Subgroups — C₂₃×#34390 #19182016

Order 5888.

2118 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#529	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2382	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2379	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2368	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2364	C₂	1	2	yes	maximal
6	C₂₃×#2353	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2349	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2346	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2343	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2334	C₂	1	4	yes	maximal, complemented
11	C₂₃×#2322	C₂	1	4	yes	maximal, complemented
12	C₂₃×#2320	C₂	1	2	yes	maximal, complemented
13	C₂₃×#2318	C₂	1	2	yes	maximal, complemented
14	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₂₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#529	C₄	1	16	yes	complemented
17	C₂×C₂₃×#126	C₂²	1	2	yes	—
18	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#123	C₂²	1	2	yes	—
21	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
25	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
27	C₂×C₂₃×#117	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×#391	C₂²	1	4	yes	—
31	C₂₃×#389	C₂²	1	4	yes	—
32	C₂₃×#389	C₂²	1	4	yes	—
33	C₂₃×#389	C₂²	1	4	yes	—
34	C₂₃×#388	C₂²	1	4	yes	—
35	C₂₃×#388	C₂²	1	4	yes	—
36	C₂₃×#388	C₂²	1	4	yes	—
37	C₂₃×#388	C₂²	1	4	yes	—
38	C₂₃×#387	C₂²	1	4	yes	—
39	C₂₃×#387	C₂²	1	4	yes	—
40	C₂₃×#387	C₂²	1	4	yes	—
41	C₂₃×#387	C₂²	1	4	yes	—
42	C₂₃×#387	C₂²	1	4	yes	—
43	C₂₃×#387	C₂²	1	4	yes	—
44	C₂₃×#386	C₂²	1	4	yes	—
45	C₂₃×#386	C₂²	1	4	yes	—
46	C₂₃×#385	C₂²	1	4	yes	—
47	C₂₃×#385	C₂²	1	4	yes	—
48	C₂₃×#385	C₂²	1	4	yes	—
49	C₂₃×#385	C₂²	1	4	yes	—
50	C₂₃×#383	C₂²	1	2	yes	—
51	C₂₃×#383	C₂²	1	2	yes	—
52	C₂₃×#382	C₂²	1	2	yes	—
53	C₂₃×#382	C₂²	1	2	yes	—
54	C₂₃×#382	C₂²	1	1	yes	characteristic
55	C₂₃×#382	C₂²	1	1	yes	characteristic
56	C₂₃×#382	C₂²	1	2	yes	—
57	C₂₃×#382	C₂²	1	2	yes	—
58	C₂₃×#382	C₂²	1	2	yes	—
59	C₂₃×#381	C₂²	1	2	yes	—
60	C₂₃×#381	C₂²	1	2	yes	—
61	C₂₃×#380	C₂²	1	1	yes	characteristic
62	C₂₃×#380	C₂²	1	1	yes	characteristic
63	C₂₃×#380	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
65	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
66	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
67	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
68	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
69	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
71	C₂₃×#376	C₂²	1	2	yes	—
72	C₂₃×#376	C₂²	1	2	yes	—
73	C₂×C₂₃×#95	C₂²	1	2	yes	—
74	C₂×C₂₃×#95	C₂²	1	2	yes	—
75	C₂×C₂₃×#95	C₂²	1	2	yes	—
76	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
78	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
81	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
82	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
85	C₂₃×#126	C₂×C₄	1	16	yes	complemented
86	C₂₃×#125	C₂×C₄	1	8	yes	complemented
87	C₂₃×#124	C₂×C₄	1	8	yes	complemented
88	C₂₃×#123	C₂×C₄	1	16	yes	complemented
89	C₂₃×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
90	C₂₃×#121	C₂×C₄	1	8	yes	—
91	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
92	C₄×D₄×C₂₃	—	4	8	no	—
93	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	complemented
94	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
95	C₂₃×#117	C₂×C₄	1	16	yes	complemented
96	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
97	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
99	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
100	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
106	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
113	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
114	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
115	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
116	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
118	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
121	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
122	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
123	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
124	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
125	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
126	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
128	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
131	C₂₃×#95	C₂³	1	2	yes	—
132	C₂₃×#95	C₂³	1	2	yes	—
133	C₂₃×#95	C₂³	1	2	yes	—
134	C₂₃×#95	C₂³	1	2	yes	—
135	C₂₃×#95	C₂³	1	2	yes	—
136	C₂₃×#95	C₂³	1	4	yes	—
137	C₂₃×#95	C₂³	1	4	yes	—
138	C₂₃×#95	C₂³	1	4	yes	—
139	C₂₃×#95	C₂³	1	2	yes	—
140	C₂₃×#95	C₂³	1	4	yes	—
141	C₂₃×#95	C₂³	1	4	yes	—
142	C₂₃×#95	C₂³	1	4	yes	—
143	C₂₃×#95	C₂³	1	4	yes	—
144	C₂₃×#95	C₂³	1	4	yes	—
145	C₂₃×#95	C₂³	1	4	yes	—
146	C₂₃×#95	C₂³	1	4	yes	—
147	C₂₃×#95	C₂³	1	4	yes	—
148	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
149	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
150	C₂×Q₈×C₂₃	—	4	2	no	—
151	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
152	C₂×D₄×C₂₃	—	2	4	no	—
153	C₂×D₄×C₂₃	—	2	4	no	—
154	C₂×D₄×C₂₃	—	4	2	no	—
155	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
156	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
157	C₂²×C₄×C₂₃	—	2	2	no	—
158	C₂²×C₄×C₂₃	—	2	2	no	—
159	C₂²×C₄×C₂₃	—	2	8	no	—
160	C₂²×C₄×C₂₃	—	4	2	no	—
161	C₂²×C₄×C₂₃	—	4	4	no	—
162	C₂²×C₄×C₂₃	—	4	2	no	—
163	C₂²×C₄×C₂₃	—	4	4	no	—
164	C₂²×C₄×C₂₃	—	4	4	no	—
165	C₂²×C₄×C₂₃	—	4	4	no	—
166	C₂²×C₄×C₂₃	—	4	4	no	—
167	C₂²×C₄×C₂₃	—	4	4	no	—
168	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
169	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
172	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
174	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
185	C₄⋊C₄×C₂₃	—	4	2	no	—
186	C₄⋊C₄×C₂₃	—	4	4	no	—
187	C₄⋊C₄×C₂₃	—	4	2	no	—
188	C₄⋊C₄×C₂₃	—	4	4	no	—
189	C₄⋊C₄×C₂₃	—	4	4	no	—
190	C₄⋊C₄×C₂₃	—	4	2	no	—
191	C₄⋊C₄×C₂₃	—	4	2	no	—
192	C₄⋊C₄×C₂₃	—	4	4	no	—
193	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
194	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
195	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
196	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
197	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
198	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
199	C₂²⋊C₄×C₂₃	—	2	4	no	—
200	C₂²⋊C₄×C₂₃	—	2	4	no	—
201	C₂²⋊C₄×C₂₃	—	4	4	no	—
202	C₂²⋊C₄×C₂₃	—	4	4	no	—
203	C₂²⋊C₄×C₂₃	—	4	4	no	—
204	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
205	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
206	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
207	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
208	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
209	C₄²×C₂₃	—	4	4	no	—
210	C₄²×C₂₃	—	4	4	no	—
211	C₄²×C₂₃	—	4	4	no	—
212	C₄²×C₂₃	—	4	4	no	—
213	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
214	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
215	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
216	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
217	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
218	#34390	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂³×C₂₃	—	2	2	no	—
220	C₂³×C₂₃	—	2	2	no	—
221	C₂³×C₂₃	—	2	2	no	—
222	C₂³×C₂₃	—	2	2	no	—
223	C₂³×C₂₃	—	2	4	no	—
224	C₂³×C₂₃	—	2	4	no	—
225	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
226	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
227	C₂³×C₂₃	H(5,𝔽₂)	1	2	yes	—
228	C₂³×C₂₃	2₋¹⁺⁴	1	2	yes	—
229	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
230	Q₈×C₂₃	—	4	4	no	—
231	D₄×C₂₃	—	2	16	no	—
232	D₄×C₂₃	—	4	4	no	—
233	C₂×C₄×C₂₃	—	2	8	no	—
234	C₂×C₄×C₂₃	—	2	2	no	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	2	no	—
240	C₂×C₄×C₂₃	—	2	2	no	—
241	C₂×C₄×C₂₃	—	2	4	no	—
242	C₂×C₄×C₂₃	—	2	4	no	—
243	C₂×C₄×C₂₃	—	2	2	no	—
244	C₂×C₄×C₂₃	—	2	2	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	2	4	no	—
247	C₂×C₄×C₂₃	—	2	4	no	—
248	C₂×C₄×C₂₃	—	2	4	no	—
249	C₂×C₄×C₂₃	—	2	4	no	—
250	C₂×C₄×C₂₃	—	2	4	no	—
251	C₂×C₄×C₂₃	—	2	4	no	—
252	C₂×C₄×C₂₃	—	2	4	no	—
253	C₂×C₄×C₂₃	—	2	4	no	—
254	C₂×C₄×C₂₃	—	2	4	no	—
255	C₂×C₄×C₂₃	—	2	4	no	—
256	C₂×C₄×C₂₃	—	2	4	no	—
257	C₂×C₄×C₂₃	—	2	4	no	—
258	C₂×C₄×C₂₃	—	4	4	no	—
259	C₂×C₄×C₂₃	—	4	8	no	—
260	C₂×C₄×C₂₃	—	4	4	no	—
261	C₂×C₄×C₂₃	—	8	8	no	—
262	C₂×C₄×C₂₃	—	8	8	no	—
263	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
264	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
265	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
266	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
267	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
268	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
269	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
270	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
271	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
272	C₂×#529	C₂×C₂₃	1	1	yes	characteristic
273	#2382	C₂×C₂₃	1	2	yes	complemented
274	#2379	C₂×C₂₃	1	2	yes	complemented
275	#2368	C₂×C₂₃	1	2	yes	complemented
276	#2364	C₂×C₂₃	1	2	yes	—
277	#2353	C₂×C₂₃	1	2	yes	complemented
278	#2349	C₂×C₂₃	1	2	yes	complemented
279	#2346	C₂×C₂₃	1	2	yes	complemented
280	#2343	C₂×C₂₃	1	2	yes	complemented
281	#2334	C₂×C₂₃	1	4	yes	complemented
282	#2322	C₂×C₂₃	1	4	yes	complemented
283	#2320	C₂×C₂₃	1	2	yes	complemented
284	#2318	C₂×C₂₃	1	2	yes	complemented
285	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
286	C₂×#382	C₂×C₂₃	1	1	yes	characteristic, complemented
287	C₂²×C₂₃	—	2	2	no	—
288	C₂²×C₂₃	—	2	2	no	—
289	C₂²×C₂₃	—	2	4	no	—
290	C₂²×C₂₃	—	2	4	no	—
291	C₂²×C₂₃	—	2	4	no	—
292	C₂²×C₂₃	—	2	4	no	—
293	C₂²×C₂₃	—	2	4	no	—
294	C₂²×C₂₃	—	4	4	no	—
295	C₂²×C₂₃	—	4	4	no	—
296	C₂²×C₂₃	#520	1	4	yes	—
297	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
298	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
299	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
300	C₂²×C₂₃	#585	1	2	yes	—
301	C₂²×C₂₃	#585	1	2	yes	—
302	C₄×C₂₃	—	2	2	no	—
303	C₄×C₂₃	—	2	2	no	—
304	C₄×C₂₃	—	2	4	no	—
305	C₄×C₂₃	—	2	2	no	—
306	C₄×C₂₃	—	2	2	no	—
307	C₄×C₂₃	—	2	4	no	—
308	C₄×C₂₃	—	2	2	no	—
309	C₄×C₂₃	—	2	2	no	—
310	C₄×C₂₃	—	2	4	no	—
311	C₄×C₂₃	—	4	4	no	—
312	C₄×C₂₃	—	4	4	no	—
313	C₄×C₂₃	—	4	4	no	—
314	C₄×C₂₃	—	4	4	no	—
315	#529	C₄×C₂₃	1	16	yes	complemented
316	C₂×#126	C₂²×C₂₃	1	2	yes	—
317	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
318	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
319	C₂×#123	C₂²×C₂₃	1	2	yes	—
320	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
323	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
324	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
325	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
326	C₂×#117	C₂²×C₂₃	1	2	yes	—
327	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
328	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
329	#391	C₂²×C₂₃	1	4	yes	—
330	#389	C₂²×C₂₃	1	4	yes	—
331	#389	C₂²×C₂₃	1	4	yes	—
332	#389	C₂²×C₂₃	1	4	yes	—
333	#388	C₂²×C₂₃	1	4	yes	—
334	#388	C₂²×C₂₃	1	4	yes	—
335	#388	C₂²×C₂₃	1	4	yes	—
336	#388	C₂²×C₂₃	1	4	yes	—
337	#387	C₂²×C₂₃	1	4	yes	—
338	#387	C₂²×C₂₃	1	4	yes	—
339	#387	C₂²×C₂₃	1	4	yes	—
340	#387	C₂²×C₂₃	1	4	yes	—
341	#387	C₂²×C₂₃	1	4	yes	—
342	#387	C₂²×C₂₃	1	4	yes	—
343	#386	C₂²×C₂₃	1	4	yes	—
344	#386	C₂²×C₂₃	1	4	yes	—
345	#385	C₂²×C₂₃	1	4	yes	—
346	#385	C₂²×C₂₃	1	4	yes	—
347	#385	C₂²×C₂₃	1	4	yes	—
348	#385	C₂²×C₂₃	1	4	yes	—
349	#383	C₂²×C₂₃	1	2	yes	—
350	#383	C₂²×C₂₃	1	2	yes	—
351	#382	C₂²×C₂₃	1	2	yes	—
352	#382	C₂²×C₂₃	1	2	yes	—
353	#382	C₂²×C₂₃	1	2	yes	—
354	#382	C₂²×C₂₃	1	1	yes	characteristic
355	#382	C₂²×C₂₃	1	1	yes	characteristic
356	#382	C₂²×C₂₃	1	2	yes	—
357	#382	C₂²×C₂₃	1	2	yes	—
358	#381	C₂²×C₂₃	1	2	yes	—
359	#381	C₂²×C₂₃	1	2	yes	—
360	#380	C₂²×C₂₃	1	2	yes	—
361	#380	C₂²×C₂₃	1	1	yes	characteristic
362	#380	C₂²×C₂₃	1	1	yes	characteristic
363	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
364	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
365	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
366	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
367	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
368	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
369	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
370	#376	C₂²×C₂₃	1	2	yes	—
371	#376	C₂²×C₂₃	1	2	yes	—
372	C₂×#95	C₂²×C₂₃	1	2	yes	—
373	C₂×#95	C₂²×C₂₃	1	2	yes	—
374	C₂×#95	C₂²×C₂₃	1	2	yes	—
375	C₂×C₂₃	—	2	4	no	—
376	C₂×C₂₃	—	4	4	no	—
377	C₂×C₂₃	C₂×#520	1	1	yes	characteristic
378	C₂×C₂₃	#3428	1	1	yes	characteristic
379	C₂×C₂₃	#3428	1	1	yes	characteristic
380	C₂×C₂₃	#3545	1	2	yes	—
381	C₂×C₂₃	#3553	1	2	yes	—
382	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
383	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
384	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
385	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
386	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
387	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
388	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
389	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
390	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
391	#126	C₂×C₄×C₂₃	1	16	yes	complemented
392	#125	C₂×C₄×C₂₃	1	8	yes	complemented
393	#124	C₂×C₄×C₂₃	1	8	yes	complemented
394	#123	C₂×C₄×C₂₃	1	16	yes	complemented
395	C₂²⋊Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
396	#121	C₂×C₄×C₂₃	1	8	yes	—
397	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
398	C₄×D₄	—	4	8	no	—
399	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	complemented
400	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
401	#117	C₂×C₄×C₂₃	1	16	yes	complemented
402	C₂×C₄⋊C₄	—	4	4	no	—
403	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
405	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
406	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
407	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
409	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
410	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
411	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
412	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
413	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
414	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
416	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
427	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
428	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
429	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
430	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
431	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
432	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
434	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
435	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
437	#95	C₂³×C₂₃	1	2	yes	—
438	#95	C₂³×C₂₃	1	2	yes	—
439	#95	C₂³×C₂₃	1	2	yes	—
440	#95	C₂³×C₂₃	1	2	yes	—
441	#95	C₂³×C₂₃	1	2	yes	—
442	#95	C₂³×C₂₃	1	4	yes	—
443	#95	C₂³×C₂₃	1	4	yes	—
444	#95	C₂³×C₂₃	1	4	yes	—
445	#95	C₂³×C₂₃	1	2	yes	—
446	#95	C₂³×C₂₃	1	4	yes	—
447	#95	C₂³×C₂₃	1	4	yes	—
448	#95	C₂³×C₂₃	1	4	yes	—
449	#95	C₂³×C₂₃	1	4	yes	—
450	#95	C₂³×C₂₃	1	4	yes	—
451	#95	C₂³×C₂₃	1	4	yes	—
452	#95	C₂³×C₂₃	1	4	yes	—
453	#95	C₂³×C₂₃	1	4	yes	—
454	C₂₃	#34390	1	1	yes	characteristic, direct product factor, Sylow
455	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
456	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
457	C₂×Q₈	—	4	2	no	—
458	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	4	2	no	—
462	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
463	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
464	C₂²×C₄	—	2	2	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	8	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	—	4	4	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	—	4	4	no	—
471	C₂²×C₄	—	4	4	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	—	4	4	no	—
474	C₂²×C₄	—	4	4	no	—
475	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
476	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
477	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
478	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
479	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
480	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
481	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
482	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
483	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
487	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
488	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
489	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
490	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
491	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
492	C₄⋊C₄	—	4	2	no	—
493	C₄⋊C₄	—	4	4	no	—
494	C₄⋊C₄	—	4	2	no	—
495	C₄⋊C₄	—	4	4	no	—
496	C₄⋊C₄	—	4	4	no	—
497	C₄⋊C₄	—	4	4	no	—
498	C₄⋊C₄	—	4	2	no	—
499	C₄⋊C₄	—	4	2	no	—
500	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
501	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
502	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
503	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
504	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
505	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	4	no	—
508	C₂²⋊C₄	—	4	4	no	—
509	C₂²⋊C₄	—	4	4	no	—
510	C₂²⋊C₄	—	4	4	no	—
511	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
512	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
513	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
514	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
515	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
516	C₄²	—	4	4	no	—
517	C₄²	—	4	4	no	—
518	C₄²	—	4	4	no	—
519	C₄²	—	4	4	no	—
520	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
521	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
522	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
523	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
524	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	2	2	no	—
528	C₂³	—	2	2	no	—
529	C₂³	—	2	4	no	—
530	C₂³	—	2	4	no	—
531	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
532	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
533	C₂³	C₂₃×H(5,𝔽₂)	1	2	yes	—
534	C₂³	C₂₃×2₋¹⁺⁴	1	2	yes	—
535	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
536	Q₈	—	4	4	no	—
537	D₄	—	2	16	no	—
538	D₄	—	4	4	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	8	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	4	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	4	4	no	—
565	C₂×C₄	—	4	8	no	—
566	C₂×C₄	—	4	4	no	—
567	C₂×C₄	—	8	8	no	—
568	C₂×C₄	—	8	8	no	—
569	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
570	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
571	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
572	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
573	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
574	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
575	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
576	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
577	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	2	no	—
580	C₂²	—	2	4	no	—
581	C₂²	—	2	4	no	—
582	C₂²	—	2	4	no	—
583	C₂²	—	2	4	no	—
584	C₂²	—	2	4	no	—
585	C₂²	—	4	4	no	—
586	C₂²	—	4	4	no	—
587	C₂²	C₂₃×#520	1	4	yes	—
588	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
589	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
590	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
591	C₂²	C₂₃×#585	1	2	yes	—
592	C₂²	C₂₃×#585	1	2	yes	—
593	C₄	—	2	2	no	—
594	C₄	—	2	2	no	—
595	C₄	—	2	4	no	—
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	2	4	no	—
600	C₄	—	2	2	no	—
601	C₄	—	2	4	no	—
602	C₄	—	4	4	no	—
603	C₄	—	4	4	no	—
604	C₄	—	4	4	no	—
605	C₄	—	4	4	no	—
606	C₂	—	2	4	no	—
607	C₂	—	4	4	no	—
608	C₂	C₂×C₂₃×#520	1	1	yes	characteristic
609	C₂	C₂₃×#3428	1	1	yes	characteristic
610	C₂	C₂₃×#3428	1	1	yes	characteristic
611	C₂	C₂₃×#3545	1	2	yes	—
612	C₂	C₂₃×#3553	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#121	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2382	C₂	semidirect
3. C₂₃×#2379	C₂	semidirect
4. C₂₃×#2368	C₂	semidirect
6. C₂₃×#2353	C₂	semidirect
7. C₂₃×#2349	C₂	semidirect
8. C₂₃×#2346	C₂	semidirect
9. C₂₃×#2343	C₂	semidirect
10. C₂₃×#2334	C₂	semidirect
11. C₂₃×#2322	C₂	semidirect
12. C₂₃×#2320	C₂	semidirect
13. C₂₃×#2318	C₂	semidirect
14. C₂₃×#2283	C₂	semidirect
15. C₂×C₂₃×#382	C₂	semidirect
16. C₂₃×#529	C₄	semidirect
85. C₂₃×#126	C₂×C₄	semidirect
86. C₂₃×#125	C₂×C₄	semidirect
87. C₂₃×#124	C₂×C₄	semidirect
88. C₂₃×#123	C₂×C₄	semidirect
89. C₂₃×C₂²⋊Q₈	C₂×C₄	semidirect
91. C₄×Q₈×C₂₃	C₂×C₄	semidirect
93. C₄×D₄×C₂₃	C₂×C₄	semidirect
94. C₄×D₄×C₂₃	C₂×C₄	semidirect
95. C₂₃×#117	C₂×C₄	semidirect
121. C₂×C₄²×C₂₃	C₂×C₄	semidirect
218. #34390	C₂₃	direct (N × H)
273. #2382	C₂×C₂₃	semidirect
274. #2379	C₂×C₂₃	semidirect
275. #2368	C₂×C₂₃	semidirect
277. #2353	C₂×C₂₃	semidirect
278. #2349	C₂×C₂₃	semidirect
279. #2346	C₂×C₂₃	semidirect
280. #2343	C₂×C₂₃	semidirect
281. #2334	C₂×C₂₃	semidirect
282. #2322	C₂×C₂₃	semidirect
283. #2320	C₂×C₂₃	semidirect
284. #2318	C₂×C₂₃	semidirect
285. #2283	C₂×C₂₃	semidirect
286. C₂×#382	C₂×C₂₃	semidirect
315. #529	C₄×C₂₃	semidirect
391. #126	C₂×C₄×C₂₃	semidirect
392. #125	C₂×C₄×C₂₃	semidirect
393. #124	C₂×C₄×C₂₃	semidirect
394. #123	C₂×C₄×C₂₃	semidirect
395. C₂²⋊Q₈	C₂×C₄×C₂₃	semidirect
397. C₄×Q₈	C₂×C₄×C₂₃	semidirect
399. C₄×D₄	C₂×C₄×C₂₃	semidirect
400. C₄×D₄	C₂×C₄×C₂₃	semidirect
401. #117	C₂×C₄×C₂₃	semidirect
427. C₂×C₄²	C₂×C₄×C₂₃	semidirect
454. C₂₃	#34390	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×#529	2
C₂₃×#2382	2
C₂₃×#2379	2
C₂₃×#2368	2
C₂₃×#2364	2
C₂₃×#2353	2
C₂₃×#2349	2
C₂₃×#2346	2
C₂₃×#2343	2
C₂₃×#2334	2
C₂₃×#2322	2
C₂₃×#2320	2
C₂₃×#2318	2
C₂₃×#2283	2
C₂×C₂₃×#382	2
#34390	23

Subgroups — C23×#34390 #19182016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34390 #19182016