Subgroups — C₂₃×#34398 #19182024

Order 5888.

2538 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2379	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2361	C₂	1	4	yes	maximal, complemented
4	C₂₃×#2357	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2349	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2339	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2331	C₂	1	2	yes	maximal, complemented
8	C₂₃×#2328	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2325	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2318	C₂	1	4	yes	maximal, complemented
11	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#532	C₄	1	16	yes	complemented
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#124	C₂²	1	2	yes	—
18	C₂×C₂₃×#121	C₂²	1	2	yes	—
19	C₂×C₂₃×#121	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₂₃	C₂²	1	8	yes	—
21	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
22	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#391	C₂²	1	8	yes	—
27	C₂₃×#390	C₂²	1	4	yes	—
28	C₂₃×#389	C₂²	1	4	yes	—
29	C₂₃×#388	C₂²	1	8	yes	—
30	C₂₃×#388	C₂²	1	8	yes	—
31	C₂₃×#387	C₂²	1	8	yes	—
32	C₂₃×#386	C₂²	1	8	yes	—
33	C₂₃×#386	C₂²	1	8	yes	—
34	C₂₃×#386	C₂²	1	8	yes	—
35	C₂₃×#385	C₂²	1	8	yes	—
36	C₂₃×#385	C₂²	1	8	yes	—
37	C₂₃×#383	C₂²	1	2	yes	—
38	C₂₃×#382	C₂²	1	2	yes	—
39	C₂₃×#382	C₂²	1	2	yes	—
40	C₂₃×#382	C₂²	1	2	yes	—
41	C₂₃×#382	C₂²	1	2	yes	—
42	C₂₃×#382	C₂²	1	2	yes	—
43	C₂₃×#381	C₂²	1	2	yes	—
44	C₂₃×#381	C₂²	1	4	yes	—
45	C₂₃×#380	C₂²	1	2	yes	—
46	C₂₃×#380	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
48	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
49	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
50	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
51	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
52	C₂₃×#376	C₂²	1	2	yes	—
53	C₂×C₂₃×#95	C₂²	1	2	yes	—
54	C₂×C₂₃×#95	C₂²	1	2	yes	—
55	C₂×C₂₃×#95	C₂²	1	4	yes	—
56	C₂×#41×C₂₃	—	4	8	no	—
57	C₂×#41×C₂₃	C₂×C₄	1	16	yes	complemented
58	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
59	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
60	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
61	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
62	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
63	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
66	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
67	C₂³×C₄×C₂₃	C₂³	1	4	yes	—
68	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂₃×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
70	C₂₃×#127	C₂×C₄	1	8	yes	complemented
71	C₂₃×#124	C₂×C₄	1	16	yes	complemented
72	C₂₃×#121	C₂×C₄	1	16	yes	complemented
73	C₂₃×#121	C₂×C₄	1	16	yes	complemented
74	C₄×D₄×C₂₃	—	4	8	no	—
75	C₄×D₄×C₂₃	—	4	16	no	—
76	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
77	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
78	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
88	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
89	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
90	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
92	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
93	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
94	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
95	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
96	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
100	C₂₃×#95	C₂³	1	4	yes	—
101	C₂₃×#95	C₂³	1	2	yes	—
102	C₂₃×#95	C₂³	1	4	yes	—
103	C₂₃×#95	C₂³	1	4	yes	—
104	C₂₃×#95	C₂³	1	4	yes	—
105	C₂₃×#95	C₂³	1	8	yes	—
106	C₂₃×#95	C₂³	1	4	yes	—
107	C₂₃×#95	C₂³	1	4	yes	—
108	C₂₃×#95	C₂³	1	2	yes	—
109	C₂₃×#95	C₂³	1	4	yes	—
110	C₂₃×#95	C₂³	1	8	yes	—
111	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
112	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
113	#41×C₂₃	—	4	32	no	—
114	C₂×Q₈×C₂₃	—	4	4	no	—
115	C₂×Q₈×C₂₃	C₂²×C₄	1	8	yes	—
116	C₂×D₄×C₂₃	—	2	16	no	—
117	C₂×D₄×C₂₃	—	2	4	no	—
118	C₂×D₄×C₂₃	—	2	4	no	—
119	C₂×D₄×C₂₃	—	4	4	no	—
120	C₂×D₄×C₂₃	—	4	4	no	—
121	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
122	C₂×D₄×C₂₃	C₂²×C₄	1	16	yes	—
123	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
124	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
125	C₂²×C₄×C₂₃	—	2	4	no	—
126	C₂²×C₄×C₂₃	—	2	2	no	—
127	C₂²×C₄×C₂₃	—	2	2	no	—
128	C₂²×C₄×C₂₃	—	2	8	no	—
129	C₂²×C₄×C₂₃	—	4	2	no	—
130	C₂²×C₄×C₂₃	—	4	8	no	—
131	C₂²×C₄×C₂₃	—	4	4	no	—
132	C₂²×C₄×C₂₃	—	4	8	no	—
133	C₂²×C₄×C₂₃	—	4	8	no	—
134	C₂²×C₄×C₂₃	—	4	4	no	—
135	C₂²×C₄×C₂₃	—	4	8	no	—
136	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
137	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
138	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
139	C₂²×C₄×C₂₃	C₂²×C₄	1	16	yes	—
140	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
141	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₂₃	C₂⁴	1	8	yes	—
143	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
146	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
150	C₄⋊C₄×C₂₃	—	2	8	no	—
151	C₄⋊C₄×C₂₃	—	4	4	no	—
152	C₄⋊C₄×C₂₃	—	4	2	no	—
153	C₄⋊C₄×C₂₃	—	4	2	no	—
154	C₄⋊C₄×C₂₃	—	4	8	no	—
155	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
156	C₄⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
157	C₂²⋊C₄×C₂₃	—	2	8	no	—
158	C₂²⋊C₄×C₂₃	—	4	8	no	—
159	C₂²⋊C₄×C₂₃	—	4	8	no	—
160	C₂²⋊C₄×C₂₃	—	4	8	no	—
161	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
162	C₂²⋊C₄×C₂₃	C₂²×C₄	1	16	yes	—
163	C₄²×C₂₃	—	2	4	no	—
164	C₄²×C₂₃	—	2	4	no	—
165	C₄²×C₂₃	—	4	8	no	—
166	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
167	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
168	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
169	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
170	#34398	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂³×C₂₃	—	2	1	no	—
172	C₂³×C₂₃	—	2	1	no	—
173	C₂³×C₂₃	—	2	2	no	—
174	C₂³×C₂₃	—	2	4	no	—
175	C₂³×C₂₃	—	2	8	no	—
176	C₂³×C₂₃	—	2	4	no	—
177	C₂³×C₂₃	—	2	8	no	—
178	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
179	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
180	C₂³×C₂₃	H(5,𝔽₂)	1	4	yes	—
181	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
182	Q₈×C₂₃	—	4	8	no	—
183	D₄×C₂₃	—	2	32	no	—
184	D₄×C₂₃	—	2	16	no	—
185	D₄×C₂₃	—	4	8	no	—
186	D₄×C₂₃	—	4	8	no	—
187	C₂×C₄×C₂₃	—	2	8	no	—
188	C₂×C₄×C₂₃	—	2	8	no	—
189	C₂×C₄×C₂₃	—	2	1	no	—
190	C₂×C₄×C₂₃	—	2	2	no	—
191	C₂×C₄×C₂₃	—	2	2	no	—
192	C₂×C₄×C₂₃	—	2	1	no	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	8	no	—
196	C₂×C₄×C₂₃	—	2	4	no	—
197	C₂×C₄×C₂₃	—	2	4	no	—
198	C₂×C₄×C₂₃	—	2	8	no	—
199	C₂×C₄×C₂₃	—	2	4	no	—
200	C₂×C₄×C₂₃	—	2	8	no	—
201	C₂×C₄×C₂₃	—	2	4	no	—
202	C₂×C₄×C₂₃	—	2	4	no	—
203	C₂×C₄×C₂₃	—	2	8	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	4	4	no	—
206	C₂×C₄×C₂₃	—	4	16	no	—
207	C₂×C₄×C₂₃	—	8	8	no	—
208	C₂×C₄×C₂₃	—	8	16	no	—
209	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
210	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
211	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
212	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
213	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
214	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
215	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
216	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
217	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
218	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
219	#2379	C₂×C₂₃	1	4	yes	complemented
220	#2361	C₂×C₂₃	1	4	yes	complemented
221	#2357	C₂×C₂₃	1	2	yes	complemented
222	#2349	C₂×C₂₃	1	2	yes	complemented
223	#2339	C₂×C₂₃	1	4	yes	complemented
224	#2331	C₂×C₂₃	1	2	yes	complemented
225	#2328	C₂×C₂₃	1	2	yes	complemented
226	#2325	C₂×C₂₃	1	4	yes	complemented
227	#2318	C₂×C₂₃	1	4	yes	complemented
228	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
229	C₂×#382	C₂×C₂₃	1	1	yes	characteristic, complemented
230	C₂²×C₂₃	—	2	2	no	—
231	C₂²×C₂₃	—	2	2	no	—
232	C₂²×C₂₃	—	2	4	no	—
233	C₂²×C₂₃	—	2	4	no	—
234	C₂²×C₂₃	—	2	8	no	—
235	C₂²×C₂₃	—	2	8	no	—
236	C₂²×C₂₃	—	2	8	no	—
237	C₂²×C₂₃	—	4	8	no	—
238	C₂²×C₂₃	—	4	8	no	—
239	C₂²×C₂₃	#518	1	4	yes	—
240	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
241	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
242	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
244	C₂²×C₂₃	#585	1	2	yes	—
245	C₄×C₂₃	—	2	2	no	—
246	C₄×C₂₃	—	2	2	no	—
247	C₄×C₂₃	—	2	8	no	—
248	C₄×C₂₃	—	2	2	no	—
249	C₄×C₂₃	—	2	2	no	—
250	C₄×C₂₃	—	2	2	no	—
251	C₄×C₂₃	—	4	4	no	—
252	C₄×C₂₃	—	4	8	no	—
253	C₄×C₂₃	—	4	4	no	—
254	C₄×C₂₃	#520	1	4	yes	—
255	C₂²×#41	C₂²×C₂₃	1	2	yes	—
256	#532	C₄×C₂₃	1	16	yes	complemented
257	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×#124	C₂²×C₂₃	1	2	yes	—
260	C₂×#121	C₂²×C₂₃	1	2	yes	—
261	C₂×#121	C₂²×C₂₃	1	2	yes	—
262	C₂×C₄×D₄	C₂²×C₂₃	1	8	yes	—
263	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
264	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
265	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
266	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
267	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
268	#391	C₂²×C₂₃	1	8	yes	—
269	#390	C₂²×C₂₃	1	4	yes	—
270	#389	C₂²×C₂₃	1	4	yes	—
271	#388	C₂²×C₂₃	1	8	yes	—
272	#388	C₂²×C₂₃	1	8	yes	—
273	#387	C₂²×C₂₃	1	8	yes	—
274	#386	C₂²×C₂₃	1	8	yes	—
275	#386	C₂²×C₂₃	1	8	yes	—
276	#386	C₂²×C₂₃	1	8	yes	—
277	#385	C₂²×C₂₃	1	8	yes	—
278	#385	C₂²×C₂₃	1	8	yes	—
279	#383	C₂²×C₂₃	1	2	yes	—
280	#382	C₂²×C₂₃	1	2	yes	—
281	#382	C₂²×C₂₃	1	2	yes	—
282	#382	C₂²×C₂₃	1	2	yes	—
283	#382	C₂²×C₂₃	1	2	yes	—
284	#382	C₂²×C₂₃	1	2	yes	—
285	#381	C₂²×C₂₃	1	4	yes	—
286	#381	C₂²×C₂₃	1	2	yes	—
287	#380	C₂²×C₂₃	1	4	yes	—
288	#380	C₂²×C₂₃	1	2	yes	—
289	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
290	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
291	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
292	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
293	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
294	#376	C₂²×C₂₃	1	2	yes	—
295	C₂×#95	C₂²×C₂₃	1	4	yes	—
296	C₂×#95	C₂²×C₂₃	1	2	yes	—
297	C₂×#95	C₂²×C₂₃	1	2	yes	—
298	C₂×C₂₃	—	2	4	no	—
299	C₂×C₂₃	—	4	8	no	—
300	C₂×C₂₃	C₂×#518	1	1	yes	characteristic
301	C₂×C₂₃	C₂×#520	1	1	yes	characteristic
302	C₂×C₂₃	#3428	1	1	yes	characteristic
303	C₂×C₂₃	#3558	1	2	yes	—
304	C₂×C₂₃	#3559	1	2	yes	—
305	C₂×#41	—	4	8	no	—
306	C₂×#41	C₂×C₄×C₂₃	1	16	yes	complemented
307	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
308	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
309	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
310	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
311	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
312	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
313	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
314	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
315	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
316	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
317	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
318	C₄⋊Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
319	#127	C₂×C₄×C₂₃	1	8	yes	complemented
320	#124	C₂×C₄×C₂₃	1	16	yes	complemented
321	#121	C₂×C₄×C₂₃	1	16	yes	complemented
322	#121	C₂×C₄×C₂₃	1	16	yes	complemented
323	C₄×D₄	—	4	8	no	—
324	C₄×D₄	—	4	16	no	—
325	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
326	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
327	C₂×C₄⋊C₄	—	4	4	no	—
328	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
329	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
330	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
331	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
332	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
333	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
334	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
335	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
336	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
337	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
338	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
339	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
340	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
341	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
342	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
343	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
344	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
345	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
347	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
348	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
349	#95	C₂³×C₂₃	1	4	yes	—
350	#95	C₂³×C₂₃	1	2	yes	—
351	#95	C₂³×C₂₃	1	4	yes	—
352	#95	C₂³×C₂₃	1	4	yes	—
353	#95	C₂³×C₂₃	1	4	yes	—
354	#95	C₂³×C₂₃	1	8	yes	—
355	#95	C₂³×C₂₃	1	4	yes	—
356	#95	C₂³×C₂₃	1	4	yes	—
357	#95	C₂³×C₂₃	1	2	yes	—
358	#95	C₂³×C₂₃	1	4	yes	—
359	#95	C₂³×C₂₃	1	8	yes	—
360	C₂₃	#34398	1	1	yes	characteristic, direct product factor, Sylow
361	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
362	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
363	#41	—	4	32	no	—
364	C₂×Q₈	—	4	4	no	—
365	C₂×Q₈	C₂²×C₄×C₂₃	1	8	yes	—
366	C₂×D₄	—	2	16	no	—
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	4	4	no	—
370	C₂×D₄	—	4	4	no	—
371	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
372	C₂×D₄	C₂²×C₄×C₂₃	1	16	yes	—
373	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
374	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	2	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	2	8	no	—
379	C₂²×C₄	—	4	8	no	—
380	C₂²×C₄	—	4	2	no	—
381	C₂²×C₄	—	4	4	no	—
382	C₂²×C₄	—	4	8	no	—
383	C₂²×C₄	—	4	8	no	—
384	C₂²×C₄	—	4	4	no	—
385	C₂²×C₄	—	4	8	no	—
386	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
387	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
388	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
389	C₂²×C₄	C₂²×C₄×C₂₃	1	16	yes	—
390	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
391	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
392	C₂²×C₄	C₂⁴×C₂₃	1	8	yes	—
393	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
394	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
395	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
396	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
397	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
398	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
399	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
400	C₄⋊C₄	—	2	8	no	—
401	C₄⋊C₄	—	4	4	no	—
402	C₄⋊C₄	—	4	8	no	—
403	C₄⋊C₄	—	4	2	no	—
404	C₄⋊C₄	—	4	2	no	—
405	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
406	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
407	C₂²⋊C₄	—	2	8	no	—
408	C₂²⋊C₄	—	4	8	no	—
409	C₂²⋊C₄	—	4	8	no	—
410	C₂²⋊C₄	—	4	8	no	—
411	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
412	C₂²⋊C₄	C₂²×C₄×C₂₃	1	16	yes	—
413	C₄²	—	2	4	no	—
414	C₄²	—	2	4	no	—
415	C₄²	—	4	8	no	—
416	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
417	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
418	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
419	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
420	C₂³	—	2	1	no	—
421	C₂³	—	2	1	no	—
422	C₂³	—	2	2	no	—
423	C₂³	—	2	4	no	—
424	C₂³	—	2	8	no	—
425	C₂³	—	2	4	no	—
426	C₂³	—	2	8	no	—
427	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
428	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
429	C₂³	C₂₃×H(5,𝔽₂)	1	4	yes	—
430	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
431	Q₈	—	4	8	no	—
432	D₄	—	2	32	no	—
433	D₄	—	2	16	no	—
434	D₄	—	4	8	no	—
435	D₄	—	4	8	no	—
436	C₂×C₄	—	2	8	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	8	no	—
444	C₂×C₄	—	2	8	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	8	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	8	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	4	16	no	—
455	C₂×C₄	—	4	4	no	—
456	C₂×C₄	—	8	8	no	—
457	C₂×C₄	—	8	16	no	—
458	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
459	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
460	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
461	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
462	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
463	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
464	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
465	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
466	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
467	C₂²	—	2	2	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	4	no	—
470	C₂²	—	2	4	no	—
471	C₂²	—	2	8	no	—
472	C₂²	—	2	8	no	—
473	C₂²	—	2	8	no	—
474	C₂²	—	4	8	no	—
475	C₂²	—	4	8	no	—
476	C₂²	C₂₃×#518	1	4	yes	—
477	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
478	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
479	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
480	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
481	C₂²	C₂₃×#585	1	2	yes	—
482	C₄	—	2	2	no	—
483	C₄	—	2	2	no	—
484	C₄	—	2	8	no	—
485	C₄	—	2	2	no	—
486	C₄	—	2	2	no	—
487	C₄	—	2	2	no	—
488	C₄	—	4	4	no	—
489	C₄	—	4	8	no	—
490	C₄	—	4	4	no	—
491	C₄	C₂₃×#520	1	4	yes	—
492	C₂	—	2	4	no	—
493	C₂	—	4	8	no	—
494	C₂	C₂×C₂₃×#518	1	1	yes	characteristic
495	C₂	C₂×C₂₃×#520	1	1	yes	characteristic
496	C₂	C₂₃×#3428	1	1	yes	characteristic
497	C₂	C₂₃×#3558	1	2	yes	—
498	C₂	C₂₃×#3559	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×#532	C₂
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×#2379	C₂	semidirect
3. C₂₃×#2361	C₂	semidirect
4. C₂₃×#2357	C₂	semidirect
5. C₂₃×#2349	C₂	semidirect
6. C₂₃×#2339	C₂	semidirect
7. C₂₃×#2331	C₂	semidirect
8. C₂₃×#2328	C₂	semidirect
9. C₂₃×#2325	C₂	semidirect
10. C₂₃×#2318	C₂	semidirect
11. C₂₃×#2283	C₂	semidirect
12. C₂×C₂₃×#382	C₂	semidirect
14. C₂₃×#532	C₄	semidirect
57. C₂×#41×C₂₃	C₂×C₄	semidirect
69. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
70. C₂₃×#127	C₂×C₄	semidirect
71. C₂₃×#124	C₂×C₄	semidirect
72. C₂₃×#121	C₂×C₄	semidirect
73. C₂₃×#121	C₂×C₄	semidirect
76. C₄×D₄×C₂₃	C₂×C₄	semidirect
77. C₄×D₄×C₂₃	C₂×C₄	semidirect
92. C₂×C₄²×C₂₃	C₂×C₄	semidirect
170. #34398	C₂₃	direct (N × H)
219. #2379	C₂×C₂₃	semidirect
220. #2361	C₂×C₂₃	semidirect
221. #2357	C₂×C₂₃	semidirect
222. #2349	C₂×C₂₃	semidirect
223. #2339	C₂×C₂₃	semidirect
224. #2331	C₂×C₂₃	semidirect
225. #2328	C₂×C₂₃	semidirect
226. #2325	C₂×C₂₃	semidirect
227. #2318	C₂×C₂₃	semidirect
228. #2283	C₂×C₂₃	semidirect
229. C₂×#382	C₂×C₂₃	semidirect
256. #532	C₄×C₂₃	semidirect
306. C₂×#41	C₂×C₄×C₂₃	semidirect
318. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
319. #127	C₂×C₄×C₂₃	semidirect
320. #124	C₂×C₄×C₂₃	semidirect
321. #121	C₂×C₄×C₂₃	semidirect
322. #121	C₂×C₄×C₂₃	semidirect
325. C₄×D₄	C₂×C₄×C₂₃	semidirect
326. C₄×D₄	C₂×C₄×C₂₃	semidirect
341. C₂×C₄²	C₂×C₄×C₂₃	semidirect
360. C₂₃	#34398	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×#532	2
C₂₃×#2379	2
C₂₃×#2361	2
C₂₃×#2357	2
C₂₃×#2349	2
C₂₃×#2339	2
C₂₃×#2331	2
C₂₃×#2328	2
C₂₃×#2325	2
C₂₃×#2318	2
C₂₃×#2283	2
C₂×C₂₃×#382	2
#34398	23

Subgroups — C23×#34398 #19182024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34398 #19182024