Subgroups — C₂₃×#21628 #19169254

Order 5888.

738 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3076	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2960	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1986	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1922	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1866	C₂	1	2	yes	maximal
8	C₂₃×#1806	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1773	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1765	C₂	1	2	yes	maximal
12	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
14	C₂₃×#533	C₂²	1	1	yes	characteristic
15	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#479	C₄	1	2	yes	complemented
17	C₂₃×#478	C₄	1	2	yes	complemented
18	C₂₃×#477	C₄	1	1	yes	characteristic, complemented
19	C₂₃×#476	C₄	1	1	yes	characteristic, complemented
20	C₂₃×#475	C₄	1	1	yes	characteristic, complemented
21	C₂₃×#474	C₄	1	1	yes	characteristic, complemented
22	C₂₃×#446	—	2	1	no	—
23	C₂₃×#446	—	2	1	no	—
24	C₈×Q₈×C₂₃	—	2	1	no	—
25	C₈×Q₈×C₂₃	—	2	1	no	—
26	C₂×C₂₃×#108	C₂²	1	2	yes	—
27	C₂₃×#427	C₂²	1	2	yes	—
28	C₂₃×#424	C₂²	1	2	yes	—
29	C₂₃×#423	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#104	C₂²	1	2	yes	—
32	C₂₃×#416	C₂²	1	2	yes	—
33	C₂₃×#405	—	2	1	no	—
34	C₂₃×#405	—	2	1	no	—
35	C₂₃×#405	C₂²	1	1	yes	characteristic
36	C₂₃×#405	C₂²	1	2	yes	—
37	C₂₃×#405	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₂₃	—	2	1	no	—
39	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×#97	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₂₃	—	2	1	no	—
42	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
44	C₂₃×#341	C₂²	1	2	yes	—
45	C₂₃×#339	C₂²	1	2	yes	—
46	C₂₃×#339	C₂²	1	2	yes	—
47	C₂₃×#339	C₂²	1	2	yes	—
48	C₂₃×#336	C₂²	1	2	yes	—
49	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
50	C₂₃×#131	—	2	2	no	—
51	C₂₃×#131	—	2	2	no	—
52	C₂₃×#131	—	2	2	no	—
53	C₂₃×#131	Q₈	1	4	yes	complemented
54	C₂×M₁₆×C₂₃	—	2	1	no	—
55	C₂×M₁₆×C₂₃	—	2	1	no	—
56	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
57	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
58	C₂×M₁₆×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₂₃	—	2	1	no	—
60	C₂²×C₈×C₂₃	—	2	1	no	—
61	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
62	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
65	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
66	C₂₃×#125	C₂×C₄	1	2	yes	—
67	C₂₃×C₂²⋊Q₈	—	2	1	no	—
68	C₂₃×C₂²⋊Q₈	—	2	1	no	—
69	C₄×Q₈×C₂₃	—	2	1	no	—
70	C₄×Q₈×C₂₃	—	2	1	no	—
71	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
72	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
73	C₄×D₄×C₂₃	—	2	1	no	—
74	C₄×D₄×C₂₃	—	2	1	no	—
75	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
76	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
77	C₂₃×#117	—	2	1	no	—
78	C₂₃×#117	C₂³	1	1	yes	characteristic
79	C₂₃×#117	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄²×C₂₃	—	2	1	no	—
82	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
83	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
84	C₂₃×#108	—	2	4	no	—
85	C₂₃×#107	C₂×C₄	1	2	yes	—
86	C₂₃×#107	C₂×C₄	1	2	yes	—
87	C₂₃×#106	C₂×C₄	1	2	yes	—
88	C₂₃×#106	C₂×C₄	1	2	yes	—
89	C₂₃×C₄⋊C₈	—	2	2	no	—
90	C₂₃×C₄⋊C₈	—	2	2	no	—
91	C₂₃×C₄⋊C₈	—	2	1	no	—
92	C₂₃×C₄⋊C₈	—	2	1	no	—
93	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
94	C₂₃×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
95	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
96	C₂₃×#104	—	2	4	no	—
97	C₂₃×#103	C₂×C₄	1	2	yes	—
98	C₂₃×#103	C₂×C₄	1	2	yes	—
99	C₂₃×#102	C₂×C₄	1	2	yes	—
100	C₂₃×#102	C₂×C₄	1	2	yes	—
101	C₂₃×#97	—	2	1	no	—
102	C₂₃×#97	—	2	1	no	—
103	C₂₃×#97	—	2	2	no	—
104	C₂₃×#97	—	2	4	no	—
105	C₂₃×#97	—	2	1	no	—
106	C₄×C₈×C₂₃	—	2	1	no	—
107	C₄×C₈×C₂₃	—	2	1	no	—
108	C₄×C₈×C₂₃	—	2	2	no	—
109	C₄×C₈×C₂₃	—	2	1	no	—
110	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
111	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
112	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
113	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
114	C₂₃×#95	—	2	1	no	—
115	#41×C₂₃	—	2	1	no	—
116	#41×C₂₃	—	2	1	no	—
117	#41×C₂₃	C₂×Q₈	1	2	yes	—
118	#41×C₂₃	#41	1	2	yes	—
119	C₂×Q₈×C₂₃	—	2	1	no	—
120	C₂×Q₈×C₂₃	—	2	1	no	—
121	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×D₄×C₂₃	—	2	1	no	—
123	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₂₃	—	2	1	no	—
125	C₂²×C₄×C₂₃	—	2	1	no	—
126	C₂²×C₄×C₂₃	—	2	1	no	—
127	C₂²×C₄×C₂₃	—	2	2	no	—
128	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
129	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
131	M₁₆×C₂₃	—	2	4	no	—
132	M₁₆×C₂₃	—	2	1	no	—
133	M₁₆×C₂₃	—	2	1	no	—
134	M₁₆×C₂₃	—	2	4	no	—
135	M₁₆×C₂₃	—	2	2	no	—
136	M₁₆×C₂₃	—	2	2	no	—
137	M₁₆×C₂₃	C₂×Q₈	1	2	yes	—
138	M₁₆×C₂₃	#41	1	2	yes	—
139	C₂×C₈×C₂₃	—	2	4	no	—
140	C₂×C₈×C₂₃	—	2	1	no	—
141	C₂×C₈×C₂₃	—	2	1	no	—
142	C₂×C₈×C₂₃	—	2	2	no	—
143	C₂×C₈×C₂₃	—	2	1	no	—
144	C₂×C₈×C₂₃	—	2	1	no	—
145	C₂×C₈×C₂₃	—	2	2	no	—
146	C₂×C₈×C₂₃	—	2	4	no	—
147	C₂×C₈×C₂₃	—	2	2	no	—
148	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
149	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
150	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
155	C₂×C₈×C₂₃	#41	1	2	yes	—
156	C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₄⋊C₄×C₂₃	—	2	1	no	—
158	C₄⋊C₄×C₂₃	—	2	2	no	—
159	C₄⋊C₄×C₂₃	—	2	2	no	—
160	C₄⋊C₄×C₂₃	—	2	1	no	—
161	C₄⋊C₄×C₂₃	—	2	1	no	—
162	C₄⋊C₄×C₂₃	—	4	2	no	—
163	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
164	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
165	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
166	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
167	C₂²⋊C₄×C₂₃	—	2	1	no	—
168	C₂²⋊C₄×C₂₃	—	2	2	no	—
169	C₂²⋊C₄×C₂₃	—	2	1	no	—
170	C₂²⋊C₄×C₂₃	—	2	1	no	—
171	C₄²×C₂₃	—	2	1	no	—
172	C₄²×C₂₃	—	2	1	no	—
173	C₄²×C₂₃	—	2	2	no	—
174	C₄²×C₂₃	—	2	1	no	—
175	C₄²×C₂₃	—	2	2	no	—
176	C₄²×C₂₃	—	2	2	no	—
177	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
179	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
181	#21628	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂³×C₂₃	—	2	1	no	—
183	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
184	Q₈×C₂₃	—	2	1	no	—
185	Q₈×C₂₃	—	4	2	no	—
186	Q₈×C₂₃	—	4	2	no	—
187	Q₈×C₂₃	C₄×Q₈	1	2	yes	—
188	D₄×C₂₃	—	2	2	no	—
189	D₄×C₂₃	—	2	2	no	—
190	D₄×C₂₃	—	2	1	no	—
191	D₄×C₂₃	C₄×Q₈	1	2	yes	—
192	C₂×C₄×C₂₃	—	2	1	no	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	1	no	—
197	C₂×C₄×C₂₃	—	2	1	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	4	1	no	—
201	C₂×C₄×C₂₃	—	4	2	no	—
202	C₂×C₄×C₂₃	—	4	2	no	—
203	C₂×C₄×C₂₃	—	4	2	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	4	2	no	—
206	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
207	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
208	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
209	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
210	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
211	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
213	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
216	C₈×C₂₃	—	2	2	no	—
217	C₈×C₂₃	—	2	2	no	—
218	C₈×C₂₃	—	2	2	no	—
219	C₈×C₂₃	—	2	1	no	—
220	C₈×C₂₃	—	2	2	no	—
221	C₈×C₂₃	—	2	1	no	—
222	C₈×C₂₃	—	4	2	no	—
223	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
224	C₈×C₂₃	C₂²⋊Q₈	1	2	yes	—
225	#3076	C₂×C₂₃	1	1	yes	characteristic
226	#2960	C₂×C₂₃	1	1	yes	characteristic, complemented
227	C₄×#131	C₂×C₂₃	1	1	yes	characteristic
228	#1986	C₂×C₂₃	1	1	yes	characteristic, complemented
229	#1950	C₂×C₂₃	1	1	yes	characteristic, complemented
230	#1922	C₂×C₂₃	1	2	yes	complemented
231	#1866	C₂×C₂₃	1	2	yes	—
232	#1806	C₂×C₂₃	1	1	yes	characteristic
233	#1773	C₂×C₂₃	1	2	yes	complemented
234	#1770	C₂×C₂₃	1	1	yes	characteristic, complemented
235	#1765	C₂×C₂₃	1	2	yes	—
236	C₂²×C₂₃	—	2	1	no	—
237	C₂²×C₂₃	—	2	1	no	—
238	C₂²×C₂₃	—	2	2	no	—
239	C₂²×C₂₃	—	4	1	no	—
240	C₂²×C₂₃	—	4	1	no	—
241	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
242	C₂²×C₂₃	#529	1	1	yes	characteristic
243	C₂²×C₂₃	#533	1	1	yes	characteristic
244	C₄×C₂₃	—	2	1	no	—
245	C₄×C₂₃	—	2	1	no	—
246	C₄×C₂₃	—	2	1	no	—
247	C₄×C₂₃	—	2	2	no	—
248	C₄×C₂₃	—	4	1	no	—
249	C₄×C₂₃	—	4	2	no	—
250	C₄×C₂₃	—	4	1	no	—
251	C₄×C₂₃	—	4	2	no	—
252	C₄×C₂₃	—	8	2	no	—
253	C₄×C₂₃	#444	1	2	yes	—
254	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
255	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
256	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
257	C₄×C₂₃	#525	1	1	yes	characteristic
258	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
260	#533	C₂²×C₂₃	1	1	yes	characteristic
261	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
262	#479	C₄×C₂₃	1	2	yes	complemented
263	#478	C₄×C₂₃	1	2	yes	complemented
264	#477	C₄×C₂₃	1	1	yes	characteristic, complemented
265	#476	C₄×C₂₃	1	1	yes	characteristic, complemented
266	#475	C₄×C₂₃	1	1	yes	characteristic, complemented
267	#474	C₄×C₂₃	1	1	yes	characteristic, complemented
268	#446	—	2	1	no	—
269	#446	—	2	1	no	—
270	C₈×Q₈	—	2	1	no	—
271	C₈×Q₈	—	2	1	no	—
272	C₂×#108	C₂²×C₂₃	1	2	yes	—
273	#427	C₂²×C₂₃	1	2	yes	—
274	#424	C₂²×C₂₃	1	2	yes	—
275	#423	C₂²×C₂₃	1	1	yes	characteristic
276	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
277	C₂×#104	C₂²×C₂₃	1	2	yes	—
278	#416	C₂²×C₂₃	1	2	yes	—
279	#405	—	2	1	no	—
280	#405	—	2	1	no	—
281	#405	C₂²×C₂₃	1	1	yes	characteristic
282	#405	C₂²×C₂₃	1	1	yes	characteristic
283	#405	C₂²×C₂₃	1	2	yes	—
284	C₄×M₁₆	—	2	1	no	—
285	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
286	C₂×#97	C₂²×C₂₃	1	2	yes	—
287	C₂×C₄×C₈	—	2	1	no	—
288	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
289	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
290	#341	C₂²×C₂₃	1	2	yes	—
291	#339	C₂²×C₂₃	1	2	yes	—
292	#339	C₂²×C₂₃	1	2	yes	—
293	#339	C₂²×C₂₃	1	2	yes	—
294	#336	C₂²×C₂₃	1	2	yes	—
295	C₂×C₂₃	—	2	1	no	—
296	C₂×C₂₃	—	2	1	no	—
297	C₂×C₂₃	—	4	2	no	—
298	C₂×C₂₃	C₄×C₂²⋊Q₈	1	1	yes	characteristic
299	C₂×C₂₃	C₂×#444	1	1	yes	characteristic
300	C₂×C₂₃	#2955	1	1	yes	characteristic
301	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
302	#131	—	2	2	no	—
303	#131	—	2	2	no	—
304	#131	—	2	2	no	—
305	#131	Q₈×C₂₃	1	4	yes	complemented
306	C₂×M₁₆	—	2	1	no	—
307	C₂×M₁₆	—	2	1	no	—
308	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
309	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
310	C₂×M₁₆	C₂³×C₂₃	1	1	yes	characteristic
311	C₂²×C₈	—	2	1	no	—
312	C₂²×C₈	—	2	1	no	—
313	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
314	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
315	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
316	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
317	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
318	#125	C₂×C₄×C₂₃	1	2	yes	—
319	C₂²⋊Q₈	—	2	1	no	—
320	C₂²⋊Q₈	—	2	1	no	—
321	C₄×Q₈	—	2	1	no	—
322	C₄×Q₈	—	2	1	no	—
323	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
324	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
325	C₄×D₄	—	2	1	no	—
326	C₄×D₄	—	2	1	no	—
327	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
328	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
329	#117	—	2	1	no	—
330	#117	C₂³×C₂₃	1	1	yes	characteristic
331	#117	C₂³×C₂₃	1	2	yes	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄²	—	2	1	no	—
334	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
335	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
336	#108	—	2	4	no	—
337	#107	C₂×C₄×C₂₃	1	2	yes	—
338	#107	C₂×C₄×C₂₃	1	2	yes	—
339	#106	C₂×C₄×C₂₃	1	2	yes	—
340	#106	C₂×C₄×C₂₃	1	2	yes	—
341	C₄⋊C₈	—	2	2	no	—
342	C₄⋊C₈	—	2	2	no	—
343	C₄⋊C₈	—	2	1	no	—
344	C₄⋊C₈	—	2	1	no	—
345	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
346	C₄⋊C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
347	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
348	#104	—	2	4	no	—
349	#103	C₂×C₄×C₂₃	1	2	yes	—
350	#103	C₂×C₄×C₂₃	1	2	yes	—
351	#102	C₂×C₄×C₂₃	1	2	yes	—
352	#102	C₂×C₄×C₂₃	1	2	yes	—
353	#97	—	2	1	no	—
354	#97	—	2	1	no	—
355	#97	—	2	2	no	—
356	#97	—	2	4	no	—
357	#97	—	2	1	no	—
358	C₄×C₈	—	2	1	no	—
359	C₄×C₈	—	2	1	no	—
360	C₄×C₈	—	2	2	no	—
361	C₄×C₈	—	2	1	no	—
362	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
363	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
364	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
365	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
366	#95	—	2	1	no	—
367	C₂₃	#21628	1	1	yes	characteristic, direct product factor, Sylow
368	#41	—	2	1	no	—
369	#41	—	2	1	no	—
370	#41	C₂×Q₈×C₂₃	1	2	yes	—
371	#41	#41×C₂₃	1	2	yes	—
372	C₂×Q₈	—	2	1	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
382	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
383	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
384	M₁₆	—	2	4	no	—
385	M₁₆	—	2	1	no	—
386	M₁₆	—	2	1	no	—
387	M₁₆	—	2	4	no	—
388	M₁₆	—	2	2	no	—
389	M₁₆	—	2	2	no	—
390	M₁₆	C₂×Q₈×C₂₃	1	2	yes	—
391	M₁₆	#41×C₂₃	1	2	yes	—
392	C₂×C₈	—	2	4	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	2	1	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	4	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
402	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
403	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
404	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
405	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
406	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
407	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
408	C₂×C₈	#41×C₂₃	1	2	yes	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	—	4	2	no	—
416	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
417	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
418	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
419	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
420	C₂²⋊C₄	—	2	1	no	—
421	C₂²⋊C₄	—	2	2	no	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₂²⋊C₄	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	—	2	2	no	—
428	C₄²	—	2	2	no	—
429	C₄²	—	2	2	no	—
430	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
431	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
432	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
433	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
434	C₂³	—	2	1	no	—
435	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
436	Q₈	—	2	1	no	—
437	Q₈	—	4	2	no	—
438	Q₈	—	4	2	no	—
439	Q₈	C₄×Q₈×C₂₃	1	2	yes	—
440	D₄	—	2	2	no	—
441	D₄	—	2	2	no	—
442	D₄	—	2	1	no	—
443	D₄	C₄×Q₈×C₂₃	1	2	yes	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
459	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
460	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
461	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
462	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
463	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
464	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
465	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
466	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
467	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
468	C₈	—	2	2	no	—
469	C₈	—	2	2	no	—
470	C₈	—	2	2	no	—
471	C₈	—	2	1	no	—
472	C₈	—	2	2	no	—
473	C₈	—	2	1	no	—
474	C₈	—	4	2	no	—
475	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
476	C₈	C₂₃×C₂²⋊Q₈	1	2	yes	—
477	C₂²	—	2	1	no	—
478	C₂²	—	2	1	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	4	1	no	—
481	C₂²	—	4	1	no	—
482	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
483	C₂²	C₂₃×#529	1	1	yes	characteristic
484	C₂²	C₂₃×#533	1	1	yes	characteristic
485	C₄	—	2	1	no	—
486	C₄	—	2	1	no	—
487	C₄	—	2	1	no	—
488	C₄	—	2	2	no	—
489	C₄	—	4	1	no	—
490	C₄	—	4	2	no	—
491	C₄	—	4	1	no	—
492	C₄	—	4	2	no	—
493	C₄	—	8	2	no	—
494	C₄	C₂₃×#444	1	2	yes	—
495	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
496	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
497	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
498	C₄	C₂₃×#525	1	1	yes	characteristic
499	C₂	—	2	1	no	—
500	C₂	—	2	1	no	—
501	C₂	—	4	2	no	—
502	C₂	C₄×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
503	C₂	C₂×C₂₃×#444	1	1	yes	characteristic
504	C₂	C₂₃×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂×#41×C₂₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2960	C₂	semidirect
4. C₂₃×#1986	C₂	semidirect
5. C₂₃×#1950	C₂	semidirect
6. C₂₃×#1922	C₂	semidirect
9. C₂₃×#1773	C₂	semidirect
10. C₂₃×#1770	C₂	semidirect
16. C₂₃×#479	C₄	semidirect
17. C₂₃×#478	C₄	semidirect
18. C₂₃×#477	C₄	semidirect
19. C₂₃×#476	C₄	semidirect
20. C₂₃×#475	C₄	semidirect
21. C₂₃×#474	C₄	semidirect
53. C₂₃×#131	Q₈	semidirect
181. #21628	C₂₃	direct (N × H)
226. #2960	C₂×C₂₃	semidirect
228. #1986	C₂×C₂₃	semidirect
229. #1950	C₂×C₂₃	semidirect
230. #1922	C₂×C₂₃	semidirect
233. #1773	C₂×C₂₃	semidirect
234. #1770	C₂×C₂₃	semidirect
262. #479	C₄×C₂₃	semidirect
263. #478	C₄×C₂₃	semidirect
264. #477	C₄×C₂₃	semidirect
265. #476	C₄×C₂₃	semidirect
266. #475	C₄×C₂₃	semidirect
267. #474	C₄×C₂₃	semidirect
305. #131	Q₈×C₂₃	semidirect
367. C₂₃	#21628	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×#3076	2
C₂₃×#2960	2
C₄×C₂₃×#131	2
C₂₃×#1986	2
C₂₃×#1950	2
C₂₃×#1922	2
C₂₃×#1866	2
C₂₃×#1806	2
C₂₃×#1773	2
C₂₃×#1770	2
C₂₃×#1765	2
#21628	23

Subgroups — C23×#21628 #19169254

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21628 #19169254