Subgroups — C₂₃×#21023 #19168649

Order 5888.

1018 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1917	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1916	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1728	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1728	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1728	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1728	C₂	1	1	yes	maximal, characteristic
10	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
15	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
18	C₂₃×#423	—	2	4	no	—
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
20	C₄×M₁₆×C₂₃	—	2	4	no	—
21	C₂×C₂₃×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
24	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
27	C₂×C₂₃×#95	C₄	1	2	yes	—
28	C₂₃×#336	C₂²	1	2	yes	—
29	C₂₃×#336	C₂²	1	4	yes	—
30	C₂₃×#336	C₂²	1	4	yes	—
31	C₂₃×#336	C₂²	1	4	yes	—
32	C₂₃×#336	C₂²	1	2	yes	—
33	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₂₃	—	2	2	no	—
39	C₂×M₁₆×C₂₃	—	2	2	no	—
40	C₂×M₁₆×C₂₃	—	2	2	no	—
41	C₂×M₁₆×C₂₃	—	2	2	no	—
42	C₂×M₁₆×C₂₃	—	2	2	no	—
43	C₂×M₁₆×C₂₃	—	2	2	no	—
44	C₂×M₁₆×C₂₃	—	2	2	no	—
45	C₂×M₁₆×C₂₃	—	2	2	no	—
46	C₂×M₁₆×C₂₃	—	2	2	no	—
47	C₂×M₁₆×C₂₃	—	2	2	no	—
48	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
49	C₂×M₁₆×C₂₃	Q₈	1	4	yes	—
50	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
52	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
55	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₂₃	—	2	2	no	—
60	C₂×C₄²×C₂₃	—	2	2	no	—
61	C₂×C₄²×C₂₃	—	2	2	no	—
62	C₂×C₄²×C₂₃	—	2	2	no	—
63	C₂×C₄²×C₂₃	—	2	2	no	—
64	C₂×C₄²×C₂₃	—	2	2	no	—
65	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
70	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
73	C₂₃×C₄⋊C₈	—	2	8	no	—
74	C₂₃×#97	—	2	4	no	—
75	C₄×C₈×C₂₃	—	2	4	no	—
76	C₂₃×#95	C₂×C₄	1	2	yes	—
77	C₂₃×#95	C₂×C₄	1	2	yes	—
78	C₂₃×#95	C₂×C₄	1	2	yes	—
79	C₂₃×#95	C₂×C₄	1	2	yes	—
80	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₂₃	—	2	2	no	—
82	C₂²×C₄×C₂₃	—	2	4	no	—
83	C₂²×C₄×C₂₃	—	2	2	no	—
84	C₂²×C₄×C₂₃	—	2	4	no	—
85	C₂²×C₄×C₂₃	—	2	4	no	—
86	C₂²×C₄×C₂₃	—	2	4	no	—
87	C₂²×C₄×C₂₃	—	2	4	no	—
88	C₂²×C₄×C₂₃	—	4	2	no	—
89	C₂²×C₄×C₂₃	—	4	2	no	—
90	C₂²×C₄×C₂₃	—	4	2	no	—
91	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
100	C₂²×C₄×C₂₃	#41	1	2	yes	—
101	C₂²×C₄×C₂₃	#41	1	2	yes	—
102	C₂²×C₄×C₂₃	#41	1	2	yes	—
103	C₂²×C₄×C₂₃	#41	1	2	yes	—
104	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
105	M₁₆×C₂₃	—	2	8	no	—
106	M₁₆×C₂₃	—	2	8	no	—
107	M₁₆×C₂₃	—	4	4	no	—
108	M₁₆×C₂₃	—	4	4	no	—
109	C₂×C₈×C₂₃	—	2	2	no	—
110	C₂×C₈×C₂₃	—	2	2	no	—
111	C₂×C₈×C₂₃	—	2	2	no	—
112	C₂×C₈×C₂₃	—	2	2	no	—
113	C₂×C₈×C₂₃	—	2	2	no	—
114	C₂×C₈×C₂₃	—	2	2	no	—
115	C₂×C₈×C₂₃	—	2	2	no	—
116	C₂×C₈×C₂₃	—	2	2	no	—
117	C₂×C₈×C₂₃	—	2	2	no	—
118	C₂×C₈×C₂₃	—	2	2	no	—
119	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
120	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
121	C₄⋊C₄×C₂₃	—	2	4	no	—
122	C₄⋊C₄×C₂₃	—	2	4	no	—
123	C₄⋊C₄×C₂₃	—	2	4	no	—
124	C₄⋊C₄×C₂₃	M₁₆	1	8	yes	complemented
125	C₄²×C₂₃	—	2	4	no	—
126	C₄²×C₂₃	—	2	2	no	—
127	C₄²×C₂₃	—	2	4	no	—
128	C₄²×C₂₃	—	2	2	no	—
129	C₄²×C₂₃	—	2	2	no	—
130	C₄²×C₂₃	—	2	1	no	—
131	C₄²×C₂₃	—	2	1	no	—
132	C₄²×C₂₃	—	2	2	no	—
133	C₄²×C₂₃	—	2	2	no	—
134	C₄²×C₂₃	—	2	1	no	—
135	C₄²×C₂₃	—	2	2	no	—
136	C₄²×C₂₃	—	2	1	no	—
137	#21023	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂³×C₂₃	—	2	2	no	—
139	C₂³×C₂₃	—	2	2	no	—
140	C₂³×C₂₃	#117	1	2	yes	—
141	C₂³×C₂₃	C₄×D₄	1	2	yes	—
142	C₂³×C₂₃	C₄×Q₈	1	2	yes	—
143	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₂₃	—	2	8	no	—
145	C₂×C₄×C₂₃	—	2	2	no	—
146	C₂×C₄×C₂₃	—	2	2	no	—
147	C₂×C₄×C₂₃	—	2	2	no	—
148	C₂×C₄×C₂₃	—	2	2	no	—
149	C₂×C₄×C₂₃	—	2	2	no	—
150	C₂×C₄×C₂₃	—	2	2	no	—
151	C₂×C₄×C₂₃	—	2	2	no	—
152	C₂×C₄×C₂₃	—	2	2	no	—
153	C₂×C₄×C₂₃	—	2	2	no	—
154	C₂×C₄×C₂₃	—	2	2	no	—
155	C₂×C₄×C₂₃	—	2	2	no	—
156	C₂×C₄×C₂₃	—	2	4	no	—
157	C₂×C₄×C₂₃	—	2	2	no	—
158	C₂×C₄×C₂₃	—	2	2	no	—
159	C₂×C₄×C₂₃	—	4	8	no	—
160	C₂×C₄×C₂₃	—	4	8	no	—
161	C₂×C₄×C₂₃	—	4	4	no	—
162	C₂×C₄×C₂₃	#117	1	2	yes	—
163	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
164	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
165	C₂×C₄×C₂₃	C₂²⋊Q₈	1	4	yes	—
166	C₂×C₄×C₂₃	#123	1	4	yes	—
167	C₂×C₄×C₂₃	#125	1	4	yes	—
168	C₂×C₄×C₂₃	#126	1	4	yes	—
169	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
170	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
171	C₂×C₄×C₂₃	C₂×M₁₆	1	4	yes	—
172	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
174	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
178	C₈×C₂₃	—	2	4	no	—
179	C₈×C₂₃	—	2	4	no	—
180	C₈×C₂₃	—	4	2	no	—
181	C₈×C₂₃	—	4	2	no	—
182	C₂×#423	C₂×C₂₃	1	1	yes	characteristic
183	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
184	C₂×C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
185	#1917	C₂×C₂₃	1	4	yes	complemented
186	#1916	C₂×C₂₃	1	4	yes	complemented
187	#1728	C₂×C₂₃	1	1	yes	characteristic
188	#1728	C₂×C₂₃	1	1	yes	characteristic
189	#1728	C₂×C₂₃	1	1	yes	characteristic
190	#1728	C₂×C₂₃	1	1	yes	characteristic
191	C₂²×C₂₃	—	2	2	no	—
192	C₂²×C₂₃	—	2	2	no	—
193	C₂²×C₂₃	—	2	4	no	—
194	C₂²×C₂₃	—	2	4	no	—
195	C₂²×C₂₃	#387	1	4	yes	—
196	C₂²×C₂₃	C₂×#117	1	1	yes	characteristic
197	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
198	C₂²×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
199	C₂²×C₂₃	C₂²×M₁₆	1	2	yes	—
200	C₂²×C₂₃	#568	1	2	yes	—
201	C₄×C₂₃	—	2	4	no	—
202	C₄×C₂₃	—	2	4	no	—
203	C₄×C₂₃	—	2	4	no	—
204	C₄×C₂₃	—	4	2	no	—
205	C₄×C₂₃	—	4	2	no	—
206	C₄×C₂₃	—	4	2	no	—
207	C₄×C₂₃	#387	1	4	yes	—
208	C₄×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
209	C₄×C₂₃	C₂×#123	1	1	yes	characteristic
210	C₄×C₂₃	C₂×#125	1	1	yes	characteristic
211	C₄×C₂₃	C₂×#126	1	1	yes	characteristic
212	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
213	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
214	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
215	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
216	C₂²×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
217	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
218	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
219	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
220	#423	—	2	4	no	—
221	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
222	C₄×M₁₆	—	2	4	no	—
223	C₂×#97	C₂²×C₂₃	1	2	yes	—
224	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
225	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
226	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
227	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
228	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
229	C₂×#95	C₄×C₂₃	1	2	yes	—
230	#336	C₂²×C₂₃	1	2	yes	—
231	#336	C₂²×C₂₃	1	4	yes	—
232	#336	C₂²×C₂₃	1	4	yes	—
233	#336	C₂²×C₂₃	1	4	yes	—
234	#336	C₂²×C₂₃	1	2	yes	—
235	C₂×C₂₃	—	2	4	no	—
236	C₂×C₂₃	C₂×#387	1	1	yes	characteristic
237	C₂×C₂₃	#2920	1	1	yes	characteristic
238	C₂×C₂₃	#2921	1	1	yes	characteristic
239	C₂×C₂₃	D₄×M₁₆	1	2	yes	—
240	C₂×C₂₃	Q₈×M₁₆	1	2	yes	—
241	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
242	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
243	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
244	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
245	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
246	C₂×M₁₆	—	2	2	no	—
247	C₂×M₁₆	—	2	2	no	—
248	C₂×M₁₆	—	2	2	no	—
249	C₂×M₁₆	—	2	2	no	—
250	C₂×M₁₆	—	2	2	no	—
251	C₂×M₁₆	—	2	2	no	—
252	C₂×M₁₆	—	2	2	no	—
253	C₂×M₁₆	—	2	2	no	—
254	C₂×M₁₆	—	2	2	no	—
255	C₂×M₁₆	—	2	2	no	—
256	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
257	C₂×M₁₆	Q₈×C₂₃	1	4	yes	—
258	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
259	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
260	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
261	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
262	C₂×C₄⋊C₄	—	2	8	no	—
263	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
264	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
265	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
266	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
267	C₂×C₄²	—	2	2	no	—
268	C₂×C₄²	—	2	2	no	—
269	C₂×C₄²	—	2	2	no	—
270	C₂×C₄²	—	2	2	no	—
271	C₂×C₄²	—	2	2	no	—
272	C₂×C₄²	—	2	2	no	—
273	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
274	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
275	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
276	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
277	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
278	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
279	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
281	C₄⋊C₈	—	2	8	no	—
282	#97	—	2	4	no	—
283	C₄×C₈	—	2	4	no	—
284	#95	C₂×C₄×C₂₃	1	2	yes	—
285	#95	C₂×C₄×C₂₃	1	2	yes	—
286	#95	C₂×C₄×C₂₃	1	2	yes	—
287	#95	C₂×C₄×C₂₃	1	2	yes	—
288	C₂₃	#21023	1	1	yes	characteristic, direct product factor, Sylow
289	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	4	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	—	4	2	no	—
300	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
301	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
302	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
303	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
304	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
305	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
308	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
309	C₂²×C₄	#41×C₂₃	1	2	yes	—
310	C₂²×C₄	#41×C₂₃	1	2	yes	—
311	C₂²×C₄	#41×C₂₃	1	2	yes	—
312	C₂²×C₄	#41×C₂₃	1	2	yes	—
313	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
314	M₁₆	—	2	8	no	—
315	M₁₆	—	2	8	no	—
316	M₁₆	—	4	4	no	—
317	M₁₆	—	4	4	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
329	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	—	2	4	no	—
333	C₄⋊C₄	M₁₆×C₂₃	1	8	yes	complemented
334	C₄²	—	2	4	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	4	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	1	no	—
339	C₄²	—	2	1	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	2	1	no	—
346	C₂³	—	2	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	C₂₃×#117	1	2	yes	—
349	C₂³	C₄×D₄×C₂₃	1	2	yes	—
350	C₂³	C₄×Q₈×C₂₃	1	2	yes	—
351	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
352	C₂×C₄	—	2	8	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	8	no	—
368	C₂×C₄	—	4	8	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	C₂₃×#117	1	2	yes	—
371	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
372	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
373	C₂×C₄	C₂₃×C₂²⋊Q₈	1	4	yes	—
374	C₂×C₄	C₂₃×#123	1	4	yes	—
375	C₂×C₄	C₂₃×#125	1	4	yes	—
376	C₂×C₄	C₂₃×#126	1	4	yes	—
377	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
378	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
379	C₂×C₄	C₂×M₁₆×C₂₃	1	4	yes	—
380	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
381	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
382	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
383	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
384	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
385	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
386	C₈	—	2	4	no	—
387	C₈	—	2	4	no	—
388	C₈	—	4	2	no	—
389	C₈	—	4	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	4	no	—
393	C₂²	—	2	4	no	—
394	C₂²	C₂₃×#387	1	4	yes	—
395	C₂²	C₂×C₂₃×#117	1	1	yes	characteristic
396	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
397	C₂²	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
398	C₂²	C₂²×M₁₆×C₂₃	1	2	yes	—
399	C₂²	C₂₃×#568	1	2	yes	—
400	C₄	—	2	4	no	—
401	C₄	—	2	4	no	—
402	C₄	—	2	4	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	C₂₃×#387	1	4	yes	—
407	C₄	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
408	C₄	C₂×C₂₃×#123	1	1	yes	characteristic
409	C₄	C₂×C₂₃×#125	1	1	yes	characteristic
410	C₄	C₂×C₂₃×#126	1	1	yes	characteristic
411	C₂	—	2	4	no	—
412	C₂	C₂×C₂₃×#387	1	1	yes	characteristic
413	C₂	C₂₃×#2920	1	1	yes	characteristic
414	C₂	C₂₃×#2921	1	1	yes	characteristic
415	C₂	D₄×M₁₆×C₂₃	1	2	yes	—
416	C₂	Q₈×M₁₆×C₂₃	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1917	C₂	semidirect
5. C₂₃×#1916	C₂	semidirect
124. C₄⋊C₄×C₂₃	M₁₆	semidirect
137. #21023	C₂₃	direct (N × H)
185. #1917	C₂×C₂₃	semidirect
186. #1916	C₂×C₂₃	semidirect
288. C₂₃	#21023	direct (N × H)
333. C₄⋊C₄	M₁₆×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×#423	2
C₂×C₄×M₁₆×C₂₃	2
C₂×C₄×C₄⋊C₄×C₂₃	2
C₂₃×#1917	2
C₂₃×#1916	2
C₂₃×#1728	2
#21023	23

Subgroups — C23×#21023 #19168649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21023 #19168649