Subgroups — C₂₃×#24697 #19172323

Order 5888.

780 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#476	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#440	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2568	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2357	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2073	C₂	1	2	yes	maximal
6	C₂₃×#2055	C₂	1	2	yes	maximal
7	C₂₃×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1949	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1940	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1927	C₂	1	2	yes	maximal, complemented
11	C₂₃×#1805	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#476	—	2	4	no	—
17	C₂₃×#440	—	2	4	no	—
18	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#106	C₂²	1	2	yes	—
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#103	C₂²	1	2	yes	—
24	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#102	C₂²	1	2	yes	—
26	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
27	C₂₃×#400	—	2	1	no	—
28	C₂₃×#400	—	2	1	no	—
29	C₂₃×#391	—	2	2	no	—
30	C₂₃×#391	C₂²	1	2	yes	—
31	C₂₃×#390	C₂²	1	2	yes	—
32	C₂₃×#388	—	2	2	no	—
33	C₂₃×#382	C₂²	1	2	yes	—
34	C₂₃×#382	C₂²	1	1	yes	characteristic
35	C₂₃×#381	—	2	1	no	—
36	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
39	C₂₃×#340	C₂²	1	2	yes	—
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂₃×#340	C₂²	1	2	yes	—
42	C₂₃×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₂₃	—	2	1	no	—
44	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₂₃	—	2	1	no	—
47	C₂×SD₁₆×C₂₃	—	2	2	no	—
48	C₂×SD₁₆×C₂₃	—	2	2	no	—
49	C₂×SD₁₆×C₂₃	—	2	2	no	—
50	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂₃×C₄⋊Q₈	—	2	2	no	—
54	C₂₃×C₄⋊Q₈	—	2	2	no	—
55	C₄×Q₈×C₂₃	—	2	2	no	—
56	C₄×Q₈×C₂₃	—	2	2	no	—
57	C₄×D₄×C₂₃	—	2	2	no	—
58	C₄×D₄×C₂₃	D₄	1	4	yes	—
59	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
62	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
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×C₂²⋊C₄×C₂₃	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
69	C₂×C₄²×C₂₃	—	2	1	no	—
70	C₂×C₄²×C₂₃	—	2	2	no	—
71	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
72	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
73	C₂₃×#107	—	2	2	no	—
74	C₂₃×#106	—	2	4	no	—
75	C₂₃×C₄⋊C₈	—	2	4	no	—
76	C₂₃×C₄⋊C₈	—	2	2	no	—
77	C₂₃×#103	—	2	4	no	—
78	C₂₃×#103	—	2	2	no	—
79	C₂₃×#102	—	2	2	no	—
80	C₂₃×#102	—	2	4	no	—
81	C₂₃×#97	—	2	2	no	—
82	C₂₃×#95	—	2	1	no	—
83	C₂₃×#95	—	2	2	no	—
84	C₂₃×#95	—	2	1	no	—
85	C₂₃×#95	—	2	2	no	—
86	C₂₃×#95	—	2	2	no	—
87	C₂₃×#95	—	2	2	no	—
88	C₂₃×#95	—	2	2	no	—
89	C₂₃×#95	D₄	1	1	yes	characteristic
90	C₂₃×#95	D₄	1	1	yes	characteristic
91	C₂⁴×C₂₃	—	2	1	no	—
92	C₂×Q₈×C₂₃	—	2	1	no	—
93	C₂×Q₈×C₂₃	—	2	1	no	—
94	C₂×Q₈×C₂₃	—	2	1	no	—
95	C₂×Q₈×C₂₃	—	2	1	no	—
96	C₂×Q₈×C₂₃	—	2	1	no	—
97	C₂×Q₈×C₂₃	—	4	4	no	—
98	C₂×Q₈×C₂₃	—	4	2	no	—
99	C₂×Q₈×C₂₃	#41	1	2	yes	—
100	C₂×D₄×C₂₃	—	2	1	no	—
101	C₂×D₄×C₂₃	—	2	1	no	—
102	C₂×D₄×C₂₃	—	2	1	no	—
103	C₂×D₄×C₂₃	—	2	1	no	—
104	C₂×D₄×C₂₃	—	2	1	no	—
105	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₂₃	—	2	1	no	—
107	C₂²×C₄×C₂₃	—	2	1	no	—
108	C₂²×C₄×C₂₃	—	2	4	no	—
109	C₂²×C₄×C₂₃	—	2	2	no	—
110	C₂²×C₄×C₂₃	—	2	1	no	—
111	C₂²×C₄×C₂₃	—	2	2	no	—
112	C₂²×C₄×C₂₃	—	8	2	no	—
113	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
114	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
118	SD₁₆×C₂₃	—	2	8	no	—
119	C₂×C₈×C₂₃	—	2	1	no	—
120	C₂×C₈×C₂₃	—	2	1	no	—
121	C₂×C₈×C₂₃	—	2	1	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	1	no	—
126	C₂×C₈×C₂₃	—	2	1	no	—
127	C₂×C₈×C₂₃	#41	1	2	yes	—
128	C₄⋊C₄×C₂₃	—	2	1	no	—
129	C₄⋊C₄×C₂₃	—	2	2	no	—
130	C₄⋊C₄×C₂₃	—	2	1	no	—
131	C₄⋊C₄×C₂₃	—	2	2	no	—
132	C₄⋊C₄×C₂₃	—	2	1	no	—
133	C₄⋊C₄×C₂₃	—	2	2	no	—
134	C₄⋊C₄×C₂₃	—	2	2	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	C₂×D₄	1	2	yes	—
139	C₂²⋊C₄×C₂₃	—	2	4	no	—
140	C₂²⋊C₄×C₂₃	—	4	4	no	—
141	C₂²⋊C₄×C₂₃	—	8	2	no	—
142	C₄²×C₂₃	—	2	2	no	—
143	C₄²×C₂₃	—	2	2	no	—
144	C₄²×C₂₃	—	4	2	no	—
145	C₄²×C₂₃	—	8	2	no	—
146	C₄²×C₂₃	C₂×D₄	1	2	yes	—
147	C₄²×C₂₃	#41	1	2	yes	—
148	#24697	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₂₃	—	2	2	no	—
150	C₂³×C₂₃	—	4	1	no	—
151	C₂³×C₂₃	—	4	1	no	—
152	C₂³×C₂₃	—	4	2	no	—
153	C₂³×C₂₃	—	4	2	no	—
154	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
155	Q₈×C₂₃	—	2	2	no	—
156	Q₈×C₂₃	—	2	2	no	—
157	Q₈×C₂₃	—	4	1	no	—
158	Q₈×C₂₃	—	4	1	no	—
159	Q₈×C₂₃	—	8	4	no	—
160	D₄×C₂₃	—	2	2	no	—
161	D₄×C₂₃	—	2	2	no	—
162	D₄×C₂₃	—	4	1	no	—
163	D₄×C₂₃	—	4	1	no	—
164	C₂×C₄×C₂₃	—	2	2	no	—
165	C₂×C₄×C₂₃	—	2	2	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	2	2	no	—
168	C₂×C₄×C₂₃	—	2	2	no	—
169	C₂×C₄×C₂₃	—	2	2	no	—
170	C₂×C₄×C₂₃	—	2	2	no	—
171	C₂×C₄×C₂₃	—	2	2	no	—
172	C₂×C₄×C₂₃	—	2	2	no	—
173	C₂×C₄×C₂₃	—	2	2	no	—
174	C₂×C₄×C₂₃	—	4	1	no	—
175	C₂×C₄×C₂₃	—	4	1	no	—
176	C₂×C₄×C₂₃	—	4	2	no	—
177	C₂×C₄×C₂₃	—	4	1	no	—
178	C₂×C₄×C₂₃	—	4	2	no	—
179	C₂×C₄×C₂₃	—	4	2	no	—
180	C₂×C₄×C₂₃	—	4	2	no	—
181	C₂×C₄×C₂₃	—	4	4	no	—
182	C₂×C₄×C₂₃	—	4	2	no	—
183	C₂×C₄×C₂₃	—	4	2	no	—
184	C₂×C₄×C₂₃	—	8	4	no	—
185	C₂×C₄×C₂₃	#121	1	2	yes	—
186	C₂×C₄×C₂₃	#121	1	2	yes	—
187	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
190	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
192	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
193	C₈×C₂₃	—	2	2	no	—
194	C₈×C₂₃	—	4	2	no	—
195	C₈×C₂₃	—	4	1	no	—
196	C₂×#476	C₂×C₂₃	1	1	yes	characteristic
197	C₂×#440	C₂×C₂₃	1	1	yes	characteristic
198	#2568	C₂×C₂₃	1	1	yes	characteristic, complemented
199	#2357	C₂×C₂₃	1	1	yes	characteristic
200	#2073	C₂×C₂₃	1	2	yes	—
201	#2055	C₂×C₂₃	1	2	yes	—
202	#1993	C₂×C₂₃	1	1	yes	characteristic, complemented
203	#1949	C₂×C₂₃	1	2	yes	complemented
204	#1940	C₂×C₂₃	1	1	yes	characteristic, complemented
205	#1927	C₂×C₂₃	1	2	yes	complemented
206	#1805	C₂×C₂₃	1	1	yes	characteristic, complemented
207	C₂²×C₂₃	—	4	2	no	—
208	C₂²×C₂₃	—	4	2	no	—
209	C₂²×C₂₃	—	4	2	no	—
210	C₂²×C₂₃	—	8	2	no	—
211	C₂²×C₂₃	—	8	2	no	—
212	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
213	C₂²×C₂₃	#547	1	1	yes	characteristic
214	C₂²×C₂₃	#548	1	1	yes	characteristic
215	C₂²×C₂₃	#575	1	2	yes	—
216	C₂²×C₂₃	#577	1	2	yes	—
217	C₄×C₂₃	—	2	2	no	—
218	C₄×C₂₃	—	2	2	no	—
219	C₄×C₂₃	—	4	2	no	—
220	C₄×C₂₃	—	4	2	no	—
221	C₄×C₂₃	—	4	2	no	—
222	C₄×C₂₃	—	4	2	no	—
223	C₄×C₂₃	—	8	2	no	—
224	C₄×C₂₃	—	8	2	no	—
225	C₄×C₂₃	—	8	1	no	—
226	C₄×C₂₃	#525	1	1	yes	characteristic
227	C₄×C₂₃	#536	1	1	yes	characteristic
228	C₄×C₂₃	#552	1	1	yes	characteristic
229	C₄×C₂₃	#556	1	1	yes	characteristic
230	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
231	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
232	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
233	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
234	#476	—	2	4	no	—
235	#440	—	2	4	no	—
236	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
237	C₂×#106	C₂²×C₂₃	1	2	yes	—
238	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
239	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
240	C₂×#103	C₂²×C₂₃	1	2	yes	—
241	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
242	C₂×#102	C₂²×C₂₃	1	2	yes	—
243	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
244	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
245	#400	—	2	1	no	—
246	#400	—	2	1	no	—
247	#391	—	2	2	no	—
248	#391	C₂²×C₂₃	1	2	yes	—
249	#390	C₂²×C₂₃	1	2	yes	—
250	#388	—	2	2	no	—
251	#382	C₂²×C₂₃	1	2	yes	—
252	#382	C₂²×C₂₃	1	1	yes	characteristic
253	#381	—	2	1	no	—
254	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
255	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
256	C₄×C₂²⋊C₄	—	2	1	no	—
257	#340	C₂²×C₂₃	1	2	yes	—
258	#340	C₂²×C₂₃	1	2	yes	—
259	#340	C₂²×C₂₃	1	2	yes	—
260	#336	C₂²×C₂₃	1	2	yes	—
261	C₂×C₂₃	—	8	2	no	—
262	C₂×C₂₃	#2415	1	1	yes	characteristic
263	C₂×C₂₃	#3040	1	1	yes	characteristic
264	C₂×C₂₃	#3042	1	1	yes	characteristic
265	C₂×C₂₃	#3350	1	2	yes	—
266	C₂×C₂₃	#3371	1	2	yes	—
267	C₂²×Q₈	—	2	1	no	—
268	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
269	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
270	C₂³×C₄	—	2	1	no	—
271	C₂×SD₁₆	—	2	2	no	—
272	C₂×SD₁₆	—	2	2	no	—
273	C₂×SD₁₆	—	2	2	no	—
274	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
276	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
277	C₄⋊Q₈	—	2	2	no	—
278	C₄⋊Q₈	—	2	2	no	—
279	C₄×Q₈	—	2	2	no	—
280	C₄×Q₈	—	2	2	no	—
281	C₄×D₄	—	2	2	no	—
282	C₄×D₄	D₄×C₂₃	1	4	yes	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	1	no	—
285	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
287	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
288	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
290	C₂×C₂²⋊C₄	—	2	1	no	—
291	C₂×C₂²⋊C₄	—	2	2	no	—
292	C₂×C₂²⋊C₄	—	2	2	no	—
293	C₂×C₄²	—	2	1	no	—
294	C₂×C₄²	—	2	2	no	—
295	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
296	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
297	#107	—	2	2	no	—
298	#106	—	2	4	no	—
299	C₄⋊C₈	—	2	4	no	—
300	C₄⋊C₈	—	2	2	no	—
301	#103	—	2	4	no	—
302	#103	—	2	2	no	—
303	#102	—	2	2	no	—
304	#102	—	2	4	no	—
305	#97	—	2	2	no	—
306	#95	—	2	1	no	—
307	#95	—	2	2	no	—
308	#95	—	2	1	no	—
309	#95	—	2	2	no	—
310	#95	—	2	2	no	—
311	#95	—	2	2	no	—
312	#95	—	2	2	no	—
313	#95	D₄×C₂₃	1	1	yes	characteristic
314	#95	D₄×C₂₃	1	1	yes	characteristic
315	C₂₃	#24697	1	1	yes	characteristic, direct product factor, Sylow
316	C₂⁴	—	2	1	no	—
317	C₂×Q₈	—	2	1	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	1	no	—
321	C₂×Q₈	—	2	1	no	—
322	C₂×Q₈	—	4	4	no	—
323	C₂×Q₈	—	4	2	no	—
324	C₂×Q₈	#41×C₂₃	1	2	yes	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	—	2	1	no	—
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	8	2	no	—
338	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
339	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
340	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
341	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
342	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
343	SD₁₆	—	2	8	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	#41×C₂₃	1	2	yes	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	4	2	no	—
361	C₄⋊C₄	—	4	2	no	—
362	C₄⋊C₄	—	4	2	no	—
363	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
364	C₂²⋊C₄	—	2	4	no	—
365	C₂²⋊C₄	—	4	4	no	—
366	C₂²⋊C₄	—	8	2	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	2	2	no	—
369	C₄²	—	4	2	no	—
370	C₄²	—	8	2	no	—
371	C₄²	C₂×D₄×C₂₃	1	2	yes	—
372	C₄²	#41×C₂₃	1	2	yes	—
373	C₂³	—	2	2	no	—
374	C₂³	—	4	1	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	4	2	no	—
378	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
379	Q₈	—	2	2	no	—
380	Q₈	—	2	2	no	—
381	Q₈	—	4	1	no	—
382	Q₈	—	4	1	no	—
383	Q₈	—	8	4	no	—
384	D₄	—	2	2	no	—
385	D₄	—	2	2	no	—
386	D₄	—	4	1	no	—
387	D₄	—	4	1	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	2	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	4	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	8	4	no	—
409	C₂×C₄	C₂₃×#121	1	2	yes	—
410	C₂×C₄	C₂₃×#121	1	2	yes	—
411	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
412	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
413	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
415	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
416	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
417	C₈	—	2	2	no	—
418	C₈	—	4	2	no	—
419	C₈	—	4	1	no	—
420	C₂²	—	4	2	no	—
421	C₂²	—	4	2	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	8	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
426	C₂²	C₂₃×#547	1	1	yes	characteristic
427	C₂²	C₂₃×#548	1	1	yes	characteristic
428	C₂²	C₂₃×#575	1	2	yes	—
429	C₂²	C₂₃×#577	1	2	yes	—
430	C₄	—	2	2	no	—
431	C₄	—	2	2	no	—
432	C₄	—	4	2	no	—
433	C₄	—	4	2	no	—
434	C₄	—	4	2	no	—
435	C₄	—	4	2	no	—
436	C₄	—	8	2	no	—
437	C₄	—	8	1	no	—
438	C₄	—	8	2	no	—
439	C₄	C₂₃×#525	1	1	yes	characteristic
440	C₄	C₂₃×#536	1	1	yes	characteristic
441	C₄	C₂₃×#552	1	1	yes	characteristic
442	C₄	C₂₃×#556	1	1	yes	characteristic
443	C₂	—	8	2	no	—
444	C₂	C₂₃×#2415	1	1	yes	characteristic
445	C₂	C₂₃×#3040	1	1	yes	characteristic
446	C₂	C₂₃×#3042	1	1	yes	characteristic
447	C₂	C₂₃×#3350	1	2	yes	—
448	C₂	C₂₃×#3371	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2568	C₂	semidirect
7. C₂₃×#1993	C₂	semidirect
8. C₂₃×#1949	C₂	semidirect
9. C₂₃×#1940	C₂	semidirect
10. C₂₃×#1927	C₂	semidirect
11. C₂₃×#1805	C₂	semidirect
148. #24697	C₂₃	direct (N × H)
198. #2568	C₂×C₂₃	semidirect
202. #1993	C₂×C₂₃	semidirect
203. #1949	C₂×C₂₃	semidirect
204. #1940	C₂×C₂₃	semidirect
205. #1927	C₂×C₂₃	semidirect
206. #1805	C₂×C₂₃	semidirect
315. C₂₃	#24697	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#476	2
C₂×C₂₃×#440	2
C₂₃×#2568	2
C₂₃×#2357	2
C₂₃×#2073	2
C₂₃×#2055	2
C₂₃×#1993	2
C₂₃×#1949	2
C₂₃×#1940	2
C₂₃×#1927	2
C₂₃×#1805	2
#24697	23

Subgroups — C23×#24697 #19172323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#24697 #19172323