Subgroups — C₂₃×#32676 #19180302

Order 5888.

732 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2592	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2056	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2055	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2016	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2015	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#2015	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#2013	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1985	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1973	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1971	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1928	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1928	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1927	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#1926	C₂	1	1	yes	maximal, characteristic
16	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₂₃	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×#103	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#103	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×C₂₃×#97	C₂²	1	1	yes	characteristic
35	C₂₃×#400	—	2	1	no	—
36	C₂₃×#400	—	2	1	no	—
37	C₂₃×#397	—	2	1	no	—
38	C₂₃×#393	—	2	1	no	—
39	C₂₃×#391	—	2	1	no	—
40	C₂₃×#391	C₂²	1	1	yes	characteristic
41	C₂₃×#391	C₂²	1	1	yes	characteristic
42	C₂₃×#391	C₂²	1	1	yes	characteristic
43	C₂₃×#391	C₂²	1	1	yes	characteristic
44	C₂₃×#391	C₂²	1	1	yes	characteristic
45	C₂₃×#391	C₂²	1	1	yes	characteristic
46	C₂₃×#390	C₂²	1	1	yes	characteristic
47	C₂₃×#390	C₂²	1	1	yes	characteristic
48	C₂₃×#388	—	2	1	no	—
49	C₂₃×#388	—	2	1	no	—
50	C₂₃×#382	—	2	1	no	—
51	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
52	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#336	C₂²	1	1	yes	characteristic
56	C₂₃×#336	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	D₄	1	1	yes	characteristic
60	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×Q₁₆×C₂₃	—	2	2	no	—
65	C₂×Q₁₆×C₂₃	—	4	1	no	—
66	C₂×Q₁₆×C₂₃	—	4	1	no	—
67	C₂×Q₁₆×C₂₃	—	4	1	no	—
68	C₂×Q₁₆×C₂₃	—	4	1	no	—
69	C₂×Q₁₆×C₂₃	—	4	1	no	—
70	C₂×SD₁₆×C₂₃	—	2	2	no	—
71	C₂×SD₁₆×C₂₃	—	4	1	no	—
72	C₂×SD₁₆×C₂₃	—	4	1	no	—
73	C₂×SD₁₆×C₂₃	—	4	1	no	—
74	C₂×SD₁₆×C₂₃	—	4	1	no	—
75	C₂×SD₁₆×C₂₃	—	4	1	no	—
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₂₃×C₄⋊Q₈	—	4	1	no	—
81	C₂₃×C₄⋊Q₈	—	4	1	no	—
82	C₂₃×#124	—	4	1	no	—
83	C₂₃×#124	—	4	1	no	—
84	C₄×Q₈×C₂₃	—	4	1	no	—
85	C₄×Q₈×C₂₃	—	4	1	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
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×C₂²⋊C₄×C₂₃	—	2	1	no	—
101	C₂×C₄²×C₂₃	—	2	1	no	—
102	C₂×C₄²×C₂₃	—	2	1	no	—
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
106	C₂₃×#107	—	4	1	no	—
107	C₂₃×#106	—	4	1	no	—
108	C₂₃×C₄⋊C₈	—	4	1	no	—
109	C₂₃×#103	—	2	2	no	—
110	C₂₃×#103	—	2	2	no	—
111	C₂₃×#103	—	2	2	no	—
112	C₂₃×#103	—	4	1	no	—
113	C₂₃×#103	—	4	1	no	—
114	C₂₃×#103	—	4	1	no	—
115	C₂₃×#102	—	2	2	no	—
116	C₂₃×#102	—	4	1	no	—
117	C₂₃×#97	—	4	1	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	—	2	1	no	—
125	C₂₃×#95	—	2	1	no	—
126	C₂₃×#95	—	2	1	no	—
127	C₂₃×#95	—	2	1	no	—
128	C₂₃×#95	—	2	1	no	—
129	C₂₃×#95	—	2	1	no	—
130	C₂₃×#95	—	2	1	no	—
131	C₂₃×#95	—	2	1	no	—
132	C₂⁴×C₂₃	—	2	1	no	—
133	C₂×Q₈×C₂₃	—	2	2	no	—
134	C₂×Q₈×C₂₃	—	2	1	no	—
135	C₂×Q₈×C₂₃	—	2	1	no	—
136	C₂×Q₈×C₂₃	—	2	1	no	—
137	C₂×Q₈×C₂₃	—	2	1	no	—
138	C₂×Q₈×C₂₃	—	2	1	no	—
139	C₂×Q₈×C₂₃	—	2	1	no	—
140	C₂×Q₈×C₂₃	—	2	1	no	—
141	C₂×Q₈×C₂₃	—	2	1	no	—
142	C₂×Q₈×C₂₃	—	2	1	no	—
143	C₂×Q₈×C₂₃	—	2	1	no	—
144	C₂×Q₈×C₂₃	—	2	1	no	—
145	C₂×Q₈×C₂₃	—	2	1	no	—
146	C₂×Q₈×C₂₃	—	2	1	no	—
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	2	1	no	—
149	C₂×Q₈×C₂₃	—	2	1	no	—
150	C₂×Q₈×C₂₃	—	2	1	no	—
151	C₂×Q₈×C₂₃	—	2	1	no	—
152	C₂×Q₈×C₂₃	—	4	1	no	—
153	C₂×Q₈×C₂₃	—	4	1	no	—
154	C₂×D₄×C₂₃	—	2	1	no	—
155	C₂×D₄×C₂₃	—	2	1	no	—
156	C₂×D₄×C₂₃	—	2	1	no	—
157	C₂×D₄×C₂₃	—	2	1	no	—
158	C₂×D₄×C₂₃	—	2	1	no	—
159	C₂×D₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	2	1	no	—
164	C₂²×C₄×C₂₃	—	2	1	no	—
165	C₂²×C₄×C₂₃	—	2	1	no	—
166	C₂²×C₄×C₂₃	—	2	1	no	—
167	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
174	Q₁₆×C₂₃	—	4	4	no	—
175	Q₁₆×C₂₃	—	4	4	no	—
176	SD₁₆×C₂₃	—	4	4	no	—
177	SD₁₆×C₂₃	—	4	4	no	—
178	C₂×C₈×C₂₃	—	2	1	no	—
179	C₂×C₈×C₂₃	—	2	1	no	—
180	C₂×C₈×C₂₃	—	2	1	no	—
181	C₂×C₈×C₂₃	—	2	1	no	—
182	C₂×C₈×C₂₃	—	2	1	no	—
183	C₂×C₈×C₂₃	—	2	1	no	—
184	C₂×C₈×C₂₃	—	2	1	no	—
185	C₂×C₈×C₂₃	—	2	1	no	—
186	C₂×C₈×C₂₃	—	2	1	no	—
187	C₂×C₈×C₂₃	—	2	1	no	—
188	C₂×C₈×C₂₃	#41	1	2	yes	—
189	C₂×C₈×C₂₃	#41	1	2	yes	—
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	—	2	1	no	—
197	C₄⋊C₄×C₂₃	—	2	1	no	—
198	C₄⋊C₄×C₂₃	—	4	1	no	—
199	C₄⋊C₄×C₂₃	—	4	1	no	—
200	C₄⋊C₄×C₂₃	—	4	1	no	—
201	C₄⋊C₄×C₂₃	—	4	2	no	—
202	C₄⋊C₄×C₂₃	—	4	2	no	—
203	C₂²⋊C₄×C₂₃	—	4	2	no	—
204	C₂²⋊C₄×C₂₃	—	4	2	no	—
205	C₂²⋊C₄×C₂₃	—	4	2	no	—
206	C₂²⋊C₄×C₂₃	—	4	2	no	—
207	C₂²⋊C₄×C₂₃	—	8	1	no	—
208	C₂²⋊C₄×C₂₃	—	8	1	no	—
209	C₄²×C₂₃	—	2	1	no	—
210	C₄²×C₂₃	—	2	1	no	—
211	C₄²×C₂₃	—	2	1	no	—
212	C₄²×C₂₃	—	2	1	no	—
213	C₄²×C₂₃	—	2	1	no	—
214	C₄²×C₂₃	—	2	1	no	—
215	C₄²×C₂₃	—	4	2	no	—
216	C₄²×C₂₃	—	8	1	no	—
217	#32676	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	—	4	1	no	—
224	C₂³×C₂₃	—	4	1	no	—
225	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	Q₈×C₂₃	—	4	4	no	—
227	Q₈×C₂₃	—	4	1	no	—
228	Q₈×C₂₃	—	4	1	no	—
229	Q₈×C₂₃	—	4	1	no	—
230	Q₈×C₂₃	—	4	1	no	—
231	Q₈×C₂₃	—	4	1	no	—
232	Q₈×C₂₃	—	4	1	no	—
233	Q₈×C₂₃	—	4	1	no	—
234	Q₈×C₂₃	—	4	1	no	—
235	Q₈×C₂₃	—	4	1	no	—
236	Q₈×C₂₃	—	4	1	no	—
237	Q₈×C₂₃	—	4	1	no	—
238	Q₈×C₂₃	—	4	1	no	—
239	D₄×C₂₃	—	4	1	no	—
240	D₄×C₂₃	—	4	1	no	—
241	D₄×C₂₃	—	4	1	no	—
242	D₄×C₂₃	—	4	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	—	2	1	no	—
249	C₂×C₄×C₂₃	—	2	1	no	—
250	C₂×C₄×C₂₃	—	2	1	no	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	1	no	—
255	C₂×C₄×C₂₃	—	2	1	no	—
256	C₂×C₄×C₂₃	—	2	1	no	—
257	C₂×C₄×C₂₃	—	2	1	no	—
258	C₂×C₄×C₂₃	—	2	1	no	—
259	C₂×C₄×C₂₃	—	2	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	—	4	1	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	1	no	—
269	C₂×C₄×C₂₃	—	4	1	no	—
270	C₂×C₄×C₂₃	—	4	1	no	—
271	C₂×C₄×C₂₃	—	4	1	no	—
272	C₂×C₄×C₂₃	—	4	1	no	—
273	C₂×C₄×C₂₃	—	4	1	no	—
274	C₂×C₄×C₂₃	—	4	1	no	—
275	C₂×C₄×C₂₃	—	4	1	no	—
276	C₂×C₄×C₂₃	—	4	1	no	—
277	C₂×C₄×C₂₃	—	4	1	no	—
278	C₂×C₄×C₂₃	—	4	1	no	—
279	C₂×C₄×C₂₃	—	4	1	no	—
280	C₂×C₄×C₂₃	—	4	1	no	—
281	C₂×C₄×C₂₃	#137	1	2	yes	—
282	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
283	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
284	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
285	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
286	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
287	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
288	C₈×C₂₃	—	2	2	no	—
289	C₈×C₂₃	—	2	2	no	—
290	C₈×C₂₃	—	4	1	no	—
291	C₈×C₂₃	—	4	1	no	—
292	#2592	C₂×C₂₃	1	1	yes	characteristic
293	#2568	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#2056	C₂×C₂₃	1	1	yes	characteristic, complemented
295	#2055	C₂×C₂₃	1	1	yes	characteristic
296	#2016	C₂×C₂₃	1	1	yes	characteristic, complemented
297	#2015	C₂×C₂₃	1	1	yes	characteristic
298	#2015	C₂×C₂₃	1	1	yes	characteristic
299	#2013	C₂×C₂₃	1	1	yes	characteristic
300	#1985	C₂×C₂₃	1	1	yes	characteristic, complemented
301	#1973	C₂×C₂₃	1	1	yes	characteristic
302	#1971	C₂×C₂₃	1	1	yes	characteristic, complemented
303	#1928	C₂×C₂₃	1	1	yes	characteristic, complemented
304	#1928	C₂×C₂₃	1	1	yes	characteristic, complemented
305	#1927	C₂×C₂₃	1	1	yes	characteristic, complemented
306	#1926	C₂×C₂₃	1	1	yes	characteristic
307	C₂²×C₂₃	—	8	1	no	—
308	C₂²×C₂₃	—	8	1	no	—
309	C₂²×C₂₃	—	8	1	no	—
310	C₂²×C₂₃	—	8	1	no	—
311	C₂²×C₂₃	—	8	1	no	—
312	C₂²×C₂₃	—	8	1	no	—
313	C₂²×C₂₃	—	8	1	no	—
314	C₂²×C₂₃	#534	1	1	yes	characteristic
315	C₂²×C₂₃	#550	1	1	yes	characteristic
316	C₂²×C₂₃	#550	1	1	yes	characteristic
317	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
318	C₂²×C₂₃	#575	1	1	yes	characteristic
319	C₂²×C₂₃	#577	1	1	yes	characteristic
320	C₂²×C₂₃	#578	1	1	yes	characteristic
321	C₄×C₂₃	—	2	2	no	—
322	C₄×C₂₃	—	4	1	no	—
323	C₄×C₂₃	—	4	1	no	—
324	C₄×C₂₃	—	4	1	no	—
325	C₄×C₂₃	—	4	1	no	—
326	C₄×C₂₃	—	4	1	no	—
327	C₄×C₂₃	—	8	1	no	—
328	C₄×C₂₃	—	8	1	no	—
329	C₄×C₂₃	—	8	1	no	—
330	C₄×C₂₃	—	8	1	no	—
331	C₄×C₂₃	—	8	1	no	—
332	C₄×C₂₃	—	8	1	no	—
333	C₄×C₂₃	—	8	1	no	—
334	C₄×C₂₃	#538	1	1	yes	characteristic
335	C₄×C₂₃	#538	1	1	yes	characteristic
336	C₄×C₂₃	#555	1	1	yes	characteristic
337	C₄×C₂₃	H(3,𝔽₄)	1	1	yes	characteristic
338	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
339	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
340	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
341	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
346	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
348	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
349	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
350	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
357	#400	—	2	1	no	—
358	#400	—	2	1	no	—
359	#397	—	2	1	no	—
360	#393	—	2	1	no	—
361	#391	—	2	1	no	—
362	#391	C₂²×C₂₃	1	1	yes	characteristic
363	#391	C₂²×C₂₃	1	1	yes	characteristic
364	#391	C₂²×C₂₃	1	1	yes	characteristic
365	#391	C₂²×C₂₃	1	1	yes	characteristic
366	#391	C₂²×C₂₃	1	1	yes	characteristic
367	#391	C₂²×C₂₃	1	1	yes	characteristic
368	#390	C₂²×C₂₃	1	1	yes	characteristic
369	#390	C₂²×C₂₃	1	1	yes	characteristic
370	#388	—	2	1	no	—
371	#388	—	2	1	no	—
372	#382	—	2	1	no	—
373	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
374	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
375	#340	C₂²×C₂₃	1	1	yes	characteristic
376	#340	C₂²×C₂₃	1	1	yes	characteristic
377	#336	C₂²×C₂₃	1	1	yes	characteristic
378	#336	C₂²×C₂₃	1	1	yes	characteristic
379	#336	C₂²×C₂₃	1	1	yes	characteristic
380	#336	C₂²×C₂₃	1	1	yes	characteristic
381	C₂×C₂₃	—	16	1	no	—
382	C₂×C₂₃	#2733	1	1	yes	characteristic
383	C₂×C₂₃	#3196	1	1	yes	characteristic
384	C₂×C₂₃	#3201	1	1	yes	characteristic
385	C₂×C₂₃	#3398	1	1	yes	characteristic
386	C₂×C₂₃	#3400	1	1	yes	characteristic
387	C₂×C₂₃	#3401	1	1	yes	characteristic
388	C₂×C₂₃	#3402	1	1	yes	characteristic
389	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
390	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
391	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
392	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
393	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
394	C₂×Q₁₆	—	2	2	no	—
395	C₂×Q₁₆	—	4	1	no	—
396	C₂×Q₁₆	—	4	1	no	—
397	C₂×Q₁₆	—	4	1	no	—
398	C₂×Q₁₆	—	4	1	no	—
399	C₂×Q₁₆	—	4	1	no	—
400	C₂×SD₁₆	—	2	2	no	—
401	C₂×SD₁₆	—	4	1	no	—
402	C₂×SD₁₆	—	4	1	no	—
403	C₂×SD₁₆	—	4	1	no	—
404	C₂×SD₁₆	—	4	1	no	—
405	C₂×SD₁₆	—	4	1	no	—
406	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
407	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
408	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
409	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
410	C₄⋊Q₈	—	4	1	no	—
411	C₄⋊Q₈	—	4	1	no	—
412	#124	—	4	1	no	—
413	#124	—	4	1	no	—
414	C₄×Q₈	—	4	1	no	—
415	C₄×Q₈	—	4	1	no	—
416	C₂×C₄⋊C₄	—	2	1	no	—
417	C₂×C₄⋊C₄	—	2	1	no	—
418	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
419	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
420	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
423	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
425	C₂×C₂²⋊C₄	—	2	1	no	—
426	C₂×C₂²⋊C₄	—	2	1	no	—
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	C₂×C₂²⋊C₄	—	2	1	no	—
429	C₂×C₂²⋊C₄	—	2	1	no	—
430	C₂×C₂²⋊C₄	—	2	1	no	—
431	C₂×C₄²	—	2	1	no	—
432	C₂×C₄²	—	2	1	no	—
433	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
434	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
435	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
436	#107	—	4	1	no	—
437	#106	—	4	1	no	—
438	C₄⋊C₈	—	4	1	no	—
439	#103	—	2	2	no	—
440	#103	—	2	2	no	—
441	#103	—	2	2	no	—
442	#103	—	4	1	no	—
443	#103	—	4	1	no	—
444	#103	—	4	1	no	—
445	#102	—	2	2	no	—
446	#102	—	4	1	no	—
447	#97	—	4	1	no	—
448	#95	—	2	1	no	—
449	#95	—	2	1	no	—
450	#95	—	2	1	no	—
451	#95	—	2	1	no	—
452	#95	—	2	1	no	—
453	#95	—	2	1	no	—
454	#95	—	2	1	no	—
455	#95	—	2	1	no	—
456	#95	—	2	1	no	—
457	#95	—	2	1	no	—
458	#95	—	2	1	no	—
459	#95	—	2	1	no	—
460	#95	—	2	1	no	—
461	#95	—	2	1	no	—
462	C₂₃	#32676	1	1	yes	characteristic, direct product factor, Sylow
463	C₂⁴	—	2	1	no	—
464	C₂×Q₈	—	2	2	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	2	1	no	—
468	C₂×Q₈	—	2	1	no	—
469	C₂×Q₈	—	2	1	no	—
470	C₂×Q₈	—	2	1	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	—	2	1	no	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	2	1	no	—
478	C₂×Q₈	—	2	1	no	—
479	C₂×Q₈	—	2	1	no	—
480	C₂×Q₈	—	2	1	no	—
481	C₂×Q₈	—	2	1	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	4	1	no	—
484	C₂×Q₈	—	4	1	no	—
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	—	2	1	no	—
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	2	1	no	—
498	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
499	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
500	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
501	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
502	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
503	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
504	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
505	Q₁₆	—	4	4	no	—
506	Q₁₆	—	4	4	no	—
507	SD₁₆	—	4	4	no	—
508	SD₁₆	—	4	4	no	—
509	C₂×C₈	—	2	1	no	—
510	C₂×C₈	—	2	1	no	—
511	C₂×C₈	—	2	1	no	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	#41×C₂₃	1	2	yes	—
520	C₂×C₈	#41×C₂₃	1	2	yes	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	4	1	no	—
530	C₄⋊C₄	—	4	1	no	—
531	C₄⋊C₄	—	4	1	no	—
532	C₄⋊C₄	—	4	2	no	—
533	C₄⋊C₄	—	4	2	no	—
534	C₂²⋊C₄	—	4	2	no	—
535	C₂²⋊C₄	—	4	2	no	—
536	C₂²⋊C₄	—	4	2	no	—
537	C₂²⋊C₄	—	4	2	no	—
538	C₂²⋊C₄	—	8	1	no	—
539	C₂²⋊C₄	—	8	1	no	—
540	C₄²	—	2	1	no	—
541	C₄²	—	2	1	no	—
542	C₄²	—	2	1	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	4	2	no	—
547	C₄²	—	8	1	no	—
548	C₂³	—	4	1	no	—
549	C₂³	—	4	1	no	—
550	C₂³	—	4	1	no	—
551	C₂³	—	4	1	no	—
552	C₂³	—	4	1	no	—
553	C₂³	—	4	1	no	—
554	C₂³	—	4	1	no	—
555	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
556	Q₈	—	4	4	no	—
557	Q₈	—	4	1	no	—
558	Q₈	—	4	1	no	—
559	Q₈	—	4	1	no	—
560	Q₈	—	4	1	no	—
561	Q₈	—	4	1	no	—
562	Q₈	—	4	1	no	—
563	Q₈	—	4	1	no	—
564	Q₈	—	4	1	no	—
565	Q₈	—	4	1	no	—
566	Q₈	—	4	1	no	—
567	Q₈	—	4	1	no	—
568	Q₈	—	4	1	no	—
569	D₄	—	4	1	no	—
570	D₄	—	4	1	no	—
571	D₄	—	4	1	no	—
572	D₄	—	4	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	1	no	—
583	C₂×C₄	—	2	1	no	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	1	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	1	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	C₂₃×#137	1	2	yes	—
612	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
613	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
614	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
615	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
616	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
617	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
618	C₈	—	2	2	no	—
619	C₈	—	2	2	no	—
620	C₈	—	4	1	no	—
621	C₈	—	4	1	no	—
622	C₂²	—	8	1	no	—
623	C₂²	—	8	1	no	—
624	C₂²	—	8	1	no	—
625	C₂²	—	8	1	no	—
626	C₂²	—	8	1	no	—
627	C₂²	—	8	1	no	—
628	C₂²	—	8	1	no	—
629	C₂²	C₂₃×#534	1	1	yes	characteristic
630	C₂²	C₂₃×#550	1	1	yes	characteristic
631	C₂²	C₂₃×#550	1	1	yes	characteristic
632	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
633	C₂²	C₂₃×#575	1	1	yes	characteristic
634	C₂²	C₂₃×#577	1	1	yes	characteristic
635	C₂²	C₂₃×#578	1	1	yes	characteristic
636	C₄	—	2	2	no	—
637	C₄	—	4	1	no	—
638	C₄	—	4	1	no	—
639	C₄	—	4	1	no	—
640	C₄	—	4	1	no	—
641	C₄	—	4	1	no	—
642	C₄	—	8	1	no	—
643	C₄	—	8	1	no	—
644	C₄	—	8	1	no	—
645	C₄	—	8	1	no	—
646	C₄	—	8	1	no	—
647	C₄	—	8	1	no	—
648	C₄	—	8	1	no	—
649	C₄	C₂₃×#538	1	1	yes	characteristic
650	C₄	C₂₃×#538	1	1	yes	characteristic
651	C₄	C₂₃×#555	1	1	yes	characteristic
652	C₄	C₂₃×H(3,𝔽₄)	1	1	yes	characteristic
653	C₂	—	16	1	no	—
654	C₂	C₂₃×#2733	1	1	yes	characteristic
655	C₂	C₂₃×#3196	1	1	yes	characteristic
656	C₂	C₂₃×#3201	1	1	yes	characteristic
657	C₂	C₂₃×#3398	1	1	yes	characteristic
658	C₂	C₂₃×#3400	1	1	yes	characteristic
659	C₂	C₂₃×#3401	1	1	yes	characteristic
660	C₂	C₂₃×#3402	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2568	C₂	semidirect
3. C₂₃×#2056	C₂	semidirect
5. C₂₃×#2016	C₂	semidirect
9. C₂₃×#1985	C₂	semidirect
11. C₂₃×#1971	C₂	semidirect
12. C₂₃×#1928	C₂	semidirect
13. C₂₃×#1928	C₂	semidirect
14. C₂₃×#1927	C₂	semidirect
217. #32676	C₂₃	direct (N × H)
293. #2568	C₂×C₂₃	semidirect
294. #2056	C₂×C₂₃	semidirect
296. #2016	C₂×C₂₃	semidirect
300. #1985	C₂×C₂₃	semidirect
302. #1971	C₂×C₂₃	semidirect
303. #1928	C₂×C₂₃	semidirect
304. #1928	C₂×C₂₃	semidirect
305. #1927	C₂×C₂₃	semidirect
462. C₂₃	#32676	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×#2592	2
C₂₃×#2568	2
C₂₃×#2056	2
C₂₃×#2055	2
C₂₃×#2016	2
C₂₃×#2015	2
C₂₃×#2013	2
C₂₃×#1985	2
C₂₃×#1973	2
C₂₃×#1971	2
C₂₃×#1928	2
C₂₃×#1927	2
C₂₃×#1926	2
#32676	23

Subgroups — C23×#32676 #19180302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#32676 #19180302