Subgroups — C₂₃×#17574 #19165200

Order 5888.

1338 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1519	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1517	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1502	C₂	1	4	yes	maximal, complemented
7	C₂²×#41×C₂₃	—	2	1	no	—
8	C₂₃×#532	—	2	1	no	—
9	C₂₃×#532	—	2	1	no	—
10	C₂₃×#532	—	2	1	no	—
11	C₂₃×#532	—	2	1	no	—
12	C₂₃×#532	C₄	1	2	yes	complemented
13	C₂₃×#532	C₄	1	2	yes	complemented
14	C₂₃×#532	C₂²	1	2	yes	—
15	C₂₃×#532	C₂²	1	2	yes	—
16	C₂×C₂₃×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
17	C₂×C₂₃×#127	C₄	1	1	yes	characteristic, complemented
18	C₂×C₂₃×#124	—	2	1	no	—
19	C₂×C₂₃×#121	—	2	1	no	—
20	C₂×C₂₃×#121	C₂²	1	2	yes	—
21	C₂×C₄×D₄×C₂₃	—	2	1	no	—
22	C₂×C₄×D₄×C₂₃	C₄	1	2	yes	complemented
23	C₂×C₂₃×#117	—	2	1	no	—
24	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#431	—	2	1	no	—
26	C₂₃×#431	—	2	1	no	—
27	C₂₃×#431	C₂²	1	2	yes	complemented
28	C₂₃×#431	C₂²	1	2	yes	complemented
29	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
30	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₂₃	—	2	1	no	—
32	C₂₃×#384	—	2	1	no	—
33	C₂₃×#382	C₂²	1	2	yes	complemented
34	C₂₃×#380	—	2	1	no	—
35	C₂₃×#380	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
37	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
38	C₂₃×#327	C₂²	1	16	yes	complemented
39	C₂×#41×C₂₃	—	2	2	no	—
40	C₂×#41×C₂₃	—	2	2	no	—
41	C₂×#41×C₂₃	—	2	2	no	—
42	C₂×#41×C₂₃	—	2	2	no	—
43	C₂×#41×C₂₃	—	4	4	no	—
44	C₂×#41×C₂₃	—	4	4	no	—
45	C₂²×Q₈×C₂₃	—	2	1	no	—
46	C₂²×D₄×C₂₃	—	2	2	no	—
47	C₂²×D₄×C₂₃	—	2	1	no	—
48	C₂²×D₄×C₂₃	—	2	1	no	—
49	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
50	C₂³×C₄×C₂₃	—	2	2	no	—
51	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₂₃	—	2	2	no	—
55	C₂₃×C₄⋊Q₈	—	2	1	no	—
56	C₂₃×C₄⋊Q₈	—	2	1	no	—
57	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
58	C₂₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
59	C₂₃×#127	—	2	1	no	—
60	C₂₃×#127	—	2	1	no	—
61	C₂₃×#127	C₂×C₄	1	2	yes	—
62	C₂₃×#127	C₂×C₄	1	2	yes	—
63	C₂₃×#124	—	2	2	no	—
64	C₂₃×#124	—	2	2	no	—
65	C₂₃×#124	—	2	2	no	—
66	C₂₃×#124	—	2	2	no	—
67	C₂₃×#121	—	2	2	no	—
68	C₂₃×#121	—	2	2	no	—
69	C₂₃×#121	—	2	2	no	—
70	C₂₃×#121	—	2	2	no	—
71	C₂₃×#121	—	2	2	no	—
72	C₂₃×#121	—	2	2	no	—
73	C₂₃×#121	C₂×C₄	1	4	yes	—
74	C₂₃×#121	C₂³	1	4	yes	—
75	C₄×D₄×C₂₃	—	2	2	no	—
76	C₄×D₄×C₂₃	—	2	2	no	—
77	C₄×D₄×C₂₃	—	2	2	no	—
78	C₄×D₄×C₂₃	—	2	2	no	—
79	C₄×D₄×C₂₃	—	2	2	no	—
80	C₄×D₄×C₂₃	—	2	2	no	—
81	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
82	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
83	C₂₃×#117	—	4	4	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
89	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
92	C₂×C₄²×C₂₃	—	2	2	no	—
93	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
95	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
104	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
105	C₂₃×C₄⋊C₈	—	2	2	no	—
106	C₂₃×C₄⋊C₈	—	2	2	no	—
107	C₂₃×C₂²⋊C₈	—	2	2	no	—
108	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
109	C₄×C₈×C₂₃	—	2	2	no	—
110	C₄×C₈×C₂₃	—	2	2	no	—
111	C₂₃×#95	—	2	2	no	—
112	C₂₃×#95	C₂³	1	4	yes	—
113	C₂⁴×C₂₃	—	2	2	no	—
114	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
115	#41×C₂₃	—	4	8	no	—
116	#41×C₂₃	—	4	8	no	—
117	#41×C₂₃	—	4	4	no	—
118	#41×C₂₃	—	4	4	no	—
119	#41×C₂₃	—	4	4	no	—
120	#41×C₂₃	—	4	4	no	—
121	C₂×Q₈×C₂₃	—	2	2	no	—
122	C₂×Q₈×C₂₃	—	2	2	no	—
123	C₂×Q₈×C₂₃	—	4	4	no	—
124	C₂×D₄×C₂₃	—	2	4	no	—
125	C₂×D₄×C₂₃	—	2	4	no	—
126	C₂×D₄×C₂₃	—	2	2	no	—
127	C₂×D₄×C₂₃	—	2	2	no	—
128	C₂×D₄×C₂₃	—	2	2	no	—
129	C₂×D₄×C₂₃	—	2	2	no	—
130	C₂×D₄×C₂₃	—	2	2	no	—
131	C₂×D₄×C₂₃	—	4	4	no	—
132	C₂×D₄×C₂₃	—	4	4	no	—
133	C₂×D₄×C₂₃	—	4	4	no	—
134	C₂×D₄×C₂₃	—	4	4	no	—
135	C₂×D₄×C₂₃	—	4	4	no	—
136	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
137	C₂²×C₄×C₂₃	—	2	2	no	—
138	C₂²×C₄×C₂₃	—	2	4	no	—
139	C₂²×C₄×C₂₃	—	2	4	no	—
140	C₂²×C₄×C₂₃	—	2	4	no	—
141	C₂²×C₄×C₂₃	—	4	2	no	—
142	C₂²×C₄×C₂₃	—	4	2	no	—
143	C₂²×C₄×C₂₃	—	4	8	no	—
144	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	D₈	1	4	yes	—
153	C₂²×C₄×C₂₃	SD₁₆	1	4	yes	—
154	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
164	C₂×C₈×C₂₃	—	2	4	no	—
165	C₂×C₈×C₂₃	—	4	4	no	—
166	C₄⋊C₄×C₂₃	—	2	2	no	—
167	C₄⋊C₄×C₂₃	—	2	2	no	—
168	C₄⋊C₄×C₂₃	—	2	2	no	—
169	C₄⋊C₄×C₂₃	—	4	4	no	—
170	C₄⋊C₄×C₂₃	—	4	4	no	—
171	C₄⋊C₄×C₂₃	—	4	4	no	—
172	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
173	C₂²⋊C₄×C₂₃	—	2	4	no	—
174	C₂²⋊C₄×C₂₃	—	2	4	no	—
175	C₂²⋊C₄×C₂₃	—	2	4	no	—
176	C₂²⋊C₄×C₂₃	—	2	4	no	—
177	C₂²⋊C₄×C₂₃	—	4	4	no	—
178	C₄²×C₂₃	—	4	4	no	—
179	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
180	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
181	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
182	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
183	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
184	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
185	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
186	C₄²×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
187	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
188	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
189	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
190	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
191	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
195	#17574	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₂₃	—	2	4	no	—
197	C₂³×C₂₃	—	4	1	no	—
198	C₂³×C₂₃	—	4	1	no	—
199	C₂³×C₂₃	—	4	4	no	—
200	C₂³×C₂₃	—	4	4	no	—
201	C₂³×C₂₃	—	4	4	no	—
202	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₂³×C₂₃	C₂×D₈	1	2	yes	—
205	C₂³×C₂₃	C₂×SD₁₆	1	2	yes	—
206	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
207	Q₈×C₂₃	—	4	4	no	—
208	Q₈×C₂₃	—	4	4	no	—
209	D₄×C₂₃	—	4	4	no	—
210	D₄×C₂₃	—	4	4	no	—
211	D₄×C₂₃	—	4	4	no	—
212	D₄×C₂₃	—	4	4	no	—
213	D₄×C₂₃	—	4	8	no	—
214	D₄×C₂₃	—	4	4	no	—
215	D₄×C₂₃	—	4	4	no	—
216	D₄×C₂₃	—	4	4	no	—
217	D₄×C₂₃	—	4	4	no	—
218	C₂×C₄×C₂₃	—	2	2	no	—
219	C₂×C₄×C₂₃	—	2	2	no	—
220	C₂×C₄×C₂₃	—	2	2	no	—
221	C₂×C₄×C₂₃	—	2	2	no	—
222	C₂×C₄×C₂₃	—	2	2	no	—
223	C₂×C₄×C₂₃	—	2	2	no	—
224	C₂×C₄×C₂₃	—	2	4	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	4	no	—
232	C₂×C₄×C₂₃	—	4	8	no	—
233	C₂×C₄×C₂₃	—	4	8	no	—
234	C₂×C₄×C₂₃	—	4	4	no	—
235	C₂×C₄×C₂₃	—	4	4	no	—
236	C₂×C₄×C₂₃	—	4	4	no	—
237	C₂×C₄×C₂₃	#102	1	4	yes	—
238	C₂×C₄×C₂₃	#102	1	4	yes	—
239	C₂×C₄×C₂₃	#102	1	4	yes	—
240	C₂×C₄×C₂₃	#102	1	4	yes	—
241	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
242	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
246	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
247	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
249	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
252	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
253	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
254	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
255	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
256	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
257	C₈×C₂₃	—	4	4	no	—
258	C₂×#532	C₂×C₂₃	1	1	yes	characteristic
259	C₂×#431	C₂×C₂₃	1	1	yes	characteristic, complemented
260	#2285	C₂×C₂₃	1	1	yes	characteristic, complemented
261	#1519	C₂×C₂₃	1	4	yes	complemented
262	#1517	C₂×C₂₃	1	4	yes	complemented
263	#1502	C₂×C₂₃	1	4	yes	complemented
264	C₂²×C₂₃	—	2	1	no	—
265	C₂²×C₂₃	—	2	1	no	—
266	C₂²×C₂₃	—	2	1	no	—
267	C₂²×C₂₃	—	2	1	no	—
268	C₂²×C₂₃	—	2	1	no	—
269	C₂²×C₂₃	—	2	1	no	—
270	C₂²×C₂₃	—	4	2	no	—
271	C₂²×C₂₃	—	4	2	no	—
272	C₂²×C₂₃	—	4	4	no	—
273	C₂²×C₂₃	—	4	4	no	—
274	C₂²×C₂₃	—	4	4	no	—
275	C₂²×C₂₃	—	8	8	no	—
276	C₂²×C₂₃	C₂×#102	1	2	yes	—
277	C₂²×C₂₃	C₂×#102	1	2	yes	—
278	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
279	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
280	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
281	C₄×C₂₃	—	2	1	no	—
282	C₄×C₂₃	—	2	1	no	—
283	C₄×C₂₃	—	2	1	no	—
284	C₄×C₂₃	—	2	1	no	—
285	C₄×C₂₃	—	2	1	no	—
286	C₄×C₂₃	—	2	1	no	—
287	C₄×C₂₃	—	4	4	no	—
288	C₄×C₂₃	—	8	4	no	—
289	C₄×C₂₃	#410	1	2	yes	—
290	C₄×C₂₃	C₂×#102	1	2	yes	—
291	C₄×C₂₃	C₂×#102	1	2	yes	—
292	C₄×C₂₃	C₂×#102	1	2	yes	—
293	C₄×C₂₃	C₂×#102	1	2	yes	—
294	C₄×C₂₃	#421	1	2	yes	—
295	C₂²×#41	—	2	1	no	—
296	#532	—	2	1	no	—
297	#532	—	2	1	no	—
298	#532	—	2	1	no	—
299	#532	—	2	1	no	—
300	#532	C₄×C₂₃	1	2	yes	complemented
301	#532	C₄×C₂₃	1	2	yes	complemented
302	#532	C₂²×C₂₃	1	2	yes	—
303	#532	C₂²×C₂₃	1	2	yes	—
304	C₂×C₄⋊Q₈	C₄×C₂₃	1	1	yes	characteristic, complemented
305	C₂×#127	C₄×C₂₃	1	1	yes	characteristic, complemented
306	C₂×#124	—	2	1	no	—
307	C₂×#121	—	2	1	no	—
308	C₂×#121	C₂²×C₂₃	1	2	yes	—
309	C₂×C₄×D₄	—	2	1	no	—
310	C₂×C₄×D₄	C₄×C₂₃	1	2	yes	complemented
311	C₂×#117	—	2	1	no	—
312	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
313	#431	—	2	1	no	—
314	#431	—	2	1	no	—
315	#431	C₂²×C₂₃	1	2	yes	complemented
316	#431	C₂²×C₂₃	1	2	yes	complemented
317	C₂×C₄⋊C₈	—	2	1	no	—
318	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
319	C₂×C₄×C₈	—	2	1	no	—
320	#384	—	2	1	no	—
321	#382	C₂²×C₂₃	1	2	yes	complemented
322	#380	—	2	1	no	—
323	#380	C₂²×C₂₃	1	2	yes	—
324	C₄×C₄⋊C₄	—	2	1	no	—
325	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
326	#327	C₂²×C₂₃	1	16	yes	complemented
327	C₂×C₂₃	—	2	1	no	—
328	C₂×C₂₃	—	2	2	no	—
329	C₂×C₂₃	—	2	1	no	—
330	C₂×C₂₃	—	2	2	no	—
331	C₂×C₂₃	—	8	4	no	—
332	C₂×C₂₃	C₂×#410	1	1	yes	characteristic
333	C₂×C₂₃	C₂²×#102	1	1	yes	characteristic
334	C₂×C₂₃	C₂×#421	1	1	yes	characteristic
335	C₂×#41	—	2	2	no	—
336	C₂×#41	—	2	2	no	—
337	C₂×#41	—	2	2	no	—
338	C₂×#41	—	2	2	no	—
339	C₂×#41	—	4	4	no	—
340	C₂×#41	—	4	4	no	—
341	C₂²×Q₈	—	2	1	no	—
342	C₂²×D₄	—	2	2	no	—
343	C₂²×D₄	—	2	1	no	—
344	C₂²×D₄	—	2	1	no	—
345	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
346	C₂³×C₄	—	2	2	no	—
347	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
348	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
349	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
350	C₂²×C₈	—	2	2	no	—
351	C₄⋊Q₈	—	2	1	no	—
352	C₄⋊Q₈	—	2	1	no	—
353	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
354	C₄⋊Q₈	C₂×C₄×C₂₃	1	2	yes	—
355	#127	—	2	1	no	—
356	#127	—	2	1	no	—
357	#127	C₂×C₄×C₂₃	1	2	yes	—
358	#127	C₂×C₄×C₂₃	1	2	yes	—
359	#124	—	2	2	no	—
360	#124	—	2	2	no	—
361	#124	—	2	2	no	—
362	#124	—	2	2	no	—
363	#121	—	2	2	no	—
364	#121	—	2	2	no	—
365	#121	—	2	2	no	—
366	#121	—	2	2	no	—
367	#121	—	2	2	no	—
368	#121	—	2	2	no	—
369	#121	C₂×C₄×C₂₃	1	4	yes	—
370	#121	C₂³×C₂₃	1	4	yes	—
371	C₄×D₄	—	2	2	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	—	2	2	no	—
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	2	no	—
377	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
378	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
379	#117	—	4	4	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	2	no	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	2	no	—
384	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	—	2	2	no	—
388	C₂×C₄²	—	2	2	no	—
389	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
390	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
391	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
392	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
393	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
394	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
395	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
396	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
397	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
398	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
400	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
401	C₄⋊C₈	—	2	2	no	—
402	C₄⋊C₈	—	2	2	no	—
403	C₂²⋊C₈	—	2	2	no	—
404	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
405	C₄×C₈	—	2	2	no	—
406	C₄×C₈	—	2	2	no	—
407	#95	—	2	2	no	—
408	#95	C₂³×C₂₃	1	4	yes	—
409	C₂₃	#17574	1	1	yes	characteristic, direct product factor, Sylow
410	C₂⁴	—	2	2	no	—
411	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
412	#41	—	4	8	no	—
413	#41	—	4	8	no	—
414	#41	—	4	4	no	—
415	#41	—	4	4	no	—
416	#41	—	4	4	no	—
417	#41	—	4	4	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	2	2	no	—
420	C₂×Q₈	—	4	4	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	2	no	—
428	C₂×D₄	—	4	4	no	—
429	C₂×D₄	—	4	4	no	—
430	C₂×D₄	—	4	4	no	—
431	C₂×D₄	—	4	4	no	—
432	C₂×D₄	—	4	4	no	—
433	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	4	2	no	—
439	C₂²×C₄	—	4	2	no	—
440	C₂²×C₄	—	4	8	no	—
441	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
442	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
443	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
444	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
445	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
446	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
447	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
448	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
449	C₂²×C₄	D₈×C₂₃	1	4	yes	—
450	C₂²×C₄	SD₁₆×C₂₃	1	4	yes	—
451	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
452	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
458	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
459	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
460	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
461	C₂×C₈	—	2	4	no	—
462	C₂×C₈	—	4	4	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	2	2	no	—
466	C₄⋊C₄	—	4	4	no	—
467	C₄⋊C₄	—	4	4	no	—
468	C₄⋊C₄	—	4	4	no	—
469	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	2	4	no	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	4	4	no	—
475	C₄²	—	4	4	no	—
476	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
477	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
478	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
479	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
480	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
481	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
482	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
483	C₄²	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
484	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
485	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
486	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
487	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
488	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
489	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
490	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
491	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
492	C₂³	—	2	4	no	—
493	C₂³	—	4	1	no	—
494	C₂³	—	4	1	no	—
495	C₂³	—	4	4	no	—
496	C₂³	—	4	4	no	—
497	C₂³	—	4	4	no	—
498	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
499	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
500	C₂³	C₂×D₈×C₂₃	1	2	yes	—
501	C₂³	C₂×SD₁₆×C₂₃	1	2	yes	—
502	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
503	Q₈	—	4	4	no	—
504	Q₈	—	4	4	no	—
505	D₄	—	4	4	no	—
506	D₄	—	4	4	no	—
507	D₄	—	4	4	no	—
508	D₄	—	4	4	no	—
509	D₄	—	4	8	no	—
510	D₄	—	4	4	no	—
511	D₄	—	4	4	no	—
512	D₄	—	4	4	no	—
513	D₄	—	4	4	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	4	no	—
528	C₂×C₄	—	4	8	no	—
529	C₂×C₄	—	4	8	no	—
530	C₂×C₄	—	4	4	no	—
531	C₂×C₄	—	4	4	no	—
532	C₂×C₄	—	4	4	no	—
533	C₂×C₄	C₂₃×#102	1	4	yes	—
534	C₂×C₄	C₂₃×#102	1	4	yes	—
535	C₂×C₄	C₂₃×#102	1	4	yes	—
536	C₂×C₄	C₂₃×#102	1	4	yes	—
537	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
538	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
539	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
540	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
541	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
542	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
543	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
544	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
545	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
546	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
547	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
548	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
549	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
550	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
551	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
552	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
553	C₈	—	4	4	no	—
554	C₂²	—	2	1	no	—
555	C₂²	—	2	1	no	—
556	C₂²	—	2	1	no	—
557	C₂²	—	2	1	no	—
558	C₂²	—	2	1	no	—
559	C₂²	—	2	1	no	—
560	C₂²	—	4	2	no	—
561	C₂²	—	4	2	no	—
562	C₂²	—	4	4	no	—
563	C₂²	—	4	4	no	—
564	C₂²	—	4	4	no	—
565	C₂²	—	8	8	no	—
566	C₂²	C₂×C₂₃×#102	1	2	yes	—
567	C₂²	C₂×C₂₃×#102	1	2	yes	—
568	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
569	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
570	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
571	C₄	—	2	1	no	—
572	C₄	—	2	1	no	—
573	C₄	—	2	1	no	—
574	C₄	—	2	1	no	—
575	C₄	—	2	1	no	—
576	C₄	—	2	1	no	—
577	C₄	—	4	4	no	—
578	C₄	—	8	4	no	—
579	C₄	C₂₃×#410	1	2	yes	—
580	C₄	C₂×C₂₃×#102	1	2	yes	—
581	C₄	C₂×C₂₃×#102	1	2	yes	—
582	C₄	C₂×C₂₃×#102	1	2	yes	—
583	C₄	C₂×C₂₃×#102	1	2	yes	—
584	C₄	C₂₃×#421	1	2	yes	—
585	C₂	—	2	1	no	—
586	C₂	—	2	2	no	—
587	C₂	—	2	1	no	—
588	C₂	—	2	2	no	—
589	C₂	—	8	4	no	—
590	C₂	C₂×C₂₃×#410	1	1	yes	characteristic
591	C₂	C₂²×C₂₃×#102	1	1	yes	characteristic
592	C₂	C₂×C₂₃×#421	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂²×C₂²⋊C₄
prime radical	C₂×C₂₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#431	C₂	semidirect
3. C₂₃×#2285	C₂	semidirect
4. C₂₃×#1519	C₂	semidirect
5. C₂₃×#1517	C₂	semidirect
6. C₂₃×#1502	C₂	semidirect
12. C₂₃×#532	C₄	semidirect
13. C₂₃×#532	C₄	semidirect
16. C₂×C₂₃×C₄⋊Q₈	C₄	semidirect
17. C₂×C₂₃×#127	C₄	semidirect
22. C₂×C₄×D₄×C₂₃	C₄	semidirect
27. C₂₃×#431	C₂²	semidirect
28. C₂₃×#431	C₂²	semidirect
33. C₂₃×#382	C₂²	semidirect
37. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
38. C₂₃×#327	C₂²	semidirect
195. #17574	C₂₃	direct (N × H)
259. C₂×#431	C₂×C₂₃	semidirect
260. #2285	C₂×C₂₃	semidirect
261. #1519	C₂×C₂₃	semidirect
262. #1517	C₂×C₂₃	semidirect
263. #1502	C₂×C₂₃	semidirect
300. #532	C₄×C₂₃	semidirect
301. #532	C₄×C₂₃	semidirect
304. C₂×C₄⋊Q₈	C₄×C₂₃	semidirect
305. C₂×#127	C₄×C₂₃	semidirect
310. C₂×C₄×D₄	C₄×C₂₃	semidirect
315. #431	C₂²×C₂₃	semidirect
316. #431	C₂²×C₂₃	semidirect
321. #382	C₂²×C₂₃	semidirect
325. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
326. #327	C₂²×C₂₃	semidirect
409. C₂₃	#17574	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#532	2
C₂×C₂₃×#431	2
C₂₃×#2285	2
C₂₃×#1519	2
C₂₃×#1517	2
C₂₃×#1502	2
#17574	23

Subgroups — C23×#17574 #19165200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#17574 #19165200