Subgroups — C₂₃×#60621 #19208247

Order 5888.

2078 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3449	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3142	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3141	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3137	C₂	1	8	yes	maximal, complemented
6	C₂₃×#3136	C₂	1	8	yes	maximal, complemented
7	C₂₃×#2896	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#417	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2873	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	2	yes	—
11	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#544	—	2	4	no	—
13	C₂₃×#544	C₂²	1	8	yes	complemented
14	C₂₃×#537	—	2	1	no	—
15	C₂₃×#537	—	2	1	no	—
16	C₂₃×#536	C₂²	1	4	yes	—
17	C₂₃×#535	C₂²	1	4	yes	—
18	C₂₃×#534	—	2	2	no	—
19	C₂₃×#532	C₂²	1	4	yes	—
20	C₂×C₂₃×C₄⋊Q₈	—	2	1	no	—
21	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
22	C₂×C₂₃×#125	—	2	1	no	—
23	C₂×C₂₃×#125	C₂²	1	2	yes	complemented
24	C₂₃×#525	—	2	4	no	—
25	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
26	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#491	C₂²	1	8	yes	complemented
30	C₂₃×#490	C₂²	1	8	yes	complemented
31	C₂₃×#489	C₂²	1	16	yes	complemented
32	C₂₃×#488	C₂²	1	16	yes	complemented
33	C₂₃×#487	C₂²	1	8	yes	complemented
34	C₂₃×#486	C₂²	1	8	yes	complemented
35	C₂₃×#419	C₂²	1	4	yes	complemented
36	C₂₃×#418	C₂²	1	4	yes	complemented
37	C₂₃×#417	C₂²	1	8	yes	—
38	C₂₃×#417	C₂²	1	4	yes	complemented
39	C₂₃×#417	C₂²	1	8	yes	complemented
40	C₂₃×#417	C₂²	1	4	yes	complemented
41	C₂₃×#416	C₂²	1	8	yes	—
42	C₂×C₂₃×#103	C₂²	1	4	yes	complemented
43	C₂×C₂₃×#102	C₂²	1	4	yes	complemented
44	C₂₃×#405	C₂²	1	8	yes	complemented
45	C₄×M₁₆×C₂₃	C₂²	1	4	yes	complemented
46	C₂×#41×C₂₃	—	2	2	no	—
47	C₂×#41×C₂₃	—	2	2	no	—
48	C₂×#41×C₂₃	—	2	8	no	—
49	C₂×#41×C₂₃	—	2	8	no	—
50	C₂×#41×C₂₃	C₂³	1	4	yes	—
51	C₂×#41×C₂₃	C₂³	1	4	yes	—
52	C₂²×Q₈×C₂₃	—	2	2	no	—
53	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
54	C₂²×D₄×C₂₃	—	2	2	no	—
55	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
56	C₂³×C₄×C₂₃	—	2	1	no	—
57	C₂³×C₄×C₂₃	—	2	2	no	—
58	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₂₃	—	2	8	no	—
60	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
61	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
62	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
63	C₂₃×C₄⋊Q₈	—	2	8	no	—
64	C₂₃×C₄⋊Q₈	—	2	4	no	—
65	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
66	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
67	C₂₃×#127	C₂³	1	4	yes	—
68	C₂₃×#125	—	2	8	no	—
69	C₂₃×#125	—	2	4	no	—
70	C₂₃×#125	C₂³	1	8	yes	—
71	C₂₃×#124	C₂³	1	8	yes	—
72	C₂₃×#123	—	2	8	no	—
73	C₂₃×C₂²⋊Q₈	—	2	4	no	—
74	C₂₃×C₂²⋊Q₈	—	2	4	no	—
75	C₂₃×C₂²⋊Q₈	—	2	16	no	—
76	C₂₃×#121	—	2	4	no	—
77	C₂₃×#121	—	2	4	no	—
78	C₂₃×#121	—	2	4	no	—
79	C₂₃×#121	—	2	4	no	—
80	C₂₃×#120	—	2	8	no	—
81	C₄×D₄×C₂₃	—	2	4	no	—
82	C₄×D₄×C₂₃	—	2	4	no	—
83	C₂₃×#117	—	2	2	no	—
84	C₂₃×#117	D₄	1	4	yes	—
85	C₂₃×#117	C₂³	1	4	yes	—
86	C₂₃×#117	C₂³	1	4	yes	—
87	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
90	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
92	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
100	C₂×C₄²×C₂₃	—	2	1	no	—
101	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
102	C₂₃×#103	C₂³	1	16	yes	—
103	C₂₃×#103	C₂³	1	8	yes	—
104	C₂₃×#103	C₂³	1	8	yes	—
105	C₂₃×#102	C₂³	1	16	yes	—
106	C₂₃×#102	C₂³	1	8	yes	—
107	C₂₃×#102	C₂³	1	8	yes	—
108	C₂₃×#97	C₂³	1	8	yes	—
109	C₄×C₈×C₂₃	C₂³	1	8	yes	—
110	C₂⁴×C₂₃	—	2	2	no	—
111	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
112	#41×C₂₃	—	2	8	no	—
113	#41×C₂₃	—	2	8	no	—
114	#41×C₂₃	—	2	8	no	—
115	#41×C₂₃	—	2	8	no	—
116	#41×C₂₃	—	4	8	no	—
117	#41×C₂₃	—	4	8	no	—
118	C₂×Q₈×C₂₃	—	2	2	no	—
119	C₂×Q₈×C₂₃	—	2	8	no	—
120	C₂×Q₈×C₂₃	—	2	4	no	—
121	C₂×Q₈×C₂₃	—	2	4	no	—
122	C₂×Q₈×C₂₃	—	8	4	no	—
123	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
124	C₂×D₄×C₂₃	—	2	4	no	—
125	C₂×D₄×C₂₃	—	2	4	no	—
126	C₂×D₄×C₂₃	—	2	2	no	—
127	C₂×D₄×C₂₃	—	2	4	no	—
128	C₂×D₄×C₂₃	—	2	4	no	—
129	C₂×D₄×C₂₃	—	2	4	no	—
130	C₂×D₄×C₂₃	—	2	4	no	—
131	C₂×D₄×C₂₃	—	4	8	no	—
132	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	2	2	no	—
136	C₂²×C₄×C₂₃	—	2	2	no	—
137	C₂²×C₄×C₂₃	—	2	4	no	—
138	C₂²×C₄×C₂₃	—	2	4	no	—
139	C₂²×C₄×C₂₃	—	2	4	no	—
140	C₂²×C₄×C₂₃	—	4	8	no	—
141	C₂²×C₄×C₂₃	—	8	2	no	—
142	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
146	C₂²×C₄×C₂₃	#41	1	8	yes	—
147	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
149	M₁₆×C₂₃	—	2	8	no	—
150	M₁₆×C₂₃	—	2	8	no	—
151	C₂×C₈×C₂₃	—	2	8	no	—
152	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
153	C₄⋊C₄×C₂₃	—	2	4	no	—
154	C₄⋊C₄×C₂₃	—	2	2	no	—
155	C₄⋊C₄×C₂₃	—	2	8	no	—
156	C₄⋊C₄×C₂₃	—	2	8	no	—
157	C₄⋊C₄×C₂₃	—	2	8	no	—
158	C₄⋊C₄×C₂₃	—	2	1	no	—
159	C₄⋊C₄×C₂₃	—	2	1	no	—
160	C₄⋊C₄×C₂₃	—	2	4	no	—
161	C₄⋊C₄×C₂₃	—	2	4	no	—
162	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
163	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
165	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
166	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
167	C₂²⋊C₄×C₂₃	—	2	4	no	—
168	C₂²⋊C₄×C₂₃	—	2	8	no	—
169	C₂²⋊C₄×C₂₃	—	2	8	no	—
170	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
171	C₄²×C₂₃	—	2	4	no	—
172	C₄²×C₂₃	C₂×D₄	1	4	yes	—
173	C₄²×C₂₃	C₂⁴	1	4	yes	—
174	#60621	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂³×C₂₃	—	2	4	no	—
176	C₂³×C₂₃	—	4	1	no	—
177	C₂³×C₂₃	—	4	1	no	—
178	C₂³×C₂₃	—	4	4	no	—
179	C₂³×C₂₃	—	8	2	no	—
180	C₂³×C₂₃	—	8	2	no	—
181	C₂³×C₂₃	#124	1	4	yes	—
182	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
183	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
184	Q₈×C₂₃	—	2	4	no	—
185	Q₈×C₂₃	—	2	4	no	—
186	Q₈×C₂₃	—	4	4	no	—
187	Q₈×C₂₃	—	8	8	no	—
188	D₄×C₂₃	—	2	4	no	—
189	D₄×C₂₃	—	2	4	no	—
190	D₄×C₂₃	—	4	8	no	—
191	D₄×C₂₃	—	4	8	no	—
192	D₄×C₂₃	—	4	4	no	—
193	D₄×C₂₃	—	8	4	no	—
194	D₄×C₂₃	—	8	4	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	2	4	no	—
198	C₂×C₄×C₂₃	—	2	4	no	—
199	C₂×C₄×C₂₃	—	4	2	no	—
200	C₂×C₄×C₂₃	—	4	2	no	—
201	C₂×C₄×C₂₃	—	4	2	no	—
202	C₂×C₄×C₂₃	—	4	4	no	—
203	C₂×C₄×C₂₃	—	4	2	no	—
204	C₂×C₄×C₂₃	—	4	4	no	—
205	C₂×C₄×C₂₃	—	4	8	no	—
206	C₂×C₄×C₂₃	—	4	8	no	—
207	C₂×C₄×C₂₃	—	8	4	no	—
208	C₂×C₄×C₂₃	#124	1	8	yes	—
209	C₂×C₄×C₂₃	#124	1	4	yes	—
210	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
211	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
212	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
213	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
215	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
216	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
217	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
218	C₈×C₂₃	—	2	8	no	—
219	C₂×#544	C₂×C₂₃	1	1	yes	characteristic, complemented
220	#3449	C₂×C₂₃	1	1	yes	characteristic
221	#3142	C₂×C₂₃	1	4	yes	complemented
222	#3141	C₂×C₂₃	1	4	yes	complemented
223	#3137	C₂×C₂₃	1	8	yes	complemented
224	#3136	C₂×C₂₃	1	8	yes	complemented
225	#2896	C₂×C₂₃	1	2	yes	complemented
226	C₂×#417	C₂×C₂₃	1	2	yes	complemented
227	#2873	C₂×C₂₃	1	1	yes	characteristic, complemented
228	C₂²×C₂₃	—	2	1	no	—
229	C₂²×C₂₃	—	2	2	no	—
230	C₂²×C₂₃	—	2	1	no	—
231	C₂²×C₂₃	—	2	2	no	—
232	C₂²×C₂₃	—	4	4	no	—
233	C₂²×C₂₃	—	4	4	no	—
234	C₂²×C₂₃	—	4	4	no	—
235	C₂²×C₂₃	—	4	4	no	—
236	C₂²×C₂₃	—	8	8	no	—
237	C₂²×C₂₃	C₂×#124	1	4	yes	—
238	C₂²×C₂₃	C₂×#124	1	2	yes	—
239	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
240	C₄×C₂₃	—	2	4	no	—
241	C₄×C₂₃	—	2	4	no	—
242	C₄×C₂₃	—	4	4	no	—
243	C₄×C₂₃	—	4	2	no	—
244	C₄×C₂₃	—	4	2	no	—
245	C₄×C₂₃	—	4	4	no	—
246	C₄×C₂₃	C₂×#124	1	2	yes	—
247	C₄×C₂₃	C₂×#124	1	4	yes	—
248	C₄×C₂₃	C₂²×#41	1	2	yes	—
249	C₂²×#41	C₂²×C₂₃	1	2	yes	—
250	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
251	#544	—	2	4	no	—
252	#544	C₂²×C₂₃	1	8	yes	complemented
253	#537	—	2	1	no	—
254	#537	—	2	1	no	—
255	#536	C₂²×C₂₃	1	4	yes	—
256	#535	C₂²×C₂₃	1	4	yes	—
257	#534	—	2	2	no	—
258	#532	C₂²×C₂₃	1	4	yes	—
259	C₂×C₄⋊Q₈	—	2	1	no	—
260	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
261	C₂×#125	—	2	1	no	—
262	C₂×#125	C₂²×C₂₃	1	2	yes	complemented
263	#525	—	2	4	no	—
264	C₂×C₂²⋊Q₈	—	2	2	no	—
265	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
266	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
267	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
268	#491	C₂²×C₂₃	1	8	yes	complemented
269	#490	C₂²×C₂₃	1	8	yes	complemented
270	#489	C₂²×C₂₃	1	16	yes	complemented
271	#488	C₂²×C₂₃	1	16	yes	complemented
272	#487	C₂²×C₂₃	1	8	yes	complemented
273	#486	C₂²×C₂₃	1	8	yes	complemented
274	#419	C₂²×C₂₃	1	4	yes	complemented
275	#418	C₂²×C₂₃	1	4	yes	complemented
276	#417	C₂²×C₂₃	1	8	yes	—
277	#417	C₂²×C₂₃	1	4	yes	complemented
278	#417	C₂²×C₂₃	1	8	yes	complemented
279	#417	C₂²×C₂₃	1	4	yes	complemented
280	#416	C₂²×C₂₃	1	8	yes	—
281	C₂×#103	C₂²×C₂₃	1	4	yes	complemented
282	C₂×#102	C₂²×C₂₃	1	4	yes	complemented
283	#405	C₂²×C₂₃	1	8	yes	complemented
284	C₄×M₁₆	C₂²×C₂₃	1	4	yes	complemented
285	C₂×C₂₃	—	2	4	no	—
286	C₂×C₂₃	—	2	2	no	—
287	C₂×C₂₃	—	8	4	no	—
288	C₂×C₂₃	C₂²×#124	1	1	yes	characteristic
289	C₂×C₂₃	#3585	1	1	yes	characteristic
290	C₂×C₂₃	#3586	1	1	yes	characteristic
291	C₂×#41	—	2	2	no	—
292	C₂×#41	—	2	2	no	—
293	C₂×#41	—	2	8	no	—
294	C₂×#41	—	2	8	no	—
295	C₂×#41	C₂³×C₂₃	1	4	yes	—
296	C₂×#41	C₂³×C₂₃	1	4	yes	—
297	C₂²×Q₈	—	2	2	no	—
298	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
299	C₂²×D₄	—	2	2	no	—
300	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
301	C₂³×C₄	—	2	1	no	—
302	C₂³×C₄	—	2	2	no	—
303	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
304	C₂×M₁₆	—	2	8	no	—
305	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
306	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
307	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
308	C₄⋊Q₈	—	2	8	no	—
309	C₄⋊Q₈	—	2	4	no	—
310	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
311	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
312	#127	C₂³×C₂₃	1	4	yes	—
313	#125	—	2	8	no	—
314	#125	—	2	4	no	—
315	#125	C₂³×C₂₃	1	8	yes	—
316	#124	C₂³×C₂₃	1	8	yes	—
317	#123	—	2	8	no	—
318	C₂²⋊Q₈	—	2	4	no	—
319	C₂²⋊Q₈	—	2	4	no	—
320	C₂²⋊Q₈	—	2	16	no	—
321	#121	—	2	4	no	—
322	#121	—	2	4	no	—
323	#121	—	2	4	no	—
324	#121	—	2	4	no	—
325	#120	—	2	8	no	—
326	C₄×D₄	—	2	4	no	—
327	C₄×D₄	—	2	4	no	—
328	#117	—	2	2	no	—
329	#117	D₄×C₂₃	1	4	yes	—
330	#117	C₂³×C₂₃	1	4	yes	—
331	#117	C₂³×C₂₃	1	4	yes	—
332	C₂×C₄⋊C₄	—	2	1	no	—
333	C₂×C₄⋊C₄	—	2	2	no	—
334	C₂×C₄⋊C₄	—	2	2	no	—
335	C₂×C₄⋊C₄	—	2	2	no	—
336	C₂×C₄⋊C₄	—	2	2	no	—
337	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
338	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
339	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
340	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
341	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
342	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
343	C₂×C₂²⋊C₄	—	2	1	no	—
344	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
345	C₂×C₄²	—	2	1	no	—
346	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
347	#103	C₂³×C₂₃	1	16	yes	—
348	#103	C₂³×C₂₃	1	8	yes	—
349	#103	C₂³×C₂₃	1	8	yes	—
350	#102	C₂³×C₂₃	1	16	yes	—
351	#102	C₂³×C₂₃	1	8	yes	—
352	#102	C₂³×C₂₃	1	8	yes	—
353	#97	C₂³×C₂₃	1	8	yes	—
354	C₄×C₈	C₂³×C₂₃	1	8	yes	—
355	C₂₃	#60621	1	1	yes	characteristic, direct product factor, Sylow
356	C₂⁴	—	2	2	no	—
357	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
358	#41	—	2	8	no	—
359	#41	—	2	8	no	—
360	#41	—	2	8	no	—
361	#41	—	2	8	no	—
362	#41	—	4	8	no	—
363	#41	—	4	8	no	—
364	C₂×Q₈	—	2	2	no	—
365	C₂×Q₈	—	2	8	no	—
366	C₂×Q₈	—	2	4	no	—
367	C₂×Q₈	—	2	4	no	—
368	C₂×Q₈	—	8	4	no	—
369	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	4	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	4	no	—
374	C₂×D₄	—	2	4	no	—
375	C₂×D₄	—	2	4	no	—
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	4	8	no	—
378	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	4	no	—
385	C₂²×C₄	—	2	4	no	—
386	C₂²×C₄	—	4	8	no	—
387	C₂²×C₄	—	8	2	no	—
388	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
390	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
391	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
392	C₂²×C₄	#41×C₂₃	1	8	yes	—
393	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
394	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
395	M₁₆	—	2	8	no	—
396	M₁₆	—	2	8	no	—
397	C₂×C₈	—	2	8	no	—
398	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
399	C₄⋊C₄	—	2	4	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	2	8	no	—
402	C₄⋊C₄	—	2	8	no	—
403	C₄⋊C₄	—	2	8	no	—
404	C₄⋊C₄	—	2	1	no	—
405	C₄⋊C₄	—	2	1	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	4	no	—
408	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
409	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
410	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
411	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
412	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
413	C₂²⋊C₄	—	2	4	no	—
414	C₂²⋊C₄	—	2	8	no	—
415	C₂²⋊C₄	—	2	8	no	—
416	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
417	C₄²	—	2	4	no	—
418	C₄²	C₂×D₄×C₂₃	1	4	yes	—
419	C₄²	C₂⁴×C₂₃	1	4	yes	—
420	C₂³	—	2	4	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	1	no	—
423	C₂³	—	4	4	no	—
424	C₂³	—	8	2	no	—
425	C₂³	—	8	2	no	—
426	C₂³	C₂₃×#124	1	4	yes	—
427	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
428	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
429	Q₈	—	2	4	no	—
430	Q₈	—	2	4	no	—
431	Q₈	—	4	4	no	—
432	Q₈	—	8	8	no	—
433	D₄	—	2	4	no	—
434	D₄	—	2	4	no	—
435	D₄	—	4	8	no	—
436	D₄	—	4	8	no	—
437	D₄	—	4	4	no	—
438	D₄	—	8	4	no	—
439	D₄	—	8	4	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	4	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	4	no	—
450	C₂×C₄	—	4	8	no	—
451	C₂×C₄	—	4	8	no	—
452	C₂×C₄	—	8	4	no	—
453	C₂×C₄	C₂₃×#124	1	8	yes	—
454	C₂×C₄	C₂₃×#124	1	4	yes	—
455	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
456	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
457	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
458	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
459	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
460	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
461	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
462	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
463	C₈	—	2	8	no	—
464	C₂²	—	2	1	no	—
465	C₂²	—	2	2	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	2	no	—
468	C₂²	—	4	4	no	—
469	C₂²	—	4	4	no	—
470	C₂²	—	4	4	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	8	8	no	—
473	C₂²	C₂×C₂₃×#124	1	4	yes	—
474	C₂²	C₂×C₂₃×#124	1	2	yes	—
475	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
476	C₄	—	2	4	no	—
477	C₄	—	2	4	no	—
478	C₄	—	4	4	no	—
479	C₄	—	4	4	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	2	no	—
482	C₄	C₂×C₂₃×#124	1	2	yes	—
483	C₄	C₂×C₂₃×#124	1	4	yes	—
484	C₄	C₂²×#41×C₂₃	1	2	yes	—
485	C₂	—	2	4	no	—
486	C₂	—	2	2	no	—
487	C₂	—	8	4	no	—
488	C₂	C₂²×C₂₃×#124	1	1	yes	characteristic
489	C₂	C₂₃×#3585	1	1	yes	characteristic
490	C₂	C₂₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#544	C₂	semidirect
3. C₂₃×#3142	C₂	semidirect
4. C₂₃×#3141	C₂	semidirect
5. C₂₃×#3137	C₂	semidirect
6. C₂₃×#3136	C₂	semidirect
7. C₂₃×#2896	C₂	semidirect
8. C₂×C₂₃×#417	C₂	semidirect
9. C₂₃×#2873	C₂	semidirect
13. C₂₃×#544	C₂²	semidirect
21. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
23. C₂×C₂₃×#125	C₂²	semidirect
29. C₂₃×#491	C₂²	semidirect
30. C₂₃×#490	C₂²	semidirect
31. C₂₃×#489	C₂²	semidirect
32. C₂₃×#488	C₂²	semidirect
33. C₂₃×#487	C₂²	semidirect
34. C₂₃×#486	C₂²	semidirect
35. C₂₃×#419	C₂²	semidirect
36. C₂₃×#418	C₂²	semidirect
38. C₂₃×#417	C₂²	semidirect
39. C₂₃×#417	C₂²	semidirect
40. C₂₃×#417	C₂²	semidirect
42. C₂×C₂₃×#103	C₂²	semidirect
43. C₂×C₂₃×#102	C₂²	semidirect
44. C₂₃×#405	C₂²	semidirect
45. C₄×M₁₆×C₂₃	C₂²	semidirect
174. #60621	C₂₃	direct (N × H)
219. C₂×#544	C₂×C₂₃	semidirect
221. #3142	C₂×C₂₃	semidirect
222. #3141	C₂×C₂₃	semidirect
223. #3137	C₂×C₂₃	semidirect
224. #3136	C₂×C₂₃	semidirect
225. #2896	C₂×C₂₃	semidirect
226. C₂×#417	C₂×C₂₃	semidirect
227. #2873	C₂×C₂₃	semidirect
252. #544	C₂²×C₂₃	semidirect
260. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
262. C₂×#125	C₂²×C₂₃	semidirect
268. #491	C₂²×C₂₃	semidirect
269. #490	C₂²×C₂₃	semidirect
270. #489	C₂²×C₂₃	semidirect
271. #488	C₂²×C₂₃	semidirect
272. #487	C₂²×C₂₃	semidirect
273. #486	C₂²×C₂₃	semidirect
274. #419	C₂²×C₂₃	semidirect
275. #418	C₂²×C₂₃	semidirect
277. #417	C₂²×C₂₃	semidirect
278. #417	C₂²×C₂₃	semidirect
279. #417	C₂²×C₂₃	semidirect
281. C₂×#103	C₂²×C₂₃	semidirect
282. C₂×#102	C₂²×C₂₃	semidirect
283. #405	C₂²×C₂₃	semidirect
284. C₄×M₁₆	C₂²×C₂₃	semidirect
355. C₂₃	#60621	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×#544	2
C₂₃×#3449	2
C₂₃×#3142	2
C₂₃×#3141	2
C₂₃×#3137	2
C₂₃×#3136	2
C₂₃×#2896	2
C₂×C₂₃×#417	2
C₂₃×#2873	2
#60621	23

Subgroups — C23×#60621 #19208247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60621 #19208247