Subgroups — C₂₃×#17581 #19165207

Order 5888.

1070 conjugacy classes of subgroups.

The subgroup lattice has 642 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	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1520	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1519	C₂	1	2	yes	maximal
6	C₂₃×#1512	C₂	1	4	yes	maximal, complemented
7	C₂₃×#1509	C₂	1	4	yes	maximal, complemented
8	C₂₃×#529	—	2	1	no	—
9	C₂₃×#529	—	2	1	no	—
10	C₂₃×#529	—	2	1	no	—
11	C₂₃×#529	—	2	1	no	—
12	C₂₃×#529	C₄	1	2	yes	complemented
13	C₂₃×#529	C₄	1	2	yes	complemented
14	C₂₃×#529	C₂²	1	2	yes	—
15	C₂₃×#529	C₂²	1	2	yes	—
16	C₂×C₂₃×#126	—	2	1	no	—
17	C₂×C₂₃×#125	C₄	1	1	yes	characteristic, complemented
18	C₂×C₂₃×#124	C₄	1	1	yes	characteristic, complemented
19	C₂×C₂₃×#123	—	2	1	no	—
20	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₂₃	C₄	1	1	yes	characteristic, complemented
23	C₂×C₄×D₄×C₂₃	—	2	1	no	—
24	C₂×C₄×D₄×C₂₃	C₄	1	1	yes	characteristic, complemented
25	C₂×C₂₃×#117	—	2	1	no	—
26	C₂×C₂₃×#117	—	2	1	no	—
27	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#432	—	2	1	no	—
29	C₂₃×#432	—	2	1	no	—
30	C₂₃×#432	C₂²	1	2	yes	complemented
31	C₂₃×#432	C₂²	1	2	yes	complemented
32	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
33	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
34	C₂×C₂₃×#97	—	2	1	no	—
35	C₂₃×#384	—	2	1	no	—
36	C₂₃×#382	C₂²	1	2	yes	complemented
37	C₂₃×#380	—	2	1	no	—
38	C₂₃×#380	C₂²	1	2	yes	—
39	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
40	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
41	C₂₃×#328	C₂²	1	8	yes	complemented
42	C₂₃×#327	C₂²	1	8	yes	—
43	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₂₃	—	2	1	no	—
45	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₂₃	—	2	1	no	—
47	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
49	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₂₃	—	2	2	no	—
51	C₂₃×#126	—	2	2	no	—
52	C₂₃×#126	—	2	2	no	—
53	C₂₃×#126	—	2	2	no	—
54	C₂₃×#126	—	2	2	no	—
55	C₂₃×#125	—	2	1	no	—
56	C₂₃×#125	—	2	1	no	—
57	C₂₃×#125	C₂×C₄	1	2	yes	—
58	C₂₃×#125	C₂×C₄	1	2	yes	—
59	C₂₃×#124	—	2	1	no	—
60	C₂₃×#124	—	2	1	no	—
61	C₂₃×#124	C₂×C₄	1	2	yes	—
62	C₂₃×#124	C₂×C₄	1	2	yes	—
63	C₂₃×#123	—	2	2	no	—
64	C₂₃×#123	—	2	2	no	—
65	C₂₃×#123	—	2	2	no	—
66	C₂₃×#123	—	2	2	no	—
67	C₂₃×C₂²⋊Q₈	—	2	1	no	—
68	C₂₃×C₂²⋊Q₈	—	2	1	no	—
69	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
70	C₂₃×C₂²⋊Q₈	C₂³	1	2	yes	—
71	C₂₃×#121	—	2	1	no	—
72	C₂₃×#121	—	2	1	no	—
73	C₂₃×#121	C₂×C₄	1	2	yes	—
74	C₂₃×#121	C₂³	1	2	yes	—
75	C₄×Q₈×C₂₃	—	2	1	no	—
76	C₄×Q₈×C₂₃	—	2	1	no	—
77	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
78	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
79	C₄×D₄×C₂₃	—	2	2	no	—
80	C₄×D₄×C₂₃	—	2	2	no	—
81	C₄×D₄×C₂₃	—	2	2	no	—
82	C₄×D₄×C₂₃	—	2	2	no	—
83	C₄×D₄×C₂₃	—	2	1	no	—
84	C₄×D₄×C₂₃	—	2	1	no	—
85	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
86	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
87	C₂₃×#117	—	2	2	no	—
88	C₂₃×#117	—	2	2	no	—
89	C₂₃×#117	—	2	2	no	—
90	C₂₃×#117	—	2	2	no	—
91	C₂₃×#117	—	4	4	no	—
92	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
106	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
107	C₂×C₄²×C₂₃	—	2	1	no	—
108	C₂×C₄²×C₂₃	—	2	2	no	—
109	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
110	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
111	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
112	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
113	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
114	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
115	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
116	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
117	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
118	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
119	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
121	C₂₃×C₄⋊C₈	—	2	2	no	—
122	C₂₃×C₄⋊C₈	—	2	2	no	—
123	C₂₃×C₂²⋊C₈	—	2	2	no	—
124	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
125	C₂₃×#97	—	2	2	no	—
126	C₂₃×#97	—	2	2	no	—
127	C₂₃×#95	—	2	2	no	—
128	C₂₃×#95	C₂³	1	4	yes	—
129	C₂⁴×C₂₃	—	2	1	no	—
130	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Q₈×C₂₃	—	2	1	no	—
132	C₂×Q₈×C₂₃	—	4	2	no	—
133	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
134	C₂×D₄×C₂₃	—	2	2	no	—
135	C₂×D₄×C₂₃	—	2	2	no	—
136	C₂×D₄×C₂₃	—	2	1	no	—
137	C₂×D₄×C₂₃	—	4	2	no	—
138	C₂×D₄×C₂₃	—	4	2	no	—
139	C₂×D₄×C₂₃	—	4	2	no	—
140	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
141	C₂²×C₄×C₂₃	—	2	2	no	—
142	C₂²×C₄×C₂₃	—	2	2	no	—
143	C₂²×C₄×C₂₃	—	2	1	no	—
144	C₂²×C₄×C₂₃	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	2	2	no	—
147	C₂²×C₄×C₂₃	—	2	2	no	—
148	C₂²×C₄×C₂₃	—	2	4	no	—
149	C₂²×C₄×C₂₃	—	4	2	no	—
150	C₂²×C₄×C₂₃	—	4	2	no	—
151	C₂²×C₄×C₂₃	—	4	4	no	—
152	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
170	C₂×C₈×C₂₃	—	2	4	no	—
171	C₂×C₈×C₂₃	—	4	4	no	—
172	C₄⋊C₄×C₂₃	—	2	2	no	—
173	C₄⋊C₄×C₂₃	—	2	2	no	—
174	C₄⋊C₄×C₂₃	—	2	2	no	—
175	C₄⋊C₄×C₂₃	—	2	2	no	—
176	C₄⋊C₄×C₂₃	—	2	2	no	—
177	C₄⋊C₄×C₂₃	—	2	2	no	—
178	C₄⋊C₄×C₂₃	—	2	2	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	2	no	—
182	C₄⋊C₄×C₂₃	—	4	4	no	—
183	C₄⋊C₄×C₂₃	—	4	4	no	—
184	C₄⋊C₄×C₂₃	—	4	4	no	—
185	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
186	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
187	C₂²⋊C₄×C₂₃	—	2	2	no	—
188	C₂²⋊C₄×C₂₃	—	2	2	no	—
189	C₂²⋊C₄×C₂₃	—	2	2	no	—
190	C₂²⋊C₄×C₂₃	—	2	2	no	—
191	C₂²⋊C₄×C₂₃	—	2	2	no	—
192	C₂²⋊C₄×C₂₃	—	2	2	no	—
193	C₂²⋊C₄×C₂₃	—	2	2	no	—
194	C₂²⋊C₄×C₂₃	—	2	2	no	—
195	C₂²⋊C₄×C₂₃	—	2	2	no	—
196	C₂²⋊C₄×C₂₃	—	2	2	no	—
197	C₂²⋊C₄×C₂₃	—	4	4	no	—
198	C₄²×C₂₃	—	2	2	no	—
199	C₄²×C₂₃	—	2	2	no	—
200	C₄²×C₂₃	—	4	4	no	—
201	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
202	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
203	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
204	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
205	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
206	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
207	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
208	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
209	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
210	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
211	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
213	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
214	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
217	#17581	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
218	C₂³×C₂₃	—	2	2	no	—
219	C₂³×C₂₃	—	4	1	no	—
220	C₂³×C₂₃	—	4	1	no	—
221	C₂³×C₂₃	—	4	2	no	—
222	C₂³×C₂₃	—	4	2	no	—
223	C₂³×C₂₃	—	4	2	no	—
224	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
225	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
226	C₂³×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
227	C₂³×C₂₃	#137	1	2	yes	—
228	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
229	Q₈×C₂₃	—	4	2	no	—
230	Q₈×C₂₃	—	4	2	no	—
231	D₄×C₂₃	—	4	2	no	—
232	D₄×C₂₃	—	4	2	no	—
233	D₄×C₂₃	—	4	2	no	—
234	D₄×C₂₃	—	4	4	no	—
235	D₄×C₂₃	—	4	2	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	1	no	—
240	C₂×C₄×C₂₃	—	2	2	no	—
241	C₂×C₄×C₂₃	—	2	2	no	—
242	C₂×C₄×C₂₃	—	2	2	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	2	no	—
248	C₂×C₄×C₂₃	—	2	2	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	2	no	—
251	C₂×C₄×C₂₃	—	2	2	no	—
252	C₂×C₄×C₂₃	—	2	2	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	4	1	no	—
258	C₂×C₄×C₂₃	—	4	1	no	—
259	C₂×C₄×C₂₃	—	4	4	no	—
260	C₂×C₄×C₂₃	—	4	2	no	—
261	C₂×C₄×C₂₃	—	4	2	no	—
262	C₂×C₄×C₂₃	—	4	2	no	—
263	C₂×C₄×C₂₃	—	4	4	no	—
264	C₂×C₄×C₂₃	—	4	4	no	—
265	C₂×C₄×C₂₃	—	4	4	no	—
266	C₂×C₄×C₂₃	—	4	4	no	—
267	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
268	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
269	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
270	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
272	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
273	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
278	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
279	C₈×C₂₃	—	4	4	no	—
280	C₂×#529	C₂×C₂₃	1	1	yes	characteristic
281	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
282	#2285	C₂×C₂₃	1	1	yes	characteristic, complemented
283	#1520	C₂×C₂₃	1	2	yes	complemented
284	#1519	C₂×C₂₃	1	2	yes	—
285	#1512	C₂×C₂₃	1	4	yes	complemented
286	#1509	C₂×C₂₃	1	4	yes	complemented
287	C₂²×C₂₃	—	2	1	no	—
288	C₂²×C₂₃	—	2	1	no	—
289	C₂²×C₂₃	—	2	1	no	—
290	C₂²×C₂₃	—	2	1	no	—
291	C₂²×C₂₃	—	2	1	no	—
292	C₂²×C₂₃	—	2	1	no	—
293	C₂²×C₂₃	—	4	2	no	—
294	C₂²×C₂₃	—	4	2	no	—
295	C₂²×C₂₃	—	4	2	no	—
296	C₂²×C₂₃	—	4	2	no	—
297	C₂²×C₂₃	—	4	2	no	—
298	C₂²×C₂₃	—	8	4	no	—
299	C₂²×C₂₃	#417	1	2	yes	—
300	C₂²×C₂₃	#417	1	2	yes	—
301	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
302	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
303	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
304	C₄×C₂₃	—	2	1	no	—
305	C₄×C₂₃	—	2	1	no	—
306	C₄×C₂₃	—	2	1	no	—
307	C₄×C₂₃	—	2	1	no	—
308	C₄×C₂₃	—	2	1	no	—
309	C₄×C₂₃	—	2	1	no	—
310	C₄×C₂₃	—	2	1	no	—
311	C₄×C₂₃	—	2	1	no	—
312	C₄×C₂₃	—	2	1	no	—
313	C₄×C₂₃	—	2	1	no	—
314	C₄×C₂₃	—	4	2	no	—
315	C₄×C₂₃	—	4	2	no	—
316	C₄×C₂₃	—	4	2	no	—
317	C₄×C₂₃	—	8	4	no	—
318	C₄×C₂₃	#410	1	2	yes	—
319	C₄×C₂₃	#421	1	2	yes	—
320	#529	—	2	1	no	—
321	#529	—	2	1	no	—
322	#529	—	2	1	no	—
323	#529	—	2	1	no	—
324	#529	C₄×C₂₃	1	2	yes	complemented
325	#529	C₄×C₂₃	1	2	yes	complemented
326	#529	C₂²×C₂₃	1	2	yes	—
327	#529	C₂²×C₂₃	1	2	yes	—
328	C₂×#126	—	2	1	no	—
329	C₂×#125	C₄×C₂₃	1	1	yes	characteristic, complemented
330	C₂×#124	C₄×C₂₃	1	1	yes	characteristic, complemented
331	C₂×#123	—	2	1	no	—
332	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
333	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
334	C₂×C₄×Q₈	C₄×C₂₃	1	1	yes	characteristic, complemented
335	C₂×C₄×D₄	—	2	1	no	—
336	C₂×C₄×D₄	C₄×C₂₃	1	1	yes	characteristic, complemented
337	C₂×#117	—	2	1	no	—
338	C₂×#117	—	2	1	no	—
339	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
340	#432	—	2	1	no	—
341	#432	—	2	1	no	—
342	#432	C₂²×C₂₃	1	2	yes	complemented
343	#432	C₂²×C₂₃	1	2	yes	complemented
344	C₂×C₄⋊C₈	—	2	1	no	—
345	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
346	C₂×#97	—	2	1	no	—
347	#384	—	2	1	no	—
348	#382	C₂²×C₂₃	1	2	yes	complemented
349	#380	—	2	1	no	—
350	#380	C₂²×C₂₃	1	2	yes	—
351	C₄×C₄⋊C₄	—	2	1	no	—
352	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
353	#328	C₂²×C₂₃	1	8	yes	complemented
354	#327	C₂²×C₂₃	1	8	yes	—
355	C₂×C₂₃	—	2	1	no	—
356	C₂×C₂₃	—	2	2	no	—
357	C₂×C₂₃	—	2	1	no	—
358	C₂×C₂₃	—	2	2	no	—
359	C₂×C₂₃	—	8	2	no	—
360	C₂×C₂₃	C₂×#410	1	1	yes	characteristic
361	C₂×C₂₃	C₂×#417	1	1	yes	characteristic
362	C₂×C₂₃	C₂×#421	1	1	yes	characteristic
363	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
364	C₂²×D₄	—	2	1	no	—
365	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
366	C₂³×C₄	—	2	1	no	—
367	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
368	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
369	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
370	C₂²×C₈	—	2	2	no	—
371	#126	—	2	2	no	—
372	#126	—	2	2	no	—
373	#126	—	2	2	no	—
374	#126	—	2	2	no	—
375	#125	—	2	1	no	—
376	#125	—	2	1	no	—
377	#125	C₂×C₄×C₂₃	1	2	yes	—
378	#125	C₂×C₄×C₂₃	1	2	yes	—
379	#124	—	2	1	no	—
380	#124	—	2	1	no	—
381	#124	C₂×C₄×C₂₃	1	2	yes	—
382	#124	C₂×C₄×C₂₃	1	2	yes	—
383	#123	—	2	2	no	—
384	#123	—	2	2	no	—
385	#123	—	2	2	no	—
386	#123	—	2	2	no	—
387	C₂²⋊Q₈	—	2	1	no	—
388	C₂²⋊Q₈	—	2	1	no	—
389	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
390	C₂²⋊Q₈	C₂³×C₂₃	1	2	yes	—
391	#121	—	2	1	no	—
392	#121	—	2	1	no	—
393	#121	C₂×C₄×C₂₃	1	2	yes	—
394	#121	C₂³×C₂₃	1	2	yes	—
395	C₄×Q₈	—	2	1	no	—
396	C₄×Q₈	—	2	1	no	—
397	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
398	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	—	2	1	no	—
405	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
406	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
407	#117	—	2	2	no	—
408	#117	—	2	2	no	—
409	#117	—	2	2	no	—
410	#117	—	2	2	no	—
411	#117	—	4	4	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	1	no	—
416	C₂×C₄⋊C₄	—	2	2	no	—
417	C₂×C₄⋊C₄	—	2	2	no	—
418	C₂×C₄⋊C₄	—	2	2	no	—
419	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
421	C₂×C₂²⋊C₄	—	2	1	no	—
422	C₂×C₂²⋊C₄	—	2	1	no	—
423	C₂×C₂²⋊C₄	—	2	1	no	—
424	C₂×C₂²⋊C₄	—	2	1	no	—
425	C₂×C₂²⋊C₄	—	2	1	no	—
426	C₂×C₂²⋊C₄	—	2	2	no	—
427	C₂×C₄²	—	2	1	no	—
428	C₂×C₄²	—	2	2	no	—
429	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
430	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
431	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
432	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
433	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
434	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
435	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
436	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
437	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
438	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
439	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
440	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
441	C₄⋊C₈	—	2	2	no	—
442	C₄⋊C₈	—	2	2	no	—
443	C₂²⋊C₈	—	2	2	no	—
444	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
445	#97	—	2	2	no	—
446	#97	—	2	2	no	—
447	#95	—	2	2	no	—
448	#95	C₂³×C₂₃	1	4	yes	—
449	C₂₃	#17581	1	1	yes	characteristic, direct product factor, Sylow
450	C₂⁴	—	2	1	no	—
451	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	—	4	2	no	—
454	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
455	C₂×D₄	—	2	2	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	1	no	—
458	C₂×D₄	—	4	2	no	—
459	C₂×D₄	—	4	2	no	—
460	C₂×D₄	—	4	2	no	—
461	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	2	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	4	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	—	4	2	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
474	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
475	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
476	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
477	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
478	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
479	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
480	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
481	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
482	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
486	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
487	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
488	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
489	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
490	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
491	C₂×C₈	—	2	4	no	—
492	C₂×C₈	—	4	4	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	4	4	no	—
504	C₄⋊C₄	—	4	4	no	—
505	C₄⋊C₄	—	4	4	no	—
506	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
507	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
508	C₂²⋊C₄	—	2	2	no	—
509	C₂²⋊C₄	—	2	2	no	—
510	C₂²⋊C₄	—	2	2	no	—
511	C₂²⋊C₄	—	2	2	no	—
512	C₂²⋊C₄	—	2	2	no	—
513	C₂²⋊C₄	—	2	2	no	—
514	C₂²⋊C₄	—	2	2	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	4	4	no	—
519	C₄²	—	2	2	no	—
520	C₄²	—	2	2	no	—
521	C₄²	—	4	4	no	—
522	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
523	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
524	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
525	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
526	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
527	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
528	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
529	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
530	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
531	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
532	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
533	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
534	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
535	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
536	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
537	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
538	C₂³	—	2	2	no	—
539	C₂³	—	4	1	no	—
540	C₂³	—	4	1	no	—
541	C₂³	—	4	2	no	—
542	C₂³	—	4	2	no	—
543	C₂³	—	4	2	no	—
544	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
545	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
546	C₂³	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
547	C₂³	C₂₃×#137	1	2	yes	—
548	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
549	Q₈	—	4	2	no	—
550	Q₈	—	4	2	no	—
551	D₄	—	4	2	no	—
552	D₄	—	4	2	no	—
553	D₄	—	4	2	no	—
554	D₄	—	4	4	no	—
555	D₄	—	4	2	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	4	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	2	no	—
582	C₂×C₄	—	4	2	no	—
583	C₂×C₄	—	4	4	no	—
584	C₂×C₄	—	4	4	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	4	no	—
587	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
588	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
589	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
590	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
591	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
592	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
593	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
594	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
595	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
596	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
597	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
598	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
599	C₈	—	4	4	no	—
600	C₂²	—	2	1	no	—
601	C₂²	—	2	1	no	—
602	C₂²	—	2	1	no	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	2	1	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	8	4	no	—
612	C₂²	C₂₃×#417	1	2	yes	—
613	C₂²	C₂₃×#417	1	2	yes	—
614	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
615	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
616	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
617	C₄	—	2	1	no	—
618	C₄	—	2	1	no	—
619	C₄	—	2	1	no	—
620	C₄	—	2	1	no	—
621	C₄	—	2	1	no	—
622	C₄	—	2	1	no	—
623	C₄	—	2	1	no	—
624	C₄	—	2	1	no	—
625	C₄	—	2	1	no	—
626	C₄	—	2	1	no	—
627	C₄	—	4	2	no	—
628	C₄	—	4	2	no	—
629	C₄	—	4	2	no	—
630	C₄	—	8	4	no	—
631	C₄	C₂₃×#410	1	2	yes	—
632	C₄	C₂₃×#421	1	2	yes	—
633	C₂	—	2	1	no	—
634	C₂	—	2	2	no	—
635	C₂	—	2	1	no	—
636	C₂	—	2	2	no	—
637	C₂	—	8	2	no	—
638	C₂	C₂×C₂₃×#410	1	1	yes	characteristic
639	C₂	C₂×C₂₃×#417	1	1	yes	characteristic
640	C₂	C₂×C₂₃×#421	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#432	C₂	semidirect
3. C₂₃×#2285	C₂	semidirect
4. C₂₃×#1520	C₂	semidirect
6. C₂₃×#1512	C₂	semidirect
7. C₂₃×#1509	C₂	semidirect
12. C₂₃×#529	C₄	semidirect
13. C₂₃×#529	C₄	semidirect
17. C₂×C₂₃×#125	C₄	semidirect
18. C₂×C₂₃×#124	C₄	semidirect
22. C₂×C₄×Q₈×C₂₃	C₄	semidirect
24. C₂×C₄×D₄×C₂₃	C₄	semidirect
30. C₂₃×#432	C₂²	semidirect
31. C₂₃×#432	C₂²	semidirect
36. C₂₃×#382	C₂²	semidirect
40. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
41. C₂₃×#328	C₂²	semidirect
217. #17581	C₂₃	direct (N × H)
281. C₂×#432	C₂×C₂₃	semidirect
282. #2285	C₂×C₂₃	semidirect
283. #1520	C₂×C₂₃	semidirect
285. #1512	C₂×C₂₃	semidirect
286. #1509	C₂×C₂₃	semidirect
324. #529	C₄×C₂₃	semidirect
325. #529	C₄×C₂₃	semidirect
329. C₂×#125	C₄×C₂₃	semidirect
330. C₂×#124	C₄×C₂₃	semidirect
334. C₂×C₄×Q₈	C₄×C₂₃	semidirect
336. C₂×C₄×D₄	C₄×C₂₃	semidirect
342. #432	C₂²×C₂₃	semidirect
343. #432	C₂²×C₂₃	semidirect
348. #382	C₂²×C₂₃	semidirect
352. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
353. #328	C₂²×C₂₃	semidirect
449. C₂₃	#17581	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
C₂×C₂₃×#432	2
C₂₃×#2285	2
C₂₃×#1520	2
C₂₃×#1519	2
C₂₃×#1512	2
C₂₃×#1509	2
#17581	23

Subgroups — C23×#17581 #19165207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17581 #19165207