Subgroups — C₂₃×#20415 #19168041

Order 5888.

1598 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#520	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#417	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#416	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2274	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1763	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1762	C₂	1	2	yes	maximal, complemented
8	C₄×C₂₃×#103	C₂	1	2	yes	maximal, complemented
9	C₄×C₂₃×#102	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1736	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#340	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#520	—	2	4	no	—
16	C₂₃×#520	C₄	1	8	yes	complemented
17	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₂₃	—	2	2	no	—
19	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₂₃×#117	—	2	1	no	—
21	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
24	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#417	C₄	1	8	yes	complemented
27	C₂₃×#416	C₄	1	8	yes	complemented
28	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#103	C₂²	1	2	yes	—
30	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#102	C₂²	1	2	yes	—
33	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
34	C₂₃×#405	—	2	4	no	—
35	C₂×C₂₃×#97	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
38	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
39	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
40	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
41	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
42	C₂₃×#340	C₂²	1	2	yes	—
43	C₂₃×#340	C₂²	1	2	yes	—
44	C₂₃×#340	C₂²	1	2	yes	—
45	C₂₃×#340	C₂²	1	2	yes	—
46	C₂×#41×C₂₃	—	2	2	no	—
47	C₂×#41×C₂₃	—	2	2	no	—
48	C₂×#41×C₂₃	—	2	2	no	—
49	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
50	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
51	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
52	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₂₃	—	2	1	no	—
54	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₂₃	—	2	1	no	—
56	C₂³×C₄×C₂₃	—	2	1	no	—
57	C₂³×C₄×C₂₃	—	2	1	no	—
58	C₂³×C₄×C₂₃	—	2	2	no	—
59	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₂₃	—	2	2	no	—
61	C₂×M₁₆×C₂₃	—	2	2	no	—
62	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
63	C₂²×C₈×C₂₃	—	2	2	no	—
64	C₂²×C₈×C₂₃	—	2	2	no	—
65	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
66	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
69	C₄×Q₈×C₂₃	—	2	4	no	—
70	C₄×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
71	C₄×D₄×C₂₃	—	2	16	no	—
72	C₄×D₄×C₂₃	—	2	4	no	—
73	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	complemented
74	C₂₃×#117	—	2	2	no	—
75	C₂₃×#117	—	2	8	no	—
76	C₂₃×#117	C₂×C₄	1	4	yes	—
77	C₂₃×#117	C₂×C₄	1	4	yes	complemented
78	C₂₃×#117	D₄	1	4	yes	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
83	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
85	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
86	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	complemented
87	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
90	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
96	C₂×C₄²×C₂₃	—	2	1	no	—
97	C₂×C₄²×C₂₃	—	2	2	no	—
98	C₂×C₄²×C₂₃	—	2	2	no	—
99	C₂×C₄²×C₂₃	—	2	2	no	—
100	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
101	C₂₃×#103	C₂×C₄	1	4	yes	complemented
102	C₂₃×#103	C₂×C₄	1	4	yes	complemented
103	C₂₃×#103	C₂×C₄	1	8	yes	complemented
104	C₂₃×#102	C₂×C₄	1	4	yes	complemented
105	C₂₃×#102	C₂×C₄	1	4	yes	complemented
106	C₂₃×#102	C₂×C₄	1	8	yes	complemented
107	C₂₃×#97	—	2	4	no	—
108	C₄×C₈×C₂₃	—	2	4	no	—
109	C₂₃×#95	—	2	2	no	—
110	C₂₃×#95	—	2	2	no	—
111	C₂⁴×C₂₃	—	2	1	no	—
112	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
113	#41×C₂₃	—	2	8	no	—
114	#41×C₂₃	—	2	8	no	—
115	#41×C₂₃	—	2	8	no	—
116	#41×C₂₃	C₄²	1	16	yes	complemented
117	C₂×Q₈×C₂₃	—	2	1	no	—
118	C₂×Q₈×C₂₃	—	2	1	no	—
119	C₂×Q₈×C₂₃	—	2	1	no	—
120	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
121	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
122	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
123	C₂×D₄×C₂₃	—	2	4	no	—
124	C₂×D₄×C₂₃	—	2	4	no	—
125	C₂×D₄×C₂₃	—	2	4	no	—
126	C₂×D₄×C₂₃	—	2	1	no	—
127	C₂×D₄×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	—	2	1	no	—
129	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
130	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
131	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
132	C₂²×C₄×C₂₃	—	2	1	no	—
133	C₂²×C₄×C₂₃	—	2	4	no	—
134	C₂²×C₄×C₂₃	—	2	4	no	—
135	C₂²×C₄×C₂₃	—	2	4	no	—
136	C₂²×C₄×C₂₃	—	2	1	no	—
137	C₂²×C₄×C₂₃	—	2	1	no	—
138	C₂²×C₄×C₂₃	—	2	2	no	—
139	C₂²×C₄×C₂₃	—	2	2	no	—
140	C₂²×C₄×C₂₃	—	2	4	no	—
141	C₂²×C₄×C₂₃	—	2	4	no	—
142	C₂²×C₄×C₂₃	—	2	8	no	—
143	C₂²×C₄×C₂₃	—	8	2	no	—
144	C₂²×C₄×C₂₃	—	8	2	no	—
145	C₂²×C₄×C₂₃	—	8	1	no	—
146	C₂²×C₄×C₂₃	—	8	1	no	—
147	C₂²×C₄×C₂₃	—	8	1	no	—
148	C₂²×C₄×C₂₃	—	8	1	no	—
149	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
150	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
151	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
152	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₂₃	#41	1	2	yes	—
157	C₂²×C₄×C₂₃	#41	1	2	yes	—
158	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
159	M₁₆×C₂₃	—	2	8	no	—
160	C₂×C₈×C₂₃	—	2	8	no	—
161	C₂×C₈×C₂₃	—	2	1	no	—
162	C₂×C₈×C₂₃	—	2	1	no	—
163	C₂×C₈×C₂₃	—	2	2	no	—
164	C₂×C₈×C₂₃	—	2	1	no	—
165	C₂×C₈×C₂₃	—	2	1	no	—
166	C₂×C₈×C₂₃	—	2	2	no	—
167	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
168	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
169	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
170	C₄⋊C₄×C₂₃	—	2	4	no	—
171	C₄⋊C₄×C₂₃	—	2	8	no	—
172	C₄⋊C₄×C₂₃	—	2	4	no	—
173	C₄⋊C₄×C₂₃	—	2	2	no	—
174	C₄⋊C₄×C₂₃	—	2	2	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
178	C₄⋊C₄×C₂₃	C₂²⋊C₄	1	4	yes	—
179	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
180	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
181	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
182	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
183	C₂²⋊C₄×C₂₃	—	2	4	no	—
184	C₂²⋊C₄×C₂₃	—	2	8	no	—
185	C₂²⋊C₄×C₂₃	C₂²⋊C₄	1	8	yes	—
186	C₄²×C₂₃	—	2	8	no	—
187	C₄²×C₂₃	—	2	4	no	—
188	C₄²×C₂₃	—	8	2	no	—
189	C₄²×C₂₃	—	8	2	no	—
190	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
191	C₄²×C₂₃	C₂×D₄	1	4	yes	—
192	#20415	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂³×C₂₃	—	2	1	no	—
194	C₂³×C₂₃	—	2	1	no	—
195	C₂³×C₂₃	—	2	1	no	—
196	C₂³×C₂₃	—	2	1	no	—
197	C₂³×C₂₃	—	2	2	no	—
198	C₂³×C₂₃	—	2	2	no	—
199	C₂³×C₂₃	—	2	2	no	—
200	C₂³×C₂₃	—	4	1	no	—
201	C₂³×C₂₃	—	4	1	no	—
202	C₂³×C₂₃	—	4	1	no	—
203	C₂³×C₂₃	—	4	1	no	—
204	C₂³×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
205	C₂³×C₂₃	C₄×D₄	1	2	yes	—
206	C₂³×C₂₃	C₄×D₄	1	2	yes	—
207	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₂₃	—	2	2	no	—
209	Q₈×C₂₃	—	2	2	no	—
210	Q₈×C₂₃	—	2	2	no	—
211	Q₈×C₂₃	C₂×C₄²	1	4	yes	—
212	D₄×C₂₃	—	2	16	no	—
213	D₄×C₂₃	—	2	2	no	—
214	D₄×C₂₃	—	2	2	no	—
215	D₄×C₂₃	—	2	2	no	—
216	D₄×C₂₃	C₂×C₄²	1	4	yes	—
217	C₂×C₄×C₂₃	—	2	2	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	16	no	—
225	C₂×C₄×C₂₃	—	2	4	no	—
226	C₂×C₄×C₂₃	—	2	2	no	—
227	C₂×C₄×C₂₃	—	2	2	no	—
228	C₂×C₄×C₂₃	—	2	4	no	—
229	C₂×C₄×C₂₃	—	4	1	no	—
230	C₂×C₄×C₂₃	—	4	2	no	—
231	C₂×C₄×C₂₃	—	4	2	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	—	4	2	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	2	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	8	8	no	—
238	C₂×C₄×C₂₃	—	8	4	no	—
239	C₂×C₄×C₂₃	—	8	4	no	—
240	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
241	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
242	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
243	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	2	yes	—
244	C₂×C₄×C₂₃	#117	1	4	yes	—
245	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
246	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
247	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
248	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
249	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
250	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
251	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
252	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
254	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
255	C₈×C₂₃	—	2	2	no	—
256	C₈×C₂₃	—	2	2	no	—
257	C₈×C₂₃	—	2	4	no	—
258	C₂×#520	C₂×C₂₃	1	1	yes	characteristic
259	C₂×#417	C₂×C₂₃	1	1	yes	characteristic
260	C₂×#416	C₂×C₂₃	1	1	yes	characteristic
261	C₂×#405	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#2274	C₂×C₂₃	1	1	yes	characteristic, complemented
263	#1763	C₂×C₂₃	1	2	yes	complemented
264	#1762	C₂×C₂₃	1	2	yes	complemented
265	C₄×#103	C₂×C₂₃	1	2	yes	complemented
266	C₄×#102	C₂×C₂₃	1	2	yes	complemented
267	#1736	C₂×C₂₃	1	1	yes	characteristic, complemented
268	C₂×#340	C₂×C₂₃	1	1	yes	characteristic, complemented
269	C₂²×C₂₃	—	2	2	no	—
270	C₂²×C₂₃	—	2	2	no	—
271	C₂²×C₂₃	—	2	2	no	—
272	C₂²×C₂₃	—	2	2	no	—
273	C₂²×C₂₃	—	2	2	no	—
274	C₂²×C₂₃	—	2	2	no	—
275	C₂²×C₂₃	—	2	4	no	—
276	C₂²×C₂₃	—	4	2	no	—
277	C₂²×C₂₃	—	4	2	no	—
278	C₂²×C₂₃	—	4	2	no	—
279	C₂²×C₂₃	—	4	2	no	—
280	C₂²×C₂₃	—	4	2	no	—
281	C₂²×C₂₃	—	4	2	no	—
282	C₂²×C₂₃	C₄×C₂²⋊C₄	1	4	yes	—
283	C₂²×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
284	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
285	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
286	C₂²×C₂₃	#576	1	1	yes	characteristic
287	C₂²×C₂₃	#577	1	1	yes	characteristic
288	C₂²×C₂₃	#577	1	1	yes	characteristic
289	C₂²×C₂₃	#578	1	1	yes	characteristic
290	C₄×C₂₃	—	2	4	no	—
291	C₄×C₂₃	—	2	4	no	—
292	C₄×C₂₃	—	2	4	no	—
293	C₄×C₂₃	—	4	4	no	—
294	C₄×C₂₃	—	4	4	no	—
295	C₄×C₂₃	—	4	2	no	—
296	C₄×C₂₃	—	4	2	no	—
297	C₄×C₂₃	C₄×C₂²⋊C₄	1	4	yes	—
298	C₄×C₂₃	C₂²×C₄²	1	1	yes	characteristic
299	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
300	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
301	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
302	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
303	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
304	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
305	#520	—	2	4	no	—
306	#520	C₄×C₂₃	1	8	yes	complemented
307	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
308	C₂×C₄×D₄	—	2	2	no	—
309	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
310	C₂×#117	—	2	1	no	—
311	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
312	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
313	C₂²×C₄⋊C₄	—	2	1	no	—
314	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
315	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
316	#417	C₄×C₂₃	1	8	yes	complemented
317	#416	C₄×C₂₃	1	8	yes	complemented
318	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
319	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×#103	C₂²×C₂₃	1	2	yes	—
321	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
323	C₂×#102	C₂²×C₂₃	1	2	yes	—
324	#405	—	2	4	no	—
325	C₂×#97	C₂²×C₂₃	1	2	yes	—
326	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
327	C₄×C₄⋊C₄	—	2	1	no	—
328	C₄×C₄⋊C₄	—	2	1	no	—
329	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
330	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
331	C₄×C₂²⋊C₄	—	2	2	no	—
332	#340	C₂²×C₂₃	1	2	yes	—
333	#340	C₂²×C₂₃	1	2	yes	—
334	#340	C₂²×C₂₃	1	2	yes	—
335	#340	C₂²×C₂₃	1	2	yes	—
336	C₂×C₂₃	—	2	4	no	—
337	C₂×C₂₃	—	4	4	no	—
338	C₂×C₂₃	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
339	C₂×C₂₃	#2897	1	1	yes	characteristic
340	C₂×C₂₃	#2898	1	1	yes	characteristic
341	C₂×C₂₃	#2946	1	1	yes	characteristic
342	C₂×C₂₃	#2947	1	1	yes	characteristic
343	C₂×C₂₃	#2951	1	1	yes	characteristic
344	C₂×C₂₃	#2952	1	1	yes	characteristic
345	C₂×#41	—	2	2	no	—
346	C₂×#41	—	2	2	no	—
347	C₂×#41	—	2	2	no	—
348	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
349	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
350	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
351	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
352	C₂²×D₄	—	2	1	no	—
353	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
354	C₂³×C₄	—	2	1	no	—
355	C₂³×C₄	—	2	1	no	—
356	C₂³×C₄	—	2	1	no	—
357	C₂³×C₄	—	2	2	no	—
358	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
359	C₂×M₁₆	—	2	2	no	—
360	C₂×M₁₆	—	2	2	no	—
361	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
362	C₂²×C₈	—	2	2	no	—
363	C₂²×C₈	—	2	2	no	—
364	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	complemented
365	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
367	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
368	C₄×Q₈	—	2	4	no	—
369	C₄×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
370	C₄×D₄	—	2	16	no	—
371	C₄×D₄	—	2	4	no	—
372	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	complemented
373	#117	—	2	2	no	—
374	#117	—	2	8	no	—
375	#117	C₂×C₄×C₂₃	1	4	yes	—
376	#117	C₂×C₄×C₂₃	1	4	yes	complemented
377	#117	D₄×C₂₃	1	4	yes	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	2	no	—
380	C₂×C₄⋊C₄	—	2	2	no	—
381	C₂×C₄⋊C₄	—	2	1	no	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	8	no	—
384	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
385	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	complemented
386	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
387	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
389	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
390	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
391	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
392	C₂×C₂²⋊C₄	—	2	1	no	—
393	C₂×C₂²⋊C₄	—	2	2	no	—
394	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
395	C₂×C₄²	—	2	2	no	—
396	C₂×C₄²	—	2	1	no	—
397	C₂×C₄²	—	2	2	no	—
398	C₂×C₄²	—	2	2	no	—
399	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
400	#103	C₂×C₄×C₂₃	1	4	yes	complemented
401	#103	C₂×C₄×C₂₃	1	4	yes	complemented
402	#103	C₂×C₄×C₂₃	1	8	yes	complemented
403	#102	C₂×C₄×C₂₃	1	4	yes	complemented
404	#102	C₂×C₄×C₂₃	1	4	yes	complemented
405	#102	C₂×C₄×C₂₃	1	8	yes	complemented
406	#97	—	2	4	no	—
407	C₄×C₈	—	2	4	no	—
408	#95	—	2	2	no	—
409	#95	—	2	2	no	—
410	C₂₃	#20415	1	1	yes	characteristic, direct product factor, Sylow
411	C₂⁴	—	2	1	no	—
412	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
413	#41	—	2	8	no	—
414	#41	—	2	8	no	—
415	#41	—	2	8	no	—
416	#41	C₄²×C₂₃	1	16	yes	complemented
417	C₂×Q₈	—	2	1	no	—
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
421	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
422	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
430	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
431	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	1	no	—
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	4	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	8	no	—
443	C₂²×C₄	—	8	2	no	—
444	C₂²×C₄	—	8	2	no	—
445	C₂²×C₄	—	8	1	no	—
446	C₂²×C₄	—	8	1	no	—
447	C₂²×C₄	—	8	1	no	—
448	C₂²×C₄	—	8	1	no	—
449	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
450	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
451	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
452	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
453	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
454	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
456	C₂²×C₄	#41×C₂₃	1	2	yes	—
457	C₂²×C₄	#41×C₂₃	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
459	M₁₆	—	2	8	no	—
460	C₂×C₈	—	2	8	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	2	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	2	no	—
467	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
468	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
469	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
470	C₄⋊C₄	—	2	4	no	—
471	C₄⋊C₄	—	2	8	no	—
472	C₄⋊C₄	—	2	4	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
478	C₄⋊C₄	C₂²⋊C₄×C₂₃	1	4	yes	—
479	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
480	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
481	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
482	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	8	no	—
485	C₂²⋊C₄	C₂²⋊C₄×C₂₃	1	8	yes	—
486	C₄²	—	2	8	no	—
487	C₄²	—	2	4	no	—
488	C₄²	—	8	2	no	—
489	C₄²	—	8	2	no	—
490	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
491	C₄²	C₂×D₄×C₂₃	1	4	yes	—
492	C₂³	—	2	1	no	—
493	C₂³	—	2	1	no	—
494	C₂³	—	2	1	no	—
495	C₂³	—	2	1	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	2	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	4	1	no	—
500	C₂³	—	4	1	no	—
501	C₂³	—	4	1	no	—
502	C₂³	—	4	1	no	—
503	C₂³	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
504	C₂³	C₄×D₄×C₂₃	1	2	yes	—
505	C₂³	C₄×D₄×C₂₃	1	2	yes	—
506	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
507	Q₈	—	2	2	no	—
508	Q₈	—	2	2	no	—
509	Q₈	—	2	2	no	—
510	Q₈	C₂×C₄²×C₂₃	1	4	yes	—
511	D₄	—	2	16	no	—
512	D₄	—	2	2	no	—
513	D₄	—	2	2	no	—
514	D₄	—	2	2	no	—
515	D₄	C₂×C₄²×C₂₃	1	4	yes	—
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	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	16	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	1	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	8	8	no	—
537	C₂×C₄	—	8	4	no	—
538	C₂×C₄	—	8	4	no	—
539	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
540	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
541	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
542	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	2	yes	—
543	C₂×C₄	C₂₃×#117	1	4	yes	—
544	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
545	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
546	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
547	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
548	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
549	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
550	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
551	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
552	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
553	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
554	C₈	—	2	2	no	—
555	C₈	—	2	2	no	—
556	C₈	—	2	4	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	4	2	no	—
570	C₂²	C₄×C₂²⋊C₄×C₂₃	1	4	yes	—
571	C₂²	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
572	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
573	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
574	C₂²	C₂₃×#576	1	1	yes	characteristic
575	C₂²	C₂₃×#577	1	1	yes	characteristic
576	C₂²	C₂₃×#577	1	1	yes	characteristic
577	C₂²	C₂₃×#578	1	1	yes	characteristic
578	C₄	—	2	4	no	—
579	C₄	—	2	4	no	—
580	C₄	—	2	4	no	—
581	C₄	—	4	4	no	—
582	C₄	—	4	2	no	—
583	C₄	—	4	2	no	—
584	C₄	—	4	4	no	—
585	C₄	C₄×C₂²⋊C₄×C₂₃	1	4	yes	—
586	C₄	C₂²×C₄²×C₂₃	1	1	yes	characteristic
587	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
588	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
589	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
590	C₂	—	2	4	no	—
591	C₂	—	4	4	no	—
592	C₂	C₂×C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
593	C₂	C₂₃×#2897	1	1	yes	characteristic
594	C₂	C₂₃×#2898	1	1	yes	characteristic
595	C₂	C₂₃×#2946	1	1	yes	characteristic
596	C₂	C₂₃×#2947	1	1	yes	characteristic
597	C₂	C₂₃×#2951	1	1	yes	characteristic
598	C₂	C₂₃×#2952	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₄	C₂²×C₄²×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₂₃×#405	C₂	semidirect
5. C₂₃×#2274	C₂	semidirect
6. C₂₃×#1763	C₂	semidirect
7. C₂₃×#1762	C₂	semidirect
8. C₄×C₂₃×#103	C₂	semidirect
9. C₄×C₂₃×#102	C₂	semidirect
10. C₂₃×#1736	C₂	semidirect
11. C₂×C₂₃×#340	C₂	semidirect
16. C₂₃×#520	C₄	semidirect
26. C₂₃×#417	C₄	semidirect
27. C₂₃×#416	C₄	semidirect
62. C₂×M₁₆×C₂₃	C₂×C₄	semidirect
65. C₂²×C₈×C₂₃	C₂×C₄	semidirect
70. C₄×Q₈×C₂₃	C₂×C₄	semidirect
73. C₄×D₄×C₂₃	C₂×C₄	semidirect
77. C₂₃×#117	C₂×C₄	semidirect
86. C₂×C₄⋊C₄×C₂₃	C₂×C₄	semidirect
101. C₂₃×#103	C₂×C₄	semidirect
102. C₂₃×#103	C₂×C₄	semidirect
103. C₂₃×#103	C₂×C₄	semidirect
104. C₂₃×#102	C₂×C₄	semidirect
105. C₂₃×#102	C₂×C₄	semidirect
106. C₂₃×#102	C₂×C₄	semidirect
116. #41×C₂₃	C₄²	semidirect
192. #20415	C₂₃	direct (N × H)
261. C₂×#405	C₂×C₂₃	semidirect
262. #2274	C₂×C₂₃	semidirect
263. #1763	C₂×C₂₃	semidirect
264. #1762	C₂×C₂₃	semidirect
265. C₄×#103	C₂×C₂₃	semidirect
266. C₄×#102	C₂×C₂₃	semidirect
267. #1736	C₂×C₂₃	semidirect
268. C₂×#340	C₂×C₂₃	semidirect
306. #520	C₄×C₂₃	semidirect
316. #417	C₄×C₂₃	semidirect
317. #416	C₄×C₂₃	semidirect
361. C₂×M₁₆	C₂×C₄×C₂₃	semidirect
364. C₂²×C₈	C₂×C₄×C₂₃	semidirect
369. C₄×Q₈	C₂×C₄×C₂₃	semidirect
372. C₄×D₄	C₂×C₄×C₂₃	semidirect
376. #117	C₂×C₄×C₂₃	semidirect
385. C₂×C₄⋊C₄	C₂×C₄×C₂₃	semidirect
400. #103	C₂×C₄×C₂₃	semidirect
401. #103	C₂×C₄×C₂₃	semidirect
402. #103	C₂×C₄×C₂₃	semidirect
403. #102	C₂×C₄×C₂₃	semidirect
404. #102	C₂×C₄×C₂₃	semidirect
405. #102	C₂×C₄×C₂₃	semidirect
410. C₂₃	#20415	direct (N × H)
416. #41	C₄²×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#520	2
C₂×C₂₃×#417	2
C₂×C₂₃×#416	2
C₂×C₂₃×#405	2
C₂₃×#2274	2
C₂₃×#1763	2
C₂₃×#1762	2
C₄×C₂₃×#103	2
C₄×C₂₃×#102	2
C₂₃×#1736	2
C₂×C₂₃×#340	2
#20415	23

Subgroups — C23×#20415 #19168041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20415 #19168041