Subgroups — C₂₃×#21505 #19169131

Order 5888.

1140 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#460	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#440	C₂	1	2	yes	maximal, complemented
3	C₄×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
4	C₄²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1982	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1925	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1880	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1803	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1762	C₂	1	2	yes	maximal, complemented
12	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₂₃	—	2	1	no	—
18	C₂×C₄×D₄×C₂₃	—	2	1	no	—
19	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
20	C₂₃×#460	C₄	1	8	yes	complemented
21	C₂₃×#440	—	2	8	no	—
22	C₂×C₂₃×#107	C₂²	1	2	yes	—
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#103	C₂²	1	2	yes	—
26	C₂×C₂₃×#102	C₂²	1	2	yes	—
27	C₂×C₂₃×#102	C₂²	1	2	yes	—
28	C₂×C₂₃×#102	C₂²	1	2	yes	—
29	C₂×C₂₃×#97	C₂²	1	2	yes	—
30	C₂₃×#390	—	2	1	no	—
31	C₂₃×#390	—	2	1	no	—
32	C₂₃×#390	C₂²	1	2	yes	—
33	C₂₃×#384	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
35	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
36	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
37	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
40	C₄³×C₂₃	—	2	1	no	—
41	C₄³×C₂₃	C₂²	1	1	yes	characteristic
42	C₂₃×#340	C₂²	1	2	yes	—
43	C₂₃×#340	C₂²	1	2	yes	—
44	C₂₃×#336	C₂²	1	2	yes	—
45	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₂₃	—	2	1	no	—
47	C₂²×D₄×C₂₃	—	2	1	no	—
48	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
49	C₂³×C₄×C₂₃	—	2	2	no	—
50	C₂×SD₁₆×C₂₃	—	2	4	no	—
51	C₂×SD₁₆×C₂₃	—	2	4	no	—
52	C₂×SD₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
53	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
55	C₂₃×#127	—	2	2	no	—
56	C₂₃×#127	C₂×C₄	1	4	yes	—
57	C₄×Q₈×C₂₃	—	2	2	no	—
58	C₄×Q₈×C₂₃	—	2	2	no	—
59	C₄×Q₈×C₂₃	—	2	2	no	—
60	C₄×Q₈×C₂₃	—	2	2	no	—
61	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	complemented
62	C₄×Q₈×C₂₃	D₄	1	4	yes	complemented
63	C₄×D₄×C₂₃	—	2	4	no	—
64	C₄×D₄×C₂₃	—	2	4	no	—
65	C₄×D₄×C₂₃	—	4	4	no	—
66	C₄×D₄×C₂₃	—	4	4	no	—
67	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
69	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
70	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
71	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
72	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
76	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
79	C₂×C₄²×C₂₃	—	2	1	no	—
80	C₂×C₄²×C₂₃	—	2	1	no	—
81	C₂×C₄²×C₂₃	—	2	1	no	—
82	C₂×C₄²×C₂₃	—	2	1	no	—
83	C₂×C₄²×C₂₃	—	2	1	no	—
84	C₂×C₄²×C₂₃	—	2	1	no	—
85	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
86	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
89	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
92	C₂₃×#107	—	2	4	no	—
93	C₂₃×C₄⋊C₈	—	2	2	no	—
94	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
95	C₂₃×#103	—	2	4	no	—
96	C₂₃×#102	—	2	4	no	—
97	C₂₃×#102	—	2	4	no	—
98	C₂₃×#102	C₂×C₄	1	8	yes	complemented
99	C₂₃×#97	—	2	4	no	—
100	C₂₃×#95	—	2	1	no	—
101	C₂₃×#95	—	2	1	no	—
102	C₂₃×#95	—	2	1	no	—
103	C₂⁴×C₂₃	—	2	2	no	—
104	C₂×Q₈×C₂₃	—	2	1	no	—
105	C₂×Q₈×C₂₃	—	2	1	no	—
106	C₂×Q₈×C₂₃	—	2	1	no	—
107	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
108	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
109	C₂×Q₈×C₂₃	#41	1	2	yes	—
110	C₂×D₄×C₂₃	—	2	4	no	—
111	C₂×D₄×C₂₃	—	2	4	no	—
112	C₂×D₄×C₂₃	—	2	2	no	—
113	C₂×D₄×C₂₃	—	2	2	no	—
114	C₂×D₄×C₂₃	—	2	2	no	—
115	C₂×D₄×C₂₃	—	2	2	no	—
116	C₂×D₄×C₂₃	—	2	2	no	—
117	C₂×D₄×C₂₃	—	4	2	no	—
118	C₂×D₄×C₂₃	—	4	2	no	—
119	C₂×D₄×C₂₃	—	4	2	no	—
120	C₂×D₄×C₂₃	—	4	2	no	—
121	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
122	C₂²×C₄×C₂₃	—	2	1	no	—
123	C₂²×C₄×C₂₃	—	2	1	no	—
124	C₂²×C₄×C₂₃	—	2	1	no	—
125	C₂²×C₄×C₂₃	—	2	1	no	—
126	C₂²×C₄×C₂₃	—	2	2	no	—
127	C₂²×C₄×C₂₃	—	4	4	no	—
128	C₂²×C₄×C₂₃	—	4	4	no	—
129	C₂²×C₄×C₂₃	—	4	4	no	—
130	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
137	SD₁₆×C₂₃	—	2	16	no	—
138	C₂×C₈×C₂₃	—	2	2	no	—
139	C₂×C₈×C₂₃	—	2	2	no	—
140	C₂×C₈×C₂₃	—	2	2	no	—
141	C₂×C₈×C₂₃	—	2	2	no	—
142	C₂×C₈×C₂₃	—	2	2	no	—
143	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
144	C₄⋊C₄×C₂₃	—	2	4	no	—
145	C₄⋊C₄×C₂₃	—	2	4	no	—
146	C₄⋊C₄×C₂₃	—	2	4	no	—
147	C₄⋊C₄×C₂₃	—	2	1	no	—
148	C₄⋊C₄×C₂₃	—	2	1	no	—
149	C₄⋊C₄×C₂₃	—	2	1	no	—
150	C₄⋊C₄×C₂₃	—	2	2	no	—
151	C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₄⋊C₄×C₂₃	—	4	2	no	—
153	C₄⋊C₄×C₂₃	—	4	2	no	—
154	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
155	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₂₃	#41	1	2	yes	—
157	C₂²⋊C₄×C₂₃	—	4	4	no	—
158	C₂²⋊C₄×C₂₃	—	4	4	no	—
159	C₂²⋊C₄×C₂₃	—	4	4	no	—
160	C₄²×C₂₃	—	2	4	no	—
161	C₄²×C₂₃	—	2	4	no	—
162	C₄²×C₂₃	—	2	4	no	—
163	C₄²×C₂₃	—	2	2	no	—
164	C₄²×C₂₃	—	2	2	no	—
165	C₄²×C₂₃	—	2	2	no	—
166	C₄²×C₂₃	—	2	1	no	—
167	C₄²×C₂₃	—	2	1	no	—
168	C₄²×C₂₃	—	4	1	no	—
169	C₄²×C₂₃	—	4	2	no	—
170	C₄²×C₂₃	—	4	2	no	—
171	C₄²×C₂₃	—	4	2	no	—
172	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
173	C₄²×C₂₃	C₂×D₄	1	2	yes	—
174	C₄²×C₂₃	#41	1	2	yes	—
175	C₄²×C₂₃	#41	1	2	yes	—
176	#21505	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₂₃	—	2	4	no	—
178	C₂³×C₂₃	—	4	2	no	—
179	C₂³×C₂₃	—	4	2	no	—
180	C₂³×C₂₃	—	4	2	no	—
181	C₂³×C₂₃	—	4	2	no	—
182	C₂³×C₂₃	—	4	2	no	—
183	C₂³×C₂₃	—	4	2	no	—
184	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
185	Q₈×C₂₃	—	2	2	no	—
186	Q₈×C₂₃	—	2	2	no	—
187	Q₈×C₂₃	—	2	2	no	—
188	Q₈×C₂₃	C₄×D₄	1	4	yes	complemented
189	D₄×C₂₃	—	2	4	no	—
190	D₄×C₂₃	—	2	4	no	—
191	D₄×C₂₃	—	4	8	no	—
192	D₄×C₂₃	—	4	2	no	—
193	D₄×C₂₃	—	4	2	no	—
194	D₄×C₂₃	—	4	8	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	2	1	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	1	no	—
200	C₂×C₄×C₂₃	—	2	1	no	—
201	C₂×C₄×C₂₃	—	2	1	no	—
202	C₂×C₄×C₂₃	—	2	1	no	—
203	C₂×C₄×C₂₃	—	2	1	no	—
204	C₂×C₄×C₂₃	—	2	1	no	—
205	C₂×C₄×C₂₃	—	2	1	no	—
206	C₂×C₄×C₂₃	—	2	2	no	—
207	C₂×C₄×C₂₃	—	2	2	no	—
208	C₂×C₄×C₂₃	—	2	1	no	—
209	C₂×C₄×C₂₃	—	2	2	no	—
210	C₂×C₄×C₂₃	—	2	1	no	—
211	C₂×C₄×C₂₃	—	2	1	no	—
212	C₂×C₄×C₂₃	—	4	1	no	—
213	C₂×C₄×C₂₃	—	4	1	no	—
214	C₂×C₄×C₂₃	—	4	1	no	—
215	C₂×C₄×C₂₃	—	4	1	no	—
216	C₂×C₄×C₂₃	—	4	1	no	—
217	C₂×C₄×C₂₃	—	4	1	no	—
218	C₂×C₄×C₂₃	—	4	2	no	—
219	C₂×C₄×C₂₃	—	4	2	no	—
220	C₂×C₄×C₂₃	—	4	2	no	—
221	C₂×C₄×C₂₃	—	8	8	no	—
222	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
223	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
224	C₂×C₄×C₂₃	#121	1	2	yes	—
225	C₂×C₄×C₂₃	#121	1	2	yes	—
226	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
227	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
228	C₂×C₄×C₂₃	#137	1	2	yes	—
229	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
230	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
231	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
232	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
233	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
234	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
235	C₈×C₂₃	—	2	4	no	—
236	C₈×C₂₃	—	4	2	no	—
237	C₂×#460	C₂×C₂₃	1	1	yes	characteristic
238	C₂×#440	C₂×C₂₃	1	2	yes	complemented
239	C₄×#127	C₂×C₂₃	1	1	yes	characteristic
240	C₄²×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
241	#1982	C₂×C₂₃	1	1	yes	characteristic
242	#1947	C₂×C₂₃	1	1	yes	characteristic, complemented
243	#1925	C₂×C₂₃	1	2	yes	complemented
244	#1880	C₂×C₂₃	1	2	yes	complemented
245	#1803	C₂×C₂₃	1	1	yes	characteristic, complemented
246	#1767	C₂×C₂₃	1	1	yes	characteristic, complemented
247	#1762	C₂×C₂₃	1	2	yes	complemented
248	C₂²×C₂₃	—	4	4	no	—
249	C₂²×C₂₃	—	4	4	no	—
250	C₂²×C₂₃	—	4	4	no	—
251	C₂²×C₂₃	—	8	2	no	—
252	C₂²×C₂₃	—	8	2	no	—
253	C₂²×C₂₃	—	8	2	no	—
254	C₂²×C₂₃	—	8	2	no	—
255	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
256	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
257	C₂²×C₂₃	#525	1	1	yes	characteristic
258	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
259	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
260	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
261	C₂²×C₂₃	#575	1	1	yes	characteristic
262	C₄×C₂₃	—	2	2	no	—
263	C₄×C₂₃	—	2	2	no	—
264	C₄×C₂₃	—	2	2	no	—
265	C₄×C₂₃	—	2	2	no	—
266	C₄×C₂₃	—	2	4	no	—
267	C₄×C₂₃	—	4	1	no	—
268	C₄×C₂₃	—	4	2	no	—
269	C₄×C₂₃	—	4	2	no	—
270	C₄×C₂₃	—	4	2	no	—
271	C₄×C₂₃	—	8	2	no	—
272	C₄×C₂₃	#440	1	4	yes	complemented
273	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
274	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
275	C₄×C₂₃	#529	1	1	yes	characteristic
276	C₄×C₂₃	#532	1	1	yes	characteristic
277	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
278	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
279	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
280	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
281	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
282	C₂×C₄×D₄	—	2	1	no	—
283	C₂×C₄×D₄	—	2	1	no	—
284	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
285	#460	C₄×C₂₃	1	8	yes	complemented
286	#440	—	2	8	no	—
287	C₂×#107	C₂²×C₂₃	1	2	yes	—
288	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
289	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
290	C₂×#103	C₂²×C₂₃	1	2	yes	—
291	C₂×#102	C₂²×C₂₃	1	2	yes	—
292	C₂×#102	C₂²×C₂₃	1	2	yes	—
293	C₂×#102	C₂²×C₂₃	1	2	yes	—
294	C₂×#97	C₂²×C₂₃	1	2	yes	—
295	#390	—	2	1	no	—
296	#390	—	2	1	no	—
297	#390	C₂²×C₂₃	1	2	yes	—
298	#384	C₂²×C₂₃	1	1	yes	characteristic
299	C₄×C₄⋊C₄	—	2	1	no	—
300	C₄×C₄⋊C₄	—	2	1	no	—
301	C₄×C₄⋊C₄	—	2	1	no	—
302	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
303	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
304	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
305	C₄³	—	2	1	no	—
306	C₄³	C₂²×C₂₃	1	1	yes	characteristic
307	#340	C₂²×C₂₃	1	2	yes	—
308	#340	C₂²×C₂₃	1	2	yes	—
309	#336	C₂²×C₂₃	1	2	yes	—
310	C₂×C₂₃	—	8	4	no	—
311	C₂×C₂₃	C₄×#121	1	1	yes	characteristic
312	C₂×C₂₃	C₂×#440	1	1	yes	characteristic
313	C₂×C₂₃	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
314	C₂×C₂₃	#3037	1	1	yes	characteristic
315	C₂×C₂₃	#3038	1	1	yes	characteristic
316	C₂×C₂₃	#3112	1	1	yes	characteristic
317	C₂×C₂₃	#3115	1	1	yes	characteristic
318	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
319	C₂²×D₄	—	2	1	no	—
320	C₂²×D₄	—	2	1	no	—
321	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
322	C₂³×C₄	—	2	2	no	—
323	C₂×SD₁₆	—	2	4	no	—
324	C₂×SD₁₆	—	2	4	no	—
325	C₂×SD₁₆	C₂×C₄×C₂₃	1	8	yes	complemented
326	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
327	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
328	#127	—	2	2	no	—
329	#127	C₂×C₄×C₂₃	1	4	yes	—
330	C₄×Q₈	—	2	2	no	—
331	C₄×Q₈	—	2	2	no	—
332	C₄×Q₈	—	2	2	no	—
333	C₄×Q₈	—	2	2	no	—
334	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
335	C₄×Q₈	D₄×C₂₃	1	4	yes	complemented
336	C₄×D₄	—	2	4	no	—
337	C₄×D₄	—	2	4	no	—
338	C₄×D₄	—	4	4	no	—
339	C₄×D₄	—	4	4	no	—
340	C₂×C₄⋊C₄	—	2	1	no	—
341	C₂×C₄⋊C₄	—	2	1	no	—
342	C₂×C₄⋊C₄	—	2	1	no	—
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
346	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
347	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
348	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
349	C₂×C₂²⋊C₄	—	2	2	no	—
350	C₂×C₂²⋊C₄	—	2	2	no	—
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₄²	—	2	1	no	—
353	C₂×C₄²	—	2	1	no	—
354	C₂×C₄²	—	2	1	no	—
355	C₂×C₄²	—	2	1	no	—
356	C₂×C₄²	—	2	1	no	—
357	C₂×C₄²	—	2	1	no	—
358	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
359	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
360	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
361	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
362	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
363	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
364	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
365	#107	—	2	4	no	—
366	C₄⋊C₈	—	2	2	no	—
367	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
368	#103	—	2	4	no	—
369	#102	—	2	4	no	—
370	#102	—	2	4	no	—
371	#102	C₂×C₄×C₂₃	1	8	yes	complemented
372	#97	—	2	4	no	—
373	#95	—	2	1	no	—
374	#95	—	2	1	no	—
375	#95	—	2	1	no	—
376	C₂₃	#21505	1	1	yes	characteristic, direct product factor, Sylow
377	C₂⁴	—	2	2	no	—
378	C₂×Q₈	—	2	1	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×Q₈	—	2	1	no	—
381	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
382	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
383	C₂×Q₈	#41×C₂₃	1	2	yes	—
384	C₂×D₄	—	2	4	no	—
385	C₂×D₄	—	2	4	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	4	2	no	—
392	C₂×D₄	—	4	2	no	—
393	C₂×D₄	—	4	2	no	—
394	C₂×D₄	—	4	2	no	—
395	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	4	4	no	—
402	C₂²×C₄	—	4	4	no	—
403	C₂²×C₄	—	4	4	no	—
404	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
405	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
406	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
408	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
409	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
410	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
411	SD₁₆	—	2	16	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
418	C₄⋊C₄	—	2	4	no	—
419	C₄⋊C₄	—	2	4	no	—
420	C₄⋊C₄	—	2	4	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	4	2	no	—
427	C₄⋊C₄	—	4	2	no	—
428	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
429	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
430	C₄⋊C₄	#41×C₂₃	1	2	yes	—
431	C₂²⋊C₄	—	4	4	no	—
432	C₂²⋊C₄	—	4	4	no	—
433	C₂²⋊C₄	—	4	4	no	—
434	C₄²	—	2	4	no	—
435	C₄²	—	2	4	no	—
436	C₄²	—	2	4	no	—
437	C₄²	—	2	2	no	—
438	C₄²	—	2	2	no	—
439	C₄²	—	2	2	no	—
440	C₄²	—	2	1	no	—
441	C₄²	—	2	1	no	—
442	C₄²	—	4	1	no	—
443	C₄²	—	4	2	no	—
444	C₄²	—	4	2	no	—
445	C₄²	—	4	2	no	—
446	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
447	C₄²	C₂×D₄×C₂₃	1	2	yes	—
448	C₄²	#41×C₂₃	1	2	yes	—
449	C₄²	#41×C₂₃	1	2	yes	—
450	C₂³	—	2	4	no	—
451	C₂³	—	4	2	no	—
452	C₂³	—	4	2	no	—
453	C₂³	—	4	2	no	—
454	C₂³	—	4	2	no	—
455	C₂³	—	4	2	no	—
456	C₂³	—	4	2	no	—
457	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
458	Q₈	—	2	2	no	—
459	Q₈	—	2	2	no	—
460	Q₈	—	2	2	no	—
461	Q₈	C₄×D₄×C₂₃	1	4	yes	complemented
462	D₄	—	2	4	no	—
463	D₄	—	2	4	no	—
464	D₄	—	4	8	no	—
465	D₄	—	4	8	no	—
466	D₄	—	4	2	no	—
467	D₄	—	4	2	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	1	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	1	no	—
478	C₂×C₄	—	2	2	no	—
479	C₂×C₄	—	2	1	no	—
480	C₂×C₄	—	2	1	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	1	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	8	8	no	—
495	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
496	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
497	C₂×C₄	C₂₃×#121	1	2	yes	—
498	C₂×C₄	C₂₃×#121	1	2	yes	—
499	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
500	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
501	C₂×C₄	C₂₃×#137	1	2	yes	—
502	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
503	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
504	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
505	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
506	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
507	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
508	C₈	—	2	4	no	—
509	C₈	—	4	2	no	—
510	C₂²	—	4	4	no	—
511	C₂²	—	4	4	no	—
512	C₂²	—	4	4	no	—
513	C₂²	—	8	2	no	—
514	C₂²	—	8	2	no	—
515	C₂²	—	8	2	no	—
516	C₂²	—	8	2	no	—
517	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
518	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
519	C₂²	C₂₃×#525	1	1	yes	characteristic
520	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
521	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
522	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
523	C₂²	C₂₃×#575	1	1	yes	characteristic
524	C₄	—	2	2	no	—
525	C₄	—	2	2	no	—
526	C₄	—	2	2	no	—
527	C₄	—	2	4	no	—
528	C₄	—	2	2	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	2	no	—
531	C₄	—	4	1	no	—
532	C₄	—	4	2	no	—
533	C₄	—	8	2	no	—
534	C₄	C₂₃×#440	1	4	yes	complemented
535	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
536	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
537	C₄	C₂₃×#529	1	1	yes	characteristic
538	C₄	C₂₃×#532	1	1	yes	characteristic
539	C₂	—	8	4	no	—
540	C₂	C₄×C₂₃×#121	1	1	yes	characteristic
541	C₂	C₂×C₂₃×#440	1	1	yes	characteristic
542	C₂	C₄×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
543	C₂	C₂₃×#3037	1	1	yes	characteristic
544	C₂	C₂₃×#3038	1	1	yes	characteristic
545	C₂	C₂₃×#3112	1	1	yes	characteristic
546	C₂	C₂₃×#3115	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#440	C₂	semidirect
4. C₄²×Q₈×C₂₃	C₂	semidirect
6. C₂₃×#1947	C₂	semidirect
7. C₂₃×#1925	C₂	semidirect
8. C₂₃×#1880	C₂	semidirect
9. C₂₃×#1803	C₂	semidirect
10. C₂₃×#1767	C₂	semidirect
11. C₂₃×#1762	C₂	semidirect
20. C₂₃×#460	C₄	semidirect
52. C₂×SD₁₆×C₂₃	C₂×C₄	semidirect
61. C₄×Q₈×C₂₃	C₂×C₄	semidirect
62. C₄×Q₈×C₂₃	D₄	semidirect
94. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
98. C₂₃×#102	C₂×C₄	semidirect
176. #21505	C₂₃	direct (N × H)
188. Q₈×C₂₃	C₄×D₄	semidirect
238. C₂×#440	C₂×C₂₃	semidirect
240. C₄²×Q₈	C₂×C₂₃	semidirect
242. #1947	C₂×C₂₃	semidirect
243. #1925	C₂×C₂₃	semidirect
244. #1880	C₂×C₂₃	semidirect
245. #1803	C₂×C₂₃	semidirect
246. #1767	C₂×C₂₃	semidirect
247. #1762	C₂×C₂₃	semidirect
272. C₄×C₂₃	#440	semidirect
285. #460	C₄×C₂₃	semidirect
325. C₂×SD₁₆	C₂×C₄×C₂₃	semidirect
334. C₄×Q₈	C₂×C₄×C₂₃	semidirect
335. C₄×Q₈	D₄×C₂₃	semidirect
367. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
371. #102	C₂×C₄×C₂₃	semidirect
376. C₂₃	#21505	direct (N × H)
461. Q₈	C₄×D₄×C₂₃	semidirect
534. C₄	C₂₃×#440	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#460	2
C₂×C₂₃×#440	2
C₄×C₂₃×#127	2
C₄²×Q₈×C₂₃	2
C₂₃×#1982	2
C₂₃×#1947	2
C₂₃×#1925	2
C₂₃×#1880	2
C₂₃×#1803	2
C₂₃×#1767	2
C₂₃×#1762	2
#21505	23

Subgroups — C23×#21505 #19169131

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21505 #19169131