Subgroups — C₂₃×#29159 #19176785

Order 5888.

828 conjugacy classes of subgroups.

The subgroup lattice has 484 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	C₄×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2099	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2096	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2058	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2024	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1919	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1892	C₂	1	2	yes	maximal
9	C₂₃×#1865	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1807	C₂	1	2	yes	maximal
11	C₂×C₂₃×#336	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
15	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#106	C₂²	1	2	yes	—
19	C₂×C₂₃×#106	C₂²	1	2	yes	—
20	C₂×C₂₃×#103	C₂²	1	2	yes	—
21	C₂×C₂₃×#103	C₂²	1	2	yes	—
22	C₂₃×#391	—	2	1	no	—
23	C₂₃×#391	C₂²	1	1	yes	characteristic
24	C₂₃×#389	—	2	1	no	—
25	C₂₃×#389	C₂²	1	2	yes	—
26	C₂₃×#389	C₂²	1	1	yes	characteristic
27	C₂₃×#387	—	2	1	no	—
28	C₂₃×#385	—	2	1	no	—
29	C₂₃×#384	C₂²	1	1	yes	characteristic
30	C₂₃×#380	—	2	1	no	—
31	C₂₃×#380	C₂²	1	2	yes	—
32	C₂₃×#380	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
36	C₂₃×#340	C₂²	1	2	yes	—
37	C₂₃×#340	C₂²	1	2	yes	—
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	—	2	1	no	—
44	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₂₃	—	2	4	no	—
49	C₂²×C₈×C₂₃	—	2	4	no	—
50	C₂²×C₈×C₂₃	—	4	2	no	—
51	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
53	C₂₃×C₄⋊Q₈	—	4	2	no	—
54	C₂₃×C₂²⋊Q₈	—	2	2	no	—
55	C₂₃×C₂²⋊Q₈	—	4	4	no	—
56	C₂₃×C₂²⋊Q₈	Q₈	1	4	yes	complemented
57	C₄×Q₈×C₂₃	—	4	2	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	1	no	—
62	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
65	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
73	C₂×C₄²×C₂₃	—	2	1	no	—
74	C₂×C₄²×C₂₃	—	2	1	no	—
75	C₂×C₄²×C₂₃	—	4	2	no	—
76	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
80	C₂₃×#106	—	2	4	no	—
81	C₂₃×#106	—	4	2	no	—
82	C₂₃×#103	—	2	4	no	—
83	C₂₃×#103	—	4	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	2	no	—
90	C₂₃×#95	—	2	2	no	—
91	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
92	C₂×Q₈×C₂₃	—	2	1	no	—
93	C₂×Q₈×C₂₃	—	2	2	no	—
94	C₂×Q₈×C₂₃	—	2	2	no	—
95	C₂×Q₈×C₂₃	—	4	2	no	—
96	C₂×Q₈×C₂₃	—	4	2	no	—
97	C₂×Q₈×C₂₃	—	4	2	no	—
98	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
99	C₂²×C₄×C₂₃	—	2	2	no	—
100	C₂²×C₄×C₂₃	—	2	2	no	—
101	C₂²×C₄×C₂₃	—	2	2	no	—
102	C₂²×C₄×C₂₃	—	2	1	no	—
103	C₂²×C₄×C₂₃	—	2	1	no	—
104	C₂²×C₄×C₂₃	—	2	2	no	—
105	C₂²×C₄×C₂₃	—	2	1	no	—
106	C₂²×C₄×C₂₃	—	2	1	no	—
107	C₂²×C₄×C₂₃	—	2	2	no	—
108	C₂²×C₄×C₂₃	—	2	2	no	—
109	C₂²×C₄×C₂₃	—	4	1	no	—
110	C₂²×C₄×C₂₃	—	4	1	no	—
111	C₂²×C₄×C₂₃	—	4	1	no	—
112	C₂²×C₄×C₂₃	SD₁₆	1	4	yes	—
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	C₂×Q₈	1	2	yes	—
119	C₂²×C₄×C₂₃	#41	1	2	yes	—
120	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
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	C₂×C₈×C₂₃	—	2	2	no	—
126	C₂×C₈×C₂₃	—	4	8	no	—
127	C₂×C₈×C₂₃	#41	1	4	yes	—
128	C₄⋊C₄×C₂₃	—	2	4	no	—
129	C₄⋊C₄×C₂₃	—	2	1	no	—
130	C₄⋊C₄×C₂₃	—	2	2	no	—
131	C₄⋊C₄×C₂₃	—	2	2	no	—
132	C₄⋊C₄×C₂₃	—	2	2	no	—
133	C₄⋊C₄×C₂₃	—	2	2	no	—
134	C₄⋊C₄×C₂₃	—	2	2	no	—
135	C₄⋊C₄×C₂₃	—	2	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	4	no	—
140	C₄⋊C₄×C₂₃	—	4	2	no	—
141	C₄⋊C₄×C₂₃	—	4	2	no	—
142	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
143	C₂²⋊C₄×C₂₃	—	2	4	no	—
144	C₂²⋊C₄×C₂₃	—	4	4	no	—
145	C₂²⋊C₄×C₂₃	—	4	2	no	—
146	C₄²×C₂₃	—	2	2	no	—
147	C₄²×C₂₃	—	2	2	no	—
148	C₄²×C₂₃	—	4	1	no	—
149	C₄²×C₂₃	—	4	1	no	—
150	C₄²×C₂₃	—	4	2	no	—
151	C₄²×C₂₃	—	4	2	no	—
152	#29159	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₂₃	—	2	1	no	—
154	C₂³×C₂₃	—	2	1	no	—
155	C₂³×C₂₃	—	2	1	no	—
156	C₂³×C₂₃	—	2	1	no	—
157	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
158	C₂³×C₂₃	C₂×SD₁₆	1	2	yes	—
159	C₂³×C₂₃	#137	1	2	yes	—
160	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₂₃	—	2	4	no	—
162	Q₈×C₂₃	—	4	2	no	—
163	Q₈×C₂₃	—	8	4	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	4	no	—
173	C₂×C₄×C₂₃	—	2	4	no	—
174	C₂×C₄×C₂₃	—	2	2	no	—
175	C₂×C₄×C₂₃	—	2	2	no	—
176	C₂×C₄×C₂₃	—	2	1	no	—
177	C₂×C₄×C₂₃	—	2	1	no	—
178	C₂×C₄×C₂₃	—	2	1	no	—
179	C₂×C₄×C₂₃	—	4	4	no	—
180	C₂×C₄×C₂₃	—	4	4	no	—
181	C₂×C₄×C₂₃	—	4	1	no	—
182	C₂×C₄×C₂₃	—	4	1	no	—
183	C₂×C₄×C₂₃	—	4	1	no	—
184	C₂×C₄×C₂₃	—	4	1	no	—
185	C₂×C₄×C₂₃	—	4	2	no	—
186	C₂×C₄×C₂₃	—	4	1	no	—
187	C₂×C₄×C₂₃	—	4	1	no	—
188	C₂×C₄×C₂₃	—	4	2	no	—
189	C₂×C₄×C₂₃	—	4	2	no	—
190	C₂×C₄×C₂₃	—	4	2	no	—
191	C₂×C₄×C₂₃	—	4	2	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	C₂²⋊Q₈	1	2	yes	—
198	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
199	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
200	C₂×C₄×C₂₃	#135	1	2	yes	—
201	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
202	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
203	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
204	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
205	C₈×C₂₃	—	2	4	no	—
206	C₈×C₂₃	—	4	2	no	—
207	#2857	C₂×C₂₃	1	1	yes	characteristic
208	C₄×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
209	#2099	C₂×C₂₃	1	1	yes	characteristic, complemented
210	#2096	C₂×C₂₃	1	1	yes	characteristic, complemented
211	#2058	C₂×C₂₃	1	1	yes	characteristic, complemented
212	#2024	C₂×C₂₃	1	1	yes	characteristic, complemented
213	#1919	C₂×C₂₃	1	2	yes	complemented
214	#1892	C₂×C₂₃	1	2	yes	—
215	#1865	C₂×C₂₃	1	2	yes	complemented
216	#1807	C₂×C₂₃	1	2	yes	—
217	C₂×#336	C₂×C₂₃	1	1	yes	characteristic
218	C₂²×C₂₃	—	2	2	no	—
219	C₂²×C₂₃	—	2	2	no	—
220	C₂²×C₂₃	—	2	2	no	—
221	C₂²×C₂₃	—	2	2	no	—
222	C₂²×C₂₃	—	2	2	no	—
223	C₂²×C₂₃	—	2	2	no	—
224	C₂²×C₂₃	#475	1	4	yes	—
225	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
226	C₂²×C₂₃	#532	1	1	yes	characteristic
227	C₂²×C₂₃	#534	1	1	yes	characteristic
228	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
229	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
230	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
231	C₂²×C₂₃	#575	1	1	yes	characteristic
232	C₄×C₂₃	—	2	2	no	—
233	C₄×C₂₃	—	2	2	no	—
234	C₄×C₂₃	—	2	2	no	—
235	C₄×C₂₃	—	2	2	no	—
236	C₄×C₂₃	—	4	2	no	—
237	C₄×C₂₃	—	4	2	no	—
238	C₄×C₂₃	—	4	1	no	—
239	C₄×C₂₃	—	8	2	no	—
240	C₄×C₂₃	—	8	1	no	—
241	C₄×C₂₃	—	8	2	no	—
242	C₄×C₂₃	—	8	1	no	—
243	C₄×C₂₃	#551	1	2	yes	—
244	C₄×C₂₃	#554	1	2	yes	—
245	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
246	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
247	C₂×C₂²⋊Q₈	—	2	1	no	—
248	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
249	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
250	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
251	C₂×#106	C₂²×C₂₃	1	2	yes	—
252	C₂×#106	C₂²×C₂₃	1	2	yes	—
253	C₂×#103	C₂²×C₂₃	1	2	yes	—
254	C₂×#103	C₂²×C₂₃	1	2	yes	—
255	#391	—	2	1	no	—
256	#391	C₂²×C₂₃	1	1	yes	characteristic
257	#389	—	2	1	no	—
258	#389	C₂²×C₂₃	1	1	yes	characteristic
259	#389	C₂²×C₂₃	1	2	yes	—
260	#387	—	2	1	no	—
261	#385	—	2	1	no	—
262	#384	C₂²×C₂₃	1	1	yes	characteristic
263	#380	—	2	1	no	—
264	#380	C₂²×C₂₃	1	1	yes	characteristic
265	#380	C₂²×C₂₃	1	2	yes	—
266	C₄×C₄⋊C₄	—	2	1	no	—
267	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
268	C₄×C₂²⋊C₄	—	2	1	no	—
269	#340	C₂²×C₂₃	1	2	yes	—
270	#340	C₂²×C₂₃	1	2	yes	—
271	#340	C₂²×C₂₃	1	2	yes	—
272	#340	C₂²×C₂₃	1	2	yes	—
273	#336	C₂²×C₂₃	1	1	yes	characteristic
274	#336	C₂²×C₂₃	1	1	yes	characteristic
275	#336	C₂²×C₂₃	1	2	yes	—
276	C₂×C₂₃	—	2	4	no	—
277	C₂×C₂₃	#2564	1	1	yes	characteristic
278	C₂×C₂₃	C₂×#475	1	1	yes	characteristic
279	C₂×C₂₃	#3076	1	1	yes	characteristic
280	C₂×C₂₃	#3167	1	1	yes	characteristic
281	C₂×C₂₃	#3181	1	1	yes	characteristic
282	C₂×C₂₃	#3187	1	1	yes	characteristic
283	C₂×C₂₃	#3190	1	1	yes	characteristic
284	C₂²×Q₈	—	2	1	no	—
285	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
286	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
287	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
288	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
289	C₂²×C₈	—	2	4	no	—
290	C₂²×C₈	—	2	4	no	—
291	C₂²×C₈	—	4	2	no	—
292	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
293	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
294	C₄⋊Q₈	—	4	2	no	—
295	C₂²⋊Q₈	—	2	2	no	—
296	C₂²⋊Q₈	—	4	4	no	—
297	C₂²⋊Q₈	Q₈×C₂₃	1	4	yes	complemented
298	C₄×Q₈	—	4	2	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	2	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
308	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
310	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
311	C₂×C₂²⋊C₄	—	2	1	no	—
312	C₂×C₂²⋊C₄	—	2	2	no	—
313	C₂×C₂²⋊C₄	—	2	1	no	—
314	C₂×C₄²	—	2	1	no	—
315	C₂×C₄²	—	2	1	no	—
316	C₂×C₄²	—	4	2	no	—
317	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
318	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
321	#106	—	2	4	no	—
322	#106	—	4	2	no	—
323	#103	—	2	4	no	—
324	#103	—	4	2	no	—
325	#95	—	2	1	no	—
326	#95	—	2	1	no	—
327	#95	—	2	1	no	—
328	#95	—	2	1	no	—
329	#95	—	2	1	no	—
330	#95	—	2	2	no	—
331	#95	—	2	2	no	—
332	C₂₃	#29159	1	1	yes	characteristic, direct product factor, Sylow
333	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
334	C₂×Q₈	—	2	1	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	4	2	no	—
338	C₂×Q₈	—	4	2	no	—
339	C₂×Q₈	—	4	2	no	—
340	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	1	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	4	1	no	—
352	C₂²×C₄	—	4	1	no	—
353	C₂²×C₄	—	4	1	no	—
354	C₂²×C₄	SD₁₆×C₂₃	1	4	yes	—
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	C₂×D₄×C₂₃	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
360	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
361	C₂²×C₄	#41×C₂₃	1	2	yes	—
362	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	4	8	no	—
369	C₂×C₈	#41×C₂₃	1	4	yes	—
370	C₄⋊C₄	—	2	4	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	2	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	—	4	2	no	—
380	C₄⋊C₄	—	4	2	no	—
381	C₄⋊C₄	—	4	4	no	—
382	C₄⋊C₄	—	4	2	no	—
383	C₄⋊C₄	—	4	2	no	—
384	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
385	C₂²⋊C₄	—	2	4	no	—
386	C₂²⋊C₄	—	4	4	no	—
387	C₂²⋊C₄	—	4	2	no	—
388	C₄²	—	2	2	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	4	1	no	—
391	C₄²	—	4	1	no	—
392	C₄²	—	4	2	no	—
393	C₄²	—	4	2	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
399	C₂³	C₂×SD₁₆×C₂₃	1	2	yes	—
400	C₂³	C₂₃×#137	1	2	yes	—
401	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
402	Q₈	—	2	4	no	—
403	Q₈	—	4	2	no	—
404	Q₈	—	8	4	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	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	4	4	no	—
421	C₂×C₄	—	4	4	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	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
439	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
440	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
441	C₂×C₄	C₂₃×#135	1	2	yes	—
442	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
443	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
444	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
445	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
446	C₈	—	2	4	no	—
447	C₈	—	4	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	2	no	—
451	C₂²	—	2	2	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	C₂₃×#475	1	4	yes	—
455	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
456	C₂²	C₂₃×#532	1	1	yes	characteristic
457	C₂²	C₂₃×#534	1	1	yes	characteristic
458	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
459	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
460	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
461	C₂²	C₂₃×#575	1	1	yes	characteristic
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	2	no	—
465	C₄	—	2	2	no	—
466	C₄	—	4	1	no	—
467	C₄	—	4	2	no	—
468	C₄	—	4	2	no	—
469	C₄	—	8	2	no	—
470	C₄	—	8	1	no	—
471	C₄	—	8	2	no	—
472	C₄	—	8	1	no	—
473	C₄	C₂₃×#551	1	2	yes	—
474	C₄	C₂₃×#554	1	2	yes	—
475	C₂	—	2	4	no	—
476	C₂	C₂₃×#2564	1	1	yes	characteristic
477	C₂	C₂×C₂₃×#475	1	1	yes	characteristic
478	C₂	C₂₃×#3076	1	1	yes	characteristic
479	C₂	C₂₃×#3167	1	1	yes	characteristic
480	C₂	C₂₃×#3181	1	1	yes	characteristic
481	C₂	C₂₃×#3187	1	1	yes	characteristic
482	C₂	C₂₃×#3190	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×#2099	C₂	semidirect
4. C₂₃×#2096	C₂	semidirect
5. C₂₃×#2058	C₂	semidirect
6. C₂₃×#2024	C₂	semidirect
7. C₂₃×#1919	C₂	semidirect
9. C₂₃×#1865	C₂	semidirect
56. C₂₃×C₂²⋊Q₈	Q₈	semidirect
152. #29159	C₂₃	direct (N × H)
209. #2099	C₂×C₂₃	semidirect
210. #2096	C₂×C₂₃	semidirect
211. #2058	C₂×C₂₃	semidirect
212. #2024	C₂×C₂₃	semidirect
213. #1919	C₂×C₂₃	semidirect
215. #1865	C₂×C₂₃	semidirect
297. C₂²⋊Q₈	Q₈×C₂₃	semidirect
332. C₂₃	#29159	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
C₄×C₂₃×C₂²⋊Q₈	2
C₂₃×#2099	2
C₂₃×#2096	2
C₂₃×#2058	2
C₂₃×#2024	2
C₂₃×#1919	2
C₂₃×#1892	2
C₂₃×#1865	2
C₂₃×#1807	2
C₂×C₂₃×#336	2
#29159	23

Subgroups — C23×#29159 #19176785

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29159 #19176785