Subgroups — C₂₃×#26124 #19173750

Order 5888.

764 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2552	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2349	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2099	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2088	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2087	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1949	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1940	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1862	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1836	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic
14	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic
15	C₂₃×#1762	C₂	1	1	yes	maximal, characteristic
16	C₂×C₂₃×#125	—	2	1	no	—
17	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
30	C₂₃×#398	—	2	1	no	—
31	C₂₃×#398	—	2	1	no	—
32	C₂₃×#389	—	2	1	no	—
33	C₂₃×#389	C₂²	1	1	yes	characteristic
34	C₂₃×#389	C₂²	1	1	yes	characteristic
35	C₂₃×#389	C₂²	1	1	yes	characteristic
36	C₂₃×#389	C₂²	1	1	yes	characteristic
37	C₂₃×#385	—	2	1	no	—
38	C₂₃×#385	—	2	1	no	—
39	C₂₃×#384	C₂²	1	1	yes	characteristic
40	C₂₃×#384	C₂²	1	1	yes	characteristic
41	C₂₃×#384	C₂²	1	1	yes	characteristic
42	C₂₃×#382	C₂²	1	1	yes	characteristic
43	C₂₃×#380	—	2	1	no	—
44	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
45	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×#95	—	2	1	no	—
47	C₂₃×#340	C₂²	1	1	yes	characteristic
48	C₂₃×#340	C₂²	1	1	yes	characteristic
49	C₂₃×#340	C₂²	1	1	yes	characteristic
50	C₂₃×#340	C₂²	1	1	yes	characteristic
51	C₂₃×#340	C₂²	1	1	yes	characteristic
52	C₂₃×#340	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#336	C₂²	1	1	yes	characteristic
58	C₂₃×#336	C₂²	1	1	yes	characteristic
59	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₂₃	—	2	1	no	—
61	C₂³×C₄×C₂₃	—	2	1	no	—
62	C₂³×C₄×C₂₃	—	2	1	no	—
63	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂₃×#125	—	4	4	no	—
68	C₄×D₄×C₂₃	—	2	2	no	—
69	C₄×D₄×C₂₃	—	2	2	no	—
70	C₄×D₄×C₂₃	—	2	2	no	—
71	C₄×D₄×C₂₃	—	4	1	no	—
72	C₄×D₄×C₂₃	—	4	1	no	—
73	C₄×D₄×C₂₃	Q₈	1	4	yes	—
74	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
96	C₂×C₄²×C₂₃	—	2	1	no	—
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×#107	—	4	1	no	—
101	C₂₃×#106	—	4	1	no	—
102	C₂₃×C₄⋊C₈	—	2	2	no	—
103	C₂₃×C₄⋊C₈	—	4	1	no	—
104	C₂₃×C₄⋊C₈	—	4	1	no	—
105	C₂₃×#102	—	2	2	no	—
106	C₂₃×#102	—	2	2	no	—
107	C₂₃×#102	—	2	2	no	—
108	C₂₃×#102	—	2	2	no	—
109	C₂₃×#97	—	2	2	no	—
110	C₂₃×#95	—	2	1	no	—
111	C₂₃×#95	—	2	1	no	—
112	C₂₃×#95	—	2	1	no	—
113	C₂₃×#95	—	2	1	no	—
114	C₂₃×#95	—	2	1	no	—
115	C₂₃×#95	—	2	1	no	—
116	C₂₃×#95	—	2	1	no	—
117	C₂₃×#95	D₄	1	1	yes	characteristic
118	C₂₃×#95	D₄	1	1	yes	characteristic
119	C₂⁴×C₂₃	—	2	1	no	—
120	C₂×D₄×C₂₃	—	2	1	no	—
121	C₂×D₄×C₂₃	—	2	1	no	—
122	C₂×D₄×C₂₃	—	2	1	no	—
123	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
124	C₂×D₄×C₂₃	#41	1	2	yes	—
125	C₂×D₄×C₂₃	#41	1	2	yes	—
126	C₂²×C₄×C₂₃	—	2	1	no	—
127	C₂²×C₄×C₂₃	—	2	4	no	—
128	C₂²×C₄×C₂₃	—	2	1	no	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	—	2	1	no	—
131	C₂²×C₄×C₂₃	—	2	1	no	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	4	2	no	—
135	C₂²×C₄×C₂₃	—	4	2	no	—
136	C₂²×C₄×C₂₃	—	4	2	no	—
137	C₂²×C₄×C₂₃	—	4	2	no	—
138	C₂²×C₄×C₂₃	—	8	1	no	—
139	C₂²×C₄×C₂₃	—	8	1	no	—
140	C₂²×C₄×C₂₃	—	8	1	no	—
141	C₂²×C₄×C₂₃	—	8	1	no	—
142	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
149	C₂×C₈×C₂₃	—	2	1	no	—
150	C₂×C₈×C₂₃	—	2	1	no	—
151	C₂×C₈×C₂₃	—	2	1	no	—
152	C₂×C₈×C₂₃	—	2	1	no	—
153	C₂×C₈×C₂₃	—	2	1	no	—
154	C₂×C₈×C₂₃	—	2	1	no	—
155	C₂×C₈×C₂₃	—	2	1	no	—
156	C₂×C₈×C₂₃	—	2	1	no	—
157	C₂×C₈×C₂₃	—	2	1	no	—
158	C₂×C₈×C₂₃	—	2	1	no	—
159	C₂×C₈×C₂₃	—	2	1	no	—
160	C₂×C₈×C₂₃	—	2	1	no	—
161	C₄⋊C₄×C₂₃	—	2	2	no	—
162	C₄⋊C₄×C₂₃	—	2	2	no	—
163	C₄⋊C₄×C₂₃	—	2	1	no	—
164	C₄⋊C₄×C₂₃	—	2	1	no	—
165	C₄⋊C₄×C₂₃	—	2	1	no	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	1	no	—
170	C₄⋊C₄×C₂₃	—	2	1	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	1	no	—
173	C₄⋊C₄×C₂₃	—	2	1	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	4	2	no	—
177	C₄⋊C₄×C₂₃	—	4	2	no	—
178	C₄⋊C₄×C₂₃	—	4	2	no	—
179	C₄⋊C₄×C₂₃	—	4	2	no	—
180	C₄⋊C₄×C₂₃	—	4	2	no	—
181	C₄⋊C₄×C₂₃	—	8	1	no	—
182	C₄⋊C₄×C₂₃	—	8	1	no	—
183	C₄⋊C₄×C₂₃	—	8	1	no	—
184	C₄⋊C₄×C₂₃	—	8	1	no	—
185	C₄⋊C₄×C₂₃	—	8	1	no	—
186	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
187	C₂²⋊C₄×C₂₃	—	2	4	no	—
188	C₂²⋊C₄×C₂₃	—	4	2	no	—
189	C₂²⋊C₄×C₂₃	—	4	2	no	—
190	C₄²×C₂₃	—	2	1	no	—
191	C₄²×C₂₃	—	2	1	no	—
192	C₄²×C₂₃	—	2	1	no	—
193	C₄²×C₂₃	—	2	1	no	—
194	C₄²×C₂₃	—	4	2	no	—
195	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
196	C₄²×C₂₃	#41	1	2	yes	—
197	#26124	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂³×C₂₃	—	2	2	no	—
199	C₂³×C₂₃	—	2	2	no	—
200	C₂³×C₂₃	—	2	2	no	—
201	C₂³×C₂₃	—	4	1	no	—
202	C₂³×C₂₃	—	4	1	no	—
203	C₂³×C₂₃	—	4	1	no	—
204	C₂³×C₂₃	—	4	1	no	—
205	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
206	D₄×C₂₃	—	2	2	no	—
207	D₄×C₂₃	—	2	2	no	—
208	D₄×C₂₃	—	2	2	no	—
209	D₄×C₂₃	#125	1	4	yes	complemented
210	C₂×C₄×C₂₃	—	2	1	no	—
211	C₂×C₄×C₂₃	—	2	1	no	—
212	C₂×C₄×C₂₃	—	2	1	no	—
213	C₂×C₄×C₂₃	—	2	1	no	—
214	C₂×C₄×C₂₃	—	2	1	no	—
215	C₂×C₄×C₂₃	—	2	1	no	—
216	C₂×C₄×C₂₃	—	2	1	no	—
217	C₂×C₄×C₂₃	—	2	1	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	1	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	1	no	—
222	C₂×C₄×C₂₃	—	2	1	no	—
223	C₂×C₄×C₂₃	—	2	1	no	—
224	C₂×C₄×C₂₃	—	2	1	no	—
225	C₂×C₄×C₂₃	—	4	1	no	—
226	C₂×C₄×C₂₃	—	4	1	no	—
227	C₂×C₄×C₂₃	—	4	1	no	—
228	C₂×C₄×C₂₃	—	4	1	no	—
229	C₂×C₄×C₂₃	—	4	1	no	—
230	C₂×C₄×C₂₃	—	4	1	no	—
231	C₂×C₄×C₂₃	—	4	1	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	—	4	1	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	8	4	no	—
246	C₂×C₄×C₂₃	—	8	4	no	—
247	C₂×C₄×C₂₃	—	8	4	no	—
248	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
249	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
250	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
251	C₂×C₄×C₂₃	#137	1	2	yes	—
252	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
253	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
255	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
256	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
257	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
258	C₈×C₂₃	—	4	1	no	—
259	C₈×C₂₃	—	4	1	no	—
260	C₈×C₂₃	—	4	1	no	—
261	C₈×C₂₃	—	4	1	no	—
262	#2552	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#2349	C₂×C₂₃	1	1	yes	characteristic
264	#2100	C₂×C₂₃	1	1	yes	characteristic, complemented
265	#2099	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#2088	C₂×C₂₃	1	1	yes	characteristic, complemented
267	#2087	C₂×C₂₃	1	1	yes	characteristic, complemented
268	#1949	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#1940	C₂×C₂₃	1	1	yes	characteristic, complemented
270	#1864	C₂×C₂₃	1	1	yes	characteristic
271	#1862	C₂×C₂₃	1	1	yes	characteristic
272	#1836	C₂×C₂₃	1	1	yes	characteristic, complemented
273	#1802	C₂×C₂₃	1	1	yes	characteristic
274	#1802	C₂×C₂₃	1	1	yes	characteristic
275	#1802	C₂×C₂₃	1	1	yes	characteristic
276	#1762	C₂×C₂₃	1	1	yes	characteristic
277	C₂²×C₂₃	—	2	4	no	—
278	C₂²×C₂₃	—	4	2	no	—
279	C₂²×C₂₃	—	4	2	no	—
280	C₂²×C₂₃	—	4	2	no	—
281	C₂²×C₂₃	—	4	2	no	—
282	C₂²×C₂₃	—	4	2	no	—
283	C₂²×C₂₃	—	4	2	no	—
284	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
285	C₂²×C₂₃	#546	1	1	yes	characteristic
286	C₂²×C₂₃	#548	1	1	yes	characteristic
287	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
288	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
289	C₂²×C₂₃	#576	1	1	yes	characteristic
290	C₂²×C₂₃	#577	1	1	yes	characteristic
291	C₄×C₂₃	—	2	2	no	—
292	C₄×C₂₃	—	2	2	no	—
293	C₄×C₂₃	—	2	2	no	—
294	C₄×C₂₃	—	2	2	no	—
295	C₄×C₂₃	—	4	2	no	—
296	C₄×C₂₃	—	4	1	no	—
297	C₄×C₂₃	—	4	1	no	—
298	C₄×C₂₃	—	8	1	no	—
299	C₄×C₂₃	—	8	1	no	—
300	C₄×C₂₃	—	8	1	no	—
301	C₄×C₂₃	—	8	1	no	—
302	C₄×C₂₃	—	8	1	no	—
303	C₄×C₂₃	—	8	1	no	—
304	C₄×C₂₃	C₂×#125	1	1	yes	characteristic
305	C₄×C₂₃	#544	1	1	yes	characteristic
306	C₄×C₂₃	#552	1	1	yes	characteristic
307	C₄×C₂₃	#553	1	1	yes	characteristic
308	C₂×#125	—	2	1	no	—
309	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
310	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
311	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
312	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
313	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
314	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
315	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
317	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
318	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
319	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
322	#398	—	2	1	no	—
323	#398	—	2	1	no	—
324	#389	—	2	1	no	—
325	#389	C₂²×C₂₃	1	1	yes	characteristic
326	#389	C₂²×C₂₃	1	1	yes	characteristic
327	#389	C₂²×C₂₃	1	1	yes	characteristic
328	#389	C₂²×C₂₃	1	1	yes	characteristic
329	#385	—	2	1	no	—
330	#385	—	2	1	no	—
331	#384	C₂²×C₂₃	1	1	yes	characteristic
332	#384	C₂²×C₂₃	1	1	yes	characteristic
333	#384	C₂²×C₂₃	1	1	yes	characteristic
334	#382	C₂²×C₂₃	1	1	yes	characteristic
335	#380	—	2	1	no	—
336	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
337	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×#95	—	2	1	no	—
339	#340	C₂²×C₂₃	1	1	yes	characteristic
340	#340	C₂²×C₂₃	1	1	yes	characteristic
341	#340	C₂²×C₂₃	1	1	yes	characteristic
342	#340	C₂²×C₂₃	1	1	yes	characteristic
343	#340	C₂²×C₂₃	1	1	yes	characteristic
344	#340	C₂²×C₂₃	1	1	yes	characteristic
345	#340	C₂²×C₂₃	1	1	yes	characteristic
346	#340	C₂²×C₂₃	1	1	yes	characteristic
347	#340	C₂²×C₂₃	1	1	yes	characteristic
348	#340	C₂²×C₂₃	1	1	yes	characteristic
349	#336	C₂²×C₂₃	1	1	yes	characteristic
350	#336	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×C₂₃	—	4	4	no	—
352	C₂×C₂₃	#2442	1	1	yes	characteristic
353	C₂×C₂₃	#3079	1	1	yes	characteristic
354	C₂×C₂₃	#3081	1	1	yes	characteristic
355	C₂×C₂₃	#3306	1	1	yes	characteristic
356	C₂×C₂₃	#3314	1	1	yes	characteristic
357	C₂×C₂₃	#3366	1	1	yes	characteristic
358	C₂×C₂₃	#3372	1	1	yes	characteristic
359	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
360	C₂³×C₄	—	2	1	no	—
361	C₂³×C₄	—	2	1	no	—
362	C₂³×C₄	—	2	1	no	—
363	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
364	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
365	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
367	#125	—	4	4	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	—	4	1	no	—
372	C₄×D₄	—	4	1	no	—
373	C₄×D₄	Q₈×C₂₃	1	4	yes	—
374	C₂×C₄⋊C₄	—	2	1	no	—
375	C₂×C₄⋊C₄	—	2	1	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
385	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
386	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
389	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
390	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
393	C₂×C₂²⋊C₄	—	2	1	no	—
394	C₂×C₂²⋊C₄	—	2	1	no	—
395	C₂×C₂²⋊C₄	—	2	1	no	—
396	C₂×C₄²	—	2	1	no	—
397	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
400	#107	—	4	1	no	—
401	#106	—	4	1	no	—
402	C₄⋊C₈	—	2	2	no	—
403	C₄⋊C₈	—	4	1	no	—
404	C₄⋊C₈	—	4	1	no	—
405	#102	—	2	2	no	—
406	#102	—	2	2	no	—
407	#102	—	2	2	no	—
408	#102	—	2	2	no	—
409	#97	—	2	2	no	—
410	#95	—	2	1	no	—
411	#95	—	2	1	no	—
412	#95	—	2	1	no	—
413	#95	—	2	1	no	—
414	#95	—	2	1	no	—
415	#95	—	2	1	no	—
416	#95	—	2	1	no	—
417	#95	D₄×C₂₃	1	1	yes	characteristic
418	#95	D₄×C₂₃	1	1	yes	characteristic
419	C₂₃	#26124	1	1	yes	characteristic, direct product factor, Sylow
420	C₂⁴	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	2	1	no	—
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
425	C₂×D₄	#41×C₂₃	1	2	yes	—
426	C₂×D₄	#41×C₂₃	1	2	yes	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	1	no	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	4	2	no	—
436	C₂²×C₄	—	4	2	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	—	8	1	no	—
440	C₂²×C₄	—	8	1	no	—
441	C₂²×C₄	—	8	1	no	—
442	C₂²×C₄	—	8	1	no	—
443	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
444	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
445	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
447	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
449	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	1	no	—
454	C₂×C₈	—	2	1	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	1	no	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	4	2	no	—
478	C₄⋊C₄	—	4	2	no	—
479	C₄⋊C₄	—	4	2	no	—
480	C₄⋊C₄	—	4	2	no	—
481	C₄⋊C₄	—	4	2	no	—
482	C₄⋊C₄	—	8	1	no	—
483	C₄⋊C₄	—	8	1	no	—
484	C₄⋊C₄	—	8	1	no	—
485	C₄⋊C₄	—	8	1	no	—
486	C₄⋊C₄	—	8	1	no	—
487	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₂²⋊C₄	—	4	2	no	—
490	C₂²⋊C₄	—	4	2	no	—
491	C₄²	—	2	1	no	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	2	1	no	—
495	C₄²	—	4	2	no	—
496	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
497	C₄²	#41×C₂₃	1	2	yes	—
498	C₂³	—	2	2	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	4	1	no	—
502	C₂³	—	4	1	no	—
503	C₂³	—	4	1	no	—
504	C₂³	—	4	1	no	—
505	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
506	D₄	—	2	2	no	—
507	D₄	—	2	2	no	—
508	D₄	—	2	2	no	—
509	D₄	C₂₃×#125	1	4	yes	complemented
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	1	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	8	4	no	—
546	C₂×C₄	—	8	4	no	—
547	C₂×C₄	—	8	4	no	—
548	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
549	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
550	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
551	C₂×C₄	C₂₃×#137	1	2	yes	—
552	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
553	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
554	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
555	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
556	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
557	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
558	C₈	—	4	1	no	—
559	C₈	—	4	1	no	—
560	C₈	—	4	1	no	—
561	C₈	—	4	1	no	—
562	C₂²	—	2	4	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	2	no	—
569	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
570	C₂²	C₂₃×#546	1	1	yes	characteristic
571	C₂²	C₂₃×#548	1	1	yes	characteristic
572	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
573	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
574	C₂²	C₂₃×#576	1	1	yes	characteristic
575	C₂²	C₂₃×#577	1	1	yes	characteristic
576	C₄	—	2	2	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	2	no	—
579	C₄	—	2	2	no	—
580	C₄	—	4	1	no	—
581	C₄	—	4	1	no	—
582	C₄	—	4	2	no	—
583	C₄	—	8	1	no	—
584	C₄	—	8	1	no	—
585	C₄	—	8	1	no	—
586	C₄	—	8	1	no	—
587	C₄	—	8	1	no	—
588	C₄	—	8	1	no	—
589	C₄	C₂×C₂₃×#125	1	1	yes	characteristic
590	C₄	C₂₃×#544	1	1	yes	characteristic
591	C₄	C₂₃×#552	1	1	yes	characteristic
592	C₄	C₂₃×#553	1	1	yes	characteristic
593	C₂	—	4	4	no	—
594	C₂	C₂₃×#2442	1	1	yes	characteristic
595	C₂	C₂₃×#3079	1	1	yes	characteristic
596	C₂	C₂₃×#3081	1	1	yes	characteristic
597	C₂	C₂₃×#3306	1	1	yes	characteristic
598	C₂	C₂₃×#3314	1	1	yes	characteristic
599	C₂	C₂₃×#3366	1	1	yes	characteristic
600	C₂	C₂₃×#3372	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2552	C₂	semidirect
3. C₂₃×#2100	C₂	semidirect
4. C₂₃×#2099	C₂	semidirect
5. C₂₃×#2088	C₂	semidirect
6. C₂₃×#2087	C₂	semidirect
7. C₂₃×#1949	C₂	semidirect
8. C₂₃×#1940	C₂	semidirect
11. C₂₃×#1836	C₂	semidirect
197. #26124	C₂₃	direct (N × H)
209. D₄×C₂₃	#125	semidirect
262. #2552	C₂×C₂₃	semidirect
264. #2100	C₂×C₂₃	semidirect
265. #2099	C₂×C₂₃	semidirect
266. #2088	C₂×C₂₃	semidirect
267. #2087	C₂×C₂₃	semidirect
268. #1949	C₂×C₂₃	semidirect
269. #1940	C₂×C₂₃	semidirect
272. #1836	C₂×C₂₃	semidirect
419. C₂₃	#26124	direct (N × H)
509. D₄	C₂₃×#125	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2552	2
C₂₃×#2349	2
C₂₃×#2100	2
C₂₃×#2099	2
C₂₃×#2088	2
C₂₃×#2087	2
C₂₃×#1949	2
C₂₃×#1940	2
C₂₃×#1864	2
C₂₃×#1862	2
C₂₃×#1836	2
C₂₃×#1802	2
C₂₃×#1762	2
#26124	23

Subgroups — C23×#26124 #19173750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#26124 #19173750