Subgroups — C₂₃×#17512 #19165138

Order 5888.

1004 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	C₂₃×#3475	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2965	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2906	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1495	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1489	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1483	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1481	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1477	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#131	—	2	1	no	—
13	C₂₃×#556	C₄	1	1	yes	characteristic
14	C₂₃×#555	C₄	1	1	yes	characteristic
15	C₂₃×#554	C₄	1	1	yes	characteristic
16	C₂₃×#553	—	2	1	no	—
17	C₂₃×#552	—	2	1	no	—
18	C₂₃×#552	C₄	1	2	yes	—
19	C₂₃×#551	—	2	2	no	—
20	C₂₃×#547	C₄	1	1	yes	characteristic
21	C₂₃×#546	C₄	1	2	yes	—
22	C₂₃×#533	C₂²	1	1	yes	characteristic
23	C₂₃×#529	—	2	2	no	—
24	C₂₃×#529	C₂²	1	2	yes	—
25	C₂₃×#525	—	2	1	no	—
26	C₄×#41×C₂₃	C₂²	1	2	yes	—
27	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#117	—	2	1	no	—
30	C₂₃×#446	—	2	1	no	—
31	C₂₃×#436	—	2	1	no	—
32	C₂₃×#435	—	2	2	no	—
33	C₂₃×#431	C₂²	1	1	yes	characteristic
34	C₂₃×#431	C₂²	1	2	yes	complemented
35	C₂₃×#424	—	2	2	no	—
36	C₂₃×#424	—	2	1	no	—
37	C₂₃×#423	C₂²	1	1	yes	characteristic
38	C₂₃×#423	C₂²	1	2	yes	complemented
39	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
41	C₄×M₁₆×C₂₃	C₂²	1	2	yes	complemented
42	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
43	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
44	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
45	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
46	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
47	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
48	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
49	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₂₃×D₄⋊C₈	C₂²	1	1	yes	characteristic
51	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
52	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
54	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
55	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
56	C₂³×C₄×C₂₃	—	2	1	no	—
57	C₂₃×#131	—	4	4	no	—
58	C₂×M₁₆×C₂₃	—	2	1	no	—
59	C₂×M₁₆×C₂₃	—	2	2	no	—
60	C₂×M₁₆×C₂₃	—	2	2	no	—
61	C₂²×C₈×C₂₃	—	2	1	no	—
62	C₂²×C₈×C₂₃	—	2	2	no	—
63	C₂²×C₈×C₂₃	—	2	1	no	—
64	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
65	C₂₃×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
66	C₂₃×#126	—	2	2	no	—
67	C₂₃×#126	—	2	2	no	—
68	C₂₃×#126	—	2	2	no	—
69	C₂₃×#125	—	2	2	no	—
70	C₂₃×#125	—	2	1	no	—
71	C₂₃×#125	C₂×C₄	1	2	yes	—
72	C₂₃×#124	—	2	2	no	—
73	C₂₃×#124	C₂×C₄	1	2	yes	—
74	C₂₃×#123	—	2	2	no	—
75	C₂₃×#123	—	2	2	no	—
76	C₂₃×#123	—	2	2	no	—
77	C₂₃×#123	—	2	2	no	—
78	C₂₃×C₂²⋊Q₈	—	2	1	no	—
79	C₂₃×C₂²⋊Q₈	—	2	1	no	—
80	C₂₃×C₂²⋊Q₈	—	2	1	no	—
81	C₂₃×C₂²⋊Q₈	—	2	2	no	—
82	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
83	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
84	C₂₃×#121	—	2	1	no	—
85	C₂₃×#121	—	2	1	no	—
86	C₂₃×#121	—	2	2	no	—
87	C₂₃×#121	C₂×C₄	1	2	yes	—
88	C₂₃×#120	—	2	2	no	—
89	C₄×Q₈×C₂₃	—	2	1	no	—
90	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
91	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
92	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
93	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
94	C₄×D₄×C₂₃	—	2	2	no	—
95	C₄×D₄×C₂₃	—	2	2	no	—
96	C₄×D₄×C₂₃	—	2	2	no	—
97	C₄×D₄×C₂₃	—	2	1	no	—
98	C₄×D₄×C₂₃	—	2	1	no	—
99	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
100	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
101	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
102	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
103	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
104	C₄×D₄×C₂₃	D₄	1	2	yes	—
105	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
106	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
107	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
108	C₂₃×#117	—	2	1	no	—
109	C₂₃×#117	—	2	1	no	—
110	C₂₃×#117	—	2	1	no	—
111	C₂₃×#117	—	2	2	no	—
112	C₂₃×#117	—	2	2	no	—
113	C₂₃×#117	—	2	2	no	—
114	C₂₃×#117	C₂×C₄	1	2	yes	—
115	C₂₃×#117	D₄	1	1	yes	characteristic
116	C₂₃×#117	D₄	1	2	yes	—
117	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
118	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
119	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
120	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
121	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
122	C₂×C₄²×C₂₃	—	2	2	no	—
123	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
125	C₂₃×C₄⋊C₈	—	2	1	no	—
126	C₂₃×C₄⋊C₈	—	2	2	no	—
127	C₂₃×C₄⋊C₈	—	2	1	no	—
128	C₂₃×C₄⋊C₈	—	2	2	no	—
129	C₂₃×C₄⋊C₈	—	2	1	no	—
130	C₂₃×C₄⋊C₈	—	2	2	no	—
131	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
132	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
133	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
134	C₂₃×C₂²⋊C₈	—	2	1	no	—
135	C₂₃×C₂²⋊C₈	—	2	2	no	—
136	C₂₃×#97	—	2	2	no	—
137	C₄×C₈×C₂₃	—	2	1	no	—
138	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
139	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₂₃	C₂³	1	2	yes	—
141	C₂⁴×C₂₃	—	2	1	no	—
142	#41×C₂₃	—	2	2	no	—
143	#41×C₂₃	—	2	2	no	—
144	#41×C₂₃	—	2	4	no	—
145	#41×C₂₃	—	4	2	no	—
146	C₂×Q₈×C₂₃	—	2	1	no	—
147	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
148	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
149	C₂×D₄×C₂₃	—	2	2	no	—
150	C₂×D₄×C₂₃	—	2	2	no	—
151	C₂×D₄×C₂₃	—	2	2	no	—
152	C₂×D₄×C₂₃	—	2	1	no	—
153	C₂×D₄×C₂₃	—	4	1	no	—
154	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
155	C₂×D₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
156	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
157	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	—	2	1	no	—
159	C₂²×C₄×C₂₃	—	2	2	no	—
160	C₂²×C₄×C₂₃	—	2	1	no	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	1	no	—
163	C₂²×C₄×C₂₃	—	2	1	no	—
164	C₂²×C₄×C₂₃	—	2	2	no	—
165	C₂²×C₄×C₂₃	—	4	2	no	—
166	C₂²×C₄×C₂₃	—	4	2	no	—
167	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
169	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	M₁₆×C₂₃	—	4	4	no	—
174	M₁₆×C₂₃	—	4	4	no	—
175	M₁₆×C₂₃	—	4	2	no	—
176	C₂×C₈×C₂₃	—	2	1	no	—
177	C₂×C₈×C₂₃	—	2	1	no	—
178	C₂×C₈×C₂₃	—	2	2	no	—
179	C₂×C₈×C₂₃	—	2	1	no	—
180	C₂×C₈×C₂₃	—	2	1	no	—
181	C₂×C₈×C₂₃	—	2	2	no	—
182	C₂×C₈×C₂₃	—	4	2	no	—
183	C₂×C₈×C₂₃	—	4	4	no	—
184	C₂×C₈×C₂₃	—	4	2	no	—
185	C₄⋊C₄×C₂₃	—	2	1	no	—
186	C₄⋊C₄×C₂₃	—	2	1	no	—
187	C₄⋊C₄×C₂₃	—	2	2	no	—
188	C₄⋊C₄×C₂₃	—	2	2	no	—
189	C₄⋊C₄×C₂₃	—	2	1	no	—
190	C₄⋊C₄×C₂₃	—	2	2	no	—
191	C₄⋊C₄×C₂₃	—	2	2	no	—
192	C₄⋊C₄×C₂₃	—	2	2	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	—	2	2	no	—
195	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
197	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
198	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
199	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
200	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
201	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
202	C₂²⋊C₄×C₂₃	—	2	2	no	—
203	C₂²⋊C₄×C₂₃	—	2	2	no	—
204	C₂²⋊C₄×C₂₃	—	2	2	no	—
205	C₂²⋊C₄×C₂₃	—	2	2	no	—
206	C₂²⋊C₄×C₂₃	—	2	2	no	—
207	C₂²⋊C₄×C₂₃	—	2	1	no	—
208	C₂²⋊C₄×C₂₃	—	2	1	no	—
209	C₂²⋊C₄×C₂₃	—	2	2	no	—
210	C₂²⋊C₄×C₂₃	—	2	2	no	—
211	C₂²⋊C₄×C₂₃	—	2	1	no	—
212	C₂²⋊C₄×C₂₃	—	2	1	no	—
213	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
214	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
215	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
216	C₄²×C₂₃	—	2	2	no	—
217	C₄²×C₂₃	—	2	2	no	—
218	C₄²×C₂₃	—	2	2	no	—
219	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
220	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
221	C₄²×C₂₃	C₂×D₄	1	2	yes	—
222	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
223	C₄²×C₂₃	C₂×D₄	1	2	yes	—
224	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
225	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
226	#17512	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂³×C₂₃	—	2	1	no	—
228	C₂³×C₂₃	—	2	1	no	—
229	C₂³×C₂₃	—	4	2	no	—
230	C₂³×C₂₃	—	4	2	no	—
231	C₂³×C₂₃	—	8	1	no	—
232	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
233	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	Q₈×C₂₃	—	2	2	no	—
235	Q₈×C₂₃	—	2	2	no	—
236	Q₈×C₂₃	—	2	2	no	—
237	Q₈×C₂₃	—	8	1	no	—
238	D₄×C₂₃	—	2	4	no	—
239	D₄×C₂₃	—	2	1	no	—
240	D₄×C₂₃	—	2	1	no	—
241	D₄×C₂₃	—	2	1	no	—
242	D₄×C₂₃	—	2	2	no	—
243	D₄×C₂₃	—	4	2	no	—
244	D₄×C₂₃	—	4	4	no	—
245	D₄×C₂₃	—	8	2	no	—
246	D₄×C₂₃	#131	1	2	yes	complemented
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	2	no	—
249	C₂×C₄×C₂₃	—	2	1	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	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	2	no	—
255	C₂×C₄×C₂₃	—	2	2	no	—
256	C₂×C₄×C₂₃	—	2	1	no	—
257	C₂×C₄×C₂₃	—	4	1	no	—
258	C₂×C₄×C₂₃	—	4	2	no	—
259	C₂×C₄×C₂₃	—	4	4	no	—
260	C₂×C₄×C₂₃	—	4	2	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	1	no	—
265	C₂×C₄×C₂₃	—	4	2	no	—
266	C₂×C₄×C₂₃	—	4	2	no	—
267	C₂×C₄×C₂₃	—	8	2	no	—
268	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
269	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
270	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
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	AGL(1,ℤ/8)	1	2	yes	—
275	C₂×C₄×C₂₃	#137	1	2	yes	—
276	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
277	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
279	C₈×C₂₃	—	2	2	no	—
280	C₈×C₂₃	—	4	1	no	—
281	C₈×C₂₃	—	4	2	no	—
282	C₈×C₂₃	—	4	1	no	—
283	C₈×C₂₃	—	4	2	no	—
284	C₈×C₂₃	—	4	1	no	—
285	#3475	C₂×C₂₃	1	1	yes	characteristic
286	#2965	C₂×C₂₃	1	1	yes	characteristic, complemented
287	#2906	C₂×C₂₃	1	1	yes	characteristic, complemented
288	#1495	C₂×C₂₃	1	2	yes	complemented
289	#1493	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#1489	C₂×C₂₃	1	2	yes	complemented
291	#1486	C₂×C₂₃	1	1	yes	characteristic, complemented
292	#1483	C₂×C₂₃	1	2	yes	complemented
293	#1481	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#1477	C₂×C₂₃	1	2	yes	complemented
295	C₂×D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
296	C₂²×C₂₃	—	2	1	no	—
297	C₂²×C₂₃	—	2	1	no	—
298	C₂²×C₂₃	—	2	2	no	—
299	C₂²×C₂₃	—	2	2	no	—
300	C₂²×C₂₃	—	2	2	no	—
301	C₂²×C₂₃	—	2	2	no	—
302	C₂²×C₂₃	—	4	1	no	—
303	C₂²×C₂₃	—	4	1	no	—
304	C₂²×C₂₃	—	4	1	no	—
305	C₂²×C₂₃	—	4	1	no	—
306	C₂²×C₂₃	—	4	1	no	—
307	C₂²×C₂₃	—	8	4	no	—
308	C₂²×C₂₃	#417	1	2	yes	—
309	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
310	C₄×C₂₃	—	2	2	no	—
311	C₄×C₂₃	—	2	1	no	—
312	C₄×C₂₃	—	2	2	no	—
313	C₄×C₂₃	—	2	1	no	—
314	C₄×C₂₃	—	2	2	no	—
315	C₄×C₂₃	—	2	1	no	—
316	C₄×C₂₃	—	4	2	no	—
317	C₄×C₂₃	—	4	1	no	—
318	C₄×C₂₃	—	4	1	no	—
319	C₄×C₂₃	—	4	2	no	—
320	C₄×C₂₃	#417	1	2	yes	—
321	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
322	C₄×C₂₃	#568	1	1	yes	characteristic
323	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
324	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
325	C₂×#131	—	2	1	no	—
326	#556	C₄×C₂₃	1	1	yes	characteristic
327	#555	C₄×C₂₃	1	1	yes	characteristic
328	#554	C₄×C₂₃	1	1	yes	characteristic
329	#553	—	2	1	no	—
330	#552	—	2	1	no	—
331	#552	C₄×C₂₃	1	2	yes	—
332	#551	—	2	2	no	—
333	#547	C₄×C₂₃	1	1	yes	characteristic
334	#546	C₄×C₂₃	1	2	yes	—
335	#533	C₂²×C₂₃	1	1	yes	characteristic
336	#529	—	2	2	no	—
337	#529	C₂²×C₂₃	1	2	yes	—
338	#525	—	2	1	no	—
339	C₄×#41	C₂²×C₂₃	1	2	yes	—
340	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×#117	—	2	1	no	—
343	#446	—	2	1	no	—
344	#436	—	2	1	no	—
345	#435	—	2	2	no	—
346	#431	C₂²×C₂₃	1	1	yes	characteristic
347	#431	C₂²×C₂₃	1	2	yes	complemented
348	#424	—	2	2	no	—
349	#424	—	2	1	no	—
350	#423	C₂²×C₂₃	1	1	yes	characteristic
351	#423	C₂²×C₂₃	1	2	yes	complemented
352	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
354	C₄×M₁₆	C₂²×C₂₃	1	2	yes	complemented
355	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
356	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	complemented
357	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	complemented
358	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	complemented
359	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	complemented
360	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
361	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
362	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
363	D₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
364	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
365	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
366	C₂×C₂₃	—	2	2	no	—
367	C₂×C₂₃	—	4	2	no	—
368	C₂×C₂₃	—	4	2	no	—
369	C₂×C₂₃	—	8	1	no	—
370	C₂×C₂₃	#2880	1	1	yes	characteristic
371	C₂×C₂₃	C₂×#417	1	1	yes	characteristic
372	C₂×C₂₃	#2901	1	1	yes	characteristic
373	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
374	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
375	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
376	C₂³×C₄	—	2	1	no	—
377	#131	—	4	4	no	—
378	C₂×M₁₆	—	2	1	no	—
379	C₂×M₁₆	—	2	2	no	—
380	C₂×M₁₆	—	2	2	no	—
381	C₂²×C₈	—	2	1	no	—
382	C₂²×C₈	—	2	2	no	—
383	C₂²×C₈	—	2	1	no	—
384	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
385	C₄⋊Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
386	#126	—	2	2	no	—
387	#126	—	2	2	no	—
388	#126	—	2	2	no	—
389	#125	—	2	2	no	—
390	#125	—	2	1	no	—
391	#125	C₂×C₄×C₂₃	1	2	yes	—
392	#124	—	2	2	no	—
393	#124	C₂×C₄×C₂₃	1	2	yes	—
394	#123	—	2	2	no	—
395	#123	—	2	2	no	—
396	#123	—	2	2	no	—
397	#123	—	2	2	no	—
398	C₂²⋊Q₈	—	2	1	no	—
399	C₂²⋊Q₈	—	2	1	no	—
400	C₂²⋊Q₈	—	2	1	no	—
401	C₂²⋊Q₈	—	2	2	no	—
402	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
403	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
404	#121	—	2	1	no	—
405	#121	—	2	1	no	—
406	#121	—	2	2	no	—
407	#121	C₂×C₄×C₂₃	1	2	yes	—
408	#120	—	2	2	no	—
409	C₄×Q₈	—	2	1	no	—
410	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
411	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
412	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
413	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
414	C₄×D₄	—	2	2	no	—
415	C₄×D₄	—	2	2	no	—
416	C₄×D₄	—	2	2	no	—
417	C₄×D₄	—	2	1	no	—
418	C₄×D₄	—	2	1	no	—
419	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
420	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
421	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
422	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
423	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
424	C₄×D₄	D₄×C₂₃	1	2	yes	—
425	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
426	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
427	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
428	#117	—	2	1	no	—
429	#117	—	2	1	no	—
430	#117	—	2	1	no	—
431	#117	—	2	2	no	—
432	#117	—	2	2	no	—
433	#117	—	2	2	no	—
434	#117	C₂×C₄×C₂₃	1	2	yes	—
435	#117	D₄×C₂₃	1	1	yes	characteristic
436	#117	D₄×C₂₃	1	2	yes	—
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	—	2	1	no	—
439	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
440	C₂×C₂²⋊C₄	—	2	1	no	—
441	C₂×C₂²⋊C₄	—	2	2	no	—
442	C₂×C₄²	—	2	2	no	—
443	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
444	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
445	C₄⋊C₈	—	2	1	no	—
446	C₄⋊C₈	—	2	2	no	—
447	C₄⋊C₈	—	2	1	no	—
448	C₄⋊C₈	—	2	2	no	—
449	C₄⋊C₈	—	2	1	no	—
450	C₄⋊C₈	—	2	2	no	—
451	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
452	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
453	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
454	C₂²⋊C₈	—	2	1	no	—
455	C₂²⋊C₈	—	2	2	no	—
456	#97	—	2	2	no	—
457	C₄×C₈	—	2	1	no	—
458	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
459	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
460	C₄×C₈	C₂³×C₂₃	1	2	yes	—
461	C₂₃	#17512	1	1	yes	characteristic, direct product factor, Sylow
462	C₂⁴	—	2	1	no	—
463	#41	—	2	2	no	—
464	#41	—	2	2	no	—
465	#41	—	2	4	no	—
466	#41	—	4	2	no	—
467	C₂×Q₈	—	2	1	no	—
468	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
469	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
470	C₂×D₄	—	2	2	no	—
471	C₂×D₄	—	2	2	no	—
472	C₂×D₄	—	2	2	no	—
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	4	1	no	—
475	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
476	C₂×D₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
477	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
478	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	2	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	4	2	no	—
487	C₂²×C₄	—	4	2	no	—
488	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
489	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
490	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
491	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
493	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
494	M₁₆	—	4	4	no	—
495	M₁₆	—	4	4	no	—
496	M₁₆	—	4	2	no	—
497	C₂×C₈	—	2	1	no	—
498	C₂×C₈	—	2	1	no	—
499	C₂×C₈	—	2	2	no	—
500	C₂×C₈	—	2	1	no	—
501	C₂×C₈	—	2	1	no	—
502	C₂×C₈	—	2	2	no	—
503	C₂×C₈	—	4	2	no	—
504	C₂×C₈	—	4	4	no	—
505	C₂×C₈	—	4	2	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
517	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
518	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
519	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
520	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
521	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
522	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
523	C₂²⋊C₄	—	2	2	no	—
524	C₂²⋊C₄	—	2	2	no	—
525	C₂²⋊C₄	—	2	2	no	—
526	C₂²⋊C₄	—	2	2	no	—
527	C₂²⋊C₄	—	2	2	no	—
528	C₂²⋊C₄	—	2	1	no	—
529	C₂²⋊C₄	—	2	1	no	—
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	2	no	—
532	C₂²⋊C₄	—	2	1	no	—
533	C₂²⋊C₄	—	2	1	no	—
534	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
535	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
536	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
537	C₄²	—	2	2	no	—
538	C₄²	—	2	2	no	—
539	C₄²	—	2	2	no	—
540	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
541	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
542	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
543	C₄²	C₂×D₄×C₂₃	1	2	yes	—
544	C₄²	C₂×D₄×C₂₃	1	2	yes	—
545	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
546	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
547	C₂³	—	2	1	no	—
548	C₂³	—	2	1	no	—
549	C₂³	—	4	2	no	—
550	C₂³	—	4	2	no	—
551	C₂³	—	8	1	no	—
552	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
553	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
554	Q₈	—	2	2	no	—
555	Q₈	—	2	2	no	—
556	Q₈	—	2	2	no	—
557	Q₈	—	8	1	no	—
558	D₄	—	2	4	no	—
559	D₄	—	2	1	no	—
560	D₄	—	2	1	no	—
561	D₄	—	2	1	no	—
562	D₄	—	2	2	no	—
563	D₄	—	4	2	no	—
564	D₄	—	4	4	no	—
565	D₄	—	8	2	no	—
566	D₄	C₂₃×#131	1	2	yes	complemented
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	2	no	—
579	C₂×C₄	—	4	4	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	8	2	no	—
588	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
589	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
590	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
591	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
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	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
595	C₂×C₄	C₂₃×#137	1	2	yes	—
596	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
597	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
598	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
599	C₈	—	2	2	no	—
600	C₈	—	4	1	no	—
601	C₈	—	4	2	no	—
602	C₈	—	4	1	no	—
603	C₈	—	4	2	no	—
604	C₈	—	4	1	no	—
605	C₂²	—	2	1	no	—
606	C₂²	—	2	1	no	—
607	C₂²	—	2	2	no	—
608	C₂²	—	2	2	no	—
609	C₂²	—	2	2	no	—
610	C₂²	—	2	2	no	—
611	C₂²	—	4	1	no	—
612	C₂²	—	4	1	no	—
613	C₂²	—	4	1	no	—
614	C₂²	—	4	1	no	—
615	C₂²	—	4	1	no	—
616	C₂²	—	8	4	no	—
617	C₂²	C₂₃×#417	1	2	yes	—
618	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
619	C₄	—	2	2	no	—
620	C₄	—	2	1	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	1	no	—
623	C₄	—	2	2	no	—
624	C₄	—	2	1	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	1	no	—
627	C₄	—	4	1	no	—
628	C₄	—	4	2	no	—
629	C₄	C₂₃×#417	1	2	yes	—
630	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
631	C₄	C₂₃×#568	1	1	yes	characteristic
632	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
633	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
634	C₂	—	2	2	no	—
635	C₂	—	4	2	no	—
636	C₂	—	4	2	no	—
637	C₂	—	8	1	no	—
638	C₂	C₂₃×#2880	1	1	yes	characteristic
639	C₂	C₂×C₂₃×#417	1	1	yes	characteristic
640	C₂	C₂₃×#2901	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
prime radical	C₂₃×#3475	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₄×#41×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2965	C₂	semidirect
3. C₂₃×#2906	C₂	semidirect
4. C₂₃×#1495	C₂	semidirect
5. C₂₃×#1493	C₂	semidirect
6. C₂₃×#1489	C₂	semidirect
7. C₂₃×#1486	C₂	semidirect
8. C₂₃×#1483	C₂	semidirect
9. C₂₃×#1481	C₂	semidirect
10. C₂₃×#1477	C₂	semidirect
11. C₂×C₂₃×D₄⋊C₈	C₂	semidirect
34. C₂₃×#431	C₂²	semidirect
38. C₂₃×#423	C₂²	semidirect
40. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
41. C₄×M₁₆×C₂₃	C₂²	semidirect
43. C₂₃×Q₈⋊C₈	C₂²	semidirect
44. C₂₃×Q₈⋊C₈	C₂²	semidirect
45. C₂₃×Q₈⋊C₈	C₂²	semidirect
46. C₂₃×Q₈⋊C₈	C₂²	semidirect
226. #17512	C₂₃	direct (N × H)
246. D₄×C₂₃	#131	semidirect
286. #2965	C₂×C₂₃	semidirect
287. #2906	C₂×C₂₃	semidirect
288. #1495	C₂×C₂₃	semidirect
289. #1493	C₂×C₂₃	semidirect
290. #1489	C₂×C₂₃	semidirect
291. #1486	C₂×C₂₃	semidirect
292. #1483	C₂×C₂₃	semidirect
293. #1481	C₂×C₂₃	semidirect
294. #1477	C₂×C₂₃	semidirect
295. C₂×D₄⋊C₈	C₂×C₂₃	semidirect
347. #431	C₂²×C₂₃	semidirect
351. #423	C₂²×C₂₃	semidirect
353. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
354. C₄×M₁₆	C₂²×C₂₃	semidirect
356. Q₈⋊C₈	C₂²×C₂₃	semidirect
357. Q₈⋊C₈	C₂²×C₂₃	semidirect
358. Q₈⋊C₈	C₂²×C₂₃	semidirect
359. Q₈⋊C₈	C₂²×C₂₃	semidirect
461. C₂₃	#17512	direct (N × H)
566. D₄	C₂₃×#131	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3475	2
C₂₃×#2965	2
C₂₃×#2906	2
C₂₃×#1495	2
C₂₃×#1493	2
C₂₃×#1489	2
C₂₃×#1486	2
C₂₃×#1483	2
C₂₃×#1481	2
C₂₃×#1477	2
C₂×C₂₃×D₄⋊C₈	2
#17512	23

Subgroups — C23×#17512 #19165138

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17512 #19165138