Subgroups — C₂₃×#20387 #19168013

Order 5888.

1586 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#408	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2297	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1753	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1753	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1752	C₂	1	2	yes	maximal, complemented
9	C₈×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1729	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#131	—	2	4	no	—
14	C₂×C₂₃×#131	—	2	4	no	—
15	C₂×C₂₃×#131	C₄	1	8	yes	complemented
16	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
22	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#408	C₄	1	8	yes	complemented
25	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
26	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
27	C₂₃×#405	—	2	4	no	—
28	C₄×M₁₆×C₂₃	—	2	4	no	—
29	C₂×C₂₃×#97	C₂²	1	2	yes	—
30	C₂×C₂₃×#97	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
33	C₂₃×#391	C₄	1	2	yes	—
34	C₂₃×#390	C₄	1	2	yes	—
35	C₂₃×#386	C₄	1	4	yes	—
36	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
37	C₄×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
38	C₂₃×#336	C₂²	1	1	yes	characteristic
39	C₂₃×#336	C₂²	1	1	yes	characteristic
40	C₂₃×#336	C₂²	1	1	yes	characteristic
41	C₂₃×#336	C₂²	1	1	yes	characteristic
42	C₂₃×#336	C₂²	1	2	yes	—
43	C₂₃×#336	C₂²	1	2	yes	—
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	4	yes	—
49	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
50	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×D₄×C₂₃	C₂×C₄	1	2	yes	—
52	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
54	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
55	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂₃×#131	—	2	32	no	—
57	C₂×M₁₆×C₂₃	—	2	2	no	—
58	C₂×M₁₆×C₂₃	—	2	2	no	—
59	C₂×M₁₆×C₂₃	—	2	4	no	—
60	C₂×M₁₆×C₂₃	—	2	4	no	—
61	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
62	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
63	C₂×M₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
64	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
65	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
66	C₂²×C₈×C₂₃	—	2	2	no	—
67	C₂²×C₈×C₂₃	—	2	2	no	—
68	C₂²×C₈×C₂₃	—	2	2	no	—
69	C₂²×C₈×C₂₃	—	2	2	no	—
70	C₂²×C₈×C₂₃	—	2	4	no	—
71	C₂²×C₈×C₂₃	—	2	4	no	—
72	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
73	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
74	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	complemented
75	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
80	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
81	C₂₃×#117	—	2	2	no	—
82	C₂₃×#117	—	2	2	no	—
83	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
84	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
85	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
86	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
87	C₂×C₄²×C₂₃	—	2	2	no	—
88	C₂×C₄²×C₂₃	—	2	2	no	—
89	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
90	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
92	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
93	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	complemented
94	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	complemented
95	C₂₃×#97	—	2	4	no	—
96	C₂₃×#97	—	2	4	no	—
97	C₄×C₈×C₂₃	—	2	4	no	—
98	C₄×C₈×C₂₃	—	2	4	no	—
99	C₂₃×#95	C₂×C₄	1	4	yes	—
100	C₂₃×#95	C₂×C₄	1	4	yes	—
101	C₂⁴×C₂₃	C₂²×C₄	1	2	yes	—
102	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
103	#41×C₂₃	—	2	8	no	—
104	#41×C₂₃	—	2	8	no	—
105	#41×C₂₃	—	2	8	no	—
106	#41×C₂₃	—	2	8	no	—
107	C₂×Q₈×C₂₃	—	2	2	no	—
108	C₂×Q₈×C₂₃	—	2	2	no	—
109	C₂×Q₈×C₂₃	C₄²	1	4	yes	—
110	C₂×D₄×C₂₃	—	2	4	no	—
111	C₂×D₄×C₂₃	—	2	4	no	—
112	C₂×D₄×C₂₃	—	2	2	no	—
113	C₂×D₄×C₂₃	—	2	2	no	—
114	C₂×D₄×C₂₃	C₄²	1	8	yes	—
115	C₂×D₄×C₂₃	C₄²	1	4	yes	—
116	C₂²×C₄×C₂₃	—	2	2	no	—
117	C₂²×C₄×C₂₃	—	2	2	no	—
118	C₂²×C₄×C₂₃	—	2	2	no	—
119	C₂²×C₄×C₂₃	—	2	2	no	—
120	C₂²×C₄×C₂₃	—	2	4	no	—
121	C₂²×C₄×C₂₃	—	2	4	no	—
122	C₂²×C₄×C₂₃	—	4	2	no	—
123	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
125	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
126	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
127	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
128	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
132	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₂₃	#41	1	2	yes	—
135	C₂²×C₄×C₂₃	#41	1	2	yes	—
136	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
137	M₁₆×C₂₃	—	2	8	no	—
138	M₁₆×C₂₃	—	2	16	no	—
139	M₁₆×C₂₃	C₂²⋊C₄	1	16	yes	complemented
140	C₂×C₈×C₂₃	—	2	8	no	—
141	C₂×C₈×C₂₃	—	2	8	no	—
142	C₂×C₈×C₂₃	—	2	1	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	1	no	—
147	C₂×C₈×C₂₃	—	2	1	no	—
148	C₂×C₈×C₂₃	—	2	2	no	—
149	C₂×C₈×C₂₃	—	2	1	no	—
150	C₂×C₈×C₂₃	—	2	1	no	—
151	C₂×C₈×C₂₃	—	2	2	no	—
152	C₂×C₈×C₂₃	—	2	16	no	—
153	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
154	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
155	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
156	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
157	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
158	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
159	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
160	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
161	C₄⋊C₄×C₂₃	—	2	4	no	—
162	C₂²⋊C₄×C₂₃	—	2	4	no	—
163	C₂²⋊C₄×C₂₃	—	2	8	no	—
164	C₂²⋊C₄×C₂₃	—	4	4	no	—
165	C₄²×C₂₃	—	2	4	no	—
166	C₄²×C₂₃	—	2	1	no	—
167	C₄²×C₂₃	—	2	1	no	—
168	C₄²×C₂₃	—	2	2	no	—
169	C₄²×C₂₃	—	2	1	no	—
170	C₄²×C₂₃	—	2	1	no	—
171	C₄²×C₂₃	—	2	2	no	—
172	#20387	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₂₃	—	2	2	no	—
174	C₂³×C₂₃	—	2	2	no	—
175	C₂³×C₂₃	—	2	2	no	—
176	C₂³×C₂₃	—	2	2	no	—
177	C₂³×C₂₃	—	2	4	no	—
178	C₂³×C₂₃	—	2	4	no	—
179	C₂³×C₂₃	C₂×C₄²	1	4	yes	—
180	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
181	C₂³×C₂₃	C₄×D₄	1	2	yes	—
182	C₂³×C₂₃	C₄×D₄	1	2	yes	—
183	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
184	Q₈×C₂₃	—	2	8	no	—
185	D₄×C₂₃	—	2	16	no	—
186	D₄×C₂₃	—	2	8	no	—
187	C₂×C₄×C₂₃	—	2	2	no	—
188	C₂×C₄×C₂₃	—	2	2	no	—
189	C₂×C₄×C₂₃	—	2	2	no	—
190	C₂×C₄×C₂₃	—	2	2	no	—
191	C₂×C₄×C₂₃	—	2	4	no	—
192	C₂×C₄×C₂₃	—	2	4	no	—
193	C₂×C₄×C₂₃	—	2	4	no	—
194	C₂×C₄×C₂₃	—	2	4	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	4	no	—
197	C₂×C₄×C₂₃	—	2	4	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	2	4	no	—
201	C₂×C₄×C₂₃	—	4	8	no	—
202	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
203	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
204	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	4	yes	—
205	C₂×C₄×C₂₃	#117	1	4	yes	—
206	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
207	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
208	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
209	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
210	C₂×C₄×C₂₃	#131	1	4	yes	complemented
211	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
212	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
213	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
217	C₈×C₂₃	—	2	2	no	—
218	C₈×C₂₃	—	2	2	no	—
219	C₈×C₂₃	—	2	2	no	—
220	C₈×C₂₃	—	2	2	no	—
221	C₈×C₂₃	—	2	4	no	—
222	C₈×C₂₃	C₂×C₂²⋊C₄	1	8	yes	—
223	C₂²×#131	C₂×C₂₃	1	1	yes	characteristic
224	C₂×#408	C₂×C₂₃	1	1	yes	characteristic
225	C₂×#405	C₂×C₂₃	1	1	yes	characteristic, complemented
226	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
227	#2297	C₂×C₂₃	1	1	yes	characteristic
228	#1753	C₂×C₂₃	1	2	yes	complemented
229	#1753	C₂×C₂₃	1	2	yes	complemented
230	#1752	C₂×C₂₃	1	2	yes	complemented
231	C₈×C₂²⋊C₄	C₂×C₂₃	1	2	yes	complemented
232	#1729	C₂×C₂₃	1	1	yes	characteristic, complemented
233	C₂×#336	C₂×C₂₃	1	1	yes	characteristic, complemented
234	C₂²×C₂₃	—	2	2	no	—
235	C₂²×C₂₃	—	2	2	no	—
236	C₂²×C₂₃	—	2	4	no	—
237	C₂²×C₂₃	—	2	4	no	—
238	C₂²×C₂₃	—	2	4	no	—
239	C₂²×C₂₃	—	2	4	no	—
240	C₂²×C₂₃	—	2	4	no	—
241	C₂²×C₂₃	—	4	4	no	—
242	C₂²×C₂₃	—	4	4	no	—
243	C₂²×C₂₃	C₄×C₂²⋊C₄	1	4	yes	—
244	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
245	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
246	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
247	C₂²×C₂₃	C₂×#131	1	2	yes	—
248	C₂²×C₂₃	#568	1	2	yes	—
249	C₄×C₂₃	—	2	4	no	—
250	C₄×C₂₃	—	2	4	no	—
251	C₄×C₂₃	—	4	2	no	—
252	C₄×C₂₃	—	4	4	no	—
253	C₄×C₂₃	C₄×C₂²⋊C₄	1	4	yes	—
254	C₄×C₂₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
255	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
256	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
257	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
258	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×#131	—	2	4	no	—
260	C₂×#131	—	2	4	no	—
261	C₂×#131	C₄×C₂₃	1	8	yes	complemented
262	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
263	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
264	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
265	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
266	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
267	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
268	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
269	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
270	#408	C₄×C₂₃	1	8	yes	complemented
271	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
272	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
273	#405	—	2	4	no	—
274	C₄×M₁₆	—	2	4	no	—
275	C₂×#97	C₂²×C₂₃	1	2	yes	—
276	C₂×#97	C₂²×C₂₃	1	2	yes	—
277	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
278	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
279	#391	C₄×C₂₃	1	2	yes	—
280	#390	C₄×C₂₃	1	2	yes	—
281	#386	C₄×C₂₃	1	4	yes	—
282	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
283	C₄×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
284	#336	C₂²×C₂₃	1	1	yes	characteristic
285	#336	C₂²×C₂₃	1	1	yes	characteristic
286	#336	C₂²×C₂₃	1	1	yes	characteristic
287	#336	C₂²×C₂₃	1	1	yes	characteristic
288	#336	C₂²×C₂₃	1	2	yes	—
289	#336	C₂²×C₂₃	1	2	yes	—
290	C₂×C₂₃	—	2	4	no	—
291	C₂×C₂₃	—	4	4	no	—
292	C₂×C₂₃	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
293	C₂×C₂₃	C₄×#131	1	1	yes	characteristic
294	C₂×C₂₃	#2875	1	1	yes	characteristic
295	C₂×C₂₃	#2933	1	2	yes	—
296	C₂×C₂₃	#2935	1	2	yes	—
297	C₂×#41	—	2	2	no	—
298	C₂×#41	—	2	2	no	—
299	C₂×#41	—	2	2	no	—
300	C₂×#41	—	2	2	no	—
301	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
302	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
303	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
304	C₂²×D₄	C₂×C₄×C₂₃	1	2	yes	—
305	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
306	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
307	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
308	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
309	#131	—	2	32	no	—
310	C₂×M₁₆	—	2	2	no	—
311	C₂×M₁₆	—	2	2	no	—
312	C₂×M₁₆	—	2	4	no	—
313	C₂×M₁₆	—	2	4	no	—
314	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
315	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
316	C₂×M₁₆	C₂×C₄×C₂₃	1	8	yes	complemented
317	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
318	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
319	C₂²×C₈	—	2	2	no	—
320	C₂²×C₈	—	2	2	no	—
321	C₂²×C₈	—	2	2	no	—
322	C₂²×C₈	—	2	2	no	—
323	C₂²×C₈	—	2	4	no	—
324	C₂²×C₈	—	2	4	no	—
325	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
326	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
327	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	complemented
328	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
329	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
331	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
332	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
333	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
334	#117	—	2	2	no	—
335	#117	—	2	2	no	—
336	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
337	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
338	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
339	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
340	C₂×C₄²	—	2	2	no	—
341	C₂×C₄²	—	2	2	no	—
342	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
343	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
344	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
345	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
346	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	complemented
347	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	complemented
348	#97	—	2	4	no	—
349	#97	—	2	4	no	—
350	C₄×C₈	—	2	4	no	—
351	C₄×C₈	—	2	4	no	—
352	#95	C₂×C₄×C₂₃	1	4	yes	—
353	#95	C₂×C₄×C₂₃	1	4	yes	—
354	C₂₃	#20387	1	1	yes	characteristic, direct product factor, Sylow
355	C₂⁴	C₂²×C₄×C₂₃	1	2	yes	—
356	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
357	#41	—	2	8	no	—
358	#41	—	2	8	no	—
359	#41	—	2	8	no	—
360	#41	—	2	8	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	C₄²×C₂₃	1	4	yes	—
364	C₂×D₄	—	2	4	no	—
365	C₂×D₄	—	2	4	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	C₄²×C₂₃	1	8	yes	—
369	C₂×D₄	C₄²×C₂₃	1	4	yes	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	—	2	4	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	4	2	no	—
377	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
378	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
379	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
380	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
381	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
382	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
383	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
384	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
385	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
386	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
388	C₂²×C₄	#41×C₂₃	1	2	yes	—
389	C₂²×C₄	#41×C₂₃	1	2	yes	—
390	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
391	M₁₆	—	2	8	no	—
392	M₁₆	—	2	16	no	—
393	M₁₆	C₂²⋊C₄×C₂₃	1	16	yes	complemented
394	C₂×C₈	—	2	8	no	—
395	C₂×C₈	—	2	8	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	1	no	—
399	C₂×C₈	—	2	1	no	—
400	C₂×C₈	—	2	1	no	—
401	C₂×C₈	—	2	1	no	—
402	C₂×C₈	—	2	1	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	16	no	—
407	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
408	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
409	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
410	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
411	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
412	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
413	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
414	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
415	C₄⋊C₄	—	2	4	no	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	—	2	8	no	—
418	C₂²⋊C₄	—	4	4	no	—
419	C₄²	—	2	4	no	—
420	C₄²	—	2	1	no	—
421	C₄²	—	2	2	no	—
422	C₄²	—	2	1	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	2	no	—
426	C₂³	—	2	2	no	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	4	no	—
429	C₂³	—	2	4	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	2	no	—
432	C₂³	C₂×C₄²×C₂₃	1	4	yes	—
433	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
434	C₂³	C₄×D₄×C₂₃	1	2	yes	—
435	C₂³	C₄×D₄×C₂₃	1	2	yes	—
436	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
437	Q₈	—	2	8	no	—
438	D₄	—	2	16	no	—
439	D₄	—	2	8	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	4	8	no	—
455	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
456	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
457	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	4	yes	—
458	C₂×C₄	C₂₃×#117	1	4	yes	—
459	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
460	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
461	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
462	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
463	C₂×C₄	C₂₃×#131	1	4	yes	complemented
464	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
465	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
466	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
467	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
468	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
469	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
470	C₈	—	2	2	no	—
471	C₈	—	2	2	no	—
472	C₈	—	2	2	no	—
473	C₈	—	2	2	no	—
474	C₈	—	2	4	no	—
475	C₈	C₂×C₂²⋊C₄×C₂₃	1	8	yes	—
476	C₂²	—	2	4	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	2	2	no	—
479	C₂²	—	2	2	no	—
480	C₂²	—	2	4	no	—
481	C₂²	—	2	4	no	—
482	C₂²	—	2	4	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	4	4	no	—
485	C₂²	C₄×C₂²⋊C₄×C₂₃	1	4	yes	—
486	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
487	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
488	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
489	C₂²	C₂×C₂₃×#131	1	2	yes	—
490	C₂²	C₂₃×#568	1	2	yes	—
491	C₄	—	2	4	no	—
492	C₄	—	2	4	no	—
493	C₄	—	4	2	no	—
494	C₄	—	4	4	no	—
495	C₄	C₄×C₂²⋊C₄×C₂₃	1	4	yes	—
496	C₄	C₂²×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
497	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
498	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
499	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
500	C₂	—	2	4	no	—
501	C₂	—	4	4	no	—
502	C₂	C₂×C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
503	C₂	C₄×C₂₃×#131	1	1	yes	characteristic
504	C₂	C₂₃×#2875	1	1	yes	characteristic
505	C₂	C₂₃×#2933	1	2	yes	—
506	C₂	C₂₃×#2935	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂×C₂₃×#405	C₂	semidirect
4. C₂×C₄×M₁₆×C₂₃	C₂	semidirect
6. C₂₃×#1753	C₂	semidirect
7. C₂₃×#1753	C₂	semidirect
8. C₂₃×#1752	C₂	semidirect
9. C₈×C₂²⋊C₄×C₂₃	C₂	semidirect
10. C₂₃×#1729	C₂	semidirect
11. C₂×C₂₃×#336	C₂	semidirect
15. C₂×C₂₃×#131	C₄	semidirect
24. C₂₃×#408	C₄	semidirect
63. C₂×M₁₆×C₂₃	C₂×C₄	semidirect
74. C₂²×C₈×C₂₃	C₂×C₄	semidirect
93. C₂₃×C₂²⋊C₈	C₂×C₄	semidirect
94. C₂₃×C₂²⋊C₈	C₂×C₄	semidirect
139. M₁₆×C₂₃	C₂²⋊C₄	semidirect
172. #20387	C₂₃	direct (N × H)
210. C₂×C₄×C₂₃	#131	semidirect
225. C₂×#405	C₂×C₂₃	semidirect
226. C₂×C₄×M₁₆	C₂×C₂₃	semidirect
228. #1753	C₂×C₂₃	semidirect
229. #1753	C₂×C₂₃	semidirect
230. #1752	C₂×C₂₃	semidirect
231. C₈×C₂²⋊C₄	C₂×C₂₃	semidirect
232. #1729	C₂×C₂₃	semidirect
233. C₂×#336	C₂×C₂₃	semidirect
261. C₂×#131	C₄×C₂₃	semidirect
270. #408	C₄×C₂₃	semidirect
316. C₂×M₁₆	C₂×C₄×C₂₃	semidirect
327. C₂²×C₈	C₂×C₄×C₂₃	semidirect
346. C₂²⋊C₈	C₂×C₄×C₂₃	semidirect
347. C₂²⋊C₈	C₂×C₄×C₂₃	semidirect
354. C₂₃	#20387	direct (N × H)
393. M₁₆	C₂²⋊C₄×C₂₃	semidirect
463. C₂×C₄	C₂₃×#131	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#131	2
C₂×C₂₃×#408	2
C₂×C₂₃×#405	2
C₂×C₄×M₁₆×C₂₃	2
C₂₃×#2297	2
C₂₃×#1753	2
C₂₃×#1752	2
C₈×C₂²⋊C₄×C₂₃	2
C₂₃×#1729	2
C₂×C₂₃×#336	2
#20387	23

Subgroups — C23×#20387 #19168013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20387 #19168013