Subgroups — C₂₃×#18060 #19165686

Order 5888.

668 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2984	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2974	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1592	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1583	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1582	C₂	1	2	yes	maximal, complemented
7	C₂₃×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1491	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1490	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₂₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#446	—	2	2	no	—
16	C₂₃×#446	C₂²	1	2	yes	complemented
17	C₈×Q₈×C₂₃	C₂²	1	2	yes	complemented
18	C₄×SD₁₆×C₂₃	C₄	1	1	yes	characteristic
19	C₄×SD₁₆×C₂₃	C₄	1	1	yes	characteristic
20	C₄×SD₁₆×C₂₃	C₄	1	1	yes	characteristic
21	C₄×SD₁₆×C₂₃	C₄	1	1	yes	characteristic
22	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
23	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
24	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
25	C₄×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#436	C₂²	1	2	yes	—
27	C₂₃×#435	—	2	2	no	—
28	C₈×D₄×C₂₃	C₂²	1	2	yes	—
29	C₂₃×#432	—	2	1	no	—
30	C₂₃×#432	C₂²	1	1	yes	characteristic
31	C₂₃×#432	C₂²	1	1	yes	characteristic
32	C₂₃×#431	—	2	1	no	—
33	C₂₃×#431	C₂²	1	1	yes	characteristic
34	C₂×C₂₃×#106	C₄	1	1	yes	characteristic
35	C₂×C₂₃×#103	C₄	1	1	yes	characteristic
36	C₂×C₂₃×#102	C₄	1	1	yes	characteristic
37	C₂₃×#407	—	2	1	no	—
38	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
39	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	C₂₃×#335	C₂²	1	2	yes	complemented
42	C₂₃×#334	C₂²	1	2	yes	complemented
43	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
44	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	complemented
45	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
46	C₂₃×D₄⋊C₈	C₂²	1	2	yes	—
47	C₂₃×#322	C₂²	1	2	yes	complemented
48	C₂₃×#322	C₂²	1	2	yes	complemented
49	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
51	C₂³×C₄×C₂₃	—	2	1	no	—
52	C₂×SD₁₆×C₂₃	—	2	4	no	—
53	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
54	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
55	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
56	C₂×SD₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
57	C₂×M₁₆×C₂₃	—	2	1	no	—
58	C₂×M₁₆×C₂₃	—	2	2	no	—
59	C₂²×C₈×C₂₃	—	2	2	no	—
60	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₄×Q₈×C₂₃	—	2	1	no	—
63	C₄×Q₈×C₂₃	—	2	1	no	—
64	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₂₃	—	2	1	no	—
69	C₄×D₄×C₂₃	—	2	1	no	—
70	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
71	C₄×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
72	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₄×D₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
78	C₂×C₄²×C₂₃	—	2	1	no	—
79	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
80	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
81	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
82	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
83	C₂₃×#106	C₂×C₄	1	1	yes	characteristic
84	C₂₃×C₄⋊C₈	—	2	2	no	—
85	C₂₃×C₄⋊C₈	—	2	2	no	—
86	C₂₃×C₄⋊C₈	D₄	1	2	yes	—
87	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
88	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
89	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
90	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
91	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
92	C₂₃×#103	C₂×C₄	1	1	yes	characteristic
93	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
94	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
95	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
96	C₂₃×#102	C₂×C₄	1	1	yes	characteristic
97	C₂₃×C₂²⋊C₈	—	2	1	no	—
98	C₂₃×C₂²⋊C₈	—	2	1	no	—
99	C₂₃×C₂²⋊C₈	—	2	2	no	—
100	C₂₃×C₂²⋊C₈	—	2	2	no	—
101	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
102	C₂₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
103	C₂₃×#97	—	2	2	no	—
104	C₂₃×#97	—	2	1	no	—
105	C₂₃×#97	—	2	1	no	—
106	C₂₃×#97	—	2	1	no	—
107	C₄×C₈×C₂₃	—	2	2	no	—
108	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
109	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
110	C₄×C₈×C₂₃	C₂³	1	2	yes	—
111	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
112	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₂₃	C₂³	1	2	yes	—
114	C₂⁴×C₂₃	—	2	1	no	—
115	C₂×Q₈×C₂₃	—	2	1	no	—
116	C₂×Q₈×C₂₃	—	2	2	no	—
117	C₂×Q₈×C₂₃	—	4	1	no	—
118	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×Q₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×D₄×C₂₃	—	2	2	no	—
121	C₂×D₄×C₂₃	—	2	1	no	—
122	C₂×D₄×C₂₃	—	4	1	no	—
123	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×D₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₂₃	—	2	1	no	—
126	C₂²×C₄×C₂₃	—	2	1	no	—
127	C₂²×C₄×C₂₃	—	2	1	no	—
128	C₂²×C₄×C₂₃	—	2	1	no	—
129	C₂²×C₄×C₂₃	—	2	1	no	—
130	C₂²×C₄×C₂₃	—	2	1	no	—
131	C₂²×C₄×C₂₃	—	2	1	no	—
132	C₂²×C₄×C₂₃	—	8	2	no	—
133	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
134	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
136	SD₁₆×C₂₃	—	2	2	no	—
137	SD₁₆×C₂₃	—	2	2	no	—
138	SD₁₆×C₂₃	—	2	2	no	—
139	SD₁₆×C₂₃	M₁₆	1	4	yes	complemented
140	M₁₆×C₂₃	—	4	2	no	—
141	M₁₆×C₂₃	—	8	2	no	—
142	C₂×C₈×C₂₃	—	2	2	no	—
143	C₂×C₈×C₂₃	—	2	2	no	—
144	C₂×C₈×C₂₃	—	2	2	no	—
145	C₂×C₈×C₂₃	—	2	2	no	—
146	C₂×C₈×C₂₃	—	4	1	no	—
147	C₂×C₈×C₂₃	—	8	2	no	—
148	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
149	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
153	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
154	C₄⋊C₄×C₂₃	—	2	1	no	—
155	C₄⋊C₄×C₂₃	—	2	1	no	—
156	C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₄⋊C₄×C₂₃	—	2	1	no	—
158	C₄⋊C₄×C₂₃	—	2	1	no	—
159	C₄⋊C₄×C₂₃	—	2	1	no	—
160	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
161	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
162	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
163	C₄⋊C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
164	C₂²⋊C₄×C₂₃	—	2	1	no	—
165	C₂²⋊C₄×C₂₃	—	2	1	no	—
166	C₂²⋊C₄×C₂₃	—	2	1	no	—
167	C₂²⋊C₄×C₂₃	—	2	1	no	—
168	C₄²×C₂₃	—	2	1	no	—
169	C₄²×C₂₃	—	2	1	no	—
170	C₄²×C₂₃	—	2	1	no	—
171	C₄²×C₂₃	—	2	1	no	—
172	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₂₃	#41	1	1	yes	characteristic
175	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
176	#18060	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂³×C₂₃	—	2	1	no	—
178	C₂³×C₂₃	—	2	1	no	—
179	C₂³×C₂₃	—	4	2	no	—
180	C₂³×C₂₃	—	4	2	no	—
181	C₂³×C₂₃	—	8	1	no	—
182	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
183	Q₈×C₂₃	—	2	1	no	—
184	Q₈×C₂₃	—	2	1	no	—
185	Q₈×C₂₃	—	2	1	no	—
186	Q₈×C₂₃	—	4	2	no	—
187	Q₈×C₂₃	C₂×M₁₆	1	2	yes	—
188	D₄×C₂₃	—	2	1	no	—
189	D₄×C₂₃	—	2	1	no	—
190	D₄×C₂₃	—	2	1	no	—
191	D₄×C₂₃	—	4	2	no	—
192	D₄×C₂₃	C₂×M₁₆	1	2	yes	—
193	C₂×C₄×C₂₃	—	2	1	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	1	no	—
196	C₂×C₄×C₂₃	—	2	1	no	—
197	C₂×C₄×C₂₃	—	2	2	no	—
198	C₂×C₄×C₂₃	—	2	1	no	—
199	C₂×C₄×C₂₃	—	2	1	no	—
200	C₂×C₄×C₂₃	—	2	1	no	—
201	C₂×C₄×C₂₃	—	2	1	no	—
202	C₂×C₄×C₂₃	—	2	1	no	—
203	C₂×C₄×C₂₃	—	4	2	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	4	2	no	—
206	C₂×C₄×C₂₃	—	4	2	no	—
207	C₂×C₄×C₂₃	—	8	1	no	—
208	C₂×C₄×C₂₃	—	8	1	no	—
209	C₂×C₄×C₂₃	—	8	1	no	—
210	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
211	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
212	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
213	C₂×C₄×C₂₃	C₂³×C₄	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	C₈×C₂₃	—	2	1	no	—
217	C₈×C₂₃	—	2	1	no	—
218	C₈×C₂₃	—	2	1	no	—
219	C₈×C₂₃	—	2	2	no	—
220	C₈×C₂₃	—	2	2	no	—
221	C₈×C₂₃	—	4	2	no	—
222	C₈×C₂₃	—	4	2	no	—
223	C₈×C₂₃	—	4	2	no	—
224	C₈×C₂₃	C₂×M₁₆	1	2	yes	—
225	#2984	C₂×C₂₃	1	1	yes	characteristic, complemented
226	#2974	C₂×C₂₃	1	1	yes	characteristic
227	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
228	#1592	C₂×C₂₃	1	2	yes	complemented
229	#1583	C₂×C₂₃	1	2	yes	complemented
230	#1582	C₂×C₂₃	1	2	yes	complemented
231	SD₁₆⋊C₈	C₂×C₂₃	1	2	yes	complemented
232	#1568	C₂×C₂₃	1	1	yes	characteristic, complemented
233	#1491	C₂×C₂₃	1	1	yes	characteristic, complemented
234	#1490	C₂×C₂₃	1	1	yes	characteristic
235	#1451	C₂×C₂₃	1	1	yes	characteristic, complemented
236	C₂²×C₂₃	—	2	1	no	—
237	C₂²×C₂₃	—	2	1	no	—
238	C₂²×C₂₃	—	2	2	no	—
239	C₂²×C₂₃	—	4	1	no	—
240	C₂²×C₂₃	—	4	1	no	—
241	C₂²×C₂₃	—	4	1	no	—
242	C₂²×C₂₃	—	4	1	no	—
243	C₂²×C₂₃	—	4	1	no	—
244	C₂²×C₂₃	—	8	4	no	—
245	C₂²×C₂₃	#444	1	2	yes	—
246	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
247	C₄×C₂₃	—	2	1	no	—
248	C₄×C₂₃	—	2	2	no	—
249	C₄×C₂₃	—	2	1	no	—
250	C₄×C₂₃	—	2	1	no	—
251	C₄×C₂₃	—	2	1	no	—
252	C₄×C₂₃	—	4	1	no	—
253	C₄×C₂₃	—	4	1	no	—
254	C₄×C₂₃	—	4	1	no	—
255	C₄×C₂₃	—	4	1	no	—
256	C₄×C₂₃	—	4	1	no	—
257	C₄×C₂₃	C₂²×M₁₆	1	1	yes	characteristic
258	C₄×C₂₃	#568	1	1	yes	characteristic
259	C₄×C₂₃	#577	1	2	yes	—
260	C₂²×SD₁₆	C₄×C₂₃	1	1	yes	characteristic
261	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
262	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
263	#446	—	2	2	no	—
264	#446	C₂²×C₂₃	1	2	yes	complemented
265	C₈×Q₈	C₂²×C₂₃	1	2	yes	complemented
266	C₄×SD₁₆	C₄×C₂₃	1	1	yes	characteristic
267	C₄×SD₁₆	C₄×C₂₃	1	1	yes	characteristic
268	C₄×SD₁₆	C₄×C₂₃	1	1	yes	characteristic
269	C₄×SD₁₆	C₄×C₂₃	1	1	yes	characteristic
270	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
271	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
272	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
273	C₄×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
274	#436	C₂²×C₂₃	1	2	yes	—
275	#435	—	2	2	no	—
276	C₈×D₄	C₂²×C₂₃	1	2	yes	—
277	#432	—	2	1	no	—
278	#432	C₂²×C₂₃	1	1	yes	characteristic
279	#432	C₂²×C₂₃	1	1	yes	characteristic
280	#431	—	2	1	no	—
281	#431	C₂²×C₂₃	1	1	yes	characteristic
282	C₂×#106	C₄×C₂₃	1	1	yes	characteristic
283	C₂×#103	C₄×C₂₃	1	1	yes	characteristic
284	C₂×#102	C₄×C₂₃	1	1	yes	characteristic
285	#407	—	2	1	no	—
286	C₂×C₂²⋊C₈	—	2	1	no	—
287	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
288	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
289	#335	C₂²×C₂₃	1	2	yes	complemented
290	#334	C₂²×C₂₃	1	2	yes	complemented
291	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	complemented
292	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	complemented
293	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
294	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
295	#322	C₂²×C₂₃	1	2	yes	complemented
296	#322	C₂²×C₂₃	1	2	yes	complemented
297	C₂×C₂₃	—	2	2	no	—
298	C₂×C₂₃	—	4	2	no	—
299	C₂×C₂₃	—	8	1	no	—
300	C₂×C₂₃	D₄×M₁₆	1	1	yes	characteristic
301	C₂×C₂₃	#2949	1	1	yes	characteristic
302	C₂×C₂₃	C₂×#444	1	1	yes	characteristic
303	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
304	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
305	C₂³×C₄	—	2	1	no	—
306	C₂×SD₁₆	—	2	4	no	—
307	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
308	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
309	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
310	C₂×SD₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
311	C₂×M₁₆	—	2	1	no	—
312	C₂×M₁₆	—	2	2	no	—
313	C₂²×C₈	—	2	2	no	—
314	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
315	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
316	C₄×Q₈	—	2	1	no	—
317	C₄×Q₈	—	2	1	no	—
318	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
319	C₄×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
320	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
321	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
322	C₄×D₄	—	2	1	no	—
323	C₄×D₄	—	2	1	no	—
324	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
325	C₄×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
326	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
327	C₄×D₄	C₂³×C₂₃	1	1	yes	characteristic
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
331	C₂×C₂²⋊C₄	—	2	1	no	—
332	C₂×C₄²	—	2	1	no	—
333	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
334	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
335	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
336	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
337	#106	C₂×C₄×C₂₃	1	1	yes	characteristic
338	C₄⋊C₈	—	2	2	no	—
339	C₄⋊C₈	—	2	2	no	—
340	C₄⋊C₈	D₄×C₂₃	1	2	yes	—
341	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
342	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
343	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
344	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
345	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
346	#103	C₂×C₄×C₂₃	1	1	yes	characteristic
347	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
348	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
349	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
350	#102	C₂×C₄×C₂₃	1	1	yes	characteristic
351	C₂²⋊C₈	—	2	1	no	—
352	C₂²⋊C₈	—	2	1	no	—
353	C₂²⋊C₈	—	2	2	no	—
354	C₂²⋊C₈	—	2	2	no	—
355	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
356	C₂²⋊C₈	D₄×C₂₃	1	1	yes	characteristic
357	#97	—	2	2	no	—
358	#97	—	2	1	no	—
359	#97	—	2	1	no	—
360	#97	—	2	1	no	—
361	C₄×C₈	—	2	2	no	—
362	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
363	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
364	C₄×C₈	C₂³×C₂₃	1	2	yes	—
365	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
366	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
367	C₄×C₈	C₂³×C₂₃	1	2	yes	—
368	C₂₃	#18060	1	1	yes	characteristic, direct product factor, Sylow
369	C₂⁴	—	2	1	no	—
370	C₂×Q₈	—	2	1	no	—
371	C₂×Q₈	—	2	2	no	—
372	C₂×Q₈	—	4	1	no	—
373	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
374	C₂×Q₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	4	1	no	—
378	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
379	C₂×D₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	1	no	—
387	C₂²×C₄	—	8	2	no	—
388	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
390	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
391	SD₁₆	—	2	2	no	—
392	SD₁₆	—	2	2	no	—
393	SD₁₆	—	2	2	no	—
394	SD₁₆	M₁₆×C₂₃	1	4	yes	complemented
395	M₁₆	—	4	2	no	—
396	M₁₆	—	8	2	no	—
397	C₂×C₈	—	2	2	no	—
398	C₂×C₈	—	2	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	4	1	no	—
402	C₂×C₈	—	8	2	no	—
403	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
404	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
405	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
406	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
407	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
408	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
409	C₄⋊C₄	—	2	1	no	—
410	C₄⋊C₄	—	2	1	no	—
411	C₄⋊C₄	—	2	1	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	1	no	—
414	C₄⋊C₄	—	2	1	no	—
415	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
416	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
417	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
418	C₄⋊C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
419	C₂²⋊C₄	—	2	1	no	—
420	C₂²⋊C₄	—	2	1	no	—
421	C₂²⋊C₄	—	2	1	no	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₄²	—	2	1	no	—
424	C₄²	—	2	1	no	—
425	C₄²	—	2	1	no	—
426	C₄²	—	2	1	no	—
427	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
428	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
429	C₄²	#41×C₂₃	1	1	yes	characteristic
430	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
431	C₂³	—	2	1	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	8	1	no	—
436	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
437	Q₈	—	2	1	no	—
438	Q₈	—	2	1	no	—
439	Q₈	—	2	1	no	—
440	Q₈	—	4	2	no	—
441	Q₈	C₂×M₁₆×C₂₃	1	2	yes	—
442	D₄	—	2	1	no	—
443	D₄	—	2	1	no	—
444	D₄	—	2	1	no	—
445	D₄	—	4	2	no	—
446	D₄	C₂×M₁₆×C₂₃	1	2	yes	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	8	1	no	—
462	C₂×C₄	—	8	1	no	—
463	C₂×C₄	—	8	1	no	—
464	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
465	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
466	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
467	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
468	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
469	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
470	C₈	—	2	1	no	—
471	C₈	—	2	1	no	—
472	C₈	—	2	1	no	—
473	C₈	—	2	2	no	—
474	C₈	—	2	2	no	—
475	C₈	—	4	2	no	—
476	C₈	—	4	2	no	—
477	C₈	—	4	2	no	—
478	C₈	C₂×M₁₆×C₂₃	1	2	yes	—
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	4	1	no	—
483	C₂²	—	4	1	no	—
484	C₂²	—	4	1	no	—
485	C₂²	—	4	1	no	—
486	C₂²	—	4	1	no	—
487	C₂²	—	8	4	no	—
488	C₂²	C₂₃×#444	1	2	yes	—
489	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
490	C₄	—	2	1	no	—
491	C₄	—	2	2	no	—
492	C₄	—	2	1	no	—
493	C₄	—	2	1	no	—
494	C₄	—	2	1	no	—
495	C₄	—	4	1	no	—
496	C₄	—	4	1	no	—
497	C₄	—	4	1	no	—
498	C₄	—	4	1	no	—
499	C₄	—	4	1	no	—
500	C₄	C₂²×M₁₆×C₂₃	1	1	yes	characteristic
501	C₄	C₂₃×#568	1	1	yes	characteristic
502	C₄	C₂₃×#577	1	2	yes	—
503	C₂	—	2	2	no	—
504	C₂	—	4	2	no	—
505	C₂	—	8	1	no	—
506	C₂	D₄×M₁₆×C₂₃	1	1	yes	characteristic
507	C₂	C₂₃×#2949	1	1	yes	characteristic
508	C₂	C₂×C₂₃×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₄²	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂×C₄×D₄
prime radical	C₂²×D₄×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂₃×#431 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2984	C₂	semidirect
4. C₂₃×#1592	C₂	semidirect
5. C₂₃×#1583	C₂	semidirect
6. C₂₃×#1582	C₂	semidirect
7. C₂₃×SD₁₆⋊C₈	C₂	semidirect
8. C₂₃×#1568	C₂	semidirect
9. C₂₃×#1491	C₂	semidirect
11. C₂₃×#1451	C₂	semidirect
16. C₂₃×#446	C₂²	semidirect
17. C₈×Q₈×C₂₃	C₂²	semidirect
41. C₂₃×#335	C₂²	semidirect
42. C₂₃×#334	C₂²	semidirect
43. C₂₃×Q₈⋊C₈	C₂²	semidirect
44. C₂₃×Q₈⋊C₈	C₂²	semidirect
47. C₂₃×#322	C₂²	semidirect
48. C₂₃×#322	C₂²	semidirect
139. SD₁₆×C₂₃	M₁₆	semidirect
176. #18060	C₂₃	direct (N × H)
225. #2984	C₂×C₂₃	semidirect
228. #1592	C₂×C₂₃	semidirect
229. #1583	C₂×C₂₃	semidirect
230. #1582	C₂×C₂₃	semidirect
231. SD₁₆⋊C₈	C₂×C₂₃	semidirect
232. #1568	C₂×C₂₃	semidirect
233. #1491	C₂×C₂₃	semidirect
235. #1451	C₂×C₂₃	semidirect
264. #446	C₂²×C₂₃	semidirect
265. C₈×Q₈	C₂²×C₂₃	semidirect
289. #335	C₂²×C₂₃	semidirect
290. #334	C₂²×C₂₃	semidirect
291. Q₈⋊C₈	C₂²×C₂₃	semidirect
292. Q₈⋊C₈	C₂²×C₂₃	semidirect
295. #322	C₂²×C₂₃	semidirect
296. #322	C₂²×C₂₃	semidirect
368. C₂₃	#18060	direct (N × H)
394. SD₁₆	M₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2984	2
C₂₃×#2974	2
C₂×C₄×SD₁₆×C₂₃	2
C₂₃×#1592	2
C₂₃×#1583	2
C₂₃×#1582	2
C₂₃×SD₁₆⋊C₈	2
C₂₃×#1568	2
C₂₃×#1491	2
C₂₃×#1490	2
C₂₃×#1451	2
#18060	23

Subgroups — C23×#18060 #19165686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18060 #19165686