Subgroups — C₂₃×#33606 #19181232

Order 5888.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3050	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2220	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2219	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2217	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2216	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#2139	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#135	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#525	—	2	1	no	—
16	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
17	C₂₃×#510	—	2	4	no	—
18	C₂×C₂₃×Q₃₂	C₂²	1	2	yes	complemented
19	C₂×C₂₃×SD₃₂	C₂²	1	2	yes	complemented
20	C₂×C₂₃×M₃₂	—	2	4	no	—
21	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
23	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
25	C₂₃×#467	—	2	1	no	—
26	C₂₃×#467	—	2	1	no	—
27	C₂₃×#467	C₂²	1	1	yes	characteristic
28	C₂₃×#467	C₂²	1	2	yes	—
29	C₂₃×#467	C₂²	1	1	yes	characteristic
30	C₂₃×#467	C₂²	1	2	yes	—
31	C₂₃×#466	C₂²	1	1	yes	characteristic
32	C₂₃×#466	C₂²	1	1	yes	characteristic
33	C₂₃×#465	—	2	1	no	—
34	C₂₃×#465	—	2	1	no	—
35	C₂₃×#427	C₂²	1	1	yes	characteristic
36	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
37	C₂₃×#416	—	2	1	no	—
38	C₂×C₂₃×#103	—	2	1	no	—
39	C₂₃×#367	C₂²	1	2	yes	complemented
40	C₂₃×#366	C₂²	1	2	yes	complemented
41	C₂₃×#358	C₂²	1	2	yes	complemented
42	C₂₃×#358	C₂²	1	2	yes	complemented
43	C₂₃×#358	C₂²	1	2	yes	complemented
44	C₂₃×#357	C₂²	1	2	yes	complemented
45	C₂×#41×C₂₃	—	2	1	no	—
46	C₂²×Q₈×C₂₃	—	2	1	no	—
47	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
48	C₂₃×#135	—	2	2	no	—
49	C₂₃×#135	—	2	2	no	—
50	C₂×Q₁₆×C₂₃	—	2	2	no	—
51	C₂×Q₁₆×C₂₃	—	2	1	no	—
52	C₂×Q₁₆×C₂₃	—	2	2	no	—
53	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
54	C₂×Q₁₆×C₂₃	C₂³	1	1	yes	characteristic
55	C₂×SD₁₆×C₂₃	—	2	1	no	—
56	C₂×D₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₂₃	—	2	2	no	—
58	C₂²×C₈×C₂₃	—	2	2	no	—
59	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂₃×#123	—	4	1	no	—
66	C₂₃×C₂²⋊Q₈	—	2	1	no	—
67	C₂₃×C₂²⋊Q₈	—	2	1	no	—
68	C₂₃×C₂²⋊Q₈	—	2	2	no	—
69	C₂₃×C₂²⋊Q₈	—	2	2	no	—
70	C₂₃×C₂²⋊Q₈	—	4	1	no	—
71	C₂₃×C₂²⋊Q₈	—	4	1	no	—
72	C₂₃×#121	—	2	1	no	—
73	C₂₃×#121	—	2	1	no	—
74	C₂₃×#120	—	4	1	no	—
75	C₄×D₄×C₂₃	—	4	1	no	—
76	C₄×D₄×C₂₃	—	4	1	no	—
77	C₂₃×#117	—	2	1	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
80	C₂×C₂²⋊C₄×C₂₃	—	4	1	no	—
81	C₂₃×Q₃₂	—	2	4	no	—
82	C₂₃×SD₃₂	—	2	4	no	—
83	C₂₃×M₃₂	—	2	2	no	—
84	C₂₃×M₃₂	—	2	4	no	—
85	C₂₃×M₃₂	D₄	1	4	yes	complemented
86	C₂×C₁₆×C₂₃	—	2	4	no	—
87	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
88	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
89	C₂₃×#107	—	2	1	no	—
90	C₂₃×#107	C₂³	1	2	yes	—
91	C₂₃×#107	C₂³	1	1	yes	characteristic
92	C₂₃×#107	C₂³	1	1	yes	characteristic
93	C₂₃×#106	—	2	1	no	—
94	C₂₃×#103	—	2	2	no	—
95	C₂₃×#103	—	2	2	no	—
96	C₂₃×#103	—	2	1	no	—
97	C₂₃×#103	—	2	1	no	—
98	C₂₃×#102	—	2	1	no	—
99	C₂₃×#102	—	2	1	no	—
100	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
101	#41×C₂₃	—	4	2	no	—
102	#41×C₂₃	—	4	2	no	—
103	C₂×Q₈×C₂₃	—	2	1	no	—
104	C₂×Q₈×C₂₃	—	2	2	no	—
105	C₂×Q₈×C₂₃	—	4	1	no	—
106	C₂×Q₈×C₂₃	—	4	2	no	—
107	C₂×Q₈×C₂₃	—	4	2	no	—
108	C₂×D₄×C₂₃	—	2	1	no	—
109	C₂×D₄×C₂₃	—	4	1	no	—
110	C₂×D₄×C₂₃	—	4	1	no	—
111	C₂×D₄×C₂₃	—	4	1	no	—
112	C₂²×C₄×C₂₃	—	2	2	no	—
113	C₂²×C₄×C₂₃	—	2	2	no	—
114	C₂²×C₄×C₂₃	—	4	1	no	—
115	C₂²×C₄×C₂₃	—	4	1	no	—
116	C₂²×C₄×C₂₃	—	4	1	no	—
117	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
118	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
119	C₂²×C₄×C₂₃	D₈	1	1	yes	characteristic
120	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
123	Q₁₆×C₂₃	—	2	2	no	—
124	Q₁₆×C₂₃	—	2	1	no	—
125	Q₁₆×C₂₃	—	2	2	no	—
126	Q₁₆×C₂₃	—	2	1	no	—
127	Q₁₆×C₂₃	—	4	2	no	—
128	SD₁₆×C₂₃	—	4	2	no	—
129	D₈×C₂₃	—	2	1	no	—
130	D₈×C₂₃	—	2	1	no	—
131	C₂×C₈×C₂₃	—	2	2	no	—
132	C₂×C₈×C₂₃	—	2	2	no	—
133	C₂×C₈×C₂₃	—	2	2	no	—
134	C₂×C₈×C₂₃	—	2	1	no	—
135	C₂×C₈×C₂₃	—	2	1	no	—
136	C₂×C₈×C₂₃	—	2	1	no	—
137	C₂×C₈×C₂₃	—	2	1	no	—
138	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₂₃	#41	1	2	yes	—
143	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
144	C₄⋊C₄×C₂₃	—	2	1	no	—
145	C₄⋊C₄×C₂₃	—	2	1	no	—
146	C₄⋊C₄×C₂₃	—	2	2	no	—
147	C₄⋊C₄×C₂₃	—	4	1	no	—
148	C₄⋊C₄×C₂₃	—	4	2	no	—
149	C₄⋊C₄×C₂₃	—	4	2	no	—
150	C₄⋊C₄×C₂₃	—	4	1	no	—
151	C₄⋊C₄×C₂₃	—	4	1	no	—
152	C₂²⋊C₄×C₂₃	—	4	2	no	—
153	C₂²⋊C₄×C₂₃	—	4	2	no	—
154	C₂²⋊C₄×C₂₃	—	4	1	no	—
155	C₂²⋊C₄×C₂₃	—	4	1	no	—
156	C₂²⋊C₄×C₂₃	—	4	1	no	—
157	C₂²⋊C₄×C₂₃	—	4	1	no	—
158	C₂²⋊C₄×C₂₃	—	4	1	no	—
159	C₄²×C₂₃	—	4	1	no	—
160	C₁₆×C₂₃	—	2	2	no	—
161	C₁₆×C₂₃	C₂×D₄	1	4	yes	—
162	#33606	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₂₃	—	2	2	no	—
164	C₂³×C₂₃	—	2	2	no	—
165	C₂³×C₂₃	—	4	1	no	—
166	C₂³×C₂₃	—	4	1	no	—
167	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
168	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
169	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
170	Q₈×C₂₃	—	4	1	no	—
171	Q₈×C₂₃	—	4	2	no	—
172	Q₈×C₂₃	—	4	2	no	—
173	Q₈×C₂₃	—	4	1	no	—
174	Q₈×C₂₃	—	8	2	no	—
175	D₄×C₂₃	—	4	1	no	—
176	D₄×C₂₃	—	4	1	no	—
177	D₄×C₂₃	—	8	2	no	—
178	D₄×C₂₃	—	8	2	no	—
179	D₄×C₂₃	—	8	2	no	—
180	C₂×C₄×C₂₃	—	2	2	no	—
181	C₂×C₄×C₂₃	—	2	1	no	—
182	C₂×C₄×C₂₃	—	2	1	no	—
183	C₂×C₄×C₂₃	—	2	1	no	—
184	C₂×C₄×C₂₃	—	2	1	no	—
185	C₂×C₄×C₂₃	—	2	2	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	4	1	no	—
188	C₂×C₄×C₂₃	—	4	1	no	—
189	C₂×C₄×C₂₃	—	4	2	no	—
190	C₂×C₄×C₂₃	—	4	1	no	—
191	C₂×C₄×C₂₃	—	4	2	no	—
192	C₂×C₄×C₂₃	—	8	2	no	—
193	C₂×C₄×C₂₃	—	8	2	no	—
194	C₂×C₄×C₂₃	—	8	2	no	—
195	C₂×C₄×C₂₃	#121	1	2	yes	—
196	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
197	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
198	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
199	C₂×C₄×C₂₃	#135	1	2	yes	—
200	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
201	C₈×C₂₃	—	2	1	no	—
202	C₈×C₂₃	—	2	1	no	—
203	C₈×C₂₃	#121	1	2	yes	—
204	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
205	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
206	C₂×#510	C₂×C₂₃	1	1	yes	characteristic, complemented
207	C₂²×M₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
208	#3050	C₂×C₂₃	1	1	yes	characteristic
209	C₂×#467	C₂×C₂₃	1	1	yes	characteristic, complemented
210	#2220	C₂×C₂₃	1	2	yes	complemented
211	#2219	C₂×C₂₃	1	2	yes	complemented
212	#2217	C₂×C₂₃	1	2	yes	complemented
213	#2216	C₂×C₂₃	1	2	yes	complemented
214	#2159	C₂×C₂₃	1	1	yes	characteristic, complemented
215	#2141	C₂×C₂₃	1	1	yes	characteristic, complemented
216	#2139	C₂×C₂₃	1	1	yes	characteristic, complemented
217	C₂²×C₂₃	—	2	1	no	—
218	C₂²×C₂₃	—	2	1	no	—
219	C₂²×C₂₃	—	2	1	no	—
220	C₂²×C₂₃	—	2	1	no	—
221	C₂²×C₂₃	—	2	1	no	—
222	C₂²×C₂₃	—	2	1	no	—
223	C₂²×C₂₃	—	2	1	no	—
224	C₂²×C₂₃	—	4	4	no	—
225	C₂²×C₂₃	—	8	1	no	—
226	C₂²×C₂₃	—	8	1	no	—
227	C₂²×C₂₃	—	8	1	no	—
228	C₂²×C₂₃	#466	1	2	yes	—
229	C₂²×C₂₃	#510	1	2	yes	complemented
230	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
231	C₄×C₂₃	—	2	1	no	—
232	C₄×C₂₃	—	2	1	no	—
233	C₄×C₂₃	—	8	1	no	—
234	C₄×C₂₃	—	8	2	no	—
235	C₄×C₂₃	—	8	1	no	—
236	C₄×C₂₃	—	8	1	no	—
237	C₄×C₂₃	—	8	2	no	—
238	C₄×C₂₃	#466	1	2	yes	—
239	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
240	C₄×C₂₃	C₂×#135	1	1	yes	characteristic
241	C₂×#135	C₂²×C₂₃	1	1	yes	characteristic
242	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
243	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
244	#525	—	2	1	no	—
245	C₂×C₂²⋊Q₈	—	2	1	no	—
246	#510	—	2	4	no	—
247	C₂×Q₃₂	C₂²×C₂₃	1	2	yes	complemented
248	C₂×SD₃₂	C₂²×C₂₃	1	2	yes	complemented
249	C₂×M₃₂	—	2	4	no	—
250	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
251	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
252	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
253	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
254	#467	—	2	1	no	—
255	#467	—	2	1	no	—
256	#467	C₂²×C₂₃	1	1	yes	characteristic
257	#467	C₂²×C₂₃	1	2	yes	—
258	#467	C₂²×C₂₃	1	1	yes	characteristic
259	#467	C₂²×C₂₃	1	2	yes	—
260	#466	C₂²×C₂₃	1	1	yes	characteristic
261	#466	C₂²×C₂₃	1	1	yes	characteristic
262	#465	—	2	1	no	—
263	#465	—	2	1	no	—
264	#427	C₂²×C₂₃	1	1	yes	characteristic
265	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
266	#416	—	2	1	no	—
267	C₂×#103	—	2	1	no	—
268	#367	C₂²×C₂₃	1	2	yes	complemented
269	#366	C₂²×C₂₃	1	2	yes	complemented
270	#358	C₂²×C₂₃	1	2	yes	complemented
271	#358	C₂²×C₂₃	1	2	yes	complemented
272	#358	C₂²×C₂₃	1	2	yes	complemented
273	#357	C₂²×C₂₃	1	2	yes	complemented
274	C₂×C₂₃	—	2	2	no	—
275	C₂×C₂₃	—	2	2	no	—
276	C₂×C₂₃	—	4	1	no	—
277	C₂×C₂₃	—	16	1	no	—
278	C₂×C₂₃	C₂×#466	1	1	yes	characteristic
279	C₂×C₂₃	C₂×#510	1	1	yes	characteristic
280	C₂×C₂₃	#3414	1	1	yes	characteristic
281	C₂×#41	—	2	1	no	—
282	C₂²×Q₈	—	2	1	no	—
283	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
284	#135	—	2	2	no	—
285	#135	—	2	2	no	—
286	C₂×Q₁₆	—	2	2	no	—
287	C₂×Q₁₆	—	2	1	no	—
288	C₂×Q₁₆	—	2	2	no	—
289	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
290	C₂×Q₁₆	C₂³×C₂₃	1	1	yes	characteristic
291	C₂×SD₁₆	—	2	1	no	—
292	C₂×D₈	C₂³×C₂₃	1	1	yes	characteristic
293	C₂²×C₈	—	2	2	no	—
294	C₂²×C₈	—	2	2	no	—
295	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
296	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
297	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
298	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
299	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
300	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
301	#123	—	4	1	no	—
302	C₂²⋊Q₈	—	2	2	no	—
303	C₂²⋊Q₈	—	2	2	no	—
304	C₂²⋊Q₈	—	2	1	no	—
305	C₂²⋊Q₈	—	2	1	no	—
306	C₂²⋊Q₈	—	4	1	no	—
307	C₂²⋊Q₈	—	4	1	no	—
308	#121	—	2	1	no	—
309	#121	—	2	1	no	—
310	#120	—	4	1	no	—
311	C₄×D₄	—	4	1	no	—
312	C₄×D₄	—	4	1	no	—
313	#117	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	4	1	no	—
316	C₂×C₂²⋊C₄	—	4	1	no	—
317	Q₃₂	—	2	4	no	—
318	SD₃₂	—	2	4	no	—
319	M₃₂	—	2	2	no	—
320	M₃₂	—	2	4	no	—
321	M₃₂	D₄×C₂₃	1	4	yes	complemented
322	C₂×C₁₆	—	2	4	no	—
323	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
324	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
325	#107	—	2	1	no	—
326	#107	C₂³×C₂₃	1	2	yes	—
327	#107	C₂³×C₂₃	1	1	yes	characteristic
328	#107	C₂³×C₂₃	1	1	yes	characteristic
329	#106	—	2	1	no	—
330	#103	—	2	2	no	—
331	#103	—	2	2	no	—
332	#103	—	2	1	no	—
333	#103	—	2	1	no	—
334	#102	—	2	1	no	—
335	#102	—	2	1	no	—
336	C₂₃	#33606	1	1	yes	characteristic, direct product factor, Sylow
337	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
338	#41	—	4	2	no	—
339	#41	—	4	2	no	—
340	C₂×Q₈	—	2	1	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	4	1	no	—
343	C₂×Q₈	—	4	2	no	—
344	C₂×Q₈	—	4	2	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	4	1	no	—
347	C₂×D₄	—	4	1	no	—
348	C₂×D₄	—	4	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	D₈×C₂₃	1	1	yes	characteristic
355	C₂²×C₄	D₈×C₂₃	1	1	yes	characteristic
356	C₂²×C₄	D₈×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	Q₁₆	—	2	2	no	—
361	Q₁₆	—	2	1	no	—
362	Q₁₆	—	2	2	no	—
363	Q₁₆	—	2	1	no	—
364	Q₁₆	—	4	2	no	—
365	SD₁₆	—	4	2	no	—
366	D₈	—	2	1	no	—
367	D₈	—	2	1	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
376	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
377	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
378	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
379	C₂×C₈	#41×C₂₃	1	2	yes	—
380	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	4	1	no	—
385	C₄⋊C₄	—	4	2	no	—
386	C₄⋊C₄	—	4	2	no	—
387	C₄⋊C₄	—	4	1	no	—
388	C₄⋊C₄	—	4	1	no	—
389	C₂²⋊C₄	—	4	2	no	—
390	C₂²⋊C₄	—	4	2	no	—
391	C₂²⋊C₄	—	4	1	no	—
392	C₂²⋊C₄	—	4	1	no	—
393	C₂²⋊C₄	—	4	1	no	—
394	C₂²⋊C₄	—	4	1	no	—
395	C₂²⋊C₄	—	4	1	no	—
396	C₄²	—	4	1	no	—
397	C₁₆	—	2	2	no	—
398	C₁₆	C₂×D₄×C₂₃	1	4	yes	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	4	1	no	—
402	C₂³	—	4	1	no	—
403	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
404	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
405	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
406	Q₈	—	4	2	no	—
407	Q₈	—	4	1	no	—
408	Q₈	—	4	1	no	—
409	Q₈	—	4	2	no	—
410	Q₈	—	8	2	no	—
411	D₄	—	4	1	no	—
412	D₄	—	4	1	no	—
413	D₄	—	8	2	no	—
414	D₄	—	8	2	no	—
415	D₄	—	8	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	8	2	no	—
429	C₂×C₄	—	8	2	no	—
430	C₂×C₄	—	8	2	no	—
431	C₂×C₄	C₂₃×#121	1	2	yes	—
432	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
433	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
434	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
435	C₂×C₄	C₂₃×#135	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
437	C₈	—	2	1	no	—
438	C₈	—	2	1	no	—
439	C₈	C₂₃×#121	1	2	yes	—
440	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
441	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
442	C₂²	—	2	1	no	—
443	C₂²	—	2	1	no	—
444	C₂²	—	2	1	no	—
445	C₂²	—	2	1	no	—
446	C₂²	—	2	1	no	—
447	C₂²	—	2	1	no	—
448	C₂²	—	2	1	no	—
449	C₂²	—	4	4	no	—
450	C₂²	—	8	1	no	—
451	C₂²	—	8	1	no	—
452	C₂²	—	8	1	no	—
453	C₂²	C₂₃×#466	1	2	yes	—
454	C₂²	C₂₃×#510	1	2	yes	complemented
455	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
456	C₄	—	2	1	no	—
457	C₄	—	2	1	no	—
458	C₄	—	8	1	no	—
459	C₄	—	8	2	no	—
460	C₄	—	8	1	no	—
461	C₄	—	8	2	no	—
462	C₄	—	8	1	no	—
463	C₄	C₂₃×#466	1	2	yes	—
464	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
465	C₄	C₂×C₂₃×#135	1	1	yes	characteristic
466	C₂	—	2	2	no	—
467	C₂	—	2	2	no	—
468	C₂	—	4	1	no	—
469	C₂	—	16	1	no	—
470	C₂	C₂×C₂₃×#466	1	1	yes	characteristic
471	C₂	C₂×C₂₃×#510	1	1	yes	characteristic
472	C₂	C₂₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂₃×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#510	C₂	semidirect
2. C₂²×C₂₃×M₃₂	C₂	semidirect
4. C₂×C₂₃×#467	C₂	semidirect
5. C₂₃×#2220	C₂	semidirect
6. C₂₃×#2219	C₂	semidirect
7. C₂₃×#2217	C₂	semidirect
8. C₂₃×#2216	C₂	semidirect
9. C₂₃×#2159	C₂	semidirect
10. C₂₃×#2141	C₂	semidirect
11. C₂₃×#2139	C₂	semidirect
18. C₂×C₂₃×Q₃₂	C₂²	semidirect
19. C₂×C₂₃×SD₃₂	C₂²	semidirect
39. C₂₃×#367	C₂²	semidirect
40. C₂₃×#366	C₂²	semidirect
41. C₂₃×#358	C₂²	semidirect
42. C₂₃×#358	C₂²	semidirect
43. C₂₃×#358	C₂²	semidirect
44. C₂₃×#357	C₂²	semidirect
85. C₂₃×M₃₂	D₄	semidirect
162. #33606	C₂₃	direct (N × H)
206. C₂×#510	C₂×C₂₃	semidirect
207. C₂²×M₃₂	C₂×C₂₃	semidirect
209. C₂×#467	C₂×C₂₃	semidirect
210. #2220	C₂×C₂₃	semidirect
211. #2219	C₂×C₂₃	semidirect
212. #2217	C₂×C₂₃	semidirect
213. #2216	C₂×C₂₃	semidirect
214. #2159	C₂×C₂₃	semidirect
215. #2141	C₂×C₂₃	semidirect
216. #2139	C₂×C₂₃	semidirect
229. C₂²×C₂₃	#510	semidirect
247. C₂×Q₃₂	C₂²×C₂₃	semidirect
248. C₂×SD₃₂	C₂²×C₂₃	semidirect
268. #367	C₂²×C₂₃	semidirect
269. #366	C₂²×C₂₃	semidirect
270. #358	C₂²×C₂₃	semidirect
271. #358	C₂²×C₂₃	semidirect
272. #358	C₂²×C₂₃	semidirect
273. #357	C₂²×C₂₃	semidirect
321. M₃₂	D₄×C₂₃	semidirect
336. C₂₃	#33606	direct (N × H)
454. C₂²	C₂₃×#510	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#510	2
C₂²×C₂₃×M₃₂	2
C₂₃×#3050	2
C₂×C₂₃×#467	2
C₂₃×#2220	2
C₂₃×#2219	2
C₂₃×#2217	2
C₂₃×#2216	2
C₂₃×#2159	2
C₂₃×#2141	2
C₂₃×#2139	2
#33606	23

Subgroups — C23×#33606 #19181232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33606 #19181232