Subgroups — C₂₃×#22063 #19169689

Order 5888.

950 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂₃×#123	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1939	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1923	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1916	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1869	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1850	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1850	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1817	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#123	C₄	1	1	yes	characteristic
16	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₈×D₄×C₂₃	—	2	4	no	—
19	C₂×C₂₃×C₄⋊C₈	—	2	4	no	—
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
30	C₂₃×#388	C₄	1	2	yes	—
31	C₂₃×#387	C₄	1	2	yes	—
32	C₂₃×#386	C₄	1	1	yes	characteristic
33	C₂₃×#385	C₄	1	1	yes	characteristic
34	C₂₃×#381	C₄	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
37	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#336	C₂²	1	2	yes	—
39	C₂₃×#336	C₂²	1	2	yes	—
40	C₂₃×#336	C₂²	1	2	yes	—
41	C₂₃×#336	C₂²	1	2	yes	—
42	C₂₃×#336	C₂²	1	2	yes	—
43	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₂₃	—	2	2	no	—
49	C₂²×C₈×C₂₃	—	2	2	no	—
50	C₂²×C₈×C₂₃	—	4	2	no	—
51	C₂²×C₈×C₂₃	—	4	2	no	—
52	C₂²×C₈×C₂₃	—	4	2	no	—
53	C₂²×C₈×C₂₃	D₄	1	4	yes	—
54	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂₃×#123	C₈	1	8	yes	complemented
61	C₄×D₄×C₂₃	—	2	2	no	—
62	C₄×D₄×C₂₃	—	2	2	no	—
63	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₂₃	—	2	2	no	—
73	C₂×C₄²×C₂₃	—	2	2	no	—
74	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
75	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
76	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
77	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
78	C₂₃×C₄⋊C₈	—	2	4	no	—
79	C₂₃×C₄⋊C₈	—	2	4	no	—
80	C₂₃×C₄⋊C₈	—	2	2	no	—
81	C₂₃×C₄⋊C₈	—	4	2	no	—
82	C₂₃×C₂²⋊C₈	—	2	2	no	—
83	C₂₃×C₂²⋊C₈	—	2	2	no	—
84	C₂₃×C₂²⋊C₈	—	2	2	no	—
85	C₂₃×C₂²⋊C₈	—	4	2	no	—
86	C₄×C₈×C₂₃	—	2	2	no	—
87	C₄×C₈×C₂₃	—	4	2	no	—
88	C₂₃×#95	C₂×C₄	1	2	yes	—
89	C₂₃×#95	C₂×C₄	1	2	yes	—
90	C₂₃×#95	C₂×C₄	1	2	yes	—
91	C₂₃×#95	C₂×C₄	1	2	yes	—
92	C₂₃×#95	C₂×C₄	1	2	yes	—
93	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
94	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×D₄×C₂₃	—	2	2	no	—
96	C₂×D₄×C₂₃	—	2	2	no	—
97	C₂×D₄×C₂₃	C₂×C₈	1	4	yes	—
98	C₂²×C₄×C₂₃	—	2	2	no	—
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	2	no	—
103	C₂²×C₄×C₂₃	—	2	1	no	—
104	C₂²×C₄×C₂₃	—	2	1	no	—
105	C₂²×C₄×C₂₃	—	2	2	no	—
106	C₂²×C₄×C₂₃	—	2	2	no	—
107	C₂²×C₄×C₂₃	—	4	2	no	—
108	C₂²×C₄×C₂₃	C₂×C₈	1	4	yes	complemented
109	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
116	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
117	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₂₃	#41	1	2	yes	—
119	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₈×C₂₃	—	2	2	no	—
121	C₂×C₈×C₂₃	—	2	2	no	—
122	C₂×C₈×C₂₃	—	2	2	no	—
123	C₂×C₈×C₂₃	—	2	1	no	—
124	C₂×C₈×C₂₃	—	2	1	no	—
125	C₂×C₈×C₂₃	—	2	2	no	—
126	C₂×C₈×C₂₃	—	2	2	no	—
127	C₂×C₈×C₂₃	—	2	2	no	—
128	C₂×C₈×C₂₃	—	2	2	no	—
129	C₂×C₈×C₂₃	—	2	2	no	—
130	C₂×C₈×C₂₃	—	2	2	no	—
131	C₂×C₈×C₂₃	—	2	2	no	—
132	C₂×C₈×C₂₃	—	4	4	no	—
133	C₂×C₈×C₂₃	—	4	4	no	—
134	C₂×C₈×C₂₃	—	4	4	no	—
135	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
136	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
137	C₄⋊C₄×C₂₃	—	2	2	no	—
138	C₄⋊C₄×C₂₃	—	4	2	no	—
139	C₄⋊C₄×C₂₃	C₂×C₈	1	8	yes	complemented
140	C₂²⋊C₄×C₂₃	—	2	2	no	—
141	C₂²⋊C₄×C₂₃	—	2	2	no	—
142	C₂²⋊C₄×C₂₃	—	2	2	no	—
143	C₂²⋊C₄×C₂₃	—	4	2	no	—
144	C₂²⋊C₄×C₂₃	C₂×C₈	1	4	yes	complemented
145	C₂²⋊C₄×C₂₃	C₂×C₈	1	8	yes	complemented
146	C₄²×C₂₃	—	2	4	no	—
147	C₄²×C₂₃	—	2	2	no	—
148	C₄²×C₂₃	—	2	1	no	—
149	C₄²×C₂₃	—	2	2	no	—
150	C₄²×C₂₃	—	2	1	no	—
151	C₄²×C₂₃	—	2	2	no	—
152	C₄²×C₂₃	—	2	2	no	—
153	#22063	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂³×C₂₃	—	2	2	no	—
155	C₂³×C₂₃	—	2	2	no	—
156	C₂³×C₂₃	—	2	1	no	—
157	C₂³×C₂₃	—	2	2	no	—
158	C₂³×C₂₃	—	2	1	no	—
159	C₂³×C₂₃	—	2	2	no	—
160	C₂³×C₂₃	C₄×D₄	1	2	yes	—
161	C₂³×C₂₃	C₂²×C₈	1	2	yes	—
162	C₂³×C₂₃	C₂²×C₈	1	2	yes	—
163	C₂³×C₂₃	#131	1	2	yes	—
164	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
165	D₄×C₂₃	—	2	8	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	2	1	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	1	no	—
172	C₂×C₄×C₂₃	—	2	2	no	—
173	C₂×C₄×C₂₃	—	2	8	no	—
174	C₂×C₄×C₂₃	—	2	2	no	—
175	C₂×C₄×C₂₃	—	2	2	no	—
176	C₂×C₄×C₂₃	—	2	4	no	—
177	C₂×C₄×C₂₃	—	2	2	no	—
178	C₂×C₄×C₂₃	—	2	1	no	—
179	C₂×C₄×C₂₃	—	2	1	no	—
180	C₂×C₄×C₂₃	—	2	2	no	—
181	C₂×C₄×C₂₃	—	2	2	no	—
182	C₂×C₄×C₂₃	—	2	2	no	—
183	C₂×C₄×C₂₃	—	2	2	no	—
184	C₂×C₄×C₂₃	—	2	2	no	—
185	C₂×C₄×C₂₃	—	2	2	no	—
186	C₂×C₄×C₂₃	—	2	2	no	—
187	C₂×C₄×C₂₃	—	4	4	no	—
188	C₂×C₄×C₂₃	—	4	1	no	—
189	C₂×C₄×C₂₃	—	4	1	no	—
190	C₂×C₄×C₂₃	—	4	2	no	—
191	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
192	C₂×C₄×C₂₃	C₂²×C₈	1	4	yes	—
193	C₂×C₄×C₂₃	C₂²×C₈	1	4	yes	—
194	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
195	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
198	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
199	C₈×C₂₃	—	2	2	no	—
200	C₈×C₂₃	—	2	2	no	—
201	C₈×C₂₃	—	4	2	no	—
202	C₈×C₂₃	—	4	1	no	—
203	C₈×C₂₃	—	4	1	no	—
204	C₈×C₂₃	—	4	2	no	—
205	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
206	C₂²×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
207	C₄×#123	C₂×C₂₃	1	1	yes	characteristic
208	#1939	C₂×C₂₃	1	2	yes	complemented
209	#1923	C₂×C₂₃	1	2	yes	complemented
210	#1916	C₂×C₂₃	1	2	yes	complemented
211	#1869	C₂×C₂₃	1	1	yes	characteristic
212	#1850	C₂×C₂₃	1	2	yes	complemented
213	#1850	C₂×C₂₃	1	1	yes	characteristic, complemented
214	#1817	C₂×C₂₃	1	1	yes	characteristic, complemented
215	C₈×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
216	C₂²×C₂₃	—	2	2	no	—
217	C₂²×C₂₃	—	2	2	no	—
218	C₂²×C₂₃	—	2	4	no	—
219	C₂²×C₂₃	—	2	4	no	—
220	C₂²×C₂₃	—	2	2	no	—
221	C₂²×C₂₃	—	2	4	no	—
222	C₂²×C₂₃	—	4	1	no	—
223	C₂²×C₂₃	—	4	1	no	—
224	C₂²×C₂₃	—	4	2	no	—
225	C₂²×C₂₃	C₈×D₄	1	4	yes	—
226	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
227	C₂²×C₂₃	#520	1	2	yes	—
228	C₂²×C₂₃	C₂³×C₈	1	1	yes	characteristic
229	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
230	C₂²×C₂₃	#568	1	2	yes	—
231	C₄×C₂₃	—	2	2	no	—
232	C₄×C₂₃	—	2	4	no	—
233	C₄×C₂₃	—	2	2	no	—
234	C₄×C₂₃	—	2	4	no	—
235	C₄×C₂₃	—	4	1	no	—
236	C₄×C₂₃	—	4	1	no	—
237	C₄×C₂₃	—	4	2	no	—
238	C₄×C₂₃	—	4	2	no	—
239	C₄×C₂₃	#534	1	1	yes	characteristic
240	C₄×C₂₃	#536	1	1	yes	characteristic
241	C₄×C₂₃	#539	1	2	yes	—
242	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
243	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
244	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
245	C₂×#123	C₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
247	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
248	C₈×D₄	—	2	4	no	—
249	C₂×C₄⋊C₈	—	2	4	no	—
250	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
251	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
252	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
253	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
254	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
255	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
256	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
257	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
258	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
260	#388	C₄×C₂₃	1	2	yes	—
261	#387	C₄×C₂₃	1	2	yes	—
262	#386	C₄×C₂₃	1	1	yes	characteristic
263	#385	C₄×C₂₃	1	1	yes	characteristic
264	#381	C₄×C₂₃	1	1	yes	characteristic
265	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
266	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
267	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
268	#336	C₂²×C₂₃	1	2	yes	—
269	#336	C₂²×C₂₃	1	2	yes	—
270	#336	C₂²×C₂₃	1	2	yes	—
271	#336	C₂²×C₂₃	1	2	yes	—
272	#336	C₂²×C₂₃	1	2	yes	—
273	C₂×C₂₃	—	2	4	no	—
274	C₂×C₂₃	—	4	2	no	—
275	C₂×C₂₃	#2322	1	1	yes	characteristic
276	C₂×C₂₃	C₂×C₈×D₄	1	1	yes	characteristic
277	C₂×C₂₃	#2930	1	1	yes	characteristic
278	C₂×C₂₃	#2988	1	2	yes	—
279	C₂×C₂₃	#2992	1	2	yes	—
280	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
281	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
282	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
283	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
284	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
285	C₂²×C₈	—	2	2	no	—
286	C₂²×C₈	—	2	2	no	—
287	C₂²×C₈	—	4	2	no	—
288	C₂²×C₈	—	4	2	no	—
289	C₂²×C₈	—	4	2	no	—
290	C₂²×C₈	D₄×C₂₃	1	4	yes	—
291	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
292	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
293	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
294	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
295	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
297	#123	C₈×C₂₃	1	8	yes	complemented
298	C₄×D₄	—	2	2	no	—
299	C₄×D₄	—	2	2	no	—
300	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
302	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
303	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
304	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
305	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
306	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
307	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
308	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
309	C₂×C₄²	—	2	2	no	—
310	C₂×C₄²	—	2	2	no	—
311	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
312	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
313	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
314	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
315	C₄⋊C₈	—	2	4	no	—
316	C₄⋊C₈	—	2	4	no	—
317	C₄⋊C₈	—	2	2	no	—
318	C₄⋊C₈	—	4	2	no	—
319	C₂²⋊C₈	—	2	2	no	—
320	C₂²⋊C₈	—	2	2	no	—
321	C₂²⋊C₈	—	2	2	no	—
322	C₂²⋊C₈	—	4	2	no	—
323	C₄×C₈	—	2	2	no	—
324	C₄×C₈	—	4	2	no	—
325	#95	C₂×C₄×C₂₃	1	2	yes	—
326	#95	C₂×C₄×C₂₃	1	2	yes	—
327	#95	C₂×C₄×C₂₃	1	2	yes	—
328	#95	C₂×C₄×C₂₃	1	2	yes	—
329	#95	C₂×C₄×C₂₃	1	2	yes	—
330	C₂₃	#22063	1	1	yes	characteristic, direct product factor, Sylow
331	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
332	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	C₂×C₈×C₂₃	1	4	yes	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	1	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	4	2	no	—
346	C₂²×C₄	C₂×C₈×C₂₃	1	4	yes	complemented
347	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
348	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
349	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
350	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
351	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
352	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
353	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
354	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
356	C₂²×C₄	#41×C₂₃	1	2	yes	—
357	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	—	2	1	no	—
362	C₂×C₈	—	2	2	no	—
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	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	4	4	no	—
371	C₂×C₈	—	4	4	no	—
372	C₂×C₈	—	4	4	no	—
373	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
374	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	4	2	no	—
377	C₄⋊C₄	C₂×C₈×C₂₃	1	8	yes	complemented
378	C₂²⋊C₄	—	2	2	no	—
379	C₂²⋊C₄	—	2	2	no	—
380	C₂²⋊C₄	—	2	2	no	—
381	C₂²⋊C₄	—	4	2	no	—
382	C₂²⋊C₄	C₂×C₈×C₂₃	1	4	yes	complemented
383	C₂²⋊C₄	C₂×C₈×C₂₃	1	8	yes	complemented
384	C₄²	—	2	4	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	2	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	2	no	—
397	C₂³	C₄×D₄×C₂₃	1	2	yes	—
398	C₂³	C₂²×C₈×C₂₃	1	2	yes	—
399	C₂³	C₂²×C₈×C₂₃	1	2	yes	—
400	C₂³	C₂₃×#131	1	2	yes	—
401	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
402	D₄	—	2	8	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	8	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	1	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
429	C₂×C₄	C₂²×C₈×C₂₃	1	4	yes	—
430	C₂×C₄	C₂²×C₈×C₂₃	1	4	yes	—
431	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
432	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
434	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
435	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
436	C₈	—	2	2	no	—
437	C₈	—	2	2	no	—
438	C₈	—	4	1	no	—
439	C₈	—	4	2	no	—
440	C₈	—	4	1	no	—
441	C₈	—	4	2	no	—
442	C₂²	—	2	2	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	2	4	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	4	no	—
448	C₂²	—	4	1	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	1	no	—
451	C₂²	C₈×D₄×C₂₃	1	4	yes	—
452	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
453	C₂²	C₂₃×#520	1	2	yes	—
454	C₂²	C₂³×C₈×C₂₃	1	1	yes	characteristic
455	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
456	C₂²	C₂₃×#568	1	2	yes	—
457	C₄	—	2	4	no	—
458	C₄	—	2	2	no	—
459	C₄	—	2	4	no	—
460	C₄	—	2	2	no	—
461	C₄	—	4	2	no	—
462	C₄	—	4	1	no	—
463	C₄	—	4	2	no	—
464	C₄	—	4	1	no	—
465	C₄	C₂₃×#534	1	1	yes	characteristic
466	C₄	C₂₃×#536	1	1	yes	characteristic
467	C₄	C₂₃×#539	1	2	yes	—
468	C₂	—	2	4	no	—
469	C₂	—	4	2	no	—
470	C₂	C₂₃×#2322	1	1	yes	characteristic
471	C₂	C₂×C₈×D₄×C₂₃	1	1	yes	characteristic
472	C₂	C₂₃×#2930	1	1	yes	characteristic
473	C₂	C₂₃×#2988	1	2	yes	—
474	C₂	C₂₃×#2992	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂³×C₄
prime radical	C₂²×D₄×C₂₃	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₂₃×C₄⋊C₈	C₂	semidirect
4. C₂₃×#1939	C₂	semidirect
5. C₂₃×#1923	C₂	semidirect
6. C₂₃×#1916	C₂	semidirect
8. C₂₃×#1850	C₂	semidirect
9. C₂₃×#1850	C₂	semidirect
10. C₂₃×#1817	C₂	semidirect
11. C₈×C₂²⋊C₄×C₂₃	C₂	semidirect
60. C₂₃×#123	C₈	semidirect
108. C₂²×C₄×C₂₃	C₂×C₈	semidirect
139. C₄⋊C₄×C₂₃	C₂×C₈	semidirect
144. C₂²⋊C₄×C₂₃	C₂×C₈	semidirect
145. C₂²⋊C₄×C₂₃	C₂×C₈	semidirect
153. #22063	C₂₃	direct (N × H)
206. C₂²×C₄⋊C₈	C₂×C₂₃	semidirect
208. #1939	C₂×C₂₃	semidirect
209. #1923	C₂×C₂₃	semidirect
210. #1916	C₂×C₂₃	semidirect
212. #1850	C₂×C₂₃	semidirect
213. #1850	C₂×C₂₃	semidirect
214. #1817	C₂×C₂₃	semidirect
215. C₈×C₂²⋊C₄	C₂×C₂₃	semidirect
297. #123	C₈×C₂₃	semidirect
330. C₂₃	#22063	direct (N × H)
346. C₂²×C₄	C₂×C₈×C₂₃	semidirect
377. C₄⋊C₄	C₂×C₈×C₂₃	semidirect
382. C₂²⋊C₄	C₂×C₈×C₂₃	semidirect
383. C₂²⋊C₄	C₂×C₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₂₃	2
C₂²×C₂₃×C₄⋊C₈	2
C₄×C₂₃×#123	2
C₂₃×#1939	2
C₂₃×#1923	2
C₂₃×#1916	2
C₂₃×#1869	2
C₂₃×#1850	2
C₂₃×#1817	2
C₈×C₂²⋊C₄×C₂₃	2
#22063	23

Subgroups — C23×#22063 #19169689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22063 #19169689