Subgroups — C₂₃×#17760 #19165386

Order 5888.

896 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3503	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2989	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2925	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1559	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1558	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1554	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1550	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1548	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1548	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1542	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#333	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈²×C₂₃	—	2	1	no	—
13	C₂₃×#557	C₄	1	2	yes	—
14	C₂₃×#554	—	2	2	no	—
15	C₂₃×#554	—	2	1	no	—
16	D₄×Q₈×C₂₃	C₄	1	2	yes	—
17	D₄×Q₈×C₂₃	C₄	1	4	yes	—
18	C₂₃×#544	—	2	1	no	—
19	C₂₃×#544	—	2	2	no	—
20	C₂₃×#533	C₂²	1	1	yes	characteristic
21	C₂₃×#533	C₂²	1	2	yes	—
22	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
24	C₂₃×#520	—	2	1	no	—
25	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#446	—	2	1	no	—
27	C₂₃×#436	—	2	1	no	—
28	C₂₃×#435	—	2	2	no	—
29	C₂₃×#433	—	2	2	no	—
30	C₂₃×#432	C₂²	1	2	yes	—
31	C₂₃×#432	C₂²	1	2	yes	—
32	C₂₃×#431	C₂²	1	2	yes	—
33	C₂₃×#431	C₂²	1	1	yes	characteristic
34	C₂₃×#423	C₂²	1	1	yes	characteristic
35	C₂₃×#423	C₂²	1	2	yes	—
36	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₂₃×#408	—	2	1	no	—
39	C₂₃×#405	—	2	1	no	—
40	C₂₃×#333	C₂²	1	2	yes	—
41	C₂₃×#333	C₂²	1	2	yes	—
42	C₂₃×#333	C₂²	1	1	yes	characteristic
43	C₂₃×#333	C₂²	1	1	yes	characteristic
44	C₂₃×#333	C₂²	1	2	yes	—
45	C₂₃×#333	C₂²	1	2	yes	—
46	C₂₃×#333	C₂²	1	2	yes	—
47	C₂₃×#332	C₂²	1	1	yes	characteristic
48	C₂₃×#332	C₂²	1	1	yes	characteristic
49	C₂₃×#332	C₂²	1	2	yes	—
50	C₂×#41×C₂₃	D₄	1	1	yes	characteristic
51	C₂²×Q₈×C₂₃	C₂×C₄	1	2	yes	—
52	C₂²×Q₈×C₂₃	C₂×C₄	1	4	yes	—
53	C₂×M₁₆×C₂₃	—	2	1	no	—
54	C₂×M₁₆×C₂₃	—	2	2	no	—
55	C₂²×C₈×C₂₃	—	2	1	no	—
56	C₂²×C₈×C₂₃	—	2	1	no	—
57	C₂₃×C₄⋊Q₈	—	2	4	no	—
58	C₂₃×C₄⋊Q₈	—	2	2	no	—
59	C₂₃×C₄⋊Q₈	—	2	1	no	—
60	C₂₃×C₄⋊Q₈	—	2	2	no	—
61	C₂₃×C₄⋊Q₈	—	2	1	no	—
62	C₂₃×C₄⋊Q₈	—	2	2	no	—
63	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
64	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
65	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₂₃×#125	—	2	4	no	—
69	C₂₃×#125	—	2	2	no	—
70	C₂₃×#125	—	2	1	no	—
71	C₂₃×#125	—	2	2	no	—
72	C₂₃×C₂²⋊Q₈	—	2	1	no	—
73	C₂₃×C₂²⋊Q₈	—	2	2	no	—
74	C₂₃×C₂²⋊Q₈	—	2	4	no	—
75	C₂₃×C₂²⋊Q₈	—	2	4	no	—
76	C₂₃×C₂²⋊Q₈	—	2	4	no	—
77	C₂₃×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
78	C₂₃×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
79	C₄×Q₈×C₂₃	—	2	2	no	—
80	C₄×Q₈×C₂₃	—	2	1	no	—
81	C₄×Q₈×C₂₃	—	2	2	no	—
82	C₄×Q₈×C₂₃	C₂×C₄	1	2	yes	—
83	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
84	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
86	C₄×D₄×C₂₃	—	2	4	no	—
87	C₄×D₄×C₂₃	—	2	2	no	—
88	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
89	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
90	C₄×D₄×C₂₃	D₄	1	2	yes	—
91	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
92	C₂₃×#117	—	2	2	no	—
93	C₂₃×#117	—	2	1	no	—
94	C₂₃×#117	D₄	1	1	yes	characteristic
95	C₂₃×#117	D₄	1	2	yes	—
96	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
97	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
102	C₂₃×C₄⋊C₈	—	2	1	no	—
103	C₂₃×C₄⋊C₈	—	2	1	no	—
104	C₂₃×C₄⋊C₈	—	2	1	no	—
105	C₂₃×C₄⋊C₈	—	2	2	no	—
106	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
107	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
108	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
109	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
110	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
111	C₂₃×C₄⋊C₈	C₂³	1	1	yes	characteristic
112	C₂₃×C₂²⋊C₈	—	2	2	no	—
113	C₂₃×C₂²⋊C₈	—	2	2	no	—
114	C₂₃×C₂²⋊C₈	—	2	1	no	—
115	C₂₃×C₂²⋊C₈	—	2	2	no	—
116	C₂₃×#97	—	2	2	no	—
117	C₂₃×#97	—	2	2	no	—
118	C₄×C₈×C₂₃	—	2	2	no	—
119	C₄×C₈×C₂₃	—	2	1	no	—
120	#41×C₂₃	—	2	2	no	—
121	#41×C₂₃	—	2	2	no	—
122	C₂×Q₈×C₂₃	—	2	2	no	—
123	C₂×Q₈×C₂₃	—	2	4	no	—
124	C₂×Q₈×C₂₃	—	2	1	no	—
125	C₂×Q₈×C₂₃	—	8	2	no	—
126	C₂×Q₈×C₂₃	—	8	4	no	—
127	C₂×Q₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
128	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
129	C₂×Q₈×C₂₃	C₂²×C₄	1	4	yes	—
130	C₂×D₄×C₂₃	C₂²⋊C₄	1	2	yes	—
131	C₂×D₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₄×C₂₃	—	2	4	no	—
133	C₂²×C₄×C₂₃	—	2	2	no	—
134	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
135	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
140	M₁₆×C₂₃	—	4	2	no	—
141	M₁₆×C₂₃	—	8	2	no	—
142	C₂×C₈×C₂₃	—	2	2	no	—
143	C₂×C₈×C₂₃	—	2	1	no	—
144	C₂×C₈×C₂₃	—	2	1	no	—
145	C₂×C₈×C₂₃	—	2	1	no	—
146	C₂×C₈×C₂₃	—	2	2	no	—
147	C₂×C₈×C₂₃	—	2	1	no	—
148	C₂×C₈×C₂₃	—	4	2	no	—
149	C₂×C₈×C₂₃	—	8	1	no	—
150	C₄⋊C₄×C₂₃	—	2	4	no	—
151	C₄⋊C₄×C₂₃	—	2	2	no	—
152	C₄⋊C₄×C₂₃	—	2	2	no	—
153	C₄⋊C₄×C₂₃	—	2	4	no	—
154	C₄⋊C₄×C₂₃	—	2	2	no	—
155	C₄⋊C₄×C₂₃	—	2	4	no	—
156	C₄⋊C₄×C₂₃	—	2	2	no	—
157	C₄⋊C₄×C₂₃	—	2	4	no	—
158	C₄⋊C₄×C₂₃	—	2	1	no	—
159	C₄⋊C₄×C₂₃	—	2	2	no	—
160	C₄⋊C₄×C₂₃	—	2	2	no	—
161	C₄⋊C₄×C₂₃	—	2	1	no	—
162	C₄⋊C₄×C₂₃	—	2	1	no	—
163	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
164	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
165	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
166	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
167	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
168	C₂²⋊C₄×C₂₃	—	2	4	no	—
169	C₂²⋊C₄×C₂₃	—	2	2	no	—
170	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
171	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
172	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
173	C₄²×C₂₃	—	2	4	no	—
174	C₄²×C₂₃	—	2	2	no	—
175	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
176	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
177	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
178	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₂₃	C₂×D₄	1	2	yes	—
180	C₄²×C₂₃	C₂×D₄	1	2	yes	—
181	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
182	#17760	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
184	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
185	Q₈×C₂₃	—	2	2	no	—
186	Q₈×C₂₃	—	4	4	no	—
187	Q₈×C₂₃	—	4	2	no	—
188	Q₈×C₂₃	—	4	1	no	—
189	Q₈×C₂₃	—	4	1	no	—
190	Q₈×C₂₃	—	8	2	no	—
191	Q₈×C₂₃	—	8	4	no	—
192	D₄×C₂₃	—	2	4	no	—
193	D₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	1	no	—
197	C₂×C₄×C₂₃	—	2	2	no	—
198	C₂×C₄×C₂₃	—	2	1	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	2	2	no	—
201	C₂×C₄×C₂₃	—	2	4	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	4	1	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	8	2	no	—
206	C₂×C₄×C₂₃	—	8	4	no	—
207	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
208	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
210	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
212	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₄×C₂₃	#137	1	2	yes	—
214	C₂×C₄×C₂₃	#137	1	2	yes	—
215	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
216	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
217	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
218	C₈×C₂₃	—	4	1	no	—
219	C₈×C₂₃	—	4	1	no	—
220	C₈×C₂₃	—	4	2	no	—
221	C₈×C₂₃	—	4	2	no	—
222	C₈×C₂₃	—	4	1	no	—
223	C₈×C₂₃	—	4	1	no	—
224	#3503	C₂×C₂₃	1	1	yes	characteristic
225	#2989	C₂×C₂₃	1	1	yes	characteristic
226	#2925	C₂×C₂₃	1	1	yes	characteristic
227	#1559	C₂×C₂₃	1	2	yes	complemented
228	#1558	C₂×C₂₃	1	2	yes	complemented
229	#1554	C₂×C₂₃	1	2	yes	complemented
230	#1550	C₂×C₂₃	1	1	yes	characteristic, complemented
231	#1548	C₂×C₂₃	1	1	yes	characteristic, complemented
232	#1548	C₂×C₂₃	1	2	yes	complemented
233	#1542	C₂×C₂₃	1	1	yes	characteristic, complemented
234	C₂×#333	C₂×C₂₃	1	1	yes	characteristic, complemented
235	C₂²×C₂₃	—	2	1	no	—
236	C₂²×C₂₃	—	2	1	no	—
237	C₂²×C₂₃	—	2	2	no	—
238	C₂²×C₂₃	—	2	2	no	—
239	C₂²×C₂₃	—	2	2	no	—
240	C₂²×C₂₃	#419	1	2	yes	—
241	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
242	C₄×C₂₃	—	2	1	no	—
243	C₄×C₂₃	—	2	1	no	—
244	C₄×C₂₃	—	2	2	no	—
245	C₄×C₂₃	—	2	2	no	—
246	C₄×C₂₃	—	2	1	no	—
247	C₄×C₂₃	—	2	2	no	—
248	C₄×C₂₃	—	4	4	no	—
249	C₄×C₂₃	—	4	2	no	—
250	C₄×C₂₃	—	4	2	no	—
251	C₄×C₂₃	#419	1	2	yes	—
252	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
253	C₄×C₂₃	C₂×#137	1	1	yes	characteristic
254	C₄×C₂₃	#577	1	1	yes	characteristic
255	C₄×C₂₃	#578	1	1	yes	characteristic
256	Q₈²	—	2	1	no	—
257	#557	C₄×C₂₃	1	2	yes	—
258	#554	—	2	2	no	—
259	#554	—	2	1	no	—
260	D₄×Q₈	C₄×C₂₃	1	2	yes	—
261	D₄×Q₈	C₄×C₂₃	1	4	yes	—
262	#544	—	2	1	no	—
263	#544	—	2	2	no	—
264	#533	C₂²×C₂₃	1	1	yes	characteristic
265	#533	C₂²×C₂₃	1	2	yes	—
266	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
267	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
268	#520	—	2	1	no	—
269	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
270	#446	—	2	1	no	—
271	#436	—	2	1	no	—
272	#435	—	2	2	no	—
273	#433	—	2	2	no	—
274	#432	C₂²×C₂₃	1	2	yes	—
275	#432	C₂²×C₂₃	1	2	yes	—
276	#431	C₂²×C₂₃	1	1	yes	characteristic
277	#431	C₂²×C₂₃	1	2	yes	—
278	#423	C₂²×C₂₃	1	1	yes	characteristic
279	#423	C₂²×C₂₃	1	2	yes	—
280	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
281	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
282	#408	—	2	1	no	—
283	#405	—	2	1	no	—
284	#333	C₂²×C₂₃	1	2	yes	—
285	#333	C₂²×C₂₃	1	2	yes	—
286	#333	C₂²×C₂₃	1	1	yes	characteristic
287	#333	C₂²×C₂₃	1	1	yes	characteristic
288	#333	C₂²×C₂₃	1	2	yes	—
289	#333	C₂²×C₂₃	1	2	yes	—
290	#333	C₂²×C₂₃	1	2	yes	—
291	#332	C₂²×C₂₃	1	1	yes	characteristic
292	#332	C₂²×C₂₃	1	1	yes	characteristic
293	#332	C₂²×C₂₃	1	2	yes	—
294	C₂×C₂₃	—	2	2	no	—
295	C₂×C₂₃	—	4	2	no	—
296	C₂×C₂₃	#2888	1	1	yes	characteristic
297	C₂×C₂₃	C₂×#419	1	1	yes	characteristic
298	C₂×C₂₃	#2898	1	1	yes	characteristic
299	C₂×#41	D₄×C₂₃	1	1	yes	characteristic
300	C₂²×Q₈	C₂×C₄×C₂₃	1	2	yes	—
301	C₂²×Q₈	C₂×C₄×C₂₃	1	4	yes	—
302	C₂×M₁₆	—	2	1	no	—
303	C₂×M₁₆	—	2	2	no	—
304	C₂²×C₈	—	2	1	no	—
305	C₂²×C₈	—	2	1	no	—
306	C₄⋊Q₈	—	2	4	no	—
307	C₄⋊Q₈	—	2	2	no	—
308	C₄⋊Q₈	—	2	2	no	—
309	C₄⋊Q₈	—	2	1	no	—
310	C₄⋊Q₈	—	2	1	no	—
311	C₄⋊Q₈	—	2	2	no	—
312	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
313	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
314	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
315	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
316	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
317	#125	—	2	4	no	—
318	#125	—	2	2	no	—
319	#125	—	2	2	no	—
320	#125	—	2	1	no	—
321	C₂²⋊Q₈	—	2	1	no	—
322	C₂²⋊Q₈	—	2	2	no	—
323	C₂²⋊Q₈	—	2	4	no	—
324	C₂²⋊Q₈	—	2	4	no	—
325	C₂²⋊Q₈	—	2	4	no	—
326	C₂²⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
327	C₂²⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
328	C₄×Q₈	—	2	2	no	—
329	C₄×Q₈	—	2	1	no	—
330	C₄×Q₈	—	2	2	no	—
331	C₄×Q₈	C₂×C₄×C₂₃	1	2	yes	—
332	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
333	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
334	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
335	C₄×D₄	—	2	4	no	—
336	C₄×D₄	—	2	2	no	—
337	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
338	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
339	C₄×D₄	D₄×C₂₃	1	2	yes	—
340	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
341	#117	—	2	2	no	—
342	#117	—	2	1	no	—
343	#117	D₄×C₂₃	1	1	yes	characteristic
344	#117	D₄×C₂₃	1	2	yes	—
345	C₂×C₄⋊C₄	—	2	2	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	—	2	2	no	—
349	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
350	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
351	C₄⋊C₈	—	2	1	no	—
352	C₄⋊C₈	—	2	1	no	—
353	C₄⋊C₈	—	2	1	no	—
354	C₄⋊C₈	—	2	2	no	—
355	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
356	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
357	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
358	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
359	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
360	C₄⋊C₈	C₂³×C₂₃	1	1	yes	characteristic
361	C₂²⋊C₈	—	2	2	no	—
362	C₂²⋊C₈	—	2	2	no	—
363	C₂²⋊C₈	—	2	1	no	—
364	C₂²⋊C₈	—	2	2	no	—
365	#97	—	2	2	no	—
366	#97	—	2	2	no	—
367	C₄×C₈	—	2	2	no	—
368	C₄×C₈	—	2	1	no	—
369	C₂₃	#17760	1	1	yes	characteristic, direct product factor, Sylow
370	#41	—	2	2	no	—
371	#41	—	2	2	no	—
372	C₂×Q₈	—	2	4	no	—
373	C₂×Q₈	—	2	2	no	—
374	C₂×Q₈	—	2	1	no	—
375	C₂×Q₈	—	8	2	no	—
376	C₂×Q₈	—	8	4	no	—
377	C₂×Q₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
378	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
379	C₂×Q₈	C₂²×C₄×C₂₃	1	4	yes	—
380	C₂×D₄	C₂²⋊C₄×C₂₃	1	2	yes	—
381	C₂×D₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	2	no	—
384	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
385	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
389	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
390	M₁₆	—	4	2	no	—
391	M₁₆	—	8	2	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	4	2	no	—
399	C₂×C₈	—	8	1	no	—
400	C₄⋊C₄	—	2	4	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	4	no	—
407	C₄⋊C₄	—	2	2	no	—
408	C₄⋊C₄	—	2	1	no	—
409	C₄⋊C₄	—	2	2	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
414	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
415	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
416	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
417	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
418	C₂²⋊C₄	—	2	4	no	—
419	C₂²⋊C₄	—	2	2	no	—
420	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
421	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
422	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
423	C₄²	—	2	4	no	—
424	C₄²	—	2	2	no	—
425	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
426	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
427	C₄²	C₂×D₄×C₂₃	1	2	yes	—
428	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
429	C₄²	C₂×D₄×C₂₃	1	2	yes	—
430	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
431	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
432	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
433	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
434	Q₈	—	2	2	no	—
435	Q₈	—	4	4	no	—
436	Q₈	—	4	2	no	—
437	Q₈	—	4	1	no	—
438	Q₈	—	4	1	no	—
439	Q₈	—	8	2	no	—
440	Q₈	—	8	4	no	—
441	D₄	—	2	4	no	—
442	D₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	8	2	no	—
455	C₂×C₄	—	8	4	no	—
456	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
457	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
458	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
459	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
460	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
461	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
462	C₂×C₄	C₂₃×#137	1	2	yes	—
463	C₂×C₄	C₂₃×#137	1	2	yes	—
464	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
465	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
466	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
467	C₈	—	4	1	no	—
468	C₈	—	4	1	no	—
469	C₈	—	4	2	no	—
470	C₈	—	4	2	no	—
471	C₈	—	4	1	no	—
472	C₈	—	4	1	no	—
473	C₂²	—	2	1	no	—
474	C₂²	—	2	1	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	2	no	—
478	C₂²	C₂₃×#419	1	2	yes	—
479	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
480	C₄	—	2	1	no	—
481	C₄	—	2	1	no	—
482	C₄	—	2	2	no	—
483	C₄	—	2	2	no	—
484	C₄	—	2	1	no	—
485	C₄	—	2	2	no	—
486	C₄	—	4	4	no	—
487	C₄	—	4	2	no	—
488	C₄	—	4	2	no	—
489	C₄	C₂₃×#419	1	2	yes	—
490	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
491	C₄	C₂×C₂₃×#137	1	1	yes	characteristic
492	C₄	C₂₃×#577	1	1	yes	characteristic
493	C₄	C₂₃×#578	1	1	yes	characteristic
494	C₂	—	2	2	no	—
495	C₂	—	4	2	no	—
496	C₂	C₂₃×#2888	1	1	yes	characteristic
497	C₂	C₂×C₂₃×#419	1	1	yes	characteristic
498	C₂	C₂₃×#2898	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1559	C₂	semidirect
5. C₂₃×#1558	C₂	semidirect
6. C₂₃×#1554	C₂	semidirect
7. C₂₃×#1550	C₂	semidirect
8. C₂₃×#1548	C₂	semidirect
9. C₂₃×#1548	C₂	semidirect
10. C₂₃×#1542	C₂	semidirect
11. C₂×C₂₃×#333	C₂	semidirect
182. #17760	C₂₃	direct (N × H)
227. #1559	C₂×C₂₃	semidirect
228. #1558	C₂×C₂₃	semidirect
229. #1554	C₂×C₂₃	semidirect
230. #1550	C₂×C₂₃	semidirect
231. #1548	C₂×C₂₃	semidirect
232. #1548	C₂×C₂₃	semidirect
233. #1542	C₂×C₂₃	semidirect
234. C₂×#333	C₂×C₂₃	semidirect
369. C₂₃	#17760	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3503	2
C₂₃×#2989	2
C₂₃×#2925	2
C₂₃×#1559	2
C₂₃×#1558	2
C₂₃×#1554	2
C₂₃×#1550	2
C₂₃×#1548	2
C₂₃×#1542	2
C₂×C₂₃×#333	2
#17760	23

Subgroups — C23×#17760 #19165386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17760 #19165386