Subgroups — C₂₃×#60644 #19208270

Order 5888.

3350 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#575	C₂	1	2	yes	maximal, complemented
2	C₂²×C₂₃×AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	C₂₃×#3449	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#535	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#3155	C₂	1	8	yes	maximal, complemented
6	C₂₃×#3154	C₂	1	8	yes	maximal, complemented
7	C₂₃×#3153	C₂	1	4	yes	maximal, complemented
8	C₂₃×#3151	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2873	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×#41×C₂₃	C₂²	1	2	yes	—
12	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#575	—	2	16	no	—
14	C₂×C₂₃×#137	C₂²	1	4	yes	complemented
15	C₂×C₂₃×#137	C₂²	1	4	yes	complemented
16	C₂×C₂₃×AGL(1,ℤ/8)	—	2	16	no	—
17	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
18	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
19	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
20	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
21	C₂×C₂₃×#135	C₂²	1	8	yes	complemented
22	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
23	C₂²×D₈×C₂₃	C₂²	1	4	yes	complemented
24	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
25	C₂₃×#537	—	2	2	no	—
26	C₂₃×#536	C₂²	1	4	yes	complemented
27	C₂₃×#535	—	2	4	no	—
28	C₂₃×#535	C₂²	1	8	yes	complemented
29	C₂₃×#535	C₂²	1	4	yes	complemented
30	C₂₃×#534	—	2	2	no	—
31	C₂₃×#532	C₂²	1	4	yes	complemented
32	C₂×C₂₃×#127	C₂²	1	2	yes	complemented
33	C₂×C₂₃×#124	C₂²	1	2	yes	complemented
34	C₂₃×#525	—	2	4	no	—
35	C₂×C₂₃×#121	—	2	2	no	—
36	C₂×C₂₃×#120	—	2	2	no	—
37	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
38	C₂₃×#497	C₂²	1	8	yes	complemented
39	C₂₃×#496	C₂²	1	8	yes	complemented
40	C₂₃×#496	C₂²	1	16	yes	complemented
41	C₂₃×#495	C₂²	1	16	yes	complemented
42	C₂₃×#493	C₂²	1	8	yes	complemented
43	C₂₃×#492	C₂²	1	8	yes	complemented
44	C₂₃×#405	C₂²	1	8	yes	complemented
45	C₄×M₁₆×C₂₃	C₂²	1	4	yes	complemented
46	C₂⁵×C₂₃	—	2	1	no	—
47	C₂×#41×C₂₃	—	2	2	no	—
48	C₂×#41×C₂₃	—	2	2	no	—
49	C₂×#41×C₂₃	—	2	2	no	—
50	C₂×#41×C₂₃	—	2	8	no	—
51	C₂×#41×C₂₃	—	2	8	no	—
52	C₂×#41×C₂₃	—	2	8	no	—
53	C₂×#41×C₂₃	C₂³	1	4	yes	—
54	C₂×#41×C₂₃	C₂³	1	4	yes	—
55	C₂×#41×C₂₃	C₂³	1	4	yes	—
56	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
57	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₂₃	—	2	1	no	—
59	C₂²×D₄×C₂₃	—	2	2	no	—
60	C₂²×D₄×C₂₃	—	2	8	no	—
61	C₂²×D₄×C₂₃	—	2	1	no	—
62	C₂²×D₄×C₂₃	—	2	2	no	—
63	C₂²×D₄×C₂₃	—	2	4	no	—
64	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
65	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
66	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
67	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂³×C₄×C₂₃	—	2	1	no	—
69	C₂³×C₄×C₂₃	—	2	2	no	—
70	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂₃×#137	—	2	16	no	—
72	C₂₃×#137	—	2	16	no	—
73	C₂₃×AGL(1,ℤ/8)	—	2	32	no	—
74	C₂₃×AGL(1,ℤ/8)	—	2	16	no	—
75	C₂₃×AGL(1,ℤ/8)	—	2	32	no	—
76	C₂₃×AGL(1,ℤ/8)	—	2	16	no	—
77	C₂₃×#135	—	2	32	no	—
78	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	complemented
79	C₂×SD₁₆×C₂₃	—	2	16	no	—
80	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
81	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	complemented
82	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	complemented
83	C₂×D₈×C₂₃	—	2	16	no	—
84	C₂×D₈×C₂₃	C₂³	1	16	yes	—
85	C₂×D₈×C₂₃	C₂³	1	8	yes	complemented
86	C₂×M₁₆×C₂₃	D₄	1	16	yes	complemented
87	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
88	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
89	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
90	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
91	C₂₃×#127	—	2	4	no	—
92	C₂₃×#127	C₂³	1	8	yes	—
93	C₂₃×#127	C₂³	1	4	yes	complemented
94	C₂₃×#124	—	2	4	no	—
95	C₂₃×#124	C₂³	1	8	yes	—
96	C₂₃×#124	C₂³	1	8	yes	complemented
97	C₂₃×#123	—	2	8	no	—
98	C₂₃×C₂²⋊Q₈	—	2	8	no	—
99	C₂₃×#121	—	2	8	no	—
100	C₂₃×#121	—	2	16	no	—
101	C₂₃×#121	—	2	8	no	—
102	C₂₃×#120	—	2	16	no	—
103	C₂₃×#120	—	2	8	no	—
104	C₄×D₄×C₂₃	—	2	8	no	—
105	C₂₃×#117	D₄	1	4	yes	—
106	C₂₃×#117	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
108	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
109	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
110	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
111	C₂₃×#97	C₂³	1	8	yes	complemented
112	C₄×C₈×C₂₃	C₂³	1	8	yes	complemented
113	C₂⁴×C₂₃	—	2	2	no	—
114	C₂⁴×C₂₃	—	2	1	no	—
115	C₂⁴×C₂₃	—	2	2	no	—
116	C₂⁴×C₂₃	—	2	4	no	—
117	C₂⁴×C₂₃	—	4	4	no	—
118	C₂⁴×C₂₃	—	8	4	no	—
119	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
120	#41×C₂₃	—	2	8	no	—
121	#41×C₂₃	—	2	8	no	—
122	#41×C₂₃	—	2	8	no	—
123	#41×C₂₃	—	2	8	no	—
124	#41×C₂₃	—	2	8	no	—
125	#41×C₂₃	—	2	8	no	—
126	#41×C₂₃	—	4	8	no	—
127	#41×C₂₃	—	4	8	no	—
128	#41×C₂₃	—	4	8	no	—
129	C₂×Q₈×C₂₃	—	2	2	no	—
130	C₂×Q₈×C₂₃	—	2	1	no	—
131	C₂×Q₈×C₂₃	—	2	4	no	—
132	C₂×Q₈×C₂₃	—	2	4	no	—
133	C₂×Q₈×C₂₃	—	2	4	no	—
134	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
135	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
136	C₂×D₄×C₂₃	—	2	4	no	—
137	C₂×D₄×C₂₃	—	2	4	no	—
138	C₂×D₄×C₂₃	—	2	4	no	—
139	C₂×D₄×C₂₃	—	2	4	no	—
140	C₂×D₄×C₂₃	—	2	8	no	—
141	C₂×D₄×C₂₃	—	2	8	no	—
142	C₂×D₄×C₂₃	—	2	8	no	—
143	C₂×D₄×C₂₃	—	2	4	no	—
144	C₂×D₄×C₂₃	—	2	4	no	—
145	C₂×D₄×C₂₃	—	2	4	no	—
146	C₂×D₄×C₂₃	—	2	2	no	—
147	C₂×D₄×C₂₃	—	2	2	no	—
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	—	2	8	no	—
150	C₂×D₄×C₂₃	—	2	16	no	—
151	C₂×D₄×C₂₃	—	4	4	no	—
152	C₂×D₄×C₂₃	—	4	8	no	—
153	C₂×D₄×C₂₃	—	4	4	no	—
154	C₂×D₄×C₂₃	—	4	8	no	—
155	C₂×D₄×C₂₃	—	8	8	no	—
156	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
157	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
158	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₂₃	—	2	1	no	—
160	C₂²×C₄×C₂₃	—	2	2	no	—
161	C₂²×C₄×C₂₃	—	2	4	no	—
162	C₂²×C₄×C₂₃	—	2	4	no	—
163	C₂²×C₄×C₂₃	—	2	4	no	—
164	C₂²×C₄×C₂₃	—	4	4	no	—
165	C₂²×C₄×C₂₃	—	4	8	no	—
166	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
167	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
170	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
172	Q₁₆×C₂₃	—	2	16	no	—
173	SD₁₆×C₂₃	—	2	32	no	—
174	SD₁₆×C₂₃	—	2	16	no	—
175	SD₁₆×C₂₃	—	2	16	no	—
176	D₈×C₂₃	—	2	32	no	—
177	D₈×C₂₃	—	2	16	no	—
178	M₁₆×C₂₃	C₂×D₄	1	16	yes	—
179	M₁₆×C₂₃	C₂×D₄	1	16	yes	complemented
180	C₂×C₈×C₂₃	C₂×D₄	1	16	yes	complemented
181	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
182	C₄⋊C₄×C₂₃	—	2	8	no	—
183	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
184	C₂²⋊C₄×C₂₃	—	2	16	no	—
185	C₂²⋊C₄×C₂₃	—	2	16	no	—
186	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
187	C₄²×C₂₃	C₂×D₄	1	4	yes	—
188	C₄²×C₂₃	C₂⁴	1	4	yes	—
189	#60644	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂³×C₂₃	—	2	4	no	—
191	C₂³×C₂₃	—	2	4	no	—
192	C₂³×C₂₃	—	2	2	no	—
193	C₂³×C₂₃	—	4	2	no	—
194	C₂³×C₂₃	—	4	4	no	—
195	C₂³×C₂₃	—	4	8	no	—
196	C₂³×C₂₃	—	4	2	no	—
197	C₂³×C₂₃	—	4	4	no	—
198	C₂³×C₂₃	—	4	4	no	—
199	C₂³×C₂₃	—	4	8	no	—
200	C₂³×C₂₃	—	8	4	no	—
201	C₂³×C₂₃	—	8	8	no	—
202	C₂³×C₂₃	—	8	16	no	—
203	C₂³×C₂₃	#127	1	4	yes	—
204	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
205	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
206	Q₈×C₂₃	—	2	4	no	—
207	Q₈×C₂₃	—	2	4	no	—
208	Q₈×C₂₃	—	2	4	no	—
209	Q₈×C₂₃	—	4	4	no	—
210	Q₈×C₂₃	—	4	2	no	—
211	D₄×C₂₃	—	2	4	no	—
212	D₄×C₂₃	—	2	8	no	—
213	D₄×C₂₃	—	2	4	no	—
214	D₄×C₂₃	—	2	4	no	—
215	D₄×C₂₃	—	4	8	no	—
216	D₄×C₂₃	—	4	8	no	—
217	D₄×C₂₃	—	4	8	no	—
218	D₄×C₂₃	—	4	8	no	—
219	D₄×C₂₃	—	4	4	no	—
220	D₄×C₂₃	—	4	4	no	—
221	D₄×C₂₃	—	4	2	no	—
222	D₄×C₂₃	—	8	8	no	—
223	D₄×C₂₃	—	8	16	no	—
224	C₂×C₄×C₂₃	—	2	2	no	—
225	C₂×C₄×C₂₃	—	2	4	no	—
226	C₂×C₄×C₂₃	—	4	4	no	—
227	C₂×C₄×C₂₃	—	4	2	no	—
228	C₂×C₄×C₂₃	—	4	4	no	—
229	C₂×C₄×C₂₃	—	4	8	no	—
230	C₂×C₄×C₂₃	—	4	8	no	—
231	C₂×C₄×C₂₃	—	4	8	no	—
232	C₂×C₄×C₂₃	#127	1	4	yes	—
233	C₂×C₄×C₂₃	#127	1	8	yes	—
234	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
235	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
237	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
238	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
240	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
241	C₈×C₂₃	C₂²×D₄	1	16	yes	—
242	C₂×#575	C₂×C₂₃	1	2	yes	complemented
243	C₂²×AGL(1,ℤ/8)	C₂×C₂₃	1	2	yes	complemented
244	#3449	C₂×C₂₃	1	1	yes	characteristic, complemented
245	C₂×#535	C₂×C₂₃	1	1	yes	characteristic, complemented
246	#3155	C₂×C₂₃	1	8	yes	complemented
247	#3154	C₂×C₂₃	1	8	yes	complemented
248	#3153	C₂×C₂₃	1	4	yes	complemented
249	#3151	C₂×C₂₃	1	4	yes	complemented
250	#2873	C₂×C₂₃	1	1	yes	characteristic, complemented
251	C₂²×C₂₃	—	2	2	no	—
252	C₂²×C₂₃	—	2	4	no	—
253	C₂²×C₂₃	—	4	4	no	—
254	C₂²×C₂₃	—	4	4	no	—
255	C₂²×C₂₃	—	4	8	no	—
256	C₂²×C₂₃	—	4	8	no	—
257	C₂²×C₂₃	—	4	4	no	—
258	C₂²×C₂₃	—	4	2	no	—
259	C₂²×C₂₃	—	4	4	no	—
260	C₂²×C₂₃	—	8	4	no	—
261	C₂²×C₂₃	—	8	8	no	—
262	C₂²×C₂₃	—	8	8	no	—
263	C₂²×C₂₃	—	8	16	no	—
264	C₂²×C₂₃	C₂×#127	1	4	yes	—
265	C₂²×C₂₃	C₂×#127	1	2	yes	—
266	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
267	C₄×C₂₃	—	2	8	no	—
268	C₄×C₂₃	—	4	2	no	—
269	C₄×C₂₃	—	4	4	no	—
270	C₄×C₂₃	C₂×#127	1	2	yes	—
271	C₄×C₂₃	C₂×#127	1	4	yes	—
272	C₄×C₂₃	C₂³×D₄	1	2	yes	—
273	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
274	C₂²×#41	C₂²×C₂₃	1	2	yes	—
275	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
276	#575	—	2	16	no	—
277	C₂×#137	C₂²×C₂₃	1	4	yes	complemented
278	C₂×#137	C₂²×C₂₃	1	4	yes	complemented
279	C₂×AGL(1,ℤ/8)	—	2	16	no	—
280	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	8	yes	complemented
281	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	8	yes	complemented
282	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	4	yes	complemented
283	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	4	yes	complemented
284	C₂×#135	C₂²×C₂₃	1	8	yes	complemented
285	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
286	C₂²×D₈	C₂²×C₂₃	1	4	yes	complemented
287	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
288	#537	—	2	2	no	—
289	#536	C₂²×C₂₃	1	4	yes	complemented
290	#535	—	2	4	no	—
291	#535	C₂²×C₂₃	1	8	yes	complemented
292	#535	C₂²×C₂₃	1	4	yes	complemented
293	#534	—	2	2	no	—
294	#532	C₂²×C₂₃	1	4	yes	complemented
295	C₂×#127	C₂²×C₂₃	1	2	yes	complemented
296	C₂×#124	C₂²×C₂₃	1	2	yes	complemented
297	#525	—	2	4	no	—
298	C₂×#121	—	2	2	no	—
299	C₂×#120	—	2	2	no	—
300	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
301	#497	C₂²×C₂₃	1	8	yes	complemented
302	#496	C₂²×C₂₃	1	8	yes	complemented
303	#496	C₂²×C₂₃	1	16	yes	complemented
304	#495	C₂²×C₂₃	1	16	yes	complemented
305	#493	C₂²×C₂₃	1	8	yes	complemented
306	#492	C₂²×C₂₃	1	8	yes	complemented
307	#405	C₂²×C₂₃	1	8	yes	complemented
308	C₄×M₁₆	C₂²×C₂₃	1	4	yes	complemented
309	C₂×C₂₃	—	2	2	no	—
310	C₂×C₂₃	—	2	4	no	—
311	C₂×C₂₃	—	8	2	no	—
312	C₂×C₂₃	—	8	4	no	—
313	C₂×C₂₃	—	8	4	no	—
314	C₂×C₂₃	C₂²×#127	1	1	yes	characteristic
315	C₂×C₂₃	#3585	1	2	yes	—
316	C₂⁵	—	2	1	no	—
317	C₂×#41	—	2	2	no	—
318	C₂×#41	—	2	2	no	—
319	C₂×#41	—	2	2	no	—
320	C₂×#41	—	2	8	no	—
321	C₂×#41	—	2	8	no	—
322	C₂×#41	—	2	8	no	—
323	C₂×#41	C₂³×C₂₃	1	4	yes	—
324	C₂×#41	C₂³×C₂₃	1	4	yes	—
325	C₂×#41	C₂³×C₂₃	1	4	yes	—
326	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
327	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
328	C₂²×D₄	—	2	1	no	—
329	C₂²×D₄	—	2	2	no	—
330	C₂²×D₄	—	2	8	no	—
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	—	2	2	no	—
333	C₂²×D₄	—	2	4	no	—
334	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
335	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
336	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
337	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
338	C₂³×C₄	—	2	1	no	—
339	C₂³×C₄	—	2	2	no	—
340	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
341	#137	—	2	16	no	—
342	#137	—	2	16	no	—
343	AGL(1,ℤ/8)	—	2	32	no	—
344	AGL(1,ℤ/8)	—	2	16	no	—
345	AGL(1,ℤ/8)	—	2	32	no	—
346	AGL(1,ℤ/8)	—	2	16	no	—
347	#135	—	2	32	no	—
348	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	complemented
349	C₂×SD₁₆	—	2	16	no	—
350	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
351	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	complemented
352	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	complemented
353	C₂×D₈	—	2	16	no	—
354	C₂×D₈	C₂³×C₂₃	1	16	yes	—
355	C₂×D₈	C₂³×C₂₃	1	8	yes	complemented
356	C₂×M₁₆	D₄×C₂₃	1	16	yes	complemented
357	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
358	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
359	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
360	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
361	#127	—	2	4	no	—
362	#127	C₂³×C₂₃	1	8	yes	—
363	#127	C₂³×C₂₃	1	4	yes	complemented
364	#124	—	2	4	no	—
365	#124	C₂³×C₂₃	1	8	yes	—
366	#124	C₂³×C₂₃	1	8	yes	complemented
367	#123	—	2	8	no	—
368	C₂²⋊Q₈	—	2	8	no	—
369	#121	—	2	8	no	—
370	#121	—	2	16	no	—
371	#121	—	2	8	no	—
372	#120	—	2	16	no	—
373	#120	—	2	8	no	—
374	C₄×D₄	—	2	8	no	—
375	#117	D₄×C₂₃	1	4	yes	—
376	#117	C₂³×C₂₃	1	4	yes	—
377	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
378	C₂×C₂²⋊C₄	—	2	4	no	—
379	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
380	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
381	#97	C₂³×C₂₃	1	8	yes	complemented
382	C₄×C₈	C₂³×C₂₃	1	8	yes	complemented
383	C₂₃	#60644	1	1	yes	characteristic, direct product factor, Sylow
384	C₂⁴	—	2	2	no	—
385	C₂⁴	—	2	1	no	—
386	C₂⁴	—	2	2	no	—
387	C₂⁴	—	2	4	no	—
388	C₂⁴	—	4	4	no	—
389	C₂⁴	—	8	4	no	—
390	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
391	#41	—	2	8	no	—
392	#41	—	2	8	no	—
393	#41	—	2	8	no	—
394	#41	—	2	8	no	—
395	#41	—	2	8	no	—
396	#41	—	2	8	no	—
397	#41	—	4	8	no	—
398	#41	—	4	8	no	—
399	#41	—	4	8	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	4	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	4	no	—
405	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
406	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	8	no	—
412	C₂×D₄	—	2	8	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	8	no	—
421	C₂×D₄	—	2	16	no	—
422	C₂×D₄	—	4	4	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	4	8	no	—
426	C₂×D₄	—	8	8	no	—
427	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
428	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
429	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	4	4	no	—
436	C₂²×C₄	—	4	8	no	—
437	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
438	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
440	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
441	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
443	Q₁₆	—	2	16	no	—
444	SD₁₆	—	2	32	no	—
445	SD₁₆	—	2	16	no	—
446	SD₁₆	—	2	16	no	—
447	D₈	—	2	32	no	—
448	D₈	—	2	16	no	—
449	M₁₆	C₂×D₄×C₂₃	1	16	yes	—
450	M₁₆	C₂×D₄×C₂₃	1	16	yes	complemented
451	C₂×C₈	C₂×D₄×C₂₃	1	16	yes	complemented
452	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
453	C₄⋊C₄	—	2	8	no	—
454	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
455	C₂²⋊C₄	—	2	16	no	—
456	C₂²⋊C₄	—	2	16	no	—
457	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
458	C₄²	C₂×D₄×C₂₃	1	4	yes	—
459	C₄²	C₂⁴×C₂₃	1	4	yes	—
460	C₂³	—	2	4	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	4	4	no	—
465	C₂³	—	4	8	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	4	4	no	—
468	C₂³	—	4	4	no	—
469	C₂³	—	4	8	no	—
470	C₂³	—	8	8	no	—
471	C₂³	—	8	4	no	—
472	C₂³	—	8	16	no	—
473	C₂³	C₂₃×#127	1	4	yes	—
474	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
475	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
476	Q₈	—	2	4	no	—
477	Q₈	—	2	4	no	—
478	Q₈	—	2	4	no	—
479	Q₈	—	4	4	no	—
480	Q₈	—	4	2	no	—
481	D₄	—	2	4	no	—
482	D₄	—	2	8	no	—
483	D₄	—	2	4	no	—
484	D₄	—	2	4	no	—
485	D₄	—	4	8	no	—
486	D₄	—	4	8	no	—
487	D₄	—	4	8	no	—
488	D₄	—	4	8	no	—
489	D₄	—	4	4	no	—
490	D₄	—	4	4	no	—
491	D₄	—	4	2	no	—
492	D₄	—	8	16	no	—
493	D₄	—	8	8	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	4	4	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	—	4	8	no	—
500	C₂×C₄	—	4	8	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	C₂₃×#127	1	4	yes	—
503	C₂×C₄	C₂₃×#127	1	8	yes	—
504	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
505	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
506	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
507	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
508	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
509	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
510	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
511	C₈	C₂²×D₄×C₂₃	1	16	yes	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	4	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	2	no	—
517	C₂²	—	4	4	no	—
518	C₂²	—	4	4	no	—
519	C₂²	—	4	8	no	—
520	C₂²	—	4	8	no	—
521	C₂²	—	8	8	no	—
522	C₂²	—	8	4	no	—
523	C₂²	—	8	16	no	—
524	C₂²	—	8	8	no	—
525	C₂²	C₂×C₂₃×#127	1	4	yes	—
526	C₂²	C₂×C₂₃×#127	1	2	yes	—
527	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
528	C₄	—	2	8	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	4	no	—
531	C₄	C₂×C₂₃×#127	1	4	yes	—
532	C₄	C₂×C₂₃×#127	1	2	yes	—
533	C₄	C₂³×D₄×C₂₃	1	2	yes	—
534	C₂	—	2	2	no	—
535	C₂	—	2	4	no	—
536	C₂	—	8	4	no	—
537	C₂	—	8	2	no	—
538	C₂	—	8	4	no	—
539	C₂	C₂²×C₂₃×#127	1	1	yes	characteristic
540	C₂	C₂₃×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#575	C₂	semidirect
2. C₂²×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
3. C₂₃×#3449	C₂	semidirect
4. C₂×C₂₃×#535	C₂	semidirect
5. C₂₃×#3155	C₂	semidirect
6. C₂₃×#3154	C₂	semidirect
7. C₂₃×#3153	C₂	semidirect
8. C₂₃×#3151	C₂	semidirect
9. C₂₃×#2873	C₂	semidirect
14. C₂×C₂₃×#137	C₂²	semidirect
15. C₂×C₂₃×#137	C₂²	semidirect
17. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
18. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
19. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
20. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
21. C₂×C₂₃×#135	C₂²	semidirect
22. C₂²×SD₁₆×C₂₃	C₂²	semidirect
23. C₂²×D₈×C₂₃	C₂²	semidirect
26. C₂₃×#536	C₂²	semidirect
28. C₂₃×#535	C₂²	semidirect
29. C₂₃×#535	C₂²	semidirect
31. C₂₃×#532	C₂²	semidirect
32. C₂×C₂₃×#127	C₂²	semidirect
33. C₂×C₂₃×#124	C₂²	semidirect
38. C₂₃×#497	C₂²	semidirect
39. C₂₃×#496	C₂²	semidirect
40. C₂₃×#496	C₂²	semidirect
41. C₂₃×#495	C₂²	semidirect
42. C₂₃×#493	C₂²	semidirect
43. C₂₃×#492	C₂²	semidirect
44. C₂₃×#405	C₂²	semidirect
45. C₄×M₁₆×C₂₃	C₂²	semidirect
78. C₂×Q₁₆×C₂₃	C₂³	semidirect
81. C₂×SD₁₆×C₂₃	C₂³	semidirect
82. C₂×SD₁₆×C₂₃	C₂³	semidirect
85. C₂×D₈×C₂₃	C₂³	semidirect
86. C₂×M₁₆×C₂₃	D₄	semidirect
90. C₂₃×C₄⋊Q₈	C₂³	semidirect
93. C₂₃×#127	C₂³	semidirect
96. C₂₃×#124	C₂³	semidirect
111. C₂₃×#97	C₂³	semidirect
112. C₄×C₈×C₂₃	C₂³	semidirect
179. M₁₆×C₂₃	C₂×D₄	semidirect
180. C₂×C₈×C₂₃	C₂×D₄	semidirect
189. #60644	C₂₃	direct (N × H)
242. C₂×#575	C₂×C₂₃	semidirect
243. C₂²×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
244. #3449	C₂×C₂₃	semidirect
245. C₂×#535	C₂×C₂₃	semidirect
246. #3155	C₂×C₂₃	semidirect
247. #3154	C₂×C₂₃	semidirect
248. #3153	C₂×C₂₃	semidirect
249. #3151	C₂×C₂₃	semidirect
250. #2873	C₂×C₂₃	semidirect
277. C₂×#137	C₂²×C₂₃	semidirect
278. C₂×#137	C₂²×C₂₃	semidirect
280. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
281. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
282. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
283. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
284. C₂×#135	C₂²×C₂₃	semidirect
285. C₂²×SD₁₆	C₂²×C₂₃	semidirect
286. C₂²×D₈	C₂²×C₂₃	semidirect
289. #536	C₂²×C₂₃	semidirect
291. #535	C₂²×C₂₃	semidirect
292. #535	C₂²×C₂₃	semidirect
294. #532	C₂²×C₂₃	semidirect
295. C₂×#127	C₂²×C₂₃	semidirect
296. C₂×#124	C₂²×C₂₃	semidirect
301. #497	C₂²×C₂₃	semidirect
302. #496	C₂²×C₂₃	semidirect
303. #496	C₂²×C₂₃	semidirect
304. #495	C₂²×C₂₃	semidirect
305. #493	C₂²×C₂₃	semidirect
306. #492	C₂²×C₂₃	semidirect
307. #405	C₂²×C₂₃	semidirect
308. C₄×M₁₆	C₂²×C₂₃	semidirect
348. C₂×Q₁₆	C₂³×C₂₃	semidirect
351. C₂×SD₁₆	C₂³×C₂₃	semidirect
352. C₂×SD₁₆	C₂³×C₂₃	semidirect
355. C₂×D₈	C₂³×C₂₃	semidirect
356. C₂×M₁₆	D₄×C₂₃	semidirect
360. C₄⋊Q₈	C₂³×C₂₃	semidirect
363. #127	C₂³×C₂₃	semidirect
366. #124	C₂³×C₂₃	semidirect
381. #97	C₂³×C₂₃	semidirect
382. C₄×C₈	C₂³×C₂₃	semidirect
383. C₂₃	#60644	direct (N × H)
450. M₁₆	C₂×D₄×C₂₃	semidirect
451. C₂×C₈	C₂×D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#575	2
C₂²×C₂₃×AGL(1,ℤ/8)	2
C₂₃×#3449	2
C₂×C₂₃×#535	2
C₂₃×#3155	2
C₂₃×#3154	2
C₂₃×#3153	2
C₂₃×#3151	2
C₂₃×#2873	2
#60644	23

Subgroups — C23×#60644 #19208270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60644 #19208270