Subgroups — C₂₃×#32777 #19180403

Order 5888.

704 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2857	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2851	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2097	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2084	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2083	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2043	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2043	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2036	C₂	1	2	yes	maximal
9	C₂₃×#2035	C₂	1	2	yes	maximal
10	C₂₃×#2026	C₂	1	2	yes	maximal, complemented
11	C₂₃×#1817	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
15	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#107	C₂²	1	2	yes	—
17	C₂×C₂₃×#106	C₂²	1	2	yes	—
18	C₂×C₂₃×#103	C₂²	1	2	yes	—
19	C₂×C₂₃×#103	C₂²	1	2	yes	—
20	C₂×C₂₃×#103	C₂²	1	2	yes	—
21	C₂×C₂₃×#103	C₂²	1	2	yes	—
22	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₂₃×#391	C₂²	1	2	yes	—
24	C₂₃×#389	—	2	1	no	—
25	C₂₃×#389	—	2	1	no	—
26	C₂₃×#389	C₂²	1	1	yes	characteristic
27	C₂₃×#389	C₂²	1	1	yes	characteristic
28	C₂₃×#387	—	2	1	no	—
29	C₂₃×#387	—	2	1	no	—
30	C₂₃×#385	—	2	1	no	—
31	C₂₃×#385	—	2	1	no	—
32	C₂₃×#384	C₂²	1	1	yes	characteristic
33	C₂₃×#382	C₂²	1	1	yes	characteristic
34	C₂₃×#380	—	2	1	no	—
35	C₂₃×#380	—	2	1	no	—
36	C₂₃×#380	C₂²	1	1	yes	characteristic
37	C₂₃×#380	C₂²	1	1	yes	characteristic
38	C₂₃×#340	C₂²	1	2	yes	—
39	C₂₃×#340	C₂²	1	2	yes	—
40	C₂₃×#336	C₂²	1	2	yes	—
41	C₂₃×#336	C₂²	1	1	yes	characteristic
42	C₂₃×#336	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
44	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
49	C₂₃×C₄⋊Q₈	—	4	2	no	—
50	C₂₃×#125	—	4	2	no	—
51	C₂₃×C₂²⋊Q₈	—	4	2	no	—
52	C₂₃×C₂²⋊Q₈	—	4	2	no	—
53	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₂×C₄²×C₂₃	—	4	2	no	—
71	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
75	C₂₃×#107	—	4	2	no	—
76	C₂₃×#106	—	4	2	no	—
77	C₂₃×#103	—	4	2	no	—
78	C₂₃×#103	—	4	2	no	—
79	C₂₃×#103	—	4	2	no	—
80	C₂₃×#103	—	4	2	no	—
81	C₂₃×C₂²⋊C₈	—	4	2	no	—
82	C₂₃×#95	—	2	2	no	—
83	C₂₃×#95	—	2	2	no	—
84	C₂₃×#95	—	2	1	no	—
85	C₂₃×#95	—	2	1	no	—
86	C₂₃×#95	—	2	1	no	—
87	C₂₃×#95	—	2	1	no	—
88	C₂₃×#95	—	2	1	no	—
89	C₂₃×#95	—	2	1	no	—
90	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
91	C₂×Q₈×C₂₃	—	2	2	no	—
92	C₂×Q₈×C₂₃	—	2	2	no	—
93	C₂×Q₈×C₂₃	—	2	2	no	—
94	C₂×Q₈×C₂₃	—	2	2	no	—
95	C₂×Q₈×C₂₃	—	2	2	no	—
96	C₂×Q₈×C₂₃	—	2	2	no	—
97	C₂²×C₄×C₂₃	—	2	2	no	—
98	C₂²×C₄×C₂₃	—	2	2	no	—
99	C₂²×C₄×C₂₃	—	2	2	no	—
100	C₂²×C₄×C₂₃	—	2	2	no	—
101	C₂²×C₄×C₂₃	—	2	1	no	—
102	C₂²×C₄×C₂₃	—	2	1	no	—
103	C₂²×C₄×C₂₃	—	2	2	no	—
104	C₂²×C₄×C₂₃	—	2	1	no	—
105	C₂²×C₄×C₂₃	—	2	1	no	—
106	C₂²×C₄×C₂₃	—	2	2	no	—
107	C₂²×C₄×C₂₃	—	2	2	no	—
108	C₂²×C₄×C₂₃	—	2	1	no	—
109	C₂²×C₄×C₂₃	—	2	1	no	—
110	C₂²×C₄×C₂₃	—	4	1	no	—
111	C₂²×C₄×C₂₃	—	4	1	no	—
112	C₂²×C₄×C₂₃	—	4	1	no	—
113	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
119	C₂×C₈×C₂₃	—	2	2	no	—
120	C₂×C₈×C₂₃	—	2	2	no	—
121	C₂×C₈×C₂₃	—	2	2	no	—
122	C₂×C₈×C₂₃	—	2	2	no	—
123	C₂×C₈×C₂₃	—	2	2	no	—
124	C₂×C₈×C₂₃	—	2	2	no	—
125	C₄⋊C₄×C₂₃	—	2	1	no	—
126	C₄⋊C₄×C₂₃	—	2	1	no	—
127	C₄⋊C₄×C₂₃	—	2	2	no	—
128	C₄⋊C₄×C₂₃	—	2	2	no	—
129	C₄⋊C₄×C₂₃	—	2	1	no	—
130	C₄⋊C₄×C₂₃	—	2	2	no	—
131	C₄⋊C₄×C₂₃	—	2	1	no	—
132	C₄⋊C₄×C₂₃	—	2	2	no	—
133	C₄⋊C₄×C₂₃	—	4	2	no	—
134	C₄⋊C₄×C₂₃	—	4	4	no	—
135	C₄⋊C₄×C₂₃	—	4	2	no	—
136	C₄⋊C₄×C₂₃	—	4	2	no	—
137	C₄⋊C₄×C₂₃	—	4	2	no	—
138	C₄⋊C₄×C₂₃	—	4	2	no	—
139	C₄⋊C₄×C₂₃	—	4	2	no	—
140	C₄⋊C₄×C₂₃	—	4	2	no	—
141	C₄⋊C₄×C₂₃	—	4	2	no	—
142	C₂²⋊C₄×C₂₃	—	4	2	no	—
143	C₂²⋊C₄×C₂₃	—	4	4	no	—
144	C₂²⋊C₄×C₂₃	—	4	2	no	—
145	C₄²×C₂₃	—	2	2	no	—
146	C₄²×C₂₃	—	2	2	no	—
147	C₄²×C₂₃	—	4	1	no	—
148	C₄²×C₂₃	—	4	1	no	—
149	#32777	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₂₃	—	2	1	no	—
151	C₂³×C₂₃	—	2	1	no	—
152	C₂³×C₂₃	—	2	1	no	—
153	C₂³×C₂₃	—	2	1	no	—
154	C₂³×C₂₃	—	2	1	no	—
155	C₂³×C₂₃	—	2	1	no	—
156	C₂³×C₂₃	#137	1	2	yes	—
157	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
158	Q₈×C₂₃	—	4	2	no	—
159	Q₈×C₂₃	—	4	2	no	—
160	Q₈×C₂₃	—	4	2	no	—
161	Q₈×C₂₃	—	4	2	no	—
162	C₂×C₄×C₂₃	—	2	1	no	—
163	C₂×C₄×C₂₃	—	2	1	no	—
164	C₂×C₄×C₂₃	—	2	1	no	—
165	C₂×C₄×C₂₃	—	2	1	no	—
166	C₂×C₄×C₂₃	—	2	1	no	—
167	C₂×C₄×C₂₃	—	2	1	no	—
168	C₂×C₄×C₂₃	—	2	1	no	—
169	C₂×C₄×C₂₃	—	2	1	no	—
170	C₂×C₄×C₂₃	—	2	1	no	—
171	C₂×C₄×C₂₃	—	2	1	no	—
172	C₂×C₄×C₂₃	—	2	1	no	—
173	C₂×C₄×C₂₃	—	2	1	no	—
174	C₂×C₄×C₂₃	—	2	1	no	—
175	C₂×C₄×C₂₃	—	4	4	no	—
176	C₂×C₄×C₂₃	—	4	4	no	—
177	C₂×C₄×C₂₃	—	4	2	no	—
178	C₂×C₄×C₂₃	—	4	2	no	—
179	C₂×C₄×C₂₃	—	4	1	no	—
180	C₂×C₄×C₂₃	—	4	1	no	—
181	C₂×C₄×C₂₃	—	4	2	no	—
182	C₂×C₄×C₂₃	—	4	2	no	—
183	C₂×C₄×C₂₃	—	4	2	no	—
184	C₂×C₄×C₂₃	—	4	1	no	—
185	C₂×C₄×C₂₃	—	4	1	no	—
186	C₂×C₄×C₂₃	—	4	2	no	—
187	C₂×C₄×C₂₃	—	4	1	no	—
188	C₂×C₄×C₂₃	—	4	2	no	—
189	C₂×C₄×C₂₃	—	4	1	no	—
190	C₂×C₄×C₂₃	—	4	1	no	—
191	C₂×C₄×C₂₃	—	4	1	no	—
192	C₂×C₄×C₂₃	—	4	1	no	—
193	C₂×C₄×C₂₃	—	4	1	no	—
194	C₂×C₄×C₂₃	—	4	1	no	—
195	C₂×C₄×C₂₃	—	4	1	no	—
196	C₂×C₄×C₂₃	—	4	2	no	—
197	C₂×C₄×C₂₃	#135	1	2	yes	—
198	C₂×C₄×C₂₃	#137	1	2	yes	—
199	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
200	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
202	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
203	C₈×C₂₃	—	4	2	no	—
204	C₈×C₂₃	—	4	2	no	—
205	#2857	C₂×C₂₃	1	1	yes	characteristic
206	#2851	C₂×C₂₃	1	1	yes	characteristic
207	#2097	C₂×C₂₃	1	2	yes	complemented
208	#2084	C₂×C₂₃	1	1	yes	characteristic, complemented
209	#2083	C₂×C₂₃	1	1	yes	characteristic, complemented
210	#2043	C₂×C₂₃	1	1	yes	characteristic, complemented
211	#2043	C₂×C₂₃	1	1	yes	characteristic, complemented
212	#2036	C₂×C₂₃	1	2	yes	—
213	#2035	C₂×C₂₃	1	2	yes	—
214	#2026	C₂×C₂₃	1	2	yes	complemented
215	#1817	C₂×C₂₃	1	1	yes	characteristic
216	C₂²×C₂₃	—	2	2	no	—
217	C₂²×C₂₃	—	2	2	no	—
218	C₂²×C₂₃	—	2	2	no	—
219	C₂²×C₂₃	—	4	1	no	—
220	C₂²×C₂₃	—	4	1	no	—
221	C₂²×C₂₃	—	4	1	no	—
222	C₂²×C₂₃	—	4	1	no	—
223	C₂²×C₂₃	#534	1	1	yes	characteristic
224	C₂²×C₂₃	#535	1	1	yes	characteristic
225	C₂²×C₂₃	#536	1	1	yes	characteristic
226	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
227	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
228	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
229	C₂²×C₂₃	#575	1	1	yes	characteristic
230	C₄×C₂₃	—	2	2	no	—
231	C₄×C₂₃	—	2	2	no	—
232	C₄×C₂₃	—	4	1	no	—
233	C₄×C₂₃	—	4	1	no	—
234	C₄×C₂₃	—	4	1	no	—
235	C₄×C₂₃	—	8	2	no	—
236	C₄×C₂₃	—	8	1	no	—
237	C₄×C₂₃	—	8	1	no	—
238	C₄×C₂₃	—	8	2	no	—
239	C₄×C₂₃	—	8	1	no	—
240	C₄×C₂₃	—	8	1	no	—
241	C₄×C₂₃	#562	1	2	yes	—
242	C₄×C₂₃	#562	1	2	yes	—
243	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
244	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
245	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
246	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
247	C₂×#107	C₂²×C₂₃	1	2	yes	—
248	C₂×#106	C₂²×C₂₃	1	2	yes	—
249	C₂×#103	C₂²×C₂₃	1	2	yes	—
250	C₂×#103	C₂²×C₂₃	1	2	yes	—
251	C₂×#103	C₂²×C₂₃	1	2	yes	—
252	C₂×#103	C₂²×C₂₃	1	2	yes	—
253	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
254	#391	C₂²×C₂₃	1	2	yes	—
255	#389	—	2	1	no	—
256	#389	—	2	1	no	—
257	#389	C₂²×C₂₃	1	1	yes	characteristic
258	#389	C₂²×C₂₃	1	1	yes	characteristic
259	#387	—	2	1	no	—
260	#387	—	2	1	no	—
261	#385	—	2	1	no	—
262	#385	—	2	1	no	—
263	#384	C₂²×C₂₃	1	1	yes	characteristic
264	#382	C₂²×C₂₃	1	1	yes	characteristic
265	#380	—	2	1	no	—
266	#380	—	2	1	no	—
267	#380	C₂²×C₂₃	1	1	yes	characteristic
268	#380	C₂²×C₂₃	1	1	yes	characteristic
269	#340	C₂²×C₂₃	1	2	yes	—
270	#340	C₂²×C₂₃	1	2	yes	—
271	#336	C₂²×C₂₃	1	1	yes	characteristic
272	#336	C₂²×C₂₃	1	1	yes	characteristic
273	#336	C₂²×C₂₃	1	2	yes	—
274	C₂×C₂₃	—	4	2	no	—
275	C₂×C₂₃	#2816	1	1	yes	characteristic
276	C₂×C₂₃	#3190	1	1	yes	characteristic
277	C₂×C₂₃	#3196	1	1	yes	characteristic
278	C₂×C₂₃	#3212	1	1	yes	characteristic
279	C₂×C₂₃	#3218	1	1	yes	characteristic
280	C₂×C₂₃	#3231	1	1	yes	characteristic
281	C₂×C₂₃	#3236	1	1	yes	characteristic
282	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
283	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
284	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
285	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
286	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
287	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
288	C₄⋊Q₈	—	4	2	no	—
289	#125	—	4	2	no	—
290	C₂²⋊Q₈	—	4	2	no	—
291	C₂²⋊Q₈	—	4	2	no	—
292	C₂×C₄⋊C₄	—	2	2	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
302	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
304	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
306	C₂×C₂²⋊C₄	—	2	2	no	—
307	C₂×C₂²⋊C₄	—	2	1	no	—
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₄²	—	4	2	no	—
310	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
311	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
314	#107	—	4	2	no	—
315	#106	—	4	2	no	—
316	#103	—	4	2	no	—
317	#103	—	4	2	no	—
318	#103	—	4	2	no	—
319	#103	—	4	2	no	—
320	C₂²⋊C₈	—	4	2	no	—
321	#95	—	2	2	no	—
322	#95	—	2	2	no	—
323	#95	—	2	1	no	—
324	#95	—	2	1	no	—
325	#95	—	2	1	no	—
326	#95	—	2	1	no	—
327	#95	—	2	1	no	—
328	#95	—	2	1	no	—
329	C₂₃	#32777	1	1	yes	characteristic, direct product factor, Sylow
330	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
331	C₂×Q₈	—	2	2	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	4	1	no	—
351	C₂²×C₄	—	4	1	no	—
352	C₂²×C₄	—	4	1	no	—
353	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
358	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	4	4	no	—
374	C₄⋊C₄	—	4	2	no	—
375	C₄⋊C₄	—	4	2	no	—
376	C₄⋊C₄	—	4	2	no	—
377	C₄⋊C₄	—	4	2	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	—	4	2	no	—
380	C₄⋊C₄	—	4	2	no	—
381	C₄⋊C₄	—	4	2	no	—
382	C₂²⋊C₄	—	4	4	no	—
383	C₂²⋊C₄	—	4	2	no	—
384	C₂²⋊C₄	—	4	2	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	4	1	no	—
388	C₄²	—	4	1	no	—
389	C₂³	—	2	1	no	—
390	C₂³	—	2	1	no	—
391	C₂³	—	2	1	no	—
392	C₂³	—	2	1	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	1	no	—
395	C₂³	C₂₃×#137	1	2	yes	—
396	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
397	Q₈	—	4	2	no	—
398	Q₈	—	4	2	no	—
399	Q₈	—	4	2	no	—
400	Q₈	—	4	2	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	4	4	no	—
415	C₂×C₄	—	4	4	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	C₂₃×#135	1	2	yes	—
437	C₂×C₄	C₂₃×#137	1	2	yes	—
438	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
439	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
440	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
441	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
442	C₈	—	4	2	no	—
443	C₈	—	4	2	no	—
444	C₂²	—	2	2	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	4	1	no	—
448	C₂²	—	4	1	no	—
449	C₂²	—	4	1	no	—
450	C₂²	—	4	1	no	—
451	C₂²	C₂₃×#534	1	1	yes	characteristic
452	C₂²	C₂₃×#535	1	1	yes	characteristic
453	C₂²	C₂₃×#536	1	1	yes	characteristic
454	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
455	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
456	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
457	C₂²	C₂₃×#575	1	1	yes	characteristic
458	C₄	—	2	2	no	—
459	C₄	—	2	2	no	—
460	C₄	—	4	1	no	—
461	C₄	—	4	1	no	—
462	C₄	—	4	1	no	—
463	C₄	—	8	1	no	—
464	C₄	—	8	1	no	—
465	C₄	—	8	1	no	—
466	C₄	—	8	2	no	—
467	C₄	—	8	1	no	—
468	C₄	—	8	2	no	—
469	C₄	C₂₃×#562	1	2	yes	—
470	C₄	C₂₃×#562	1	2	yes	—
471	C₂	—	4	2	no	—
472	C₂	C₂₃×#2816	1	1	yes	characteristic
473	C₂	C₂₃×#3190	1	1	yes	characteristic
474	C₂	C₂₃×#3196	1	1	yes	characteristic
475	C₂	C₂₃×#3212	1	1	yes	characteristic
476	C₂	C₂₃×#3218	1	1	yes	characteristic
477	C₂	C₂₃×#3231	1	1	yes	characteristic
478	C₂	C₂₃×#3236	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 ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2097	C₂	semidirect
4. C₂₃×#2084	C₂	semidirect
5. C₂₃×#2083	C₂	semidirect
6. C₂₃×#2043	C₂	semidirect
7. C₂₃×#2043	C₂	semidirect
10. C₂₃×#2026	C₂	semidirect
149. #32777	C₂₃	direct (N × H)
207. #2097	C₂×C₂₃	semidirect
208. #2084	C₂×C₂₃	semidirect
209. #2083	C₂×C₂₃	semidirect
210. #2043	C₂×C₂₃	semidirect
211. #2043	C₂×C₂₃	semidirect
214. #2026	C₂×C₂₃	semidirect
329. C₂₃	#32777	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×#2857	2
C₂₃×#2851	2
C₂₃×#2097	2
C₂₃×#2084	2
C₂₃×#2083	2
C₂₃×#2043	2
C₂₃×#2036	2
C₂₃×#2035	2
C₂₃×#2026	2
C₂₃×#1817	2
#32777	23

Subgroups — C23×#32777 #19180403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#32777 #19180403