Subgroups — C₂₃×#60801 #19208427

Order 5888.

2186 conjugacy classes of subgroups.

The subgroup lattice has 590 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	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3219	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3210	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3207	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3152	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#3151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#3138	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3040	C₂	1	4	yes	maximal, complemented
11	C₂₃×#3005	C₂	1	4	yes	maximal, complemented
12	C₂₃×#2920	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₂₃	C₂²	1	2	yes	—
14	C₂×C₂₃×#137	C₂²	1	4	yes	—
15	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	4	yes	—
16	C₂₃×#533	C₂²	1	1	yes	characteristic
17	C₂₃×#533	C₂²	1	2	yes	complemented
18	C₂₃×#532	—	2	2	no	—
19	C₂₃×#532	—	2	2	no	—
20	C₂₃×#532	C₂²	1	4	yes	—
21	C₂₃×#532	C₂²	1	4	yes	—
22	C₂₃×#532	C₂²	1	1	yes	characteristic
23	C₂₃×#532	C₂²	1	2	yes	complemented
24	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
26	C₂₃×#529	—	2	2	no	—
27	C₂₃×#529	—	2	2	no	—
28	C₂₃×#529	C₂²	1	4	yes	complemented
29	C₂×C₂₃×#124	C₂²	1	2	yes	—
30	C₂₃×#525	—	2	2	no	—
31	C₂₃×#525	C₂²	1	2	yes	—
32	C₂×C₂₃×#121	—	2	2	no	—
33	C₄×#41×C₂₃	—	2	2	no	—
34	C₄×#41×C₂₃	C₂²	1	2	yes	—
35	C₂×C₄×D₄×C₂₃	—	2	2	no	—
36	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#497	C₂²	1	4	yes	complemented
38	C₂₃×#496	C₂²	1	4	yes	complemented
39	C₂₃×#494	C₂²	1	2	yes	complemented
40	C₂₃×#493	C₂²	1	2	yes	complemented
41	C₂₃×#492	C₂²	1	2	yes	complemented
42	C₂₃×#492	C₂²	1	2	yes	complemented
43	C₂₃×#491	C₂²	1	4	yes	complemented
44	C₂₃×#490	C₂²	1	4	yes	complemented
45	C₂₃×#487	C₂²	1	4	yes	complemented
46	C₂₃×#486	C₂²	1	4	yes	complemented
47	C₂₃×#464	C₂²	1	16	yes	complemented
48	C₂₃×#463	C₂²	1	16	yes	complemented
49	C₂₃×#452	C₂²	1	16	yes	complemented
50	C₂₃×#449	C₂²	1	16	yes	complemented
51	C₂₃×#432	C₂²	1	4	yes	complemented
52	C₂₃×#431	C₂²	1	4	yes	complemented
53	C₂₃×#423	C₂²	1	2	yes	—
54	C₂₃×#419	C₂²	1	4	yes	—
55	C₂₃×#418	C₂²	1	4	yes	—
56	C₂₃×#407	C₂²	1	2	yes	—
57	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
58	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
59	C₂×#41×C₂₃	—	2	4	no	—
60	C₂×#41×C₂₃	—	2	2	no	—
61	C₂×#41×C₂₃	—	4	8	no	—
62	C₂×#41×C₂₃	C₂³	1	2	yes	—
63	C₂×#41×C₂₃	C₂³	1	8	yes	—
64	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
65	C₂²×D₄×C₂₃	—	2	2	no	—
66	C₂²×D₄×C₂₃	—	2	2	no	—
67	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
68	C₂³×C₄×C₂₃	—	2	2	no	—
69	C₂³×C₄×C₂₃	D₄	1	2	yes	—
70	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂₃×#137	—	4	8	no	—
72	C₂₃×AGL(1,ℤ/8)	—	4	8	no	—
73	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
74	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
75	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
76	C₂×D₈×C₂₃	C₂³	1	8	yes	—
77	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
78	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
79	C₂₃×C₄⋊Q₈	—	2	2	no	—
80	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
81	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
82	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
84	C₂₃×#127	—	2	2	no	—
85	C₂₃×#127	C₂³	1	2	yes	—
86	C₂₃×#127	C₂³	1	2	yes	—
87	C₂₃×#127	C₂³	1	1	yes	characteristic
88	C₂₃×#126	—	2	4	no	—
89	C₂₃×#125	—	2	2	no	—
90	C₂₃×#125	C₂³	1	2	yes	—
91	C₂₃×#124	—	2	2	no	—
92	C₂₃×#124	—	2	2	no	—
93	C₂₃×#124	—	2	2	no	—
94	C₂₃×#124	C₂³	1	8	yes	—
95	C₂₃×#124	C₂³	1	2	yes	—
96	C₂₃×#123	—	2	4	no	—
97	C₂₃×#123	—	2	2	no	—
98	C₂₃×C₂²⋊Q₈	—	2	2	no	—
99	C₂₃×C₂²⋊Q₈	—	2	2	no	—
100	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
101	C₂₃×#121	—	2	4	no	—
102	C₂₃×#121	—	2	4	no	—
103	C₂₃×#121	—	2	4	no	—
104	C₂₃×#121	—	2	4	no	—
105	C₂₃×#121	—	2	2	no	—
106	C₂₃×#121	—	2	2	no	—
107	C₂₃×#121	C₂³	1	4	yes	—
108	C₂₃×#120	—	2	2	no	—
109	C₄×Q₈×C₂₃	—	2	2	no	—
110	C₄×Q₈×C₂₃	—	2	2	no	—
111	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
112	C₄×D₄×C₂₃	—	2	4	no	—
113	C₄×D₄×C₂₃	—	2	4	no	—
114	C₄×D₄×C₂₃	—	2	4	no	—
115	C₄×D₄×C₂₃	—	2	4	no	—
116	C₄×D₄×C₂₃	—	2	4	no	—
117	C₄×D₄×C₂₃	—	2	4	no	—
118	C₄×D₄×C₂₃	—	2	4	no	—
119	C₄×D₄×C₂₃	—	2	2	no	—
120	C₄×D₄×C₂₃	—	2	2	no	—
121	C₄×D₄×C₂₃	C₂³	1	4	yes	—
122	C₂₃×#117	—	2	2	no	—
123	C₂₃×#117	—	2	4	no	—
124	C₂₃×#117	—	2	2	no	—
125	C₂₃×#117	C₂³	1	2	yes	—
126	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
127	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
128	C₂×C₄²×C₂₃	—	2	2	no	—
129	C₂×C₄²×C₂₃	D₄	1	2	yes	—
130	C₂×C₄²×C₂₃	D₄	1	2	yes	—
131	C₂×C₄²×C₂₃	D₄	1	2	yes	—
132	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
135	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
136	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
137	C₂₃×#103	C₂³	1	16	yes	—
138	C₂₃×#102	C₂³	1	16	yes	—
139	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
140	C₂₃×#97	C₂³	1	2	yes	—
141	C₂₃×#97	C₂³	1	2	yes	—
142	C₄×C₈×C₂₃	C₂³	1	2	yes	—
143	C₄×C₈×C₂₃	C₂³	1	2	yes	—
144	C₂⁴×C₂₃	—	2	2	no	—
145	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
146	#41×C₂₃	—	2	8	no	—
147	#41×C₂₃	—	4	16	no	—
148	#41×C₂₃	—	4	8	no	—
149	#41×C₂₃	—	4	16	no	—
150	C₂×Q₈×C₂₃	—	2	2	no	—
151	C₂×Q₈×C₂₃	—	2	2	no	—
152	C₂×Q₈×C₂₃	—	2	8	no	—
153	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
154	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
155	C₂×D₄×C₂₃	—	2	2	no	—
156	C₂×D₄×C₂₃	—	2	4	no	—
157	C₂×D₄×C₂₃	—	2	8	no	—
158	C₂×D₄×C₂₃	—	2	2	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	8	no	—
163	C₂×D₄×C₂₃	—	4	8	no	—
164	C₂×D₄×C₂₃	—	4	8	no	—
165	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
166	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₂₃	—	2	4	no	—
168	C₂²×C₄×C₂₃	—	2	2	no	—
169	C₂²×C₄×C₂₃	—	2	2	no	—
170	C₂²×C₄×C₂₃	—	2	4	no	—
171	C₂²×C₄×C₂₃	—	2	8	no	—
172	C₂²×C₄×C₂₃	—	4	4	no	—
173	C₂²×C₄×C₂₃	—	4	8	no	—
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	2	yes	—
178	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
180	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
181	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
182	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
184	Q₁₆×C₂₃	—	4	8	no	—
185	SD₁₆×C₂₃	—	4	8	no	—
186	SD₁₆×C₂₃	—	4	8	no	—
187	D₈×C₂₃	—	4	8	no	—
188	M₁₆×C₂₃	—	2	4	no	—
189	M₁₆×C₂₃	—	2	4	no	—
190	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
191	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
192	C₄⋊C₄×C₂₃	—	2	4	no	—
193	C₄⋊C₄×C₂₃	—	2	4	no	—
194	C₄⋊C₄×C₂₃	—	2	2	no	—
195	C₄⋊C₄×C₂₃	—	2	2	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	C₂⁴	1	4	yes	—
200	C₂²⋊C₄×C₂₃	—	2	4	no	—
201	C₂²⋊C₄×C₂₃	—	2	2	no	—
202	C₂²⋊C₄×C₂₃	—	2	8	no	—
203	C₂²⋊C₄×C₂₃	—	2	8	no	—
204	C₂²⋊C₄×C₂₃	—	2	4	no	—
205	C₂²⋊C₄×C₂₃	—	2	4	no	—
206	C₂²⋊C₄×C₂₃	—	2	4	no	—
207	C₄²×C₂₃	—	2	4	no	—
208	C₄²×C₂₃	—	2	2	no	—
209	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₂₃	C₂×D₄	1	2	yes	—
211	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₂₃	C₂×D₄	1	2	yes	—
213	C₄²×C₂₃	C₂×D₄	1	2	yes	—
214	C₄²×C₂₃	C₂×D₄	1	2	yes	—
215	C₄²×C₂₃	C₂×D₄	1	2	yes	—
216	C₄²×C₂₃	C₂⁴	1	2	yes	—
217	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
218	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
219	#60801	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂³×C₂₃	—	2	2	no	—
221	C₂³×C₂₃	—	2	2	no	—
222	C₂³×C₂₃	—	2	4	no	—
223	C₂³×C₂₃	—	4	2	no	—
224	C₂³×C₂₃	—	4	8	no	—
225	C₂³×C₂₃	—	4	4	no	—
226	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
227	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
228	Q₈×C₂₃	—	2	4	no	—
229	Q₈×C₂₃	—	2	8	no	—
230	Q₈×C₂₃	—	4	4	no	—
231	Q₈×C₂₃	—	4	4	no	—
232	D₄×C₂₃	—	2	4	no	—
233	D₄×C₂₃	—	2	8	no	—
234	D₄×C₂₃	—	4	8	no	—
235	D₄×C₂₃	—	4	16	no	—
236	D₄×C₂₃	—	4	8	no	—
237	D₄×C₂₃	—	4	4	no	—
238	D₄×C₂₃	—	4	4	no	—
239	D₄×C₂₃	—	4	8	no	—
240	D₄×C₂₃	—	4	4	no	—
241	C₂×C₄×C₂₃	—	2	2	no	—
242	C₂×C₄×C₂₃	—	2	4	no	—
243	C₂×C₄×C₂₃	—	2	4	no	—
244	C₂×C₄×C₂₃	—	2	4	no	—
245	C₂×C₄×C₂₃	—	2	4	no	—
246	C₂×C₄×C₂₃	—	2	4	no	—
247	C₂×C₄×C₂₃	—	2	2	no	—
248	C₂×C₄×C₂₃	—	2	4	no	—
249	C₂×C₄×C₂₃	—	2	4	no	—
250	C₂×C₄×C₂₃	—	4	2	no	—
251	C₂×C₄×C₂₃	—	4	2	no	—
252	C₂×C₄×C₂₃	—	4	2	no	—
253	C₂×C₄×C₂₃	—	4	16	no	—
254	C₂×C₄×C₂₃	—	4	8	no	—
255	C₂×C₄×C₂₃	—	4	4	no	—
256	C₂×C₄×C₂₃	—	4	4	no	—
257	C₂×C₄×C₂₃	C₂²×D₄	1	2	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	2	yes	—
261	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
263	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
264	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
265	C₈×C₂₃	—	2	4	no	—
266	C₈×C₂₃	—	2	4	no	—
267	#3444	C₂×C₂₃	1	1	yes	characteristic, complemented
268	C₂×#532	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#3219	C₂×C₂₃	1	8	yes	complemented
270	#3210	C₂×C₂₃	1	4	yes	complemented
271	#3207	C₂×C₂₃	1	4	yes	complemented
272	#3152	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#3151	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#3139	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#3138	C₂×C₂₃	1	1	yes	characteristic, complemented
276	#3040	C₂×C₂₃	1	4	yes	complemented
277	#3005	C₂×C₂₃	1	4	yes	complemented
278	#2920	C₂×C₂₃	1	1	yes	characteristic, complemented
279	C₂²×C₂₃	—	2	2	no	—
280	C₂²×C₂₃	—	2	2	no	—
281	C₂²×C₂₃	—	2	2	no	—
282	C₂²×C₂₃	—	2	2	no	—
283	C₂²×C₂₃	—	4	4	no	—
284	C₂²×C₂₃	—	4	4	no	—
285	C₂²×C₂₃	—	4	2	no	—
286	C₂²×C₂₃	—	4	4	no	—
287	C₂²×C₂₃	—	4	8	no	—
288	C₂²×C₂₃	—	8	8	no	—
289	C₂²×C₂₃	#535	1	2	yes	—
290	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
291	C₄×C₂₃	—	2	2	no	—
292	C₄×C₂₃	—	2	2	no	—
293	C₄×C₂₃	—	2	2	no	—
294	C₄×C₂₃	—	2	2	no	—
295	C₄×C₂₃	—	4	4	no	—
296	C₄×C₂₃	—	4	2	no	—
297	C₄×C₂₃	—	4	4	no	—
298	C₄×C₂₃	#535	1	2	yes	—
299	C₄×C₂₃	#575	1	4	yes	—
300	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
301	C₂²×#41	C₂²×C₂₃	1	2	yes	—
302	C₂×#137	C₂²×C₂₃	1	4	yes	—
303	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	4	yes	—
304	#533	C₂²×C₂₃	1	1	yes	characteristic
305	#533	C₂²×C₂₃	1	2	yes	complemented
306	#532	—	2	2	no	—
307	#532	—	2	2	no	—
308	#532	C₂²×C₂₃	1	1	yes	characteristic
309	#532	C₂²×C₂₃	1	4	yes	—
310	#532	C₂²×C₂₃	1	4	yes	—
311	#532	C₂²×C₂₃	1	2	yes	complemented
312	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
313	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
314	#529	—	2	2	no	—
315	#529	—	2	2	no	—
316	#529	C₂²×C₂₃	1	4	yes	complemented
317	C₂×#124	C₂²×C₂₃	1	2	yes	—
318	#525	—	2	2	no	—
319	#525	C₂²×C₂₃	1	2	yes	—
320	C₂×#121	—	2	2	no	—
321	C₄×#41	—	2	2	no	—
322	C₄×#41	C₂²×C₂₃	1	2	yes	—
323	C₂×C₄×D₄	—	2	2	no	—
324	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
325	#497	C₂²×C₂₃	1	4	yes	complemented
326	#496	C₂²×C₂₃	1	4	yes	complemented
327	#494	C₂²×C₂₃	1	2	yes	complemented
328	#493	C₂²×C₂₃	1	2	yes	complemented
329	#492	C₂²×C₂₃	1	2	yes	complemented
330	#492	C₂²×C₂₃	1	2	yes	complemented
331	#491	C₂²×C₂₃	1	4	yes	complemented
332	#490	C₂²×C₂₃	1	4	yes	complemented
333	#487	C₂²×C₂₃	1	4	yes	complemented
334	#486	C₂²×C₂₃	1	4	yes	complemented
335	#464	C₂²×C₂₃	1	16	yes	complemented
336	#463	C₂²×C₂₃	1	16	yes	complemented
337	#452	C₂²×C₂₃	1	16	yes	complemented
338	#449	C₂²×C₂₃	1	16	yes	complemented
339	#432	C₂²×C₂₃	1	4	yes	complemented
340	#431	C₂²×C₂₃	1	4	yes	complemented
341	#423	C₂²×C₂₃	1	2	yes	—
342	#419	C₂²×C₂₃	1	4	yes	—
343	#418	C₂²×C₂₃	1	4	yes	—
344	#407	C₂²×C₂₃	1	2	yes	—
345	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
346	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
347	C₂×C₂₃	—	2	2	no	—
348	C₂×C₂₃	—	4	2	no	—
349	C₂×C₂₃	—	8	2	no	—
350	C₂×C₂₃	—	8	4	no	—
351	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
352	C₂×C₂₃	C₂×#575	1	2	yes	—
353	C₂×#41	—	2	2	no	—
354	C₂×#41	—	2	4	no	—
355	C₂×#41	—	4	8	no	—
356	C₂×#41	C₂³×C₂₃	1	2	yes	—
357	C₂×#41	C₂³×C₂₃	1	8	yes	—
358	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
359	C₂²×D₄	—	2	2	no	—
360	C₂²×D₄	—	2	2	no	—
361	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
362	C₂³×C₄	—	2	2	no	—
363	C₂³×C₄	D₄×C₂₃	1	2	yes	—
364	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
365	#137	—	4	8	no	—
366	AGL(1,ℤ/8)	—	4	8	no	—
367	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
368	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
369	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
370	C₂×D₈	C₂³×C₂₃	1	8	yes	—
371	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
372	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
373	C₄⋊Q₈	—	2	2	no	—
374	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
375	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
376	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
377	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
378	#127	—	2	2	no	—
379	#127	C₂³×C₂₃	1	2	yes	—
380	#127	C₂³×C₂₃	1	2	yes	—
381	#127	C₂³×C₂₃	1	1	yes	characteristic
382	#126	—	2	4	no	—
383	#125	—	2	2	no	—
384	#125	C₂³×C₂₃	1	2	yes	—
385	#124	—	2	2	no	—
386	#124	—	2	2	no	—
387	#124	—	2	2	no	—
388	#124	C₂³×C₂₃	1	8	yes	—
389	#124	C₂³×C₂₃	1	2	yes	—
390	#123	—	2	4	no	—
391	#123	—	2	2	no	—
392	C₂²⋊Q₈	—	2	2	no	—
393	C₂²⋊Q₈	—	2	2	no	—
394	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
395	#121	—	2	4	no	—
396	#121	—	2	4	no	—
397	#121	—	2	4	no	—
398	#121	—	2	4	no	—
399	#121	—	2	2	no	—
400	#121	—	2	2	no	—
401	#121	C₂³×C₂₃	1	4	yes	—
402	#120	—	2	2	no	—
403	C₄×Q₈	—	2	2	no	—
404	C₄×Q₈	—	2	2	no	—
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	4	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	C₂³×C₂₃	1	4	yes	—
416	#117	—	2	2	no	—
417	#117	—	2	4	no	—
418	#117	—	2	2	no	—
419	#117	C₂³×C₂₃	1	2	yes	—
420	C₂×C₄⋊C₄	—	2	2	no	—
421	C₂×C₂²⋊C₄	—	2	4	no	—
422	C₂×C₄²	—	2	2	no	—
423	C₂×C₄²	D₄×C₂₃	1	2	yes	—
424	C₂×C₄²	D₄×C₂₃	1	2	yes	—
425	C₂×C₄²	D₄×C₂₃	1	2	yes	—
426	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
427	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
428	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
429	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
430	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
431	#103	C₂³×C₂₃	1	16	yes	—
432	#102	C₂³×C₂₃	1	16	yes	—
433	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
434	#97	C₂³×C₂₃	1	2	yes	—
435	#97	C₂³×C₂₃	1	2	yes	—
436	C₄×C₈	C₂³×C₂₃	1	2	yes	—
437	C₄×C₈	C₂³×C₂₃	1	2	yes	—
438	C₂₃	#60801	1	1	yes	characteristic, direct product factor, Sylow
439	C₂⁴	—	2	2	no	—
440	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
441	#41	—	2	8	no	—
442	#41	—	4	16	no	—
443	#41	—	4	16	no	—
444	#41	—	4	8	no	—
445	C₂×Q₈	—	2	2	no	—
446	C₂×Q₈	—	2	2	no	—
447	C₂×Q₈	—	2	8	no	—
448	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
449	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	4	no	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	8	no	—
458	C₂×D₄	—	4	8	no	—
459	C₂×D₄	—	4	8	no	—
460	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
461	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	8	no	—
467	C₂²×C₄	—	4	4	no	—
468	C₂²×C₄	—	4	8	no	—
469	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
470	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
471	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
472	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
473	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
474	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
475	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
476	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
477	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
478	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
479	Q₁₆	—	4	8	no	—
480	SD₁₆	—	4	8	no	—
481	SD₁₆	—	4	8	no	—
482	D₈	—	4	8	no	—
483	M₁₆	—	2	4	no	—
484	M₁₆	—	2	4	no	—
485	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
486	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	4	no	—
492	C₄⋊C₄	—	2	4	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	8	no	—
498	C₂²⋊C₄	—	2	8	no	—
499	C₂²⋊C₄	—	2	4	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₄²	—	2	4	no	—
503	C₄²	—	2	2	no	—
504	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
505	C₄²	C₂×D₄×C₂₃	1	2	yes	—
506	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
507	C₄²	C₂×D₄×C₂₃	1	2	yes	—
508	C₄²	C₂×D₄×C₂₃	1	2	yes	—
509	C₄²	C₂×D₄×C₂₃	1	2	yes	—
510	C₄²	C₂×D₄×C₂₃	1	2	yes	—
511	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
512	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
513	C₄²	C₂⁴×C₂₃	1	2	yes	—
514	C₂³	—	2	2	no	—
515	C₂³	—	2	2	no	—
516	C₂³	—	2	4	no	—
517	C₂³	—	4	2	no	—
518	C₂³	—	4	8	no	—
519	C₂³	—	4	4	no	—
520	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
521	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
522	Q₈	—	2	4	no	—
523	Q₈	—	2	8	no	—
524	Q₈	—	4	4	no	—
525	Q₈	—	4	4	no	—
526	D₄	—	2	4	no	—
527	D₄	—	2	8	no	—
528	D₄	—	4	8	no	—
529	D₄	—	4	16	no	—
530	D₄	—	4	8	no	—
531	D₄	—	4	4	no	—
532	D₄	—	4	8	no	—
533	D₄	—	4	4	no	—
534	D₄	—	4	4	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	4	2	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	16	no	—
548	C₂×C₄	—	4	8	no	—
549	C₂×C₄	—	4	4	no	—
550	C₂×C₄	—	4	4	no	—
551	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
552	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
553	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
554	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
555	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
556	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
557	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
558	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
559	C₈	—	2	4	no	—
560	C₈	—	2	4	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	4	4	no	—
566	C₂²	—	4	4	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	4	8	no	—
570	C₂²	—	8	8	no	—
571	C₂²	C₂₃×#535	1	2	yes	—
572	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
573	C₄	—	2	2	no	—
574	C₄	—	2	2	no	—
575	C₄	—	2	2	no	—
576	C₄	—	2	2	no	—
577	C₄	—	4	4	no	—
578	C₄	—	4	2	no	—
579	C₄	—	4	4	no	—
580	C₄	C₂₃×#535	1	2	yes	—
581	C₄	C₂₃×#575	1	4	yes	—
582	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
583	C₂	—	2	2	no	—
584	C₂	—	4	2	no	—
585	C₂	—	8	4	no	—
586	C₂	—	8	2	no	—
587	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
588	C₂	C₂×C₂₃×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. C₂×C₂₃×#532	C₂	semidirect
3. C₂₃×#3219	C₂	semidirect
4. C₂₃×#3210	C₂	semidirect
5. C₂₃×#3207	C₂	semidirect
6. C₂₃×#3152	C₂	semidirect
7. C₂₃×#3151	C₂	semidirect
8. C₂₃×#3139	C₂	semidirect
9. C₂₃×#3138	C₂	semidirect
10. C₂₃×#3040	C₂	semidirect
11. C₂₃×#3005	C₂	semidirect
12. C₂₃×#2920	C₂	semidirect
17. C₂₃×#533	C₂²	semidirect
23. C₂₃×#532	C₂²	semidirect
28. C₂₃×#529	C₂²	semidirect
37. C₂₃×#497	C₂²	semidirect
38. C₂₃×#496	C₂²	semidirect
39. C₂₃×#494	C₂²	semidirect
40. C₂₃×#493	C₂²	semidirect
41. C₂₃×#492	C₂²	semidirect
42. C₂₃×#492	C₂²	semidirect
43. C₂₃×#491	C₂²	semidirect
44. C₂₃×#490	C₂²	semidirect
45. C₂₃×#487	C₂²	semidirect
46. C₂₃×#486	C₂²	semidirect
47. C₂₃×#464	C₂²	semidirect
48. C₂₃×#463	C₂²	semidirect
49. C₂₃×#452	C₂²	semidirect
50. C₂₃×#449	C₂²	semidirect
51. C₂₃×#432	C₂²	semidirect
52. C₂₃×#431	C₂²	semidirect
219. #60801	C₂₃	direct (N × H)
267. #3444	C₂×C₂₃	semidirect
268. C₂×#532	C₂×C₂₃	semidirect
269. #3219	C₂×C₂₃	semidirect
270. #3210	C₂×C₂₃	semidirect
271. #3207	C₂×C₂₃	semidirect
272. #3152	C₂×C₂₃	semidirect
273. #3151	C₂×C₂₃	semidirect
274. #3139	C₂×C₂₃	semidirect
275. #3138	C₂×C₂₃	semidirect
276. #3040	C₂×C₂₃	semidirect
277. #3005	C₂×C₂₃	semidirect
278. #2920	C₂×C₂₃	semidirect
305. #533	C₂²×C₂₃	semidirect
311. #532	C₂²×C₂₃	semidirect
316. #529	C₂²×C₂₃	semidirect
325. #497	C₂²×C₂₃	semidirect
326. #496	C₂²×C₂₃	semidirect
327. #494	C₂²×C₂₃	semidirect
328. #493	C₂²×C₂₃	semidirect
329. #492	C₂²×C₂₃	semidirect
330. #492	C₂²×C₂₃	semidirect
331. #491	C₂²×C₂₃	semidirect
332. #490	C₂²×C₂₃	semidirect
333. #487	C₂²×C₂₃	semidirect
334. #486	C₂²×C₂₃	semidirect
335. #464	C₂²×C₂₃	semidirect
336. #463	C₂²×C₂₃	semidirect
337. #452	C₂²×C₂₃	semidirect
338. #449	C₂²×C₂₃	semidirect
339. #432	C₂²×C₂₃	semidirect
340. #431	C₂²×C₂₃	semidirect
438. C₂₃	#60801	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
C₂×C₂₃×#532	2
C₂₃×#3219	2
C₂₃×#3210	2
C₂₃×#3207	2
C₂₃×#3152	2
C₂₃×#3151	2
C₂₃×#3139	2
C₂₃×#3138	2
C₂₃×#3040	2
C₂₃×#3005	2
C₂₃×#2920	2
#60801	23

Subgroups — C23×#60801 #19208427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60801 #19208427