Subgroups — C₂₃×#20919 #19168545

Order 5888.

1682 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#421	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#421	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2879	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1876	C₂	1	8	yes	maximal, complemented
7	C₂×C₂₃×#341	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1729	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
10	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#135	—	2	4	no	—
12	C₂×C₂₃×#135	—	2	4	no	—
13	C₂×C₂₃×#135	C₄	1	4	yes	complemented
14	C₂×C₂₃×#135	C₂²	1	4	yes	—
15	C₂²×Q₁₆×C₂₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×SD₁₆×C₂₃	C₄	1	1	yes	characteristic, complemented
17	C₂²×SD₁₆×C₂₃	C₄	1	1	yes	characteristic, complemented
18	C₂²×D₈×C₂₃	C₄	1	1	yes	characteristic, complemented
19	C₂₃×#568	—	2	4	no	—
20	C₂₃×#568	—	2	4	no	—
21	C₂×C₂₃×#131	—	2	2	no	—
22	C₂×C₂₃×#131	C₂²	1	4	yes	complemented
23	C₂²×M₁₆×C₂₃	—	2	2	no	—
24	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
25	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
26	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
27	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
29	C₂₃×#421	—	2	4	no	—
30	C₂₃×#421	—	2	4	no	—
31	C₂×C₂₃×#104	C₂²	1	4	yes	complemented
32	C₂×C₂₃×#104	C₂²	1	4	yes	complemented
33	C₂₃×#408	—	2	2	no	—
34	C₂₃×#408	C₂²	1	4	yes	complemented
35	C₂₃×#407	—	2	2	no	—
36	C₂₃×#341	C₂²	1	4	yes	complemented
37	C₂₃×#336	C₂²	1	4	yes	complemented
38	C₂×#41×C₂₃	—	2	1	no	—
39	C₂×#41×C₂₃	—	2	2	no	—
40	C₂×#41×C₂₃	—	2	1	no	—
41	C₂×#41×C₂₃	—	2	1	no	—
42	C₂×#41×C₂₃	—	2	2	no	—
43	C₂×#41×C₂₃	—	2	1	no	—
44	C₂×#41×C₂₃	—	2	2	no	—
45	C₂×#41×C₂₃	—	2	2	no	—
46	C₂×#41×C₂₃	—	2	2	no	—
47	C₂×#41×C₂₃	—	2	2	no	—
48	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
49	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
50	C₂×#41×C₂₃	D₄	1	4	yes	—
51	C₂×#41×C₂₃	D₄	1	4	yes	—
52	C₂×#41×C₂₃	C₂³	1	2	yes	—
53	C₂×#41×C₂₃	C₂³	1	2	yes	—
54	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×D₄×C₂₃	—	2	1	no	—
57	C₂²×D₄×C₂₃	—	2	1	no	—
58	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
60	C₂³×C₄×C₂₃	—	2	1	no	—
61	C₂³×C₄×C₂₃	—	2	1	no	—
62	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
63	C₂₃×#135	—	2	16	no	—
64	C₂₃×#135	—	2	8	no	—
65	C₂₃×#135	—	2	8	no	—
66	C₂×Q₁₆×C₂₃	—	2	4	no	—
67	C₂×Q₁₆×C₂₃	C₂×C₄	1	4	yes	—
68	C₂×SD₁₆×C₂₃	—	2	4	no	—
69	C₂×SD₁₆×C₂₃	—	2	4	no	—
70	C₂×SD₁₆×C₂₃	C₂×C₄	1	4	yes	—
71	C₂×SD₁₆×C₂₃	C₂×C₄	1	4	yes	—
72	C₂×D₈×C₂₃	—	2	4	no	—
73	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
74	C₂₃×#131	—	2	16	no	—
75	C₂₃×#131	—	2	8	no	—
76	C₂₃×#131	—	2	8	no	—
77	C₂×M₁₆×C₂₃	—	2	16	no	—
78	C₂×M₁₆×C₂₃	—	2	1	no	—
79	C₂×M₁₆×C₂₃	—	2	2	no	—
80	C₂×M₁₆×C₂₃	—	2	2	no	—
81	C₂×M₁₆×C₂₃	—	2	2	no	—
82	C₂×M₁₆×C₂₃	—	2	1	no	—
83	C₂×M₁₆×C₂₃	—	2	4	no	—
84	C₂×M₁₆×C₂₃	—	2	2	no	—
85	C₂×M₁₆×C₂₃	—	2	2	no	—
86	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
87	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
88	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
89	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
90	C₂×M₁₆×C₂₃	C₂³	1	2	yes	—
91	C₂²×C₈×C₂₃	—	2	2	no	—
92	C₂²×C₈×C₂₃	—	2	2	no	—
93	C₂²×C₈×C₂₃	—	2	1	no	—
94	C₂²×C₈×C₂₃	—	2	4	no	—
95	C₂²×C₈×C₂₃	—	2	1	no	—
96	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
97	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
98	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
99	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
100	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
101	C₂₃×#117	—	2	2	no	—
102	C₂₃×#117	—	2	2	no	—
103	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
105	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
106	C₂₃×#104	—	2	8	no	—
107	C₂₃×#104	—	2	8	no	—
108	C₂₃×C₂²⋊C₈	—	2	8	no	—
109	C₂⁴×C₂₃	—	2	1	no	—
110	C₂⁴×C₂₃	—	2	1	no	—
111	C₂⁴×C₂₃	#41	1	1	yes	characteristic
112	#41×C₂₃	—	2	4	no	—
113	#41×C₂₃	—	2	4	no	—
114	#41×C₂₃	—	2	4	no	—
115	#41×C₂₃	—	2	4	no	—
116	#41×C₂₃	—	2	4	no	—
117	#41×C₂₃	—	2	2	no	—
118	#41×C₂₃	—	2	2	no	—
119	#41×C₂₃	—	2	4	no	—
120	#41×C₂₃	—	2	4	no	—
121	#41×C₂₃	—	2	2	no	—
122	#41×C₂₃	—	2	2	no	—
123	#41×C₂₃	—	2	2	no	—
124	#41×C₂₃	—	2	2	no	—
125	#41×C₂₃	—	4	2	no	—
126	#41×C₂₃	—	4	2	no	—
127	#41×C₂₃	—	4	2	no	—
128	#41×C₂₃	—	4	2	no	—
129	#41×C₂₃	C₂²⋊C₄	1	8	yes	—
130	#41×C₂₃	C₂×D₄	1	4	yes	—
131	#41×C₂₃	C₂×D₄	1	4	yes	—
132	C₂×Q₈×C₂₃	—	2	1	no	—
133	C₂×Q₈×C₂₃	—	2	2	no	—
134	C₂×Q₈×C₂₃	—	2	1	no	—
135	C₂×Q₈×C₂₃	—	2	2	no	—
136	C₂×Q₈×C₂₃	—	2	2	no	—
137	C₂×Q₈×C₂₃	C₂²⋊C₄	1	4	yes	—
138	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
139	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
140	C₂×D₄×C₂₃	—	2	2	no	—
141	C₂×D₄×C₂₃	—	2	4	no	—
142	C₂×D₄×C₂₃	—	2	4	no	—
143	C₂×D₄×C₂₃	—	2	2	no	—
144	C₂×D₄×C₂₃	—	2	2	no	—
145	C₂×D₄×C₂₃	—	2	2	no	—
146	C₂×D₄×C₂₃	—	2	2	no	—
147	C₂×D₄×C₂₃	—	2	2	no	—
148	C₂×D₄×C₂₃	—	2	1	no	—
149	C₂×D₄×C₂₃	—	2	2	no	—
150	C₂×D₄×C₂₃	—	2	1	no	—
151	C₂×D₄×C₂₃	—	4	2	no	—
152	C₂×D₄×C₂₃	—	4	2	no	—
153	C₂×D₄×C₂₃	C₂²⋊C₄	1	4	yes	—
154	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
155	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
156	C₂²×C₄×C₂₃	—	2	1	no	—
157	C₂²×C₄×C₂₃	—	2	2	no	—
158	C₂²×C₄×C₂₃	—	2	4	no	—
159	C₂²×C₄×C₂₃	—	2	4	no	—
160	C₂²×C₄×C₂₃	—	2	2	no	—
161	C₂²×C₄×C₂₃	—	2	1	no	—
162	C₂²×C₄×C₂₃	—	2	2	no	—
163	C₂²×C₄×C₂₃	—	2	2	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	1	yes	characteristic
169	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
175	C₂²×C₄×C₂₃	#41	1	2	yes	—
176	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
177	Q₁₆×C₂₃	—	2	8	no	—
178	SD₁₆×C₂₃	—	2	8	no	—
179	SD₁₆×C₂₃	—	2	8	no	—
180	D₈×C₂₃	—	2	8	no	—
181	M₁₆×C₂₃	—	2	16	no	—
182	M₁₆×C₂₃	—	2	4	no	—
183	M₁₆×C₂₃	—	2	4	no	—
184	M₁₆×C₂₃	—	2	4	no	—
185	M₁₆×C₂₃	—	2	2	no	—
186	M₁₆×C₂₃	—	2	2	no	—
187	M₁₆×C₂₃	—	2	2	no	—
188	M₁₆×C₂₃	—	4	2	no	—
189	M₁₆×C₂₃	C₂×D₄	1	4	yes	—
190	C₂×C₈×C₂₃	—	2	16	no	—
191	C₂×C₈×C₂₃	—	2	2	no	—
192	C₂×C₈×C₂₃	—	2	2	no	—
193	C₂×C₈×C₂₃	—	2	4	no	—
194	C₂×C₈×C₂₃	—	2	4	no	—
195	C₂×C₈×C₂₃	—	2	2	no	—
196	C₂×C₈×C₂₃	—	2	2	no	—
197	C₂×C₈×C₂₃	—	2	2	no	—
198	C₂×C₈×C₂₃	—	2	2	no	—
199	C₂×C₈×C₂₃	—	2	2	no	—
200	C₂×C₈×C₂₃	—	2	2	no	—
201	C₂×C₈×C₂₃	—	4	2	no	—
202	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
203	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
204	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
205	C₄⋊C₄×C₂₃	—	4	2	no	—
206	C₂²⋊C₄×C₂₃	—	4	2	no	—
207	C₄²×C₂₃	—	2	2	no	—
208	C₄²×C₂₃	—	2	2	no	—
209	#20919	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂³×C₂₃	—	2	2	no	—
211	C₂³×C₂₃	—	2	4	no	—
212	C₂³×C₂₃	—	2	2	no	—
213	C₂³×C₂₃	—	4	2	no	—
214	C₂³×C₂₃	—	4	2	no	—
215	C₂³×C₂₃	—	8	2	no	—
216	C₂³×C₂₃	—	8	2	no	—
217	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
218	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
219	C₂³×C₂₃	#121	1	2	yes	—
220	C₂³×C₂₃	#123	1	2	yes	—
221	C₂³×C₂₃	C₂×#41	1	1	yes	characteristic
222	Q₈×C₂₃	—	2	2	no	—
223	Q₈×C₂₃	—	2	4	no	—
224	Q₈×C₂₃	—	2	2	no	—
225	Q₈×C₂₃	—	2	2	no	—
226	Q₈×C₂₃	—	4	2	no	—
227	Q₈×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
228	D₄×C₂₃	—	2	4	no	—
229	D₄×C₂₃	—	2	4	no	—
230	D₄×C₂₃	—	2	8	no	—
231	D₄×C₂₃	—	2	2	no	—
232	D₄×C₂₃	—	2	2	no	—
233	D₄×C₂₃	—	2	2	no	—
234	D₄×C₂₃	—	2	4	no	—
235	D₄×C₂₃	—	4	2	no	—
236	D₄×C₂₃	—	4	2	no	—
237	D₄×C₂₃	—	4	4	no	—
238	D₄×C₂₃	—	4	2	no	—
239	D₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
240	C₂×C₄×C₂₃	—	2	2	no	—
241	C₂×C₄×C₂₃	—	2	4	no	—
242	C₂×C₄×C₂₃	—	2	4	no	—
243	C₂×C₄×C₂₃	—	2	4	no	—
244	C₂×C₄×C₂₃	—	2	8	no	—
245	C₂×C₄×C₂₃	—	2	2	no	—
246	C₂×C₄×C₂₃	—	4	2	no	—
247	C₂×C₄×C₂₃	—	4	2	no	—
248	C₂×C₄×C₂₃	—	4	2	no	—
249	C₂×C₄×C₂₃	—	4	4	no	—
250	C₂×C₄×C₂₃	—	4	4	no	—
251	C₂×C₄×C₂₃	—	4	2	no	—
252	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
253	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
254	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
255	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
256	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
257	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
258	C₂×C₄×C₂₃	#120	1	2	yes	—
259	C₂×C₄×C₂₃	#120	1	2	yes	—
260	C₂×C₄×C₂₃	#121	1	2	yes	—
261	C₂×C₄×C₂₃	#121	1	2	yes	—
262	C₂×C₄×C₂₃	#121	1	2	yes	—
263	C₂×C₄×C₂₃	#123	1	2	yes	—
264	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
265	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
270	C₈×C₂₃	—	2	2	no	—
271	C₈×C₂₃	—	2	2	no	—
272	C₈×C₂₃	—	2	4	no	—
273	C₈×C₂₃	—	2	4	no	—
274	C₈×C₂₃	—	4	2	no	—
275	C₂²×#135	C₂×C₂₃	1	1	yes	characteristic
276	C₂×#568	C₂×C₂₃	1	1	yes	characteristic, complemented
277	C₂×#421	C₂×C₂₃	1	1	yes	characteristic, complemented
278	C₂×#421	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#2879	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#1876	C₂×C₂₃	1	8	yes	complemented
281	C₂×#341	C₂×C₂₃	1	1	yes	characteristic, complemented
282	#1729	C₂×C₂₃	1	1	yes	characteristic, complemented
283	C₂²×C₂₃	—	2	2	no	—
284	C₂²×C₂₃	—	2	2	no	—
285	C₂²×C₂₃	—	2	2	no	—
286	C₂²×C₂₃	—	2	4	no	—
287	C₂²×C₂₃	—	4	2	no	—
288	C₂²×C₂₃	—	4	2	no	—
289	C₂²×C₂₃	—	4	4	no	—
290	C₂²×C₂₃	—	4	4	no	—
291	C₂²×C₂₃	—	4	2	no	—
292	C₂²×C₂₃	—	8	4	no	—
293	C₂²×C₂₃	—	8	4	no	—
294	C₂²×C₂₃	#386	1	2	yes	—
295	C₂²×C₂₃	#386	1	2	yes	—
296	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
297	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
298	C₂²×C₂₃	C₂×#123	1	1	yes	characteristic
299	C₄×C₂₃	—	2	4	no	—
300	C₄×C₂₃	—	4	2	no	—
301	C₄×C₂₃	—	8	2	no	—
302	C₄×C₂₃	#386	1	2	yes	—
303	C₄×C₂₃	#386	1	2	yes	—
304	C₄×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
305	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
306	C₄×C₂₃	C₂×#120	1	1	yes	characteristic
307	C₄×C₂₃	C₂×#121	1	1	yes	characteristic
308	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
309	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
310	C₂×#135	—	2	4	no	—
311	C₂×#135	—	2	4	no	—
312	C₂×#135	C₄×C₂₃	1	4	yes	complemented
313	C₂×#135	C₂²×C₂₃	1	4	yes	—
314	C₂²×Q₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
315	C₂²×SD₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
316	C₂²×SD₁₆	C₄×C₂₃	1	1	yes	characteristic, complemented
317	C₂²×D₈	C₄×C₂₃	1	1	yes	characteristic, complemented
318	#568	—	2	4	no	—
319	#568	—	2	4	no	—
320	C₂×#131	—	2	2	no	—
321	C₂×#131	C₂²×C₂₃	1	4	yes	complemented
322	C₂²×M₁₆	—	2	2	no	—
323	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
324	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
325	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
326	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
328	#421	—	2	4	no	—
329	#421	—	2	4	no	—
330	C₂×#104	C₂²×C₂₃	1	4	yes	complemented
331	C₂×#104	C₂²×C₂₃	1	4	yes	complemented
332	#408	—	2	2	no	—
333	#408	C₂²×C₂₃	1	4	yes	complemented
334	#407	—	2	2	no	—
335	#341	C₂²×C₂₃	1	4	yes	complemented
336	#336	C₂²×C₂₃	1	4	yes	complemented
337	C₂×C₂₃	—	2	1	no	—
338	C₂×C₂₃	—	2	2	no	—
339	C₂×C₂₃	—	2	1	no	—
340	C₂×C₂₃	—	2	2	no	—
341	C₂×C₂₃	—	4	4	no	—
342	C₂×C₂₃	—	8	2	no	—
343	C₂×C₂₃	C₂×#386	1	1	yes	characteristic
344	C₂×C₂₃	#2957	1	2	yes	—
345	C₂×#41	—	2	1	no	—
346	C₂×#41	—	2	2	no	—
347	C₂×#41	—	2	1	no	—
348	C₂×#41	—	2	1	no	—
349	C₂×#41	—	2	2	no	—
350	C₂×#41	—	2	1	no	—
351	C₂×#41	—	2	2	no	—
352	C₂×#41	—	2	2	no	—
353	C₂×#41	—	2	2	no	—
354	C₂×#41	—	2	2	no	—
355	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
356	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
357	C₂×#41	D₄×C₂₃	1	4	yes	—
358	C₂×#41	D₄×C₂₃	1	4	yes	—
359	C₂×#41	C₂³×C₂₃	1	2	yes	—
360	C₂×#41	C₂³×C₂₃	1	2	yes	—
361	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
362	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
363	C₂²×D₄	—	2	1	no	—
364	C₂²×D₄	—	2	1	no	—
365	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
366	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
367	C₂³×C₄	—	2	1	no	—
368	C₂³×C₄	—	2	1	no	—
369	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
370	#135	—	2	16	no	—
371	#135	—	2	8	no	—
372	#135	—	2	8	no	—
373	C₂×Q₁₆	—	2	4	no	—
374	C₂×Q₁₆	C₂×C₄×C₂₃	1	4	yes	—
375	C₂×SD₁₆	—	2	4	no	—
376	C₂×SD₁₆	—	2	4	no	—
377	C₂×SD₁₆	C₂×C₄×C₂₃	1	4	yes	—
378	C₂×SD₁₆	C₂×C₄×C₂₃	1	4	yes	—
379	C₂×D₈	—	2	4	no	—
380	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
381	#131	—	2	16	no	—
382	#131	—	2	8	no	—
383	#131	—	2	8	no	—
384	C₂×M₁₆	—	2	16	no	—
385	C₂×M₁₆	—	2	1	no	—
386	C₂×M₁₆	—	2	2	no	—
387	C₂×M₁₆	—	2	2	no	—
388	C₂×M₁₆	—	2	2	no	—
389	C₂×M₁₆	—	2	1	no	—
390	C₂×M₁₆	—	2	4	no	—
391	C₂×M₁₆	—	2	2	no	—
392	C₂×M₁₆	—	2	2	no	—
393	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
394	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
395	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
396	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
397	C₂×M₁₆	C₂³×C₂₃	1	2	yes	—
398	C₂²×C₈	—	2	2	no	—
399	C₂²×C₈	—	2	2	no	—
400	C₂²×C₈	—	2	1	no	—
401	C₂²×C₈	—	2	4	no	—
402	C₂²×C₈	—	2	1	no	—
403	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
404	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
405	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
406	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
407	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
408	#117	—	2	2	no	—
409	#117	—	2	2	no	—
410	C₂×C₄⋊C₄	—	2	1	no	—
411	C₂×C₂²⋊C₄	—	2	1	no	—
412	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
413	#104	—	2	8	no	—
414	#104	—	2	8	no	—
415	C₂²⋊C₈	—	2	8	no	—
416	C₂₃	#20919	1	1	yes	characteristic, direct product factor, Sylow
417	C₂⁴	—	2	1	no	—
418	C₂⁴	—	2	1	no	—
419	C₂⁴	#41×C₂₃	1	1	yes	characteristic
420	#41	—	2	4	no	—
421	#41	—	2	4	no	—
422	#41	—	2	4	no	—
423	#41	—	2	4	no	—
424	#41	—	2	4	no	—
425	#41	—	2	2	no	—
426	#41	—	2	2	no	—
427	#41	—	2	4	no	—
428	#41	—	2	4	no	—
429	#41	—	2	2	no	—
430	#41	—	2	2	no	—
431	#41	—	2	2	no	—
432	#41	—	2	2	no	—
433	#41	—	4	2	no	—
434	#41	—	4	2	no	—
435	#41	—	4	2	no	—
436	#41	—	4	2	no	—
437	#41	C₂²⋊C₄×C₂₃	1	8	yes	—
438	#41	C₂×D₄×C₂₃	1	4	yes	—
439	#41	C₂×D₄×C₂₃	1	4	yes	—
440	C₂×Q₈	—	2	2	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	—	2	2	no	—
444	C₂×Q₈	—	2	2	no	—
445	C₂×Q₈	C₂²⋊C₄×C₂₃	1	4	yes	—
446	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
447	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	4	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	2	2	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	2	no	—
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	4	2	no	—
460	C₂×D₄	—	4	2	no	—
461	C₂×D₄	C₂²⋊C₄×C₂₃	1	4	yes	—
462	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
463	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	2	no	—
471	C₂²×C₄	—	2	2	no	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	4	2	no	—
474	C₂²×C₄	—	4	2	no	—
475	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
476	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
478	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
479	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
480	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
482	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
483	C₂²×C₄	#41×C₂₃	1	2	yes	—
484	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
485	Q₁₆	—	2	8	no	—
486	SD₁₆	—	2	8	no	—
487	SD₁₆	—	2	8	no	—
488	D₈	—	2	8	no	—
489	M₁₆	—	2	16	no	—
490	M₁₆	—	2	4	no	—
491	M₁₆	—	2	4	no	—
492	M₁₆	—	2	4	no	—
493	M₁₆	—	2	2	no	—
494	M₁₆	—	2	2	no	—
495	M₁₆	—	2	2	no	—
496	M₁₆	—	4	2	no	—
497	M₁₆	C₂×D₄×C₂₃	1	4	yes	—
498	C₂×C₈	—	2	16	no	—
499	C₂×C₈	—	2	2	no	—
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	4	no	—
502	C₂×C₈	—	2	4	no	—
503	C₂×C₈	—	2	2	no	—
504	C₂×C₈	—	2	2	no	—
505	C₂×C₈	—	2	2	no	—
506	C₂×C₈	—	2	2	no	—
507	C₂×C₈	—	2	2	no	—
508	C₂×C₈	—	2	2	no	—
509	C₂×C₈	—	4	2	no	—
510	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
511	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
512	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
513	C₄⋊C₄	—	4	2	no	—
514	C₂²⋊C₄	—	4	2	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	2	no	—
517	C₂³	—	2	2	no	—
518	C₂³	—	2	4	no	—
519	C₂³	—	2	2	no	—
520	C₂³	—	4	2	no	—
521	C₂³	—	4	2	no	—
522	C₂³	—	8	2	no	—
523	C₂³	—	8	2	no	—
524	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
525	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
526	C₂³	C₂₃×#121	1	2	yes	—
527	C₂³	C₂₃×#123	1	2	yes	—
528	C₂³	C₂×#41×C₂₃	1	1	yes	characteristic
529	Q₈	—	2	4	no	—
530	Q₈	—	2	2	no	—
531	Q₈	—	2	2	no	—
532	Q₈	—	2	2	no	—
533	Q₈	—	4	2	no	—
534	Q₈	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
535	D₄	—	2	8	no	—
536	D₄	—	2	4	no	—
537	D₄	—	2	4	no	—
538	D₄	—	2	4	no	—
539	D₄	—	2	2	no	—
540	D₄	—	2	2	no	—
541	D₄	—	2	2	no	—
542	D₄	—	4	4	no	—
543	D₄	—	4	2	no	—
544	D₄	—	4	2	no	—
545	D₄	—	4	2	no	—
546	D₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	8	no	—
550	C₂×C₄	—	2	4	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	4	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	4	4	no	—
558	C₂×C₄	—	4	2	no	—
559	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
560	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
561	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
562	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
563	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
564	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
565	C₂×C₄	C₂₃×#120	1	2	yes	—
566	C₂×C₄	C₂₃×#120	1	2	yes	—
567	C₂×C₄	C₂₃×#121	1	2	yes	—
568	C₂×C₄	C₂₃×#121	1	2	yes	—
569	C₂×C₄	C₂₃×#121	1	2	yes	—
570	C₂×C₄	C₂₃×#123	1	2	yes	—
571	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
572	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
573	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
574	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
575	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
576	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
577	C₈	—	2	2	no	—
578	C₈	—	2	2	no	—
579	C₈	—	2	4	no	—
580	C₈	—	2	4	no	—
581	C₈	—	4	2	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	2	2	no	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	4	no	—
586	C₂²	—	4	2	no	—
587	C₂²	—	4	2	no	—
588	C₂²	—	4	4	no	—
589	C₂²	—	4	4	no	—
590	C₂²	—	4	2	no	—
591	C₂²	—	8	4	no	—
592	C₂²	—	8	4	no	—
593	C₂²	C₂₃×#386	1	2	yes	—
594	C₂²	C₂₃×#386	1	2	yes	—
595	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
596	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
597	C₂²	C₂×C₂₃×#123	1	1	yes	characteristic
598	C₄	—	2	4	no	—
599	C₄	—	4	2	no	—
600	C₄	—	8	2	no	—
601	C₄	C₂₃×#386	1	2	yes	—
602	C₄	C₂₃×#386	1	2	yes	—
603	C₄	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
604	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
605	C₄	C₂×C₂₃×#120	1	1	yes	characteristic
606	C₄	C₂×C₂₃×#121	1	1	yes	characteristic
607	C₂	—	2	1	no	—
608	C₂	—	2	2	no	—
609	C₂	—	2	2	no	—
610	C₂	—	2	1	no	—
611	C₂	—	4	4	no	—
612	C₂	—	8	2	no	—
613	C₂	C₂×C₂₃×#386	1	1	yes	characteristic
614	C₂	C₂₃×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂²×C₂₃×#135	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×M₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#568	C₂	semidirect
3. C₂×C₂₃×#421	C₂	semidirect
4. C₂×C₂₃×#421	C₂	semidirect
5. C₂₃×#2879	C₂	semidirect
6. C₂₃×#1876	C₂	semidirect
7. C₂×C₂₃×#341	C₂	semidirect
8. C₂₃×#1729	C₂	semidirect
13. C₂×C₂₃×#135	C₄	semidirect
15. C₂²×Q₁₆×C₂₃	C₄	semidirect
16. C₂²×SD₁₆×C₂₃	C₄	semidirect
17. C₂²×SD₁₆×C₂₃	C₄	semidirect
18. C₂²×D₈×C₂₃	C₄	semidirect
22. C₂×C₂₃×#131	C₂²	semidirect
31. C₂×C₂₃×#104	C₂²	semidirect
32. C₂×C₂₃×#104	C₂²	semidirect
34. C₂₃×#408	C₂²	semidirect
36. C₂₃×#341	C₂²	semidirect
37. C₂₃×#336	C₂²	semidirect
209. #20919	C₂₃	direct (N × H)
276. C₂×#568	C₂×C₂₃	semidirect
277. C₂×#421	C₂×C₂₃	semidirect
278. C₂×#421	C₂×C₂₃	semidirect
279. #2879	C₂×C₂₃	semidirect
280. #1876	C₂×C₂₃	semidirect
281. C₂×#341	C₂×C₂₃	semidirect
282. #1729	C₂×C₂₃	semidirect
312. C₂×#135	C₄×C₂₃	semidirect
314. C₂²×Q₁₆	C₄×C₂₃	semidirect
315. C₂²×SD₁₆	C₄×C₂₃	semidirect
316. C₂²×SD₁₆	C₄×C₂₃	semidirect
317. C₂²×D₈	C₄×C₂₃	semidirect
321. C₂×#131	C₂²×C₂₃	semidirect
330. C₂×#104	C₂²×C₂₃	semidirect
331. C₂×#104	C₂²×C₂₃	semidirect
333. #408	C₂²×C₂₃	semidirect
335. #341	C₂²×C₂₃	semidirect
336. #336	C₂²×C₂₃	semidirect
416. C₂₃	#20919	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#135	2
C₂×C₂₃×#568	2
C₂×C₂₃×#421	2
C₂₃×#2879	2
C₂₃×#1876	2
C₂×C₂₃×#341	2
C₂₃×#1729	2
#20919	23

Subgroups — C23×#20919 #19168545

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20919 #19168545