—
291	C₄×C₂₃	—	4	1	no	—
292	C₄×C₂₃	—	4	1	no	—
293	C₄×C₂₃	—	4	2	no	—
294	C₄×C₂₃	—	4	1	no	—
295	C₄×C₂₃	—	4	2	no	—
296	C₄×C₂₃	—	4	1	no	—
297	C₄×C₂₃	—	4	2	no	—
298	C₄×C₂₃	—	8	1	no	—
299	C₄×C₂₃	—	8	1	no	—
300	C₄×C₂₃	—	8	1	no	—
301	C₄×C₂₃	—	8	1	no	—
302	C₄×C₂₃	—	8	1	no	—
303	C₄×C₂₃	#529	1	1	yes	characteristic
304	C₄×C₂₃	#539	1	1	yes	characteristic
305	C₄×C₂₃	#553	1	1	yes	characteristic
306	C₄×C₂₃	#556	1	1	yes	characteristic
307	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
308	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
309	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
310	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
311	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
312	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
313	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
314	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
315	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
317	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
318	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
319	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
322	#401	—	2	1	no	—
323	#397	—	2	1	no	—
324	#391	C₂²×C₂₃	1	1	yes	characteristic
325	#389	—	2	1	no	—
326	#389	C₂²×C₂₃	1	1	yes	characteristic
327	#389	C₂²×C₂₃	1	1	yes	characteristic
328	#389	C₂²×C₂₃	1	1	yes	characteristic
329	#388	—	2	1	no	—
330	#387	—	2	1	no	—
331	#387	—	2	1	no	—
332	#385	—	2	1	no	—
333	#382	C₂²×C₂₃	1	1	yes	characteristic
334	#380	C₂²×C₂₃	1	1	yes	characteristic
335	#380	C₂²×C₂₃	1	1	yes	characteristic
336	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
337	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
338	C₄×C₂²⋊C₄	—	2	1	no	—
339	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
340	#340	C₂²×C₂₃	1	1	yes	characteristic
341	#340	C₂²×C₂₃	1	1	yes	characteristic
342	#340	C₂²×C₂₃	1	1	yes	characteristic
343	#340	C₂²×C₂₃	1	1	yes	characteristic
344	#340	C₂²×C₂₃	1	1	yes	characteristic
345	#340	C₂²×C₂₃	1	1	yes	characteristic
346	#340	C₂²×C₂₃	1	1	yes	characteristic
347	#340	C₂²×C₂₃	1	1	yes	characteristic
348	#336	C₂²×C₂₃	1	1	yes	characteristic
349	#336	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×C₂₃	—	4	2	no	—
351	C₂×C₂₃	#2466	1	1	yes	characteristic
352	C₂×C₂₃	#3108	1	1	yes	characteristic
353	C₂×C₂₃	#3125	1	1	yes	characteristic
354	C₂×C₂₃	#3296	1	1	yes	characteristic
355	C₂×C₂₃	#3320	1	1	yes	characteristic
356	C₂×C₂₃	#3337	1	1	yes	characteristic
357	C₂×C₂₃	#3356	1	1	yes	characteristic
358	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
359	C₂³×C₄	—	2	1	no	—
360	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
361	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
362	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
363	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
365	C₂²⋊Q₈	—	2	2	no	—
366	C₂²⋊Q₈	—	2	2	no	—
367	C₄×Q₈	—	2	2	no	—
368	C₄×Q₈	—	4	1	no	—
369	C₂×C₄⋊C₄	—	2	1	no	—
370	C₂×C₄⋊C₄	—	2	2	no	—
371	C₂×C₄⋊C₄	—	2	1	no	—
372	C₂×C₄⋊C₄	—	2	1	no	—
373	C₂×C₄⋊C₄	—	2	1	no	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	4	1	no	—
377	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
378	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
380	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
381	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₂²⋊C₄	—	2	1	no	—
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
388	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
389	C₂×C₄²	—	2	1	no	—
390	C₂×C₄²	—	2	1	no	—
391	C₂×C₄²	—	2	1	no	—
392	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
394	#107	—	2	2	no	—
395	#106	—	2	2	no	—
396	#106	—	2	2	no	—
397	#106	—	4	1	no	—
398	C₄⋊C₈	—	2	2	no	—
399	#103	—	2	2	no	—
400	#103	—	2	2	no	—
401	#103	—	2	2	no	—
402	#103	—	4	1	no	—
403	C₂²⋊C₈	—	2	2	no	—
404	C₂²⋊C₈	—	4	1	no	—
405	C₄×C₈	—	4	1	no	—
406	#95	—	2	1	no	—
407	#95	—	2	1	no	—
408	#95	—	2	1	no	—
409	#95	—	2	1	no	—
410	#95	—	2	1	no	—
411	#95	—	2	1	no	—
412	#95	—	2	1	no	—
413	#95	—	2	1	no	—
414	#95	—	2	1	no	—
415	#95	—	2	1	no	—
416	#95	—	2	1	no	—
417	#95	D₄×C₂₃	1	1	yes	characteristic
418	#95	D₄×C₂₃	1	1	yes	characteristic
419	C₂₃	#27037	1	1	yes	characteristic, direct product factor, Sylow
420	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
421	C₂×Q₈	—	2	1	no	—
422	C₂×Q₈	—	2	1	no	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	2	1	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×Q₈	#41×C₂₃	1	2	yes	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	1	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	1	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	1	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	8	1	no	—
444	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
445	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
447	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
449	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	#41×C₂₃	1	2	yes	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	4	2	no	—
476	C₄⋊C₄	—	4	2	no	—
477	C₄⋊C₄	—	4	2	no	—
478	C₄⋊C₄	—	8	1	no	—
479	C₄⋊C₄	—	8	1	no	—
480	C₄⋊C₄	#41×C₂₃	1	2	yes	—
481	C₄⋊C₄	#41×C₂₃	1	2	yes	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	2	no	—
484	C₂²⋊C₄	—	2	2	no	—
485	C₂²⋊C₄	—	4	2	no	—
486	C₂²⋊C₄	—	8	1	no	—
487	C₄²	—	2	1	no	—
488	C₄²	—	2	1	no	—
489	C₄²	—	2	1	no	—
490	C₄²	—	2	1	no	—
491	C₄²	—	4	2	no	—
492	C₄²	—	8	1	no	—
493	C₄²	—	8	1	no	—
494	C₂³	—	2	1	no	—
495	C₂³	—	2	1	no	—
496	C₂³	—	2	1	no	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	1	no	—
499	C₂³	—	2	1	no	—
500	C₂³	C₂₃×#135	1	2	yes	—
501	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
502	Q₈	—	2	2	no	—
503	Q₈	—	2	2	no	—
504	Q₈	—	4	1	no	—
505	Q₈	—	4	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	8	4	no	—
549	C₂×C₄	C₂₃×#137	1	2	yes	—
550	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
551	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
552	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
553	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
554	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
555	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
556	C₈	—	2	2	no	—
557	C₈	—	4	1	no	—
558	C₈	—	4	1	no	—
559	C₈	—	4	1	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	4	1	no	—
564	C₂²	—	4	1	no	—
565	C₂²	—	4	1	no	—
566	C₂²	—	4	1	no	—
567	C₂²	C₂₃×#525	1	1	yes	characteristic
568	C₂²	C₂₃×#546	1	1	yes	characteristic
569	C₂²	C₂₃×#547	1	1	yes	characteristic
570	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
571	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
572	C₂²	C₂₃×#577	1	1	yes	characteristic
573	C₂²	C₂₃×#577	1	1	yes	characteristic
574	C₄	—	2	2	no	—
575	C₄	—	4	1	no	—
576	C₄	—	4	2	no	—
577	C₄	—	4	1	no	—
578	C₄	—	4	2	no	—
579	C₄	—	4	1	no	—
580	C₄	—	4	1	no	—
581	C₄	—	4	2	no	—
582	C₄	—	8	1	no	—
583	C₄	—	8	1	no	—
584	C₄	—	8	1	no	—
585	C₄	—	8	1	no	—
586	C₄	—	8	1	no	—
587	C₄	C₂₃×#529	1	1	yes	characteristic
588	C₄	C₂₃×#539	1	1	yes	characteristic
589	C₄	C₂₃×#553	1	1	yes	characteristic
590	C₄	C₂₃×#556	1	1	yes	characteristic
591	C₂	—	4	2	no	—
592	C₂	C₂₃×#2466	1	1	yes	characteristic
593	C₂	C₂₃×#3108	1	1	yes	characteristic
594	C₂	C₂₃×#3125	1	1	yes	characteristic
595	C₂	C₂₃×#3296	1	1	yes	characteristic
596	C₂	C₂₃×#3320	1	1	yes	characteristic
597	C₂	C₂₃×#3337	1	1	yes	characteristic
598	C₂	C₂₃×#3356	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2074	C₂	semidirect
4. C₂₃×#2066	C₂	semidirect
7. C₂₃×#1949	C₂	semidirect
8. C₂₃×#1941	C₂	semidirect
10. C₂₃×#1921	C₂	semidirect
11. C₂₃×#1920	C₂	semidirect
12. C₂₃×#1865	C₂	semidirect
15. C₄×C₂₃×#103	C₂	semidirect
194. #27037	C₂₃	direct (N × H)
263. #2074	C₂×C₂₃	semidirect
264. #2066	C₂×C₂₃	semidirect
267. #1949	C₂×C₂₃	semidirect
268. #1941	C₂×C₂₃	semidirect
270. #1921	C₂×C₂₃	semidirect
271. #1920	C₂×C₂₃	semidirect
272. #1865	C₂×C₂₃	semidirect
275. C₄×#103	C₂×C₂₃	semidirect
419. C₂₃	#27037	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×#2494	2
C₂₃×#2360	2
C₂₃×#2074	2
C₂₃×#2066	2
C₂₃×#2036	2
C₂₃×#2035	2
C₂₃×#1949	2
C₂₃×#1941	2
C₂₃×#1923	2
C₂₃×#1921	2
C₂₃×#1920	2
C₂₃×#1865	2
C₂₃×#1853	2
C₂₃×#1823	2
C₄×C₂₃×#103	2
#27037	23

Subgroups — C23×#27037 #19174663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#27037 #19174663