Subgroups — C₂₃×#60828 #19208454

Order 5888.

2018 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3225	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3219	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3218	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3153	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3139	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#3135	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3044	C₂	1	2	yes	maximal, complemented
11	C₂₃×#3042	C₂	1	2	yes	maximal, complemented
12	C₂₃×#3011	C₂	1	2	yes	maximal, complemented
13	C₂₃×#3010	C₂	1	2	yes	maximal, complemented
14	C₂₃×#2923	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₂₃×2₋¹⁺⁴	C₂²	1	2	yes	—
16	C₂×C₂₃×#137	C₂²	1	2	yes	—
17	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	—
18	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
19	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
20	C₂₃×#559	—	2	2	no	—
21	C₂₃×#556	—	2	2	no	—
22	C₂₃×#551	—	2	4	no	—
23	D₄×Q₈×C₂₃	—	2	2	no	—
24	C₂₃×#547	—	2	4	no	—
25	C₂₃×#547	—	2	2	no	—
26	C₂₃×#542	C₂²	1	4	yes	—
27	C₂₃×#536	C₂²	1	2	yes	—
28	C₂₃×#536	C₂²	1	1	yes	characteristic
29	C₂₃×#536	C₂²	1	4	yes	complemented
30	C₂₃×#535	C₂²	1	1	yes	characteristic
31	C₂₃×#533	C₂²	1	1	yes	characteristic
32	C₂₃×#532	C₂²	1	1	yes	characteristic
33	C₂₃×#532	C₂²	1	2	yes	complemented
34	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
35	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
36	C₂₃×#529	C₂²	1	4	yes	—
37	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
38	C₂₃×#519	C₂²	1	1	yes	characteristic
39	C₂₃×#518	C₂²	1	1	yes	characteristic
40	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#497	C₂²	1	2	yes	—
42	C₂₃×#497	C₂²	1	4	yes	—
43	C₂₃×#497	C₂²	1	4	yes	—
44	C₂₃×#496	C₂²	1	2	yes	—
45	C₂₃×#495	C₂²	1	4	yes	—
46	C₂₃×#491	C₂²	1	2	yes	complemented
47	C₂₃×#491	C₂²	1	4	yes	complemented
48	C₂₃×#490	C₂²	1	2	yes	complemented
49	C₂₃×#489	C₂²	1	4	yes	complemented
50	C₂₃×#487	C₂²	1	4	yes	complemented
51	C₂₃×#464	C₂²	1	8	yes	complemented
52	C₂₃×#463	C₂²	1	8	yes	—
53	C₂₃×#462	C₂²	1	8	yes	complemented
54	C₂₃×#461	C₂²	1	8	yes	—
55	C₂₃×#452	C₂²	1	8	yes	complemented
56	C₂₃×#451	C₂²	1	8	yes	complemented
57	C₂₃×#449	C₂²	1	8	yes	—
58	C₂₃×#448	C₂²	1	8	yes	—
59	C₂₃×#433	C₂²	1	4	yes	complemented
60	C₂₃×#432	C₂²	1	4	yes	complemented
61	C₂₃×#424	C₂²	1	2	yes	—
62	C₂₃×#419	C₂²	1	2	yes	—
63	C₂₃×#419	C₂²	1	2	yes	—
64	C₂₃×#416	C₂²	1	2	yes	—
65	C₂₃×#416	C₂²	1	2	yes	—
66	C₂₃×#408	C₂²	1	2	yes	—
67	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
68	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
69	C₂₃×2₋¹⁺⁴	—	4	4	no	—
70	C₂₃×2₋¹⁺⁴	—	4	4	no	—
71	C₂×#41×C₂₃	—	2	4	no	—
72	C₂×#41×C₂₃	—	2	2	no	—
73	C₂×#41×C₂₃	C₂³	1	2	yes	—
74	C₂×#41×C₂₃	C₂³	1	4	yes	—
75	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₂₃	—	2	2	no	—
77	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
78	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
80	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₂₃×#137	—	4	4	no	—
82	C₂₃×AGL(1,ℤ/8)	—	4	4	no	—
83	C₂×Q₁₆×C₂₃	—	4	4	no	—
84	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
85	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
86	C₂×SD₁₆×C₂₃	—	4	4	no	—
87	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
88	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
89	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
90	C₂×D₈×C₂₃	C₂³	1	4	yes	—
91	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
92	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
93	C₂₃×C₄⋊Q₈	—	2	1	no	—
94	C₂₃×C₄⋊Q₈	—	2	2	no	—
95	C₂₃×C₄⋊Q₈	—	2	4	no	—
96	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
97	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
98	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
99	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
100	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
101	C₂₃×#127	—	2	1	no	—
102	C₂₃×#127	C₂³	1	1	yes	characteristic
103	C₂₃×#126	—	2	4	no	—
104	C₂₃×#126	—	2	4	no	—
105	C₂₃×#125	C₂³	1	2	yes	—
106	C₂₃×#124	—	2	1	no	—
107	C₂₃×#124	—	2	1	no	—
108	C₂₃×#124	—	2	4	no	—
109	C₂₃×#124	—	2	2	no	—
110	C₂₃×#124	C₂³	1	4	yes	—
111	C₂₃×#124	C₂³	1	2	yes	—
112	C₂₃×#124	C₂³	1	2	yes	—
113	C₂₃×#124	C₂³	1	2	yes	—
114	C₂₃×#123	—	2	2	no	—
115	C₂₃×#123	—	2	4	no	—
116	C₂₃×#123	—	2	4	no	—
117	C₂₃×#123	—	2	8	no	—
118	C₂₃×#123	—	2	4	no	—
119	C₂₃×C₂²⋊Q₈	—	2	2	no	—
120	C₂₃×C₂²⋊Q₈	—	2	4	no	—
121	C₂₃×C₂²⋊Q₈	—	2	8	no	—
122	C₂₃×C₂²⋊Q₈	—	2	4	no	—
123	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
124	C₂₃×#121	—	2	2	no	—
125	C₂₃×#121	—	2	4	no	—
126	C₂₃×#121	C₂³	1	4	yes	—
127	C₂₃×#120	—	2	2	no	—
128	C₄×Q₈×C₂₃	—	2	2	no	—
129	C₄×Q₈×C₂₃	—	2	2	no	—
130	C₄×Q₈×C₂₃	D₄	1	2	yes	—
131	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
132	C₄×D₄×C₂₃	—	2	4	no	—
133	C₄×D₄×C₂₃	—	2	2	no	—
134	C₄×D₄×C₂₃	—	2	2	no	—
135	C₄×D₄×C₂₃	—	2	4	no	—
136	C₄×D₄×C₂₃	D₄	1	4	yes	—
137	C₄×D₄×C₂₃	D₄	1	2	yes	—
138	C₄×D₄×C₂₃	C₂³	1	4	yes	—
139	C₂₃×#117	—	2	2	no	—
140	C₂₃×#117	—	2	4	no	—
141	C₂₃×#117	C₂³	1	1	yes	characteristic
142	C₂₃×#117	C₂³	1	1	yes	characteristic
143	C₂₃×#117	C₂³	1	1	yes	characteristic
144	C₂₃×#117	C₂³	1	2	yes	—
145	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
146	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
147	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
148	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
149	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
151	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
152	C₂₃×#103	C₂³	1	4	yes	—
153	C₂₃×#103	C₂³	1	4	yes	—
154	C₂₃×#103	C₂³	1	8	yes	—
155	C₂₃×#103	C₂³	1	8	yes	—
156	C₂₃×#102	C₂³	1	4	yes	—
157	C₂₃×#102	C₂³	1	4	yes	—
158	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
159	C₂₃×#97	C₂³	1	2	yes	—
160	C₂₃×#97	C₂³	1	2	yes	—
161	C₂₃×#97	C₂³	1	2	yes	—
162	C₄×C₈×C₂₃	C₂³	1	2	yes	—
163	C₂⁴×C₂₃	—	2	1	no	—
164	#41×C₂₃	—	2	4	no	—
165	#41×C₂₃	—	2	4	no	—
166	#41×C₂₃	—	2	4	no	—
167	#41×C₂₃	—	4	4	no	—
168	#41×C₂₃	—	4	4	no	—
169	#41×C₂₃	—	4	16	no	—
170	#41×C₂₃	—	4	8	no	—
171	C₂×Q₈×C₂₃	—	2	1	no	—
172	C₂×Q₈×C₂₃	—	2	4	no	—
173	C₂×Q₈×C₂₃	—	2	2	no	—
174	C₂×Q₈×C₂₃	—	2	2	no	—
175	C₂×Q₈×C₂₃	—	2	4	no	—
176	C₂×Q₈×C₂₃	—	2	4	no	—
177	C₂×Q₈×C₂₃	—	2	4	no	—
178	C₂×Q₈×C₂₃	—	4	4	no	—
179	C₂×Q₈×C₂₃	—	4	4	no	—
180	C₂×Q₈×C₂₃	—	4	8	no	—
181	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
182	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
183	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
184	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
185	C₂×D₄×C₂₃	—	2	4	no	—
186	C₂×D₄×C₂₃	—	2	2	no	—
187	C₂×D₄×C₂₃	—	2	4	no	—
188	C₂×D₄×C₂₃	—	2	2	no	—
189	C₂×D₄×C₂₃	—	2	1	no	—
190	C₂×D₄×C₂₃	—	2	4	no	—
191	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
192	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
193	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
194	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₂₃	—	2	1	no	—
196	C₂²×C₄×C₂₃	—	2	4	no	—
197	C₂²×C₄×C₂₃	—	2	2	no	—
198	C₂²×C₄×C₂₃	—	2	4	no	—
199	C₂²×C₄×C₂₃	—	2	2	no	—
200	C₂²×C₄×C₂₃	—	2	4	no	—
201	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
202	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
203	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
205	Q₁₆×C₂₃	—	4	8	no	—
206	Q₁₆×C₂₃	—	4	4	no	—
207	SD₁₆×C₂₃	—	4	8	no	—
208	SD₁₆×C₂₃	—	4	4	no	—
209	SD₁₆×C₂₃	—	4	4	no	—
210	D₈×C₂₃	—	4	4	no	—
211	M₁₆×C₂₃	—	2	4	no	—
212	C₂×C₈×C₂₃	—	2	4	no	—
213	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
214	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
215	C₄⋊C₄×C₂₃	—	2	4	no	—
216	C₄⋊C₄×C₂₃	—	2	2	no	—
217	C₄⋊C₄×C₂₃	—	2	8	no	—
218	C₄⋊C₄×C₂₃	—	2	4	no	—
219	C₄⋊C₄×C₂₃	—	2	2	no	—
220	C₄⋊C₄×C₂₃	—	2	4	no	—
221	C₄⋊C₄×C₂₃	—	2	4	no	—
222	C₄⋊C₄×C₂₃	—	2	2	no	—
223	C₄⋊C₄×C₂₃	—	2	2	no	—
224	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
225	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
226	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
227	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
228	C₂²⋊C₄×C₂₃	—	2	1	no	—
229	C₂²⋊C₄×C₂₃	—	2	1	no	—
230	C₂²⋊C₄×C₂₃	—	2	4	no	—
231	C₂²⋊C₄×C₂₃	—	2	4	no	—
232	C₂²⋊C₄×C₂₃	—	2	4	no	—
233	C₂²⋊C₄×C₂₃	—	2	4	no	—
234	C₂²⋊C₄×C₂₃	—	2	4	no	—
235	C₂²⋊C₄×C₂₃	—	2	8	no	—
236	C₂²⋊C₄×C₂₃	—	2	4	no	—
237	C₂²⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
238	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
239	C₄²×C₂₃	—	2	4	no	—
240	C₄²×C₂₃	—	2	1	no	—
241	C₄²×C₂₃	—	2	1	no	—
242	C₄²×C₂₃	C₂×D₄	1	4	yes	—
243	C₄²×C₂₃	C₂×D₄	1	2	yes	—
244	C₄²×C₂₃	C₂⁴	1	2	yes	—
245	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
246	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
247	#60828	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
248	C₂³×C₂₃	—	2	2	no	—
249	C₂³×C₂₃	—	2	2	no	—
250	C₂³×C₂₃	—	4	2	no	—
251	C₂³×C₂₃	—	8	2	no	—
252	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
253	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
254	Q₈×C₂₃	—	2	2	no	—
255	Q₈×C₂₃	—	2	4	no	—
256	Q₈×C₂₃	—	2	2	no	—
257	Q₈×C₂₃	—	2	4	no	—
258	Q₈×C₂₃	—	2	4	no	—
259	Q₈×C₂₃	—	2	2	no	—
260	Q₈×C₂₃	—	4	8	no	—
261	Q₈×C₂₃	—	4	2	no	—
262	Q₈×C₂₃	—	4	4	no	—
263	Q₈×C₂₃	—	8	2	no	—
264	D₄×C₂₃	—	2	4	no	—
265	D₄×C₂₃	—	2	2	no	—
266	D₄×C₂₃	—	2	2	no	—
267	D₄×C₂₃	—	2	4	no	—
268	D₄×C₂₃	—	2	2	no	—
269	D₄×C₂₃	—	4	4	no	—
270	D₄×C₂₃	—	4	8	no	—
271	D₄×C₂₃	—	4	4	no	—
272	D₄×C₂₃	—	4	2	no	—
273	D₄×C₂₃	—	8	4	no	—
274	C₂×C₄×C₂₃	—	2	4	no	—
275	C₂×C₄×C₂₃	—	2	2	no	—
276	C₂×C₄×C₂₃	—	2	2	no	—
277	C₂×C₄×C₂₃	—	2	2	no	—
278	C₂×C₄×C₂₃	—	2	4	no	—
279	C₂×C₄×C₂₃	—	2	2	no	—
280	C₂×C₄×C₂₃	—	4	4	no	—
281	C₂×C₄×C₂₃	—	4	2	no	—
282	C₂×C₄×C₂₃	—	4	8	no	—
283	C₂×C₄×C₂₃	—	4	8	no	—
284	C₂×C₄×C₂₃	—	4	2	no	—
285	C₂×C₄×C₂₃	—	4	4	no	—
286	C₂×C₄×C₂₃	—	4	4	no	—
287	C₂×C₄×C₂₃	—	8	4	no	—
288	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
289	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
290	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
291	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
292	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
294	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
295	C₈×C₂₃	—	2	4	no	—
296	C₈×C₂₃	—	2	4	no	—
297	#3510	C₂×C₂₃	1	1	yes	characteristic, complemented
298	#3499	C₂×C₂₃	1	1	yes	characteristic, complemented
299	#3225	C₂×C₂₃	1	8	yes	complemented
300	#3219	C₂×C₂₃	1	4	yes	complemented
301	#3218	C₂×C₂₃	1	4	yes	complemented
302	#3153	C₂×C₂₃	1	1	yes	characteristic, complemented
303	C₂×#497	C₂×C₂₃	1	1	yes	characteristic, complemented
304	#3139	C₂×C₂₃	1	1	yes	characteristic, complemented
305	#3135	C₂×C₂₃	1	1	yes	characteristic, complemented
306	#3044	C₂×C₂₃	1	2	yes	complemented
307	#3042	C₂×C₂₃	1	2	yes	complemented
308	#3011	C₂×C₂₃	1	2	yes	complemented
309	#3010	C₂×C₂₃	1	2	yes	complemented
310	#2923	C₂×C₂₃	1	1	yes	characteristic, complemented
311	C₂²×C₂₃	—	2	2	no	—
312	C₂²×C₂₃	—	2	4	no	—
313	C₂²×C₂₃	—	2	2	no	—
314	C₂²×C₂₃	—	4	4	no	—
315	C₂²×C₂₃	—	4	4	no	—
316	C₂²×C₂₃	—	4	2	no	—
317	C₂²×C₂₃	—	4	2	no	—
318	C₂²×C₂₃	—	16	2	no	—
319	C₂²×C₂₃	#535	1	2	yes	—
320	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
321	C₄×C₂₃	—	2	2	no	—
322	C₄×C₂₃	—	2	4	no	—
323	C₄×C₂₃	—	2	2	no	—
324	C₄×C₂₃	—	2	2	no	—
325	C₄×C₂₃	—	4	2	no	—
326	C₄×C₂₃	—	4	2	no	—
327	C₄×C₂₃	—	4	4	no	—
328	C₄×C₂₃	—	4	2	no	—
329	C₄×C₂₃	—	4	2	no	—
330	C₄×C₂₃	#535	1	2	yes	—
331	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
332	C₂×2₋¹⁺⁴	C₂²×C₂₃	1	2	yes	—
333	C₂×#137	C₂²×C₂₃	1	2	yes	—
334	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	2	yes	—
335	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
336	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
337	#559	—	2	2	no	—
338	#556	—	2	2	no	—
339	#551	—	2	4	no	—
340	D₄×Q₈	—	2	2	no	—
341	#547	—	2	4	no	—
342	#547	—	2	2	no	—
343	#542	C₂²×C₂₃	1	4	yes	—
344	#536	C₂²×C₂₃	1	2	yes	—
345	#536	C₂²×C₂₃	1	1	yes	characteristic
346	#536	C₂²×C₂₃	1	4	yes	complemented
347	#535	C₂²×C₂₃	1	1	yes	characteristic
348	#533	C₂²×C₂₃	1	1	yes	characteristic
349	#532	C₂²×C₂₃	1	1	yes	characteristic
350	#532	C₂²×C₂₃	1	2	yes	complemented
351	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
353	#529	C₂²×C₂₃	1	4	yes	—
354	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
355	#519	C₂²×C₂₃	1	1	yes	characteristic
356	#518	C₂²×C₂₃	1	1	yes	characteristic
357	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
358	#497	C₂²×C₂₃	1	4	yes	—
359	#497	C₂²×C₂₃	1	4	yes	—
360	#497	C₂²×C₂₃	1	2	yes	—
361	#496	C₂²×C₂₃	1	2	yes	—
362	#495	C₂²×C₂₃	1	4	yes	—
363	#491	C₂²×C₂₃	1	2	yes	complemented
364	#491	C₂²×C₂₃	1	4	yes	complemented
365	#490	C₂²×C₂₃	1	2	yes	complemented
366	#489	C₂²×C₂₃	1	4	yes	complemented
367	#487	C₂²×C₂₃	1	4	yes	complemented
368	#464	C₂²×C₂₃	1	8	yes	complemented
369	#463	C₂²×C₂₃	1	8	yes	—
370	#462	C₂²×C₂₃	1	8	yes	complemented
371	#461	C₂²×C₂₃	1	8	yes	—
372	#452	C₂²×C₂₃	1	8	yes	complemented
373	#451	C₂²×C₂₃	1	8	yes	complemented
374	#449	C₂²×C₂₃	1	8	yes	—
375	#448	C₂²×C₂₃	1	8	yes	—
376	#433	C₂²×C₂₃	1	4	yes	complemented
377	#432	C₂²×C₂₃	1	4	yes	complemented
378	#424	C₂²×C₂₃	1	2	yes	—
379	#419	C₂²×C₂₃	1	2	yes	—
380	#419	C₂²×C₂₃	1	2	yes	—
381	#416	C₂²×C₂₃	1	2	yes	—
382	#416	C₂²×C₂₃	1	2	yes	—
383	#408	C₂²×C₂₃	1	2	yes	—
384	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
385	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
386	C₂×C₂₃	—	2	2	no	—
387	C₂×C₂₃	—	4	2	no	—
388	C₂×C₂₃	—	8	2	no	—
389	C₂×C₂₃	—	8	2	no	—
390	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
391	C₂×C₂₃	#3586	1	2	yes	—
392	2₋¹⁺⁴	—	4	4	no	—
393	2₋¹⁺⁴	—	4	4	no	—
394	C₂×#41	—	2	4	no	—
395	C₂×#41	—	2	2	no	—
396	C₂×#41	C₂³×C₂₃	1	2	yes	—
397	C₂×#41	C₂³×C₂₃	1	4	yes	—
398	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
399	C₂²×Q₈	—	2	2	no	—
400	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
401	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
402	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
403	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
404	#137	—	4	4	no	—
405	AGL(1,ℤ/8)	—	4	4	no	—
406	C₂×Q₁₆	—	4	4	no	—
407	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
408	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
409	C₂×SD₁₆	—	4	4	no	—
410	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
411	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
412	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
413	C₂×D₈	C₂³×C₂₃	1	4	yes	—
414	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
415	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
416	C₄⋊Q₈	—	2	1	no	—
417	C₄⋊Q₈	—	2	2	no	—
418	C₄⋊Q₈	—	2	4	no	—
419	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
420	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
421	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
422	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
423	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
424	#127	—	2	1	no	—
425	#127	C₂³×C₂₃	1	1	yes	characteristic
426	#126	—	2	4	no	—
427	#126	—	2	4	no	—
428	#125	C₂³×C₂₃	1	2	yes	—
429	#124	—	2	1	no	—
430	#124	—	2	1	no	—
431	#124	—	2	4	no	—
432	#124	—	2	2	no	—
433	#124	C₂³×C₂₃	1	2	yes	—
434	#124	C₂³×C₂₃	1	2	yes	—
435	#124	C₂³×C₂₃	1	2	yes	—
436	#124	C₂³×C₂₃	1	4	yes	—
437	#123	—	2	2	no	—
438	#123	—	2	4	no	—
439	#123	—	2	4	no	—
440	#123	—	2	8	no	—
441	#123	—	2	4	no	—
442	C₂²⋊Q₈	—	2	2	no	—
443	C₂²⋊Q₈	—	2	4	no	—
444	C₂²⋊Q₈	—	2	8	no	—
445	C₂²⋊Q₈	—	2	4	no	—
446	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
447	#121	—	2	2	no	—
448	#121	—	2	4	no	—
449	#121	C₂³×C₂₃	1	4	yes	—
450	#120	—	2	2	no	—
451	C₄×Q₈	—	2	2	no	—
452	C₄×Q₈	—	2	2	no	—
453	C₄×Q₈	D₄×C₂₃	1	2	yes	—
454	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
455	C₄×D₄	—	2	4	no	—
456	C₄×D₄	—	2	2	no	—
457	C₄×D₄	—	2	2	no	—
458	C₄×D₄	—	2	4	no	—
459	C₄×D₄	D₄×C₂₃	1	4	yes	—
460	C₄×D₄	D₄×C₂₃	1	2	yes	—
461	C₄×D₄	C₂³×C₂₃	1	4	yes	—
462	#117	—	2	2	no	—
463	#117	—	2	4	no	—
464	#117	C₂³×C₂₃	1	1	yes	characteristic
465	#117	C₂³×C₂₃	1	1	yes	characteristic
466	#117	C₂³×C₂₃	1	1	yes	characteristic
467	#117	C₂³×C₂₃	1	2	yes	—
468	C₂×C₄⋊C₄	—	2	2	no	—
469	C₂×C₄⋊C₄	—	2	4	no	—
470	C₂×C₂²⋊C₄	—	2	2	no	—
471	C₂×C₂²⋊C₄	—	2	2	no	—
472	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
473	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
474	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
475	#103	C₂³×C₂₃	1	4	yes	—
476	#103	C₂³×C₂₃	1	4	yes	—
477	#103	C₂³×C₂₃	1	8	yes	—
478	#103	C₂³×C₂₃	1	8	yes	—
479	#102	C₂³×C₂₃	1	4	yes	—
480	#102	C₂³×C₂₃	1	4	yes	—
481	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
482	#97	C₂³×C₂₃	1	2	yes	—
483	#97	C₂³×C₂₃	1	2	yes	—
484	#97	C₂³×C₂₃	1	2	yes	—
485	C₄×C₈	C₂³×C₂₃	1	2	yes	—
486	C₂₃	#60828	1	1	yes	characteristic, direct product factor, Sylow
487	C₂⁴	—	2	1	no	—
488	#41	—	2	4	no	—
489	#41	—	2	4	no	—
490	#41	—	2	4	no	—
491	#41	—	4	4	no	—
492	#41	—	4	4	no	—
493	#41	—	4	16	no	—
494	#41	—	4	8	no	—
495	C₂×Q₈	—	2	1	no	—
496	C₂×Q₈	—	2	4	no	—
497	C₂×Q₈	—	2	2	no	—
498	C₂×Q₈	—	2	2	no	—
499	C₂×Q₈	—	2	4	no	—
500	C₂×Q₈	—	2	4	no	—
501	C₂×Q₈	—	2	4	no	—
502	C₂×Q₈	—	4	4	no	—
503	C₂×Q₈	—	4	4	no	—
504	C₂×Q₈	—	4	8	no	—
505	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
506	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
507	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
508	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
509	C₂×D₄	—	2	4	no	—
510	C₂×D₄	—	2	2	no	—
511	C₂×D₄	—	2	4	no	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	4	no	—
515	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
516	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
517	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
518	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
519	C₂²×C₄	—	2	1	no	—
520	C₂²×C₄	—	2	4	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	2	4	no	—
523	C₂²×C₄	—	2	2	no	—
524	C₂²×C₄	—	2	4	no	—
525	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
526	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
527	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
528	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
529	Q₁₆	—	4	8	no	—
530	Q₁₆	—	4	4	no	—
531	SD₁₆	—	4	8	no	—
532	SD₁₆	—	4	4	no	—
533	SD₁₆	—	4	4	no	—
534	D₈	—	4	4	no	—
535	M₁₆	—	2	4	no	—
536	C₂×C₈	—	2	4	no	—
537	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
538	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
539	C₄⋊C₄	—	2	4	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	8	no	—
542	C₄⋊C₄	—	2	4	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	4	no	—
546	C₄⋊C₄	—	2	2	no	—
547	C₄⋊C₄	—	2	2	no	—
548	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
549	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
550	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
551	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
552	C₂²⋊C₄	—	2	1	no	—
553	C₂²⋊C₄	—	2	1	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	2	4	no	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	4	no	—
558	C₂²⋊C₄	—	2	4	no	—
559	C₂²⋊C₄	—	2	8	no	—
560	C₂²⋊C₄	—	2	4	no	—
561	C₂²⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
562	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
563	C₄²	—	2	4	no	—
564	C₄²	—	2	1	no	—
565	C₄²	—	2	1	no	—
566	C₄²	C₂×D₄×C₂₃	1	4	yes	—
567	C₄²	C₂×D₄×C₂₃	1	2	yes	—
568	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
569	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
570	C₄²	C₂⁴×C₂₃	1	2	yes	—
571	C₂³	—	2	2	no	—
572	C₂³	—	2	2	no	—
573	C₂³	—	4	2	no	—
574	C₂³	—	8	2	no	—
575	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
576	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
577	Q₈	—	2	2	no	—
578	Q₈	—	2	4	no	—
579	Q₈	—	2	2	no	—
580	Q₈	—	2	4	no	—
581	Q₈	—	2	4	no	—
582	Q₈	—	2	2	no	—
583	Q₈	—	4	8	no	—
584	Q₈	—	4	4	no	—
585	Q₈	—	4	2	no	—
586	Q₈	—	8	2	no	—
587	D₄	—	2	4	no	—
588	D₄	—	2	2	no	—
589	D₄	—	2	4	no	—
590	D₄	—	2	2	no	—
591	D₄	—	2	2	no	—
592	D₄	—	4	4	no	—
593	D₄	—	4	8	no	—
594	D₄	—	4	4	no	—
595	D₄	—	4	2	no	—
596	D₄	—	8	4	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	4	4	no	—
604	C₂×C₄	—	4	2	no	—
605	C₂×C₄	—	4	8	no	—
606	C₂×C₄	—	4	8	no	—
607	C₂×C₄	—	4	4	no	—
608	C₂×C₄	—	4	2	no	—
609	C₂×C₄	—	4	4	no	—
610	C₂×C₄	—	8	4	no	—
611	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
612	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
613	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
614	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
615	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
616	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
617	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
618	C₈	—	2	4	no	—
619	C₈	—	2	4	no	—
620	C₂²	—	2	2	no	—
621	C₂²	—	2	2	no	—
622	C₂²	—	2	4	no	—
623	C₂²	—	4	2	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	4	4	no	—
626	C₂²	—	4	4	no	—
627	C₂²	—	16	2	no	—
628	C₂²	C₂₃×#535	1	2	yes	—
629	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
630	C₄	—	2	2	no	—
631	C₄	—	2	2	no	—
632	C₄	—	2	4	no	—
633	C₄	—	2	2	no	—
634	C₄	—	4	4	no	—
635	C₄	—	4	2	no	—
636	C₄	—	4	2	no	—
637	C₄	—	4	2	no	—
638	C₄	—	4	2	no	—
639	C₄	C₂₃×#535	1	2	yes	—
640	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
641	C₂	—	2	2	no	—
642	C₂	—	4	2	no	—
643	C₂	—	8	2	no	—
644	C₂	—	8	2	no	—
645	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
646	C₂	C₂₃×#3586	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 ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3510	C₂	semidirect
2. C₂₃×#3499	C₂	semidirect
3. C₂₃×#3225	C₂	semidirect
4. C₂₃×#3219	C₂	semidirect
5. C₂₃×#3218	C₂	semidirect
6. C₂₃×#3153	C₂	semidirect
7. C₂×C₂₃×#497	C₂	semidirect
8. C₂₃×#3139	C₂	semidirect
9. C₂₃×#3135	C₂	semidirect
10. C₂₃×#3044	C₂	semidirect
11. C₂₃×#3042	C₂	semidirect
12. C₂₃×#3011	C₂	semidirect
13. C₂₃×#3010	C₂	semidirect
14. C₂₃×#2923	C₂	semidirect
29. C₂₃×#536	C₂²	semidirect
33. C₂₃×#532	C₂²	semidirect
35. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
46. C₂₃×#491	C₂²	semidirect
47. C₂₃×#491	C₂²	semidirect
48. C₂₃×#490	C₂²	semidirect
49. C₂₃×#489	C₂²	semidirect
50. C₂₃×#487	C₂²	semidirect
51. C₂₃×#464	C₂²	semidirect
53. C₂₃×#462	C₂²	semidirect
55. C₂₃×#452	C₂²	semidirect
56. C₂₃×#451	C₂²	semidirect
59. C₂₃×#433	C₂²	semidirect
60. C₂₃×#432	C₂²	semidirect
247. #60828	C₂₃	direct (N × H)
297. #3510	C₂×C₂₃	semidirect
298. #3499	C₂×C₂₃	semidirect
299. #3225	C₂×C₂₃	semidirect
300. #3219	C₂×C₂₃	semidirect
301. #3218	C₂×C₂₃	semidirect
302. #3153	C₂×C₂₃	semidirect
303. C₂×#497	C₂×C₂₃	semidirect
304. #3139	C₂×C₂₃	semidirect
305. #3135	C₂×C₂₃	semidirect
306. #3044	C₂×C₂₃	semidirect
307. #3042	C₂×C₂₃	semidirect
308. #3011	C₂×C₂₃	semidirect
309. #3010	C₂×C₂₃	semidirect
310. #2923	C₂×C₂₃	semidirect
346. #536	C₂²×C₂₃	semidirect
350. #532	C₂²×C₂₃	semidirect
352. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
363. #491	C₂²×C₂₃	semidirect
364. #491	C₂²×C₂₃	semidirect
365. #490	C₂²×C₂₃	semidirect
366. #489	C₂²×C₂₃	semidirect
367. #487	C₂²×C₂₃	semidirect
368. #464	C₂²×C₂₃	semidirect
370. #462	C₂²×C₂₃	semidirect
372. #452	C₂²×C₂₃	semidirect
373. #451	C₂²×C₂₃	semidirect
376. #433	C₂²×C₂₃	semidirect
377. #432	C₂²×C₂₃	semidirect
486. C₂₃	#60828	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×#3510	2
C₂₃×#3499	2
C₂₃×#3225	2
C₂₃×#3219	2
C₂₃×#3218	2
C₂₃×#3153	2
C₂×C₂₃×#497	2
C₂₃×#3139	2
C₂₃×#3135	2
C₂₃×#3044	2
C₂₃×#3042	2
C₂₃×#3011	2
C₂₃×#3010	2
C₂₃×#2923	2
#60828	23

Subgroups — C23×#60828 #19208454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60828 #19208454