Subgroups — C₂₃×#18789 #19166415

Order 5888.

916 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#3209	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3079	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1637	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1636	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1633	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1632	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1550	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1487	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1477	C₂	1	1	yes	maximal, characteristic
12	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#557	—	2	1	no	—
14	C₂₃×#557	C₂²	1	1	yes	characteristic
15	C₂₃×#554	—	2	2	no	—
16	C₂₃×#554	—	2	1	no	—
17	C₂₃×#554	—	2	1	no	—
18	C₂₃×#554	C₂²	1	1	yes	characteristic
19	D₄×Q₈×C₂₃	—	2	1	no	—
20	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×#533	—	2	2	no	—
22	C₂₃×#533	—	2	1	no	—
23	C₂₃×#533	—	2	1	no	—
24	C₂₃×#533	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₄×#41×C₂₃	—	2	1	no	—
27	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
29	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
30	C₂₃×#494	C₂²	1	2	yes	—
31	C₂₃×#487	C₂²	1	2	yes	—
32	C₂₃×#477	—	2	1	no	—
33	C₂₃×#477	—	2	1	no	—
34	C₂₃×#477	C₂²	1	2	yes	—
35	C₂₃×#477	C₂²	1	2	yes	—
36	C₂₃×#477	C₂²	1	2	yes	—
37	C₂₃×#476	C₂²	1	2	yes	—
38	C₂₃×#475	—	2	1	no	—
39	C₂₃×#474	—	2	1	no	—
40	C₂₃×#462	—	2	2	no	—
41	C₂₃×#451	—	2	2	no	—
42	C₂₃×#431	C₂²	1	1	yes	characteristic
43	C₂₃×#428	—	2	1	no	—
44	C₂×C₂₃×#107	—	2	1	no	—
45	C₂₃×#423	C₂²	1	1	yes	characteristic
46	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
47	C₂₃×#417	—	2	1	no	—
48	C₂×C₂₃×#103	—	2	1	no	—
49	C₂₃×#333	C₂²	1	2	yes	—
50	C₂₃×#332	C₂²	1	2	yes	—
51	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
52	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
53	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
54	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
55	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
56	C₂²×Q₈×C₂₃	—	2	1	no	—
57	C₂²×Q₈×C₂₃	—	2	1	no	—
58	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
59	C₂×Q₁₆×C₂₃	—	2	4	no	—
60	C₂×M₁₆×C₂₃	—	2	1	no	—
61	C₂²×C₈×C₂₃	—	2	1	no	—
62	C₂₃×C₄⋊Q₈	—	2	2	no	—
63	C₂₃×C₄⋊Q₈	—	2	1	no	—
64	C₂₃×C₄⋊Q₈	—	2	2	no	—
65	C₂₃×C₄⋊Q₈	—	2	1	no	—
66	C₂₃×C₄⋊Q₈	—	2	1	no	—
67	C₂₃×C₄⋊Q₈	—	2	1	no	—
68	C₂₃×C₄⋊Q₈	—	2	1	no	—
69	C₂₃×C₄⋊Q₈	—	2	1	no	—
70	C₂₃×C₄⋊Q₈	—	2	1	no	—
71	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
72	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
73	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
74	C₂₃×#125	—	2	2	no	—
75	C₂₃×#125	—	2	2	no	—
76	C₂₃×#125	—	2	2	no	—
77	C₂₃×#125	—	2	1	no	—
78	C₂₃×#125	—	2	1	no	—
79	C₂₃×#125	—	2	1	no	—
80	C₂₃×C₂²⋊Q₈	—	2	2	no	—
81	C₂₃×C₂²⋊Q₈	—	2	2	no	—
82	C₂₃×C₂²⋊Q₈	—	2	2	no	—
83	C₂₃×C₂²⋊Q₈	—	2	2	no	—
84	C₂₃×C₂²⋊Q₈	—	2	2	no	—
85	C₂₃×C₂²⋊Q₈	—	2	1	no	—
86	C₂₃×C₂²⋊Q₈	—	2	2	no	—
87	C₂₃×C₂²⋊Q₈	—	2	1	no	—
88	C₂₃×C₂²⋊Q₈	—	2	1	no	—
89	C₂₃×C₂²⋊Q₈	—	2	2	no	—
90	C₂₃×C₂²⋊Q₈	D₄	1	2	yes	—
91	C₄×Q₈×C₂₃	—	2	2	no	—
92	C₄×Q₈×C₂₃	—	2	2	no	—
93	C₄×Q₈×C₂₃	—	2	2	no	—
94	C₄×Q₈×C₂₃	—	2	1	no	—
95	C₄×Q₈×C₂₃	—	2	1	no	—
96	C₄×Q₈×C₂₃	—	2	1	no	—
97	C₄×Q₈×C₂₃	—	2	1	no	—
98	C₄×Q₈×C₂₃	—	2	1	no	—
99	C₄×Q₈×C₂₃	—	2	1	no	—
100	C₄×Q₈×C₂₃	—	2	1	no	—
101	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
102	C₄×Q₈×C₂₃	D₄	1	2	yes	—
103	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
104	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
105	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
106	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
107	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
108	C₄×D₄×C₂₃	—	2	2	no	—
109	C₄×D₄×C₂₃	—	2	2	no	—
110	C₄×D₄×C₂₃	—	2	1	no	—
111	C₄×D₄×C₂₃	—	2	1	no	—
112	C₄×D₄×C₂₃	—	2	1	no	—
113	C₄×D₄×C₂₃	—	2	1	no	—
114	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
115	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
116	C₂₃×#117	—	2	2	no	—
117	C₂₃×#117	—	2	1	no	—
118	C₂₃×#117	—	2	1	no	—
119	C₂₃×#117	D₄	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
121	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
122	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
123	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
124	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
125	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₂₃	—	2	2	no	—
127	C₂×C₄²×C₂₃	—	2	1	no	—
128	C₂×C₄²×C₂₃	—	2	1	no	—
129	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
130	C₂₃×#107	—	2	2	no	—
131	C₂₃×#107	—	2	2	no	—
132	C₂₃×#107	—	2	2	no	—
133	C₂₃×#106	—	2	2	no	—
134	C₂₃×C₄⋊C₈	—	2	1	no	—
135	C₂₃×C₄⋊C₈	—	2	1	no	—
136	C₂₃×C₄⋊C₈	—	2	1	no	—
137	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
138	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
139	C₂₃×#103	—	2	2	no	—
140	C₂₃×#103	—	2	2	no	—
141	C₂₃×#103	—	2	2	no	—
142	C₂₃×#103	—	2	4	no	—
143	C₂₃×#102	—	2	2	no	—
144	C₂₃×C₂²⋊C₈	—	2	1	no	—
145	C₄×C₈×C₂₃	C₂³	1	2	yes	—
146	#41×C₂₃	—	2	2	no	—
147	#41×C₂₃	Q₁₆	1	4	yes	complemented
148	C₂×Q₈×C₂₃	—	2	2	no	—
149	C₂×Q₈×C₂₃	—	2	2	no	—
150	C₂×Q₈×C₂₃	—	2	1	no	—
151	C₂×Q₈×C₂₃	—	2	1	no	—
152	C₂×Q₈×C₂₃	—	2	2	no	—
153	C₂×Q₈×C₂₃	—	2	2	no	—
154	C₂×Q₈×C₂₃	—	4	1	no	—
155	C₂×Q₈×C₂₃	—	4	2	no	—
156	C₂×Q₈×C₂₃	—	4	2	no	—
157	C₂×Q₈×C₂₃	—	4	1	no	—
158	C₂×Q₈×C₂₃	—	8	2	no	—
159	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
163	C₂×D₄×C₂₃	—	2	1	no	—
164	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	—	2	1	no	—
166	C₂²×C₄×C₂₃	—	2	2	no	—
167	C₂²×C₄×C₂₃	—	2	1	no	—
168	C₂²×C₄×C₂₃	—	2	2	no	—
169	C₂²×C₄×C₂₃	—	2	1	no	—
170	C₂²×C₄×C₂₃	—	2	1	no	—
171	C₂²×C₄×C₂₃	—	2	1	no	—
172	C₂²×C₄×C₂₃	—	4	1	no	—
173	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
176	Q₁₆×C₂₃	—	8	4	no	—
177	M₁₆×C₂₃	—	4	2	no	—
178	C₂×C₈×C₂₃	—	2	2	no	—
179	C₂×C₈×C₂₃	—	2	2	no	—
180	C₂×C₈×C₂₃	—	2	2	no	—
181	C₂×C₈×C₂₃	—	4	2	no	—
182	C₄⋊C₄×C₂₃	—	2	2	no	—
183	C₄⋊C₄×C₂₃	—	2	2	no	—
184	C₄⋊C₄×C₂₃	—	2	2	no	—
185	C₄⋊C₄×C₂₃	—	2	2	no	—
186	C₄⋊C₄×C₂₃	—	2	2	no	—
187	C₄⋊C₄×C₂₃	—	2	2	no	—
188	C₄⋊C₄×C₂₃	—	2	2	no	—
189	C₄⋊C₄×C₂₃	—	2	2	no	—
190	C₄⋊C₄×C₂₃	—	2	1	no	—
191	C₄⋊C₄×C₂₃	—	2	2	no	—
192	C₄⋊C₄×C₂₃	—	2	1	no	—
193	C₄⋊C₄×C₂₃	—	2	1	no	—
194	C₄⋊C₄×C₂₃	—	2	1	no	—
195	C₄⋊C₄×C₂₃	—	2	2	no	—
196	C₄⋊C₄×C₂₃	—	2	1	no	—
197	C₄⋊C₄×C₂₃	—	2	1	no	—
198	C₄⋊C₄×C₂₃	—	2	1	no	—
199	C₄⋊C₄×C₂₃	—	2	1	no	—
200	C₄⋊C₄×C₂₃	—	2	1	no	—
201	C₄⋊C₄×C₂₃	—	2	1	no	—
202	C₄⋊C₄×C₂₃	—	2	2	no	—
203	C₄⋊C₄×C₂₃	—	2	2	no	—
204	C₄⋊C₄×C₂₃	—	4	1	no	—
205	C₄⋊C₄×C₂₃	—	4	1	no	—
206	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
207	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
208	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
209	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
210	C₄⋊C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
211	C₂²⋊C₄×C₂₃	—	2	2	no	—
212	C₂²⋊C₄×C₂₃	—	2	2	no	—
213	C₂²⋊C₄×C₂₃	—	2	2	no	—
214	C₂²⋊C₄×C₂₃	—	2	1	no	—
215	C₂²⋊C₄×C₂₃	—	2	1	no	—
216	C₂²⋊C₄×C₂₃	—	2	1	no	—
217	C₄²×C₂₃	—	2	2	no	—
218	C₄²×C₂₃	—	2	2	no	—
219	C₄²×C₂₃	—	2	1	no	—
220	C₄²×C₂₃	—	2	2	no	—
221	C₄²×C₂₃	—	2	1	no	—
222	C₄²×C₂₃	—	2	1	no	—
223	C₄²×C₂₃	—	2	1	no	—
224	C₄²×C₂₃	—	2	1	no	—
225	C₄²×C₂₃	—	2	1	no	—
226	C₄²×C₂₃	—	2	1	no	—
227	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
228	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
229	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
230	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
231	#18789	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂³×C₂₃	—	2	1	no	—
233	C₂³×C₂₃	#120	1	1	yes	characteristic
234	Q₈×C₂₃	—	2	1	no	—
235	Q₈×C₂₃	—	2	1	no	—
236	Q₈×C₂₃	—	2	1	no	—
237	Q₈×C₂₃	—	2	1	no	—
238	Q₈×C₂₃	—	4	2	no	—
239	Q₈×C₂₃	—	4	2	no	—
240	Q₈×C₂₃	—	4	4	no	—
241	Q₈×C₂₃	—	4	4	no	—
242	Q₈×C₂₃	—	4	1	no	—
243	Q₈×C₂₃	—	4	1	no	—
244	Q₈×C₂₃	—	8	2	no	—
245	Q₈×C₂₃	—	8	2	no	—
246	Q₈×C₂₃	C₂×Q₁₆	1	2	yes	—
247	Q₈×C₂₃	#137	1	2	yes	—
248	D₄×C₂₃	—	2	4	no	—
249	D₄×C₂₃	—	2	1	no	—
250	D₄×C₂₃	C₂×Q₁₆	1	2	yes	—
251	C₂×C₄×C₂₃	—	2	1	no	—
252	C₂×C₄×C₂₃	—	2	1	no	—
253	C₂×C₄×C₂₃	—	2	1	no	—
254	C₂×C₄×C₂₃	—	2	1	no	—
255	C₂×C₄×C₂₃	—	2	4	no	—
256	C₂×C₄×C₂₃	—	2	1	no	—
257	C₂×C₄×C₂₃	—	2	1	no	—
258	C₂×C₄×C₂₃	—	2	2	no	—
259	C₂×C₄×C₂₃	—	2	2	no	—
260	C₂×C₄×C₂₃	—	2	1	no	—
261	C₂×C₄×C₂₃	—	2	1	no	—
262	C₂×C₄×C₂₃	—	2	1	no	—
263	C₂×C₄×C₂₃	—	4	1	no	—
264	C₂×C₄×C₂₃	—	4	2	no	—
265	C₂×C₄×C₂₃	—	4	4	no	—
266	C₂×C₄×C₂₃	—	4	1	no	—
267	C₂×C₄×C₂₃	—	4	1	no	—
268	C₂×C₄×C₂₃	—	4	2	no	—
269	C₂×C₄×C₂₃	—	8	1	no	—
270	C₂×C₄×C₂₃	—	8	1	no	—
271	C₂×C₄×C₂₃	—	8	2	no	—
272	C₂×C₄×C₂₃	—	8	2	no	—
273	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
274	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
275	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
276	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
277	C₂×C₄×C₂₃	#137	1	2	yes	—
278	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
279	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
281	C₈×C₂₃	—	4	2	no	—
282	C₈×C₂₃	—	4	2	no	—
283	C₈×C₂₃	—	4	2	no	—
284	Q₈×#41	C₂×C₂₃	1	1	yes	characteristic
285	#3209	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#3079	C₂×C₂₃	1	1	yes	characteristic
287	C₂×#477	C₂×C₂₃	1	1	yes	characteristic, complemented
288	#1637	C₂×C₂₃	1	2	yes	complemented
289	#1636	C₂×C₂₃	1	2	yes	complemented
290	#1633	C₂×C₂₃	1	2	yes	complemented
291	#1632	C₂×C₂₃	1	2	yes	complemented
292	#1550	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#1487	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#1477	C₂×C₂₃	1	1	yes	characteristic
295	C₂²×C₂₃	—	2	1	no	—
296	C₂²×C₂₃	—	2	1	no	—
297	C₂²×C₂₃	—	2	2	no	—
298	C₂²×C₂₃	—	4	1	no	—
299	C₂²×C₂₃	—	4	1	no	—
300	C₂²×C₂₃	#451	1	2	yes	—
301	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
302	C₄×C₂₃	—	2	2	no	—
303	C₄×C₂₃	—	2	1	no	—
304	C₄×C₂₃	—	2	2	no	—
305	C₄×C₂₃	—	2	1	no	—
306	C₄×C₂₃	—	2	1	no	—
307	C₄×C₂₃	—	4	1	no	—
308	C₄×C₂₃	—	4	2	no	—
309	C₄×C₂₃	—	4	1	no	—
310	C₄×C₂₃	—	4	1	no	—
311	C₄×C₂₃	—	4	2	no	—
312	C₄×C₂₃	—	4	1	no	—
313	C₄×C₂₃	—	4	1	no	—
314	C₄×C₂₃	—	8	2	no	—
315	C₄×C₂₃	#451	1	2	yes	—
316	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
317	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
318	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
319	C₄×C₂₃	#575	1	1	yes	characteristic
320	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
321	#557	—	2	1	no	—
322	#557	C₂²×C₂₃	1	1	yes	characteristic
323	#554	—	2	2	no	—
324	#554	—	2	1	no	—
325	#554	—	2	1	no	—
326	#554	C₂²×C₂₃	1	1	yes	characteristic
327	D₄×Q₈	—	2	1	no	—
328	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
329	#533	—	2	2	no	—
330	#533	—	2	1	no	—
331	#533	—	2	1	no	—
332	#533	C₂²×C₂₃	1	1	yes	characteristic
333	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
334	C₄×#41	—	2	1	no	—
335	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×C₄×Q₈	—	2	1	no	—
337	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
338	#494	C₂²×C₂₃	1	2	yes	—
339	#487	C₂²×C₂₃	1	2	yes	—
340	#477	—	2	1	no	—
341	#477	—	2	1	no	—
342	#477	C₂²×C₂₃	1	2	yes	—
343	#477	C₂²×C₂₃	1	2	yes	—
344	#477	C₂²×C₂₃	1	2	yes	—
345	#476	C₂²×C₂₃	1	2	yes	—
346	#475	—	2	1	no	—
347	#474	—	2	1	no	—
348	#462	—	2	2	no	—
349	#451	—	2	2	no	—
350	#431	C₂²×C₂₃	1	1	yes	characteristic
351	#428	—	2	1	no	—
352	C₂×#107	—	2	1	no	—
353	#423	C₂²×C₂₃	1	1	yes	characteristic
354	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
355	#417	—	2	1	no	—
356	C₂×#103	—	2	1	no	—
357	#333	C₂²×C₂₃	1	2	yes	—
358	#332	C₂²×C₂₃	1	2	yes	—
359	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
360	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
361	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
362	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
363	C₂×C₂₃	—	2	2	no	—
364	C₂×C₂₃	—	4	2	no	—
365	C₂×C₂₃	C₂×#451	1	1	yes	characteristic
366	C₂×C₂₃	#3007	1	1	yes	characteristic
367	C₂×C₂₃	#3021	1	1	yes	characteristic
368	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
369	C₂²×Q₈	—	2	1	no	—
370	C₂²×Q₈	—	2	1	no	—
371	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
372	C₂×Q₁₆	—	2	4	no	—
373	C₂×M₁₆	—	2	1	no	—
374	C₂²×C₈	—	2	1	no	—
375	C₄⋊Q₈	—	2	2	no	—
376	C₄⋊Q₈	—	2	1	no	—
377	C₄⋊Q₈	—	2	2	no	—
378	C₄⋊Q₈	—	2	1	no	—
379	C₄⋊Q₈	—	2	1	no	—
380	C₄⋊Q₈	—	2	1	no	—
381	C₄⋊Q₈	—	2	1	no	—
382	C₄⋊Q₈	—	2	1	no	—
383	C₄⋊Q₈	—	2	1	no	—
384	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
385	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
386	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
387	#125	—	2	2	no	—
388	#125	—	2	2	no	—
389	#125	—	2	2	no	—
390	#125	—	2	1	no	—
391	#125	—	2	1	no	—
392	#125	—	2	1	no	—
393	C₂²⋊Q₈	—	2	2	no	—
394	C₂²⋊Q₈	—	2	2	no	—
395	C₂²⋊Q₈	—	2	2	no	—
396	C₂²⋊Q₈	—	2	2	no	—
397	C₂²⋊Q₈	—	2	2	no	—
398	C₂²⋊Q₈	—	2	1	no	—
399	C₂²⋊Q₈	—	2	2	no	—
400	C₂²⋊Q₈	—	2	1	no	—
401	C₂²⋊Q₈	—	2	1	no	—
402	C₂²⋊Q₈	—	2	2	no	—
403	C₂²⋊Q₈	D₄×C₂₃	1	2	yes	—
404	C₄×Q₈	—	2	2	no	—
405	C₄×Q₈	—	2	2	no	—
406	C₄×Q₈	—	2	2	no	—
407	C₄×Q₈	—	2	1	no	—
408	C₄×Q₈	—	2	1	no	—
409	C₄×Q₈	—	2	1	no	—
410	C₄×Q₈	—	2	1	no	—
411	C₄×Q₈	—	2	1	no	—
412	C₄×Q₈	—	2	1	no	—
413	C₄×Q₈	—	2	1	no	—
414	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
415	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
416	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
417	C₄×Q₈	D₄×C₂₃	1	2	yes	—
418	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
419	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
420	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
421	C₄×D₄	—	2	2	no	—
422	C₄×D₄	—	2	2	no	—
423	C₄×D₄	—	2	1	no	—
424	C₄×D₄	—	2	1	no	—
425	C₄×D₄	—	2	1	no	—
426	C₄×D₄	—	2	1	no	—
427	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
428	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
429	#117	—	2	2	no	—
430	#117	—	2	1	no	—
431	#117	—	2	1	no	—
432	#117	D₄×C₂₃	1	1	yes	characteristic
433	C₂×C₄⋊C₄	—	2	2	no	—
434	C₂×C₄⋊C₄	—	2	1	no	—
435	C₂×C₄⋊C₄	—	2	1	no	—
436	C₂×C₄⋊C₄	—	2	1	no	—
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
439	C₂×C₄²	—	2	2	no	—
440	C₂×C₄²	—	2	1	no	—
441	C₂×C₄²	—	2	1	no	—
442	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
443	#107	—	2	2	no	—
444	#107	—	2	2	no	—
445	#107	—	2	2	no	—
446	#106	—	2	2	no	—
447	C₄⋊C₈	—	2	1	no	—
448	C₄⋊C₈	—	2	1	no	—
449	C₄⋊C₈	—	2	1	no	—
450	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
451	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
452	#103	—	2	2	no	—
453	#103	—	2	2	no	—
454	#103	—	2	2	no	—
455	#103	—	2	4	no	—
456	#102	—	2	2	no	—
457	C₂²⋊C₈	—	2	1	no	—
458	C₄×C₈	C₂³×C₂₃	1	2	yes	—
459	C₂₃	#18789	1	1	yes	characteristic, direct product factor, Sylow
460	#41	—	2	2	no	—
461	#41	Q₁₆×C₂₃	1	4	yes	complemented
462	C₂×Q₈	—	2	2	no	—
463	C₂×Q₈	—	2	2	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	2	no	—
467	C₂×Q₈	—	2	2	no	—
468	C₂×Q₈	—	4	1	no	—
469	C₂×Q₈	—	4	2	no	—
470	C₂×Q₈	—	4	2	no	—
471	C₂×Q₈	—	4	1	no	—
472	C₂×Q₈	—	8	2	no	—
473	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
474	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
475	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
476	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	C₂×D₄×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	2	no	—
483	C₂²×C₄	—	2	1	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	4	1	no	—
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	Q₁₆	—	8	4	no	—
491	M₁₆	—	4	2	no	—
492	C₂×C₈	—	2	2	no	—
493	C₂×C₈	—	2	2	no	—
494	C₂×C₈	—	2	2	no	—
495	C₂×C₈	—	4	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	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	2	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	2	1	no	—
513	C₄⋊C₄	—	2	1	no	—
514	C₄⋊C₄	—	2	1	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	2	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	4	1	no	—
519	C₄⋊C₄	—	4	1	no	—
520	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
521	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
522	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
523	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
524	C₄⋊C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
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	1	no	—
531	C₄²	—	2	2	no	—
532	C₄²	—	2	2	no	—
533	C₄²	—	2	1	no	—
534	C₄²	—	2	2	no	—
535	C₄²	—	2	1	no	—
536	C₄²	—	2	1	no	—
537	C₄²	—	2	1	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	2	1	no	—
540	C₄²	—	2	1	no	—
541	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
542	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
543	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
544	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
545	C₂³	—	2	1	no	—
546	C₂³	C₂₃×#120	1	1	yes	characteristic
547	Q₈	—	2	1	no	—
548	Q₈	—	2	1	no	—
549	Q₈	—	2	1	no	—
550	Q₈	—	2	1	no	—
551	Q₈	—	4	2	no	—
552	Q₈	—	4	2	no	—
553	Q₈	—	4	4	no	—
554	Q₈	—	4	4	no	—
555	Q₈	—	4	1	no	—
556	Q₈	—	4	1	no	—
557	Q₈	—	8	2	no	—
558	Q₈	—	8	2	no	—
559	Q₈	C₂×Q₁₆×C₂₃	1	2	yes	—
560	Q₈	C₂₃×#137	1	2	yes	—
561	D₄	—	2	4	no	—
562	D₄	—	2	1	no	—
563	D₄	C₂×Q₁₆×C₂₃	1	2	yes	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	1	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	2	no	—
578	C₂×C₄	—	4	4	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	2	no	—
582	C₂×C₄	—	8	1	no	—
583	C₂×C₄	—	8	1	no	—
584	C₂×C₄	—	8	2	no	—
585	C₂×C₄	—	8	2	no	—
586	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
587	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
588	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
589	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
590	C₂×C₄	C₂₃×#137	1	2	yes	—
591	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
592	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
593	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
594	C₈	—	4	2	no	—
595	C₈	—	4	2	no	—
596	C₈	—	4	2	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	1	no	—
599	C₂²	—	2	2	no	—
600	C₂²	—	4	1	no	—
601	C₂²	—	4	1	no	—
602	C₂²	C₂₃×#451	1	2	yes	—
603	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
604	C₄	—	2	2	no	—
605	C₄	—	2	1	no	—
606	C₄	—	2	2	no	—
607	C₄	—	2	1	no	—
608	C₄	—	2	1	no	—
609	C₄	—	4	1	no	—
610	C₄	—	4	2	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	1	no	—
613	C₄	—	4	2	no	—
614	C₄	—	4	1	no	—
615	C₄	—	4	1	no	—
616	C₄	—	8	2	no	—
617	C₄	C₂₃×#451	1	2	yes	—
618	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
619	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
620	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
621	C₄	C₂₃×#575	1	1	yes	characteristic
622	C₂	—	2	2	no	—
623	C₂	—	4	2	no	—
624	C₂	C₂×C₂₃×#451	1	1	yes	characteristic
625	C₂	C₂₃×#3007	1	1	yes	characteristic
626	C₂	C₂₃×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×#41×C₂₃	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#3209	C₂	semidirect
4. C₂×C₂₃×#477	C₂	semidirect
5. C₂₃×#1637	C₂	semidirect
6. C₂₃×#1636	C₂	semidirect
7. C₂₃×#1633	C₂	semidirect
8. C₂₃×#1632	C₂	semidirect
9. C₂₃×#1550	C₂	semidirect
10. C₂₃×#1487	C₂	semidirect
147. #41×C₂₃	Q₁₆	semidirect
231. #18789	C₂₃	direct (N × H)
285. #3209	C₂×C₂₃	semidirect
287. C₂×#477	C₂×C₂₃	semidirect
288. #1637	C₂×C₂₃	semidirect
289. #1636	C₂×C₂₃	semidirect
290. #1633	C₂×C₂₃	semidirect
291. #1632	C₂×C₂₃	semidirect
292. #1550	C₂×C₂₃	semidirect
293. #1487	C₂×C₂₃	semidirect
459. C₂₃	#18789	direct (N × H)
461. #41	Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#41×C₂₃	2
C₂₃×#3209	2
C₂₃×#3079	2
C₂×C₂₃×#477	2
C₂₃×#1637	2
C₂₃×#1636	2
C₂₃×#1633	2
C₂₃×#1632	2
C₂₃×#1550	2
C₂₃×#1487	2
C₂₃×#1477	2
#18789	23

Subgroups — C23×#18789 #19166415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18789 #19166415