Subgroups — C₂₃×#17882 #19165508

Order 5888.

660 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2989	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×#135	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1587	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1586	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1583	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1582	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1477	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1477	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1452	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#135	C₄	1	1	yes	characteristic
13	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
14	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#446	C₂²	1	2	yes	—
16	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
17	C₄×Q₁₆×C₂₃	C₄	1	1	yes	characteristic
18	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₄×SD₁₆×C₂₃	C₄	1	1	yes	characteristic
20	C₄×SD₁₆×C₂₃	C₄	1	1	yes	characteristic
21	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
22	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
23	C₄×D₈×C₂₃	C₄	1	1	yes	characteristic
24	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#436	C₂²	1	2	yes	—
26	C₂₃×#435	—	2	2	no	—
27	C₈×D₄×C₂₃	—	2	2	no	—
28	C₈×D₄×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#432	—	2	1	no	—
30	C₂₃×#431	—	2	1	no	—
31	C₂₃×#431	C₂²	1	1	yes	characteristic
32	C₂₃×#431	C₂²	1	1	yes	characteristic
33	C₂₃×#427	C₄	1	1	yes	characteristic
34	C₂₃×#423	—	2	1	no	—
35	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
36	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₂₃×#416	C₄	1	1	yes	characteristic
39	C₂₃×#416	C₄	1	1	yes	characteristic
40	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#334	C₂²	1	2	yes	—
42	C₂₃×#334	C₂²	1	2	yes	—
43	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
45	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
46	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
47	C₂₃×#322	C₂²	1	2	yes	—
48	C₂₃×#322	C₂²	1	2	yes	—
49	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₂₃×#135	—	2	4	no	—
52	C₂×Q₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂×D₈×C₂₃	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₂₃	—	2	2	no	—
57	C₂²×C₈×C₂₃	—	2	1	no	—
58	C₂²×C₈×C₂₃	—	2	1	no	—
59	C₂²×C₈×C₂₃	—	2	2	no	—
60	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
63	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
64	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₄×D₄×C₂₃	—	2	1	no	—
67	C₄×D₄×C₂₃	—	2	1	no	—
68	C₄×D₄×C₂₃	—	2	1	no	—
69	C₄×D₄×C₂₃	—	2	1	no	—
70	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
72	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂₃×#117	—	2	1	no	—
75	C₂₃×#117	—	2	1	no	—
76	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
77	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₂₃	—	2	1	no	—
79	C₂×C₄²×C₂₃	—	2	1	no	—
80	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
82	C₂₃×#107	C₂×C₄	1	1	yes	characteristic
83	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
84	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
85	C₂₃×C₄⋊C₈	—	2	2	no	—
86	C₂₃×C₄⋊C₈	—	2	2	no	—
87	C₂₃×C₄⋊C₈	—	2	1	no	—
88	C₂₃×C₄⋊C₈	—	2	1	no	—
89	C₂₃×C₄⋊C₈	—	2	1	no	—
90	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
91	C₂₃×C₄⋊C₈	D₄	1	1	yes	characteristic
92	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
93	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
94	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
95	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
96	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
97	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
98	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
99	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
100	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
101	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
102	C₂₃×C₂²⋊C₈	—	2	1	no	—
103	C₂₃×C₂²⋊C₈	—	2	2	no	—
104	C₂₃×C₂²⋊C₈	—	2	2	no	—
105	C₂₃×C₂²⋊C₈	D₄	1	2	yes	—
106	C₂₃×#97	—	2	2	no	—
107	C₄×C₈×C₂₃	—	2	2	no	—
108	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
109	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
110	C₄×C₈×C₂₃	C₂³	1	2	yes	—
111	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
112	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₂₃	C₂³	1	2	yes	—
114	#41×C₂₃	—	2	2	no	—
115	#41×C₂₃	—	2	2	no	—
116	#41×C₂₃	—	2	2	no	—
117	#41×C₂₃	M₁₆	1	4	yes	complemented
118	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×D₄×C₂₃	—	2	1	no	—
121	C₂×D₄×C₂₃	—	2	1	no	—
122	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
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	1	no	—
128	C₂²×C₄×C₂₃	—	2	1	no	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
133	Q₁₆×C₂₃	—	2	2	no	—
134	SD₁₆×C₂₃	—	2	2	no	—
135	SD₁₆×C₂₃	—	2	2	no	—
136	D₈×C₂₃	—	2	2	no	—
137	M₁₆×C₂₃	—	8	2	no	—
138	C₂×C₈×C₂₃	—	2	2	no	—
139	C₂×C₈×C₂₃	—	2	2	no	—
140	C₂×C₈×C₂₃	—	2	2	no	—
141	C₂×C₈×C₂₃	—	2	2	no	—
142	C₂×C₈×C₂₃	—	2	2	no	—
143	C₂×C₈×C₂₃	—	8	2	no	—
144	C₂×C₈×C₂₃	—	8	1	no	—
145	C₂×C₈×C₂₃	—	8	1	no	—
146	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
148	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
151	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
152	C₄⋊C₄×C₂₃	—	2	1	no	—
153	C₄⋊C₄×C₂₃	—	2	1	no	—
154	C₄⋊C₄×C₂₃	—	2	1	no	—
155	C₄⋊C₄×C₂₃	—	2	1	no	—
156	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
157	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
158	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
159	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
160	C₂²⋊C₄×C₂₃	—	2	1	no	—
161	C₂²⋊C₄×C₂₃	—	2	1	no	—
162	C₂²⋊C₄×C₂₃	—	2	1	no	—
163	C₂²⋊C₄×C₂₃	—	2	1	no	—
164	C₂²⋊C₄×C₂₃	—	2	1	no	—
165	C₂²⋊C₄×C₂₃	—	2	1	no	—
166	C₄²×C₂₃	—	2	1	no	—
167	C₄²×C₂₃	—	2	1	no	—
168	C₄²×C₂₃	—	2	1	no	—
169	C₄²×C₂₃	—	2	1	no	—
170	C₄²×C₂₃	—	2	1	no	—
171	C₄²×C₂₃	—	2	1	no	—
172	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₂₃	#41	1	1	yes	characteristic
175	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
176	#17882	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₂₃	—	2	1	no	—
178	C₂³×C₂₃	—	2	1	no	—
179	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
180	Q₈×C₂₃	—	2	1	no	—
181	Q₈×C₂₃	—	2	1	no	—
182	Q₈×C₂₃	—	2	1	no	—
183	Q₈×C₂₃	C₂×M₁₆	1	2	yes	—
184	D₄×C₂₃	—	2	4	no	—
185	D₄×C₂₃	—	2	1	no	—
186	D₄×C₂₃	—	2	1	no	—
187	D₄×C₂₃	—	2	1	no	—
188	D₄×C₂₃	—	4	2	no	—
189	D₄×C₂₃	C₂×M₁₆	1	2	yes	—
190	C₂×C₄×C₂₃	—	2	1	no	—
191	C₂×C₄×C₂₃	—	2	1	no	—
192	C₂×C₄×C₂₃	—	2	1	no	—
193	C₂×C₄×C₂₃	—	2	4	no	—
194	C₂×C₄×C₂₃	—	2	1	no	—
195	C₂×C₄×C₂₃	—	2	1	no	—
196	C₂×C₄×C₂₃	—	2	1	no	—
197	C₂×C₄×C₂₃	—	2	1	no	—
198	C₂×C₄×C₂₃	—	2	1	no	—
199	C₂×C₄×C₂₃	—	4	1	no	—
200	C₂×C₄×C₂₃	—	4	1	no	—
201	C₂×C₄×C₂₃	—	4	1	no	—
202	C₂×C₄×C₂₃	—	4	1	no	—
203	C₂×C₄×C₂₃	—	4	1	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	8	1	no	—
206	C₂×C₄×C₂₃	—	8	1	no	—
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	C₂×M₁₆	1	2	yes	—
211	C₂×C₄×C₂₃	#131	1	2	yes	—
212	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
215	C₈×C₂₃	—	2	1	no	—
216	C₈×C₂₃	—	2	1	no	—
217	C₈×C₂₃	—	2	1	no	—
218	C₈×C₂₃	—	2	1	no	—
219	C₈×C₂₃	—	2	2	no	—
220	C₈×C₂₃	—	2	2	no	—
221	C₈×C₂₃	—	4	2	no	—
222	C₈×C₂₃	—	4	2	no	—
223	C₈×C₂₃	—	4	2	no	—
224	#2989	C₂×C₂₃	1	1	yes	characteristic, complemented
225	#2988	C₂×C₂₃	1	1	yes	characteristic, complemented
226	C₄×#135	C₂×C₂₃	1	1	yes	characteristic
227	#1587	C₂×C₂₃	1	2	yes	complemented
228	#1586	C₂×C₂₃	1	2	yes	complemented
229	#1583	C₂×C₂₃	1	2	yes	complemented
230	#1582	C₂×C₂₃	1	2	yes	complemented
231	C₂×#334	C₂×C₂₃	1	1	yes	characteristic, complemented
232	#1477	C₂×C₂₃	1	1	yes	characteristic, complemented
233	#1477	C₂×C₂₃	1	1	yes	characteristic, complemented
234	#1452	C₂×C₂₃	1	1	yes	characteristic, complemented
235	C₂²×C₂₃	—	2	1	no	—
236	C₂²×C₂₃	—	2	1	no	—
237	C₂²×C₂₃	—	2	2	no	—
238	C₂²×C₂₃	—	4	1	no	—
239	C₂²×C₂₃	—	4	1	no	—
240	C₂²×C₂₃	—	4	1	no	—
241	C₂²×C₂₃	—	4	1	no	—
242	C₂²×C₂₃	—	4	1	no	—
243	C₂²×C₂₃	#435	1	2	yes	—
244	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
245	C₄×C₂₃	—	2	2	no	—
246	C₄×C₂₃	—	2	1	no	—
247	C₄×C₂₃	—	2	1	no	—
248	C₄×C₂₃	—	2	1	no	—
249	C₄×C₂₃	—	4	1	no	—
250	C₄×C₂₃	—	4	1	no	—
251	C₄×C₂₃	—	4	1	no	—
252	C₄×C₂₃	—	4	1	no	—
253	C₄×C₂₃	—	4	1	no	—
254	C₄×C₂₃	#435	1	2	yes	—
255	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
256	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
257	C₄×C₂₃	#577	1	1	yes	characteristic
258	C₄×C₂₃	#577	1	1	yes	characteristic
259	C₂×#135	C₄×C₂₃	1	1	yes	characteristic
260	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
261	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
262	#446	C₂²×C₂₃	1	2	yes	—
263	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
264	C₄×Q₁₆	C₄×C₂₃	1	1	yes	characteristic
265	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
266	C₄×SD₁₆	C₄×C₂₃	1	1	yes	characteristic
267	C₄×SD₁₆	C₄×C₂₃	1	1	yes	characteristic
268	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
269	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
270	C₄×D₈	C₄×C₂₃	1	1	yes	characteristic
271	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
272	#436	C₂²×C₂₃	1	2	yes	—
273	#435	—	2	2	no	—
274	C₈×D₄	—	2	2	no	—
275	C₈×D₄	C₂²×C₂₃	1	2	yes	—
276	#432	—	2	1	no	—
277	#431	—	2	1	no	—
278	#431	C₂²×C₂₃	1	1	yes	characteristic
279	#431	C₂²×C₂₃	1	1	yes	characteristic
280	#427	C₄×C₂₃	1	1	yes	characteristic
281	#423	—	2	1	no	—
282	C₂×C₄⋊C₈	—	2	1	no	—
283	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
284	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
285	#416	C₄×C₂₃	1	1	yes	characteristic
286	#416	C₄×C₂₃	1	1	yes	characteristic
287	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
288	#334	C₂²×C₂₃	1	2	yes	—
289	#334	C₂²×C₂₃	1	2	yes	—
290	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
291	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
292	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
293	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
294	#322	C₂²×C₂₃	1	2	yes	—
295	#322	C₂²×C₂₃	1	2	yes	—
296	C₂×C₂₃	—	2	2	no	—
297	C₂×C₂₃	—	4	2	no	—
298	C₂×C₂₃	—	8	1	no	—
299	C₂×C₂₃	C₂×#435	1	1	yes	characteristic
300	C₂×C₂₃	#2952	1	1	yes	characteristic
301	C₂×C₂₃	#2957	1	1	yes	characteristic
302	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
303	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
304	#135	—	2	4	no	—
305	C₂×Q₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
306	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
307	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
308	C₂×D₈	C₂×C₄×C₂₃	1	1	yes	characteristic
309	C₂×M₁₆	—	2	2	no	—
310	C₂²×C₈	—	2	1	no	—
311	C₂²×C₈	—	2	1	no	—
312	C₂²×C₈	—	2	2	no	—
313	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
314	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
315	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
316	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
317	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
318	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
319	C₄×D₄	—	2	1	no	—
320	C₄×D₄	—	2	1	no	—
321	C₄×D₄	—	2	1	no	—
322	C₄×D₄	—	2	1	no	—
323	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
324	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
325	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
326	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
327	#117	—	2	1	no	—
328	#117	—	2	1	no	—
329	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
330	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
331	C₂×C₄²	—	2	1	no	—
332	C₂×C₄²	—	2	1	no	—
333	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
334	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
335	#107	C₂×C₄×C₂₃	1	1	yes	characteristic
336	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
337	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
338	C₄⋊C₈	—	2	2	no	—
339	C₄⋊C₈	—	2	2	no	—
340	C₄⋊C₈	—	2	1	no	—
341	C₄⋊C₈	—	2	1	no	—
342	C₄⋊C₈	—	2	1	no	—
343	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
344	C₄⋊C₈	D₄×C₂₃	1	1	yes	characteristic
345	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
346	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
347	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
348	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
349	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
350	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
351	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
352	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
353	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
354	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
355	C₂²⋊C₈	—	2	1	no	—
356	C₂²⋊C₈	—	2	2	no	—
357	C₂²⋊C₈	—	2	2	no	—
358	C₂²⋊C₈	D₄×C₂₃	1	2	yes	—
359	#97	—	2	2	no	—
360	C₄×C₈	—	2	2	no	—
361	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
362	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
363	C₄×C₈	C₂³×C₂₃	1	2	yes	—
364	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
365	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
366	C₄×C₈	C₂³×C₂₃	1	2	yes	—
367	C₂₃	#17882	1	1	yes	characteristic, direct product factor, Sylow
368	#41	—	2	2	no	—
369	#41	—	2	2	no	—
370	#41	—	2	2	no	—
371	#41	M₁₆×C₂₃	1	4	yes	complemented
372	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
373	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
374	C₂×D₄	—	2	1	no	—
375	C₂×D₄	—	2	1	no	—
376	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
377	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
378	C₂²×C₄	—	2	1	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
385	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
386	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
387	Q₁₆	—	2	2	no	—
388	SD₁₆	—	2	2	no	—
389	SD₁₆	—	2	2	no	—
390	D₈	—	2	2	no	—
391	M₁₆	—	8	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	2	2	no	—
397	C₂×C₈	—	8	2	no	—
398	C₂×C₈	—	8	1	no	—
399	C₂×C₈	—	8	1	no	—
400	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
401	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
402	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
403	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
404	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
405	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	1	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
411	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
412	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
413	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
414	C₂²⋊C₄	—	2	1	no	—
415	C₂²⋊C₄	—	2	1	no	—
416	C₂²⋊C₄	—	2	1	no	—
417	C₂²⋊C₄	—	2	1	no	—
418	C₂²⋊C₄	—	2	1	no	—
419	C₂²⋊C₄	—	2	1	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	2	1	no	—
422	C₄²	—	2	1	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
427	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
428	C₄²	#41×C₂₃	1	1	yes	characteristic
429	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
430	C₂³	—	2	1	no	—
431	C₂³	—	2	1	no	—
432	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
433	Q₈	—	2	1	no	—
434	Q₈	—	2	1	no	—
435	Q₈	—	2	1	no	—
436	Q₈	C₂×M₁₆×C₂₃	1	2	yes	—
437	D₄	—	2	4	no	—
438	D₄	—	2	1	no	—
439	D₄	—	2	1	no	—
440	D₄	—	2	1	no	—
441	D₄	—	4	2	no	—
442	D₄	C₂×M₁₆×C₂₃	1	2	yes	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	8	1	no	—
459	C₂×C₄	—	8	1	no	—
460	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
461	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
462	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
463	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
464	C₂×C₄	C₂₃×#131	1	2	yes	—
465	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
466	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
467	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
468	C₈	—	2	1	no	—
469	C₈	—	2	1	no	—
470	C₈	—	2	1	no	—
471	C₈	—	2	1	no	—
472	C₈	—	2	2	no	—
473	C₈	—	2	2	no	—
474	C₈	—	4	2	no	—
475	C₈	—	4	2	no	—
476	C₈	—	4	2	no	—
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	—	4	1	no	—
483	C₂²	—	4	1	no	—
484	C₂²	—	4	1	no	—
485	C₂²	C₂₃×#435	1	2	yes	—
486	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
487	C₄	—	2	2	no	—
488	C₄	—	2	1	no	—
489	C₄	—	2	1	no	—
490	C₄	—	2	1	no	—
491	C₄	—	4	1	no	—
492	C₄	—	4	1	no	—
493	C₄	—	4	1	no	—
494	C₄	—	4	1	no	—
495	C₄	—	4	1	no	—
496	C₄	C₂₃×#435	1	2	yes	—
497	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
498	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
499	C₄	C₂₃×#577	1	1	yes	characteristic
500	C₄	C₂₃×#577	1	1	yes	characteristic
501	C₂	—	2	2	no	—
502	C₂	—	4	2	no	—
503	C₂	—	8	1	no	—
504	C₂	C₂×C₂₃×#435	1	1	yes	characteristic
505	C₂	C₂₃×#2952	1	1	yes	characteristic
506	C₂	C₂₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂×C₂₃×#135	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2989	C₂	semidirect
2. C₂₃×#2988	C₂	semidirect
4. C₂₃×#1587	C₂	semidirect
5. C₂₃×#1586	C₂	semidirect
6. C₂₃×#1583	C₂	semidirect
7. C₂₃×#1582	C₂	semidirect
8. C₂×C₂₃×#334	C₂	semidirect
9. C₂₃×#1477	C₂	semidirect
10. C₂₃×#1477	C₂	semidirect
11. C₂₃×#1452	C₂	semidirect
117. #41×C₂₃	M₁₆	semidirect
176. #17882	C₂₃	direct (N × H)
224. #2989	C₂×C₂₃	semidirect
225. #2988	C₂×C₂₃	semidirect
227. #1587	C₂×C₂₃	semidirect
228. #1586	C₂×C₂₃	semidirect
229. #1583	C₂×C₂₃	semidirect
230. #1582	C₂×C₂₃	semidirect
231. C₂×#334	C₂×C₂₃	semidirect
232. #1477	C₂×C₂₃	semidirect
233. #1477	C₂×C₂₃	semidirect
234. #1452	C₂×C₂₃	semidirect
367. C₂₃	#17882	direct (N × H)
371. #41	M₁₆×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#2989	2
C₂₃×#2988	2
C₄×C₂₃×#135	2
C₂₃×#1587	2
C₂₃×#1586	2
C₂₃×#1583	2
C₂₃×#1582	2
C₂×C₂₃×#334	2
C₂₃×#1477	2
C₂₃×#1452	2
#17882	23

Subgroups — C23×#17882 #19165508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17882 #19165508