Subgroups — C₂₃×#34350 #19181976

Order 5888.

2106 conjugacy classes of subgroups.

The subgroup lattice has 620 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×#2383	C₂	1	4	yes	maximal, complemented
3	C₂₃×#2382	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2379	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2378	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2377	C₂	1	2	yes	maximal
7	C₂₃×#2376	C₂	1	4	yes	maximal, complemented
8	C₂₃×#2298	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2296	C₂	1	2	yes	maximal, complemented
10	C₂₃×#2295	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2293	C₂	1	2	yes	maximal, complemented
12	C₂₃×#2290	C₂	1	2	yes	maximal, complemented
13	C₂₃×#2290	C₂	1	2	yes	maximal, complemented
14	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#2283	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	1	yes	characteristic
25	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
26	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#117	C₂²	1	2	yes	—
29	C₂×C₂₃×#117	C₂²	1	2	yes	—
30	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
31	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#391	C₂²	1	4	yes	—
33	C₂₃×#390	C₂²	1	4	yes	—
34	C₂₃×#389	C₂²	1	4	yes	—
35	C₂₃×#389	C₂²	1	4	yes	—
36	C₂₃×#389	C₂²	1	4	yes	—
37	C₂₃×#389	C₂²	1	4	yes	—
38	C₂₃×#388	C₂²	1	4	yes	—
39	C₂₃×#388	C₂²	1	4	yes	—
40	C₂₃×#388	C₂²	1	4	yes	—
41	C₂₃×#388	C₂²	1	4	yes	—
42	C₂₃×#388	C₂²	1	4	yes	—
43	C₂₃×#388	C₂²	1	4	yes	—
44	C₂₃×#387	C₂²	1	4	yes	—
45	C₂₃×#387	C₂²	1	4	yes	—
46	C₂₃×#387	C₂²	1	4	yes	—
47	C₂₃×#387	C₂²	1	4	yes	—
48	C₂₃×#387	C₂²	1	4	yes	—
49	C₂₃×#387	C₂²	1	4	yes	—
50	C₂₃×#386	C₂²	1	4	yes	—
51	C₂₃×#386	C₂²	1	4	yes	—
52	C₂₃×#385	C₂²	1	4	yes	—
53	C₂₃×#385	C₂²	1	4	yes	—
54	C₂₃×#385	C₂²	1	4	yes	—
55	C₂₃×#385	C₂²	1	4	yes	—
56	C₂₃×#384	C₂²	1	2	yes	—
57	C₂₃×#383	C₂²	1	2	yes	—
58	C₂₃×#382	C₂²	1	2	yes	—
59	C₂₃×#382	C₂²	1	2	yes	—
60	C₂₃×#381	C₂²	1	2	yes	—
61	C₂₃×#380	C₂²	1	2	yes	—
62	C₂₃×#380	C₂²	1	2	yes	—
63	C₂₃×#380	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
65	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
66	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
67	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
68	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
69	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
70	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
71	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
72	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
73	C₂₃×#376	C₂²	1	2	yes	—
74	C₂₃×#376	C₂²	1	2	yes	—
75	C₄³×C₂₃	C₂²	1	2	yes	—
76	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
79	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
80	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
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×#126	C₂×C₄	1	16	yes	complemented
85	C₂₃×#125	C₂×C₄	1	8	yes	complemented
86	C₂₃×#124	C₂×C₄	1	8	yes	—
87	C₂₃×#123	C₂×C₄	1	16	yes	complemented
88	C₂₃×C₂²⋊Q₈	C₂×C₄	1	8	yes	complemented
89	C₂₃×#121	C₂×C₄	1	8	yes	—
90	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
91	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	—
92	C₄×D₄×C₂₃	C₂×C₄	1	16	yes	complemented
93	C₂₃×#117	—	4	2	no	—
94	C₂₃×#117	—	4	4	no	—
95	C₂₃×#117	—	4	2	no	—
96	C₂₃×#117	C₂×C₄	1	16	yes	complemented
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	4	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	2	yes	—
105	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	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×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
112	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	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	2	yes	—
116	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₂₃	C₂³	1	4	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×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
123	C₂×C₄²×C₂₃	—	4	4	no	—
124	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
125	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
126	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
127	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
128	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
129	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
130	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
131	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
132	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
135	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
137	C₂₃×#95	C₂³	1	4	yes	—
138	C₂₃×#95	C₂³	1	4	yes	—
139	C₂₃×#95	C₂³	1	4	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	2	yes	—
147	C₂₃×#95	C₂³	1	2	yes	—
148	C₂₃×#95	C₂³	1	4	yes	—
149	C₂₃×#95	C₂³	1	4	yes	—
150	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
151	C₂⁴×C₂₃	C₂⁴	1	2	yes	—
152	C₂×Q₈×C₂₃	—	4	2	no	—
153	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
154	C₂×D₄×C₂₃	—	4	4	no	—
155	C₂×D₄×C₂₃	—	4	2	no	—
156	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
157	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
158	C₂²×C₄×C₂₃	—	2	4	no	—
159	C₂²×C₄×C₂₃	—	2	2	no	—
160	C₂²×C₄×C₂₃	—	2	4	no	—
161	C₂²×C₄×C₂₃	—	2	2	no	—
162	C₂²×C₄×C₂₃	—	4	4	no	—
163	C₂²×C₄×C₂₃	—	4	2	no	—
164	C₂²×C₄×C₂₃	—	4	2	no	—
165	C₂²×C₄×C₂₃	—	4	4	no	—
166	C₂²×C₄×C₂₃	—	4	4	no	—
167	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
168	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
169	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
170	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₂₃	C₂⁴	1	4	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	2	yes	—
176	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
178	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
184	C₄⋊C₄×C₂₃	—	4	4	no	—
185	C₄⋊C₄×C₂₃	—	4	4	no	—
186	C₄⋊C₄×C₂₃	—	4	2	no	—
187	C₄⋊C₄×C₂₃	—	4	4	no	—
188	C₄⋊C₄×C₂₃	—	4	4	no	—
189	C₄⋊C₄×C₂₃	—	4	2	no	—
190	C₄⋊C₄×C₂₃	—	4	4	no	—
191	C₄⋊C₄×C₂₃	—	4	2	no	—
192	C₄⋊C₄×C₂₃	—	4	2	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	—	4	4	no	—
200	C₂²⋊C₄×C₂₃	—	4	4	no	—
201	C₂²⋊C₄×C₂₃	—	4	4	no	—
202	C₂²⋊C₄×C₂₃	—	4	2	no	—
203	C₂²⋊C₄×C₂₃	—	4	4	no	—
204	C₂²⋊C₄×C₂₃	—	4	2	no	—
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	C₂²⋊C₄×C₂₃	C₂²×C₄	1	8	yes	—
210	C₄²×C₂₃	—	2	4	no	—
211	C₄²×C₂₃	—	2	4	no	—
212	C₄²×C₂₃	—	2	4	no	—
213	C₄²×C₂₃	—	2	4	no	—
214	C₄²×C₂₃	—	4	4	no	—
215	C₄²×C₂₃	—	4	4	no	—
216	C₄²×C₂₃	—	4	4	no	—
217	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
218	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
219	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
220	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
221	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
222	#34350	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
223	C₂³×C₂₃	—	2	1	no	—
224	C₂³×C₂₃	—	2	2	no	—
225	C₂³×C₂₃	—	2	1	no	—
226	C₂³×C₂₃	—	2	2	no	—
227	C₂³×C₂₃	—	2	2	no	—
228	C₂³×C₂₃	—	2	4	no	—
229	C₂³×C₂₃	—	2	4	no	—
230	C₂³×C₂₃	—	2	2	no	—
231	C₂³×C₂₃	C₂³×C₄	1	4	yes	—
232	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
233	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
234	Q₈×C₂₃	—	4	4	no	—
235	D₄×C₂₃	—	4	4	no	—
236	D₄×C₂₃	—	4	8	no	—
237	C₂×C₄×C₂₃	—	2	2	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	2	no	—
240	C₂×C₄×C₂₃	—	2	1	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	2	no	—
247	C₂×C₄×C₂₃	—	2	16	no	—
248	C₂×C₄×C₂₃	—	2	8	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	—	2	4	no	—
259	C₂×C₄×C₂₃	—	2	4	no	—
260	C₂×C₄×C₂₃	—	2	4	no	—
261	C₂×C₄×C₂₃	—	4	4	no	—
262	C₂×C₄×C₂₃	—	4	4	no	—
263	C₂×C₄×C₂₃	—	4	16	no	—
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×C₄×C₂₃	C₂³×C₄	1	2	yes	—
273	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
274	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
275	C₂×#529	C₂×C₂₃	1	1	yes	characteristic
276	#2383	C₂×C₂₃	1	4	yes	complemented
277	#2382	C₂×C₂₃	1	2	yes	complemented
278	#2379	C₂×C₂₃	1	2	yes	complemented
279	#2378	C₂×C₂₃	1	2	yes	complemented
280	#2377	C₂×C₂₃	1	2	yes	—
281	#2376	C₂×C₂₃	1	4	yes	complemented
282	#2298	C₂×C₂₃	1	2	yes	complemented
283	#2296	C₂×C₂₃	1	2	yes	complemented
284	#2295	C₂×C₂₃	1	2	yes	complemented
285	#2293	C₂×C₂₃	1	2	yes	complemented
286	#2290	C₂×C₂₃	1	2	yes	complemented
287	#2290	C₂×C₂₃	1	2	yes	complemented
288	#2285	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#2283	C₂×C₂₃	1	1	yes	characteristic, complemented
290	C₂²×C₂₃	—	2	2	no	—
291	C₂²×C₂₃	—	2	4	no	—
292	C₂²×C₂₃	—	2	2	no	—
293	C₂²×C₂₃	—	2	2	no	—
294	C₂²×C₂₃	—	2	4	no	—
295	C₂²×C₂₃	—	2	4	no	—
296	C₂²×C₂₃	—	4	4	no	—
297	C₂²×C₂₃	—	4	2	no	—
298	C₂²×C₂₃	—	4	2	no	—
299	C₂²×C₂₃	—	4	4	no	—
300	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
301	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
302	C₂²×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
303	C₂²×C₂₃	#585	1	2	yes	—
304	C₂²×C₂₃	#585	1	2	yes	—
305	C₄×C₂₃	—	2	4	no	—
306	C₄×C₂₃	—	2	2	no	—
307	C₄×C₂₃	—	2	2	no	—
308	C₄×C₂₃	—	2	2	no	—
309	C₄×C₂₃	—	2	2	no	—
310	C₄×C₂₃	—	2	2	no	—
311	C₄×C₂₃	—	2	4	no	—
312	C₄×C₂₃	—	2	4	no	—
313	C₄×C₂₃	—	4	4	no	—
314	C₄×C₂₃	—	4	4	no	—
315	C₄×C₂₃	—	4	4	no	—
316	C₄×C₂₃	—	4	4	no	—
317	C₄×C₂₃	#520	1	4	yes	—
318	#529	C₄×C₂₃	1	16	yes	complemented
319	C₂×#126	C₂²×C₂₃	1	2	yes	—
320	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×#123	C₂²×C₂₃	1	2	yes	—
323	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
325	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
326	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
328	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
329	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
330	C₂×#117	C₂²×C₂₃	1	2	yes	—
331	C₂×#117	C₂²×C₂₃	1	2	yes	—
332	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
333	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
334	#391	C₂²×C₂₃	1	4	yes	—
335	#390	C₂²×C₂₃	1	4	yes	—
336	#389	C₂²×C₂₃	1	4	yes	—
337	#389	C₂²×C₂₃	1	4	yes	—
338	#389	C₂²×C₂₃	1	4	yes	—
339	#389	C₂²×C₂₃	1	4	yes	—
340	#388	C₂²×C₂₃	1	4	yes	—
341	#388	C₂²×C₂₃	1	4	yes	—
342	#388	C₂²×C₂₃	1	4	yes	—
343	#388	C₂²×C₂₃	1	4	yes	—
344	#388	C₂²×C₂₃	1	4	yes	—
345	#388	C₂²×C₂₃	1	4	yes	—
346	#387	C₂²×C₂₃	1	4	yes	—
347	#387	C₂²×C₂₃	1	4	yes	—
348	#387	C₂²×C₂₃	1	4	yes	—
349	#387	C₂²×C₂₃	1	4	yes	—
350	#387	C₂²×C₂₃	1	4	yes	—
351	#387	C₂²×C₂₃	1	4	yes	—
352	#386	C₂²×C₂₃	1	4	yes	—
353	#386	C₂²×C₂₃	1	4	yes	—
354	#385	C₂²×C₂₃	1	4	yes	—
355	#385	C₂²×C₂₃	1	4	yes	—
356	#385	C₂²×C₂₃	1	4	yes	—
357	#385	C₂²×C₂₃	1	4	yes	—
358	#384	C₂²×C₂₃	1	2	yes	—
359	#383	C₂²×C₂₃	1	2	yes	—
360	#382	C₂²×C₂₃	1	2	yes	—
361	#382	C₂²×C₂₃	1	2	yes	—
362	#381	C₂²×C₂₃	1	2	yes	—
363	#380	C₂²×C₂₃	1	2	yes	—
364	#380	C₂²×C₂₃	1	2	yes	—
365	#380	C₂²×C₂₃	1	2	yes	—
366	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
367	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
368	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
369	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
370	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
371	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
372	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
373	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
374	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
375	#376	C₂²×C₂₃	1	2	yes	—
376	#376	C₂²×C₂₃	1	2	yes	—
377	C₄³	C₂²×C₂₃	1	2	yes	—
378	C₂×C₂₃	—	4	2	no	—
379	C₂×C₂₃	—	4	4	no	—
380	C₂×C₂₃	C₂×#520	1	1	yes	characteristic
381	C₂×C₂₃	#3428	1	1	yes	characteristic
382	C₂×C₂₃	#3428	1	1	yes	characteristic
383	C₂×C₂₃	#3529	1	2	yes	—
384	C₂×C₂₃	#3531	1	2	yes	—
385	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
386	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
387	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
388	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
389	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
390	C₂³×C₄	C₂³×C₂₃	1	4	yes	—
391	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
392	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
393	#126	C₂×C₄×C₂₃	1	16	yes	complemented
394	#125	C₂×C₄×C₂₃	1	8	yes	complemented
395	#124	C₂×C₄×C₂₃	1	8	yes	—
396	#123	C₂×C₄×C₂₃	1	16	yes	complemented
397	C₂²⋊Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
398	#121	C₂×C₄×C₂₃	1	8	yes	—
399	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
400	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	—
401	C₄×D₄	C₂×C₄×C₂₃	1	16	yes	complemented
402	#117	—	4	2	no	—
403	#117	—	4	4	no	—
404	#117	—	4	2	no	—
405	#117	C₂×C₄×C₂₃	1	16	yes	complemented
406	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
407	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
409	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
410	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
411	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
412	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
413	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
414	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
415	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
416	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
419	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
421	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
422	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
423	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
424	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
425	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
426	C₂×C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
427	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
432	C₂×C₄²	—	4	4	no	—
433	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
434	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
435	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
436	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
437	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
438	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
439	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
440	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
441	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
442	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
443	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
445	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
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	#95	C₂³×C₂₃	1	4	yes	—
455	#95	C₂³×C₂₃	1	2	yes	—
456	#95	C₂³×C₂₃	1	2	yes	—
457	#95	C₂³×C₂₃	1	4	yes	—
458	#95	C₂³×C₂₃	1	4	yes	—
459	C₂₃	#34350	1	1	yes	characteristic, direct product factor, Sylow
460	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
461	C₂⁴	C₂⁴×C₂₃	1	2	yes	—
462	C₂×Q₈	—	4	2	no	—
463	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
464	C₂×D₄	—	4	4	no	—
465	C₂×D₄	—	4	2	no	—
466	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
467	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
468	C₂²×C₄	—	2	4	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	4	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	—	4	2	no	—
474	C₂²×C₄	—	4	2	no	—
475	C₂²×C₄	—	4	4	no	—
476	C₂²×C₄	—	4	4	no	—
477	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
478	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
479	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
480	C₂²×C₄	C₂²×C₄×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	4	yes	—
484	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
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	2	yes	—
489	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
490	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
492	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
493	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
494	C₄⋊C₄	—	4	4	no	—
495	C₄⋊C₄	—	4	2	no	—
496	C₄⋊C₄	—	4	2	no	—
497	C₄⋊C₄	—	4	4	no	—
498	C₄⋊C₄	—	4	4	no	—
499	C₄⋊C₄	—	4	2	no	—
500	C₄⋊C₄	—	4	4	no	—
501	C₄⋊C₄	—	4	4	no	—
502	C₄⋊C₄	—	4	2	no	—
503	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
504	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
505	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
506	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
507	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
508	C₄⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
509	C₂²⋊C₄	—	4	2	no	—
510	C₂²⋊C₄	—	4	4	no	—
511	C₂²⋊C₄	—	4	2	no	—
512	C₂²⋊C₄	—	4	4	no	—
513	C₂²⋊C₄	—	4	4	no	—
514	C₂²⋊C₄	—	4	4	no	—
515	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
516	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
517	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
518	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
519	C₂²⋊C₄	C₂²×C₄×C₂₃	1	8	yes	—
520	C₄²	—	2	4	no	—
521	C₄²	—	2	4	no	—
522	C₄²	—	2	4	no	—
523	C₄²	—	2	4	no	—
524	C₄²	—	4	4	no	—
525	C₄²	—	4	4	no	—
526	C₄²	—	4	4	no	—
527	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
528	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
529	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
530	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
531	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
532	C₂³	—	2	1	no	—
533	C₂³	—	2	2	no	—
534	C₂³	—	2	1	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	2	4	no	—
538	C₂³	—	2	4	no	—
539	C₂³	—	2	2	no	—
540	C₂³	C₂³×C₄×C₂₃	1	4	yes	—
541	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
542	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
543	Q₈	—	4	4	no	—
544	D₄	—	4	4	no	—
545	D₄	—	4	8	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	16	no	—
557	C₂×C₄	—	2	8	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	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	4	4	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	16	no	—
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	2	yes	—
576	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
577	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
578	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
579	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
580	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
581	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
582	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
583	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	4	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	4	no	—
589	C₂²	—	2	4	no	—
590	C₂²	—	4	4	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	4	no	—
594	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
595	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
596	C₂²	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
597	C₂²	C₂₃×#585	1	2	yes	—
598	C₂²	C₂₃×#585	1	2	yes	—
599	C₄	—	2	4	no	—
600	C₄	—	2	2	no	—
601	C₄	—	2	2	no	—
602	C₄	—	2	2	no	—
603	C₄	—	2	2	no	—
604	C₄	—	2	2	no	—
605	C₄	—	2	4	no	—
606	C₄	—	2	4	no	—
607	C₄	—	4	4	no	—
608	C₄	—	4	4	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	4	no	—
611	C₄	C₂₃×#520	1	4	yes	—
612	C₂	—	4	2	no	—
613	C₂	—	4	4	no	—
614	C₂	C₂×C₂₃×#520	1	1	yes	characteristic
615	C₂	C₂₃×#3428	1	1	yes	characteristic
616	C₂	C₂₃×#3428	1	1	yes	characteristic
617	C₂	C₂₃×#3529	1	2	yes	—
618	C₂	C₂₃×#3531	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
socle	C₂³×C₂₃	C₂⁵
prime radical	C₂×C₂₃×#121	C₂²
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. C₂₃×#2383	C₂	semidirect
3. C₂₃×#2382	C₂	semidirect
4. C₂₃×#2379	C₂	semidirect
5. C₂₃×#2378	C₂	semidirect
7. C₂₃×#2376	C₂	semidirect
8. C₂₃×#2298	C₂	semidirect
9. C₂₃×#2296	C₂	semidirect
10. C₂₃×#2295	C₂	semidirect
11. C₂₃×#2293	C₂	semidirect
12. C₂₃×#2290	C₂	semidirect
13. C₂₃×#2290	C₂	semidirect
14. C₂₃×#2285	C₂	semidirect
15. C₂₃×#2283	C₂	semidirect
16. C₂₃×#529	C₄	semidirect
84. C₂₃×#126	C₂×C₄	semidirect
85. C₂₃×#125	C₂×C₄	semidirect
87. C₂₃×#123	C₂×C₄	semidirect
88. C₂₃×C₂²⋊Q₈	C₂×C₄	semidirect
90. C₄×Q₈×C₂₃	C₂×C₄	semidirect
92. C₄×D₄×C₂₃	C₂×C₄	semidirect
96. C₂₃×#117	C₂×C₄	semidirect
124. C₂×C₄²×C₂₃	C₂×C₄	semidirect
222. #34350	C₂₃	direct (N × H)
276. #2383	C₂×C₂₃	semidirect
277. #2382	C₂×C₂₃	semidirect
278. #2379	C₂×C₂₃	semidirect
279. #2378	C₂×C₂₃	semidirect
281. #2376	C₂×C₂₃	semidirect
282. #2298	C₂×C₂₃	semidirect
283. #2296	C₂×C₂₃	semidirect
284. #2295	C₂×C₂₃	semidirect
285. #2293	C₂×C₂₃	semidirect
286. #2290	C₂×C₂₃	semidirect
287. #2290	C₂×C₂₃	semidirect
288. #2285	C₂×C₂₃	semidirect
289. #2283	C₂×C₂₃	semidirect
318. #529	C₄×C₂₃	semidirect
393. #126	C₂×C₄×C₂₃	semidirect
394. #125	C₂×C₄×C₂₃	semidirect
396. #123	C₂×C₄×C₂₃	semidirect
397. C₂²⋊Q₈	C₂×C₄×C₂₃	semidirect
399. C₄×Q₈	C₂×C₄×C₂₃	semidirect
401. C₄×D₄	C₂×C₄×C₂₃	semidirect
405. #117	C₂×C₄×C₂₃	semidirect
433. C₂×C₄²	C₂×C₄×C₂₃	semidirect
459. C₂₃	#34350	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×#2383	2
C₂₃×#2382	2
C₂₃×#2379	2
C₂₃×#2378	2
C₂₃×#2377	2
C₂₃×#2376	2
C₂₃×#2298	2
C₂₃×#2296	2
C₂₃×#2295	2
C₂₃×#2293	2
C₂₃×#2290	2
C₂₃×#2285	2
C₂₃×#2283	2
#34350	23

Subgroups — C23×#34350 #19181976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34350 #19181976