Subgroups — C₂₃×#22735 #19170361

Order 5888.

740 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2295	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1940	C₂	1	2	yes	maximal
5	C₂₃×#1939	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1939	C₂	1	2	yes	maximal
7	C₂₃×#1939	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1832	C₂	1	2	yes	maximal
9	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#1768	C₂	1	2	yes	maximal
11	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#446	—	2	4	no	—
14	C₈×Q₈×C₂₃	—	2	4	no	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
22	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#97	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×#391	C₄	1	1	yes	characteristic
28	C₂₃×#391	C₄	1	1	yes	characteristic
29	C₂₃×#389	C₄	1	1	yes	characteristic
30	C₂₃×#389	C₄	1	1	yes	characteristic
31	C₂₃×#387	C₄	1	2	yes	—
32	C₂₃×#387	C₄	1	2	yes	—
33	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
35	C₂₃×#376	C₂²	1	2	yes	—
36	C₄³×C₂₃	C₂²	1	1	yes	characteristic
37	C₂₃×#336	C₂²	1	2	yes	—
38	C₂₃×#336	C₂²	1	2	yes	—
39	C₂₃×#336	C₂²	1	2	yes	—
40	C₂₃×#336	C₂²	1	1	yes	characteristic
41	C₂₃×#336	C₂²	1	1	yes	characteristic
42	C₂₃×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
50	C₄×Q₈×C₂₃	D₄	1	4	yes	—
51	C₄×Q₈×C₂₃	Q₈	1	4	yes	—
52	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
58	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
65	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
67	C₂₃×C₄⋊C₈	—	2	2	no	—
68	C₂₃×C₄⋊C₈	—	2	4	no	—
69	C₂₃×C₄⋊C₈	—	2	4	no	—
70	C₂₃×C₄⋊C₈	—	2	2	no	—
71	C₂₃×C₄⋊C₈	—	2	2	no	—
72	C₂₃×C₄⋊C₈	—	2	4	no	—
73	C₂₃×C₄⋊C₈	—	2	2	no	—
74	C₂₃×C₄⋊C₈	—	2	4	no	—
75	C₂₃×#97	—	2	4	no	—
76	C₄×C₈×C₂₃	—	2	2	no	—
77	C₄×C₈×C₂₃	—	2	4	no	—
78	C₄×C₈×C₂₃	—	2	2	no	—
79	C₂₃×#95	C₂×C₄	1	2	yes	—
80	C₂₃×#95	C₂×C₄	1	2	yes	—
81	C₂₃×#95	C₂×C₄	1	2	yes	—
82	C₂₃×#95	C₂×C₄	1	2	yes	—
83	C₂₃×#95	C₂×C₄	1	2	yes	—
84	C₂₃×#95	C₂×C₄	1	2	yes	—
85	C₂₃×#95	C₂×C₄	1	2	yes	—
86	C₂×Q₈×C₂₃	—	2	2	no	—
87	C₂×Q₈×C₂₃	—	2	2	no	—
88	C₂×Q₈×C₂₃	C₄⋊C₄	1	4	yes	—
89	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
100	C₂×C₈×C₂₃	—	2	1	no	—
101	C₂×C₈×C₂₃	—	2	1	no	—
102	C₂×C₈×C₂₃	—	2	1	no	—
103	C₂×C₈×C₂₃	—	2	1	no	—
104	C₂×C₈×C₂₃	—	2	2	no	—
105	C₂×C₈×C₂₃	—	2	1	no	—
106	C₂×C₈×C₂₃	—	2	1	no	—
107	C₂×C₈×C₂₃	—	2	2	no	—
108	C₂×C₈×C₂₃	—	2	1	no	—
109	C₂×C₈×C₂₃	—	2	2	no	—
110	C₂×C₈×C₂₃	—	2	1	no	—
111	C₂×C₈×C₂₃	—	2	2	no	—
112	C₂×C₈×C₂₃	—	2	1	no	—
113	C₂×C₈×C₂₃	—	2	2	no	—
114	C₂×C₈×C₂₃	—	2	1	no	—
115	C₂×C₈×C₂₃	—	2	2	no	—
116	C₂×C₈×C₂₃	#41	1	2	yes	—
117	C₂×C₈×C₂₃	#41	1	2	yes	—
118	C₄⋊C₄×C₂₃	—	2	4	no	—
119	C₄⋊C₄×C₂₃	—	2	2	no	—
120	C₄⋊C₄×C₂₃	—	2	2	no	—
121	C₄⋊C₄×C₂₃	—	4	2	no	—
122	C₄⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
123	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
124	C₄²×C₂₃	—	2	2	no	—
125	C₄²×C₂₃	—	2	2	no	—
126	C₄²×C₂₃	—	2	2	no	—
127	C₄²×C₂₃	—	2	1	no	—
128	C₄²×C₂₃	—	2	1	no	—
129	C₄²×C₂₃	—	2	2	no	—
130	C₄²×C₂₃	—	2	2	no	—
131	C₄²×C₂₃	—	2	1	no	—
132	C₄²×C₂₃	—	2	1	no	—
133	C₄²×C₂₃	—	4	1	no	—
134	C₄²×C₂₃	C₂×D₄	1	2	yes	—
135	C₄²×C₂₃	C₂×D₄	1	4	yes	—
136	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
137	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
138	#22735	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
140	Q₈×C₂₃	—	2	8	no	—
141	C₂×C₄×C₂₃	—	2	1	no	—
142	C₂×C₄×C₂₃	—	2	1	no	—
143	C₂×C₄×C₂₃	—	2	1	no	—
144	C₂×C₄×C₂₃	—	2	2	no	—
145	C₂×C₄×C₂₃	—	2	2	no	—
146	C₂×C₄×C₂₃	—	2	2	no	—
147	C₂×C₄×C₂₃	—	2	1	no	—
148	C₂×C₄×C₂₃	—	2	2	no	—
149	C₂×C₄×C₂₃	—	2	2	no	—
150	C₂×C₄×C₂₃	—	2	1	no	—
151	C₂×C₄×C₂₃	—	2	2	no	—
152	C₂×C₄×C₂₃	—	2	1	no	—
153	C₂×C₄×C₂₃	—	2	1	no	—
154	C₂×C₄×C₂₃	—	2	1	no	—
155	C₂×C₄×C₂₃	—	2	1	no	—
156	C₂×C₄×C₂₃	—	2	1	no	—
157	C₂×C₄×C₂₃	—	2	1	no	—
158	C₂×C₄×C₂₃	—	2	2	no	—
159	C₂×C₄×C₂₃	—	2	2	no	—
160	C₂×C₄×C₂₃	—	2	2	no	—
161	C₂×C₄×C₂₃	—	2	1	no	—
162	C₂×C₄×C₂₃	—	2	1	no	—
163	C₂×C₄×C₂₃	—	2	2	no	—
164	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
165	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
166	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
167	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
168	C₂×C₄×C₂₃	#131	1	2	yes	—
169	C₂×C₄×C₂₃	#131	1	2	yes	—
170	C₂×C₄×C₂₃	#131	1	2	yes	—
171	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
173	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
176	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
177	C₈×C₂₃	—	2	2	no	—
178	C₈×C₂₃	—	2	2	no	—
179	C₈×C₂₃	—	4	2	no	—
180	C₈×C₂₃	—	4	1	no	—
181	C₈×C₂₃	—	4	1	no	—
182	C₈×C₂₃	—	4	2	no	—
183	C₂×#446	C₂×C₂₃	1	1	yes	characteristic
184	C₂×C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic
185	#2295	C₂×C₂₃	1	1	yes	characteristic
186	#1940	C₂×C₂₃	1	2	yes	—
187	#1939	C₂×C₂₃	1	2	yes	—
188	#1939	C₂×C₂₃	1	1	yes	characteristic
189	#1939	C₂×C₂₃	1	1	yes	characteristic
190	#1832	C₂×C₂₃	1	2	yes	—
191	#1831	C₂×C₂₃	1	1	yes	characteristic
192	#1768	C₂×C₂₃	1	2	yes	—
193	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
194	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
195	C₂²×C₂₃	C₄×#41	1	1	yes	characteristic
196	C₂²×C₂₃	#519	1	1	yes	characteristic
197	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
198	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
199	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
200	C₂²×C₂₃	#568	1	1	yes	characteristic
201	C₄×C₂₃	—	2	4	no	—
202	C₄×C₂₃	—	2	2	no	—
203	C₄×C₂₃	—	2	4	no	—
204	C₄×C₂₃	—	2	2	no	—
205	C₄×C₂₃	—	4	2	no	—
206	C₄×C₂₃	—	4	1	no	—
207	C₄×C₂₃	—	4	2	no	—
208	C₄×C₂₃	—	4	1	no	—
209	C₄×C₂₃	—	4	1	no	—
210	C₄×C₂₃	#424	1	4	yes	—
211	C₄×C₂₃	#548	1	1	yes	characteristic
212	C₄×C₂₃	#548	1	1	yes	characteristic
213	C₄×C₂₃	#557	1	1	yes	characteristic
214	C₄×C₂₃	#557	1	1	yes	characteristic
215	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
216	#446	—	2	4	no	—
217	C₈×Q₈	—	2	4	no	—
218	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
219	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
220	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
221	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
222	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
223	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
224	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
225	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
226	C₂×#97	C₂²×C₂₃	1	2	yes	—
227	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
228	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
229	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
230	#391	C₄×C₂₃	1	1	yes	characteristic
231	#391	C₄×C₂₃	1	1	yes	characteristic
232	#389	C₄×C₂₃	1	1	yes	characteristic
233	#389	C₄×C₂₃	1	1	yes	characteristic
234	#387	C₄×C₂₃	1	2	yes	—
235	#387	C₄×C₂₃	1	2	yes	—
236	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
237	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
238	#376	C₂²×C₂₃	1	2	yes	—
239	C₄³	C₂²×C₂₃	1	1	yes	characteristic
240	#336	C₂²×C₂₃	1	2	yes	—
241	#336	C₂²×C₂₃	1	2	yes	—
242	#336	C₂²×C₂₃	1	1	yes	characteristic
243	#336	C₂²×C₂₃	1	2	yes	—
244	#336	C₂²×C₂₃	1	2	yes	—
245	#336	C₂²×C₂₃	1	1	yes	characteristic
246	C₂×C₂₃	#2351	1	1	yes	characteristic
247	C₂×C₂₃	C₂×#424	1	1	yes	characteristic
248	C₂×C₂₃	#2906	1	1	yes	characteristic
249	C₂×C₂₃	#2966	1	1	yes	characteristic
250	C₂×C₂₃	#2969	1	1	yes	characteristic
251	C₂×C₂₃	#2985	1	1	yes	characteristic
252	C₂×C₂₃	#2986	1	1	yes	characteristic
253	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
254	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
255	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
256	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
258	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
260	C₄×Q₈	D₄×C₂₃	1	4	yes	—
261	C₄×Q₈	Q₈×C₂₃	1	4	yes	—
262	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
263	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
264	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
265	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
266	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
267	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
268	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
269	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
270	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
271	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
272	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
273	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
275	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
276	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
277	C₄⋊C₈	—	2	2	no	—
278	C₄⋊C₈	—	2	4	no	—
279	C₄⋊C₈	—	2	4	no	—
280	C₄⋊C₈	—	2	2	no	—
281	C₄⋊C₈	—	2	2	no	—
282	C₄⋊C₈	—	2	4	no	—
283	C₄⋊C₈	—	2	2	no	—
284	C₄⋊C₈	—	2	4	no	—
285	#97	—	2	4	no	—
286	C₄×C₈	—	2	2	no	—
287	C₄×C₈	—	2	4	no	—
288	C₄×C₈	—	2	2	no	—
289	#95	C₂×C₄×C₂₃	1	2	yes	—
290	#95	C₂×C₄×C₂₃	1	2	yes	—
291	#95	C₂×C₄×C₂₃	1	2	yes	—
292	#95	C₂×C₄×C₂₃	1	2	yes	—
293	#95	C₂×C₄×C₂₃	1	2	yes	—
294	#95	C₂×C₄×C₂₃	1	2	yes	—
295	#95	C₂×C₄×C₂₃	1	2	yes	—
296	C₂₃	#22735	1	1	yes	characteristic, direct product factor, Sylow
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	C₄⋊C₄×C₂₃	1	4	yes	—
300	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
301	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
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	1	yes	characteristic
305	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
307	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
308	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
310	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	1	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	#41×C₂₃	1	2	yes	—
328	C₂×C₈	#41×C₂₃	1	2	yes	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	4	2	no	—
333	C₄⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
334	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	2	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	1	no	—
343	C₄²	—	2	1	no	—
344	C₄²	—	4	1	no	—
345	C₄²	C₂×D₄×C₂₃	1	2	yes	—
346	C₄²	C₂×D₄×C₂₃	1	4	yes	—
347	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
348	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
349	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
350	Q₈	—	2	8	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
375	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
376	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
377	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
378	C₂×C₄	C₂₃×#131	1	2	yes	—
379	C₂×C₄	C₂₃×#131	1	2	yes	—
380	C₂×C₄	C₂₃×#131	1	2	yes	—
381	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
382	C₂×C₄	C₂²×Q₈×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	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
386	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
387	C₈	—	2	2	no	—
388	C₈	—	2	2	no	—
389	C₈	—	4	1	no	—
390	C₈	—	4	2	no	—
391	C₈	—	4	1	no	—
392	C₈	—	4	2	no	—
393	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
394	C₂²	C₄×#41×C₂₃	1	1	yes	characteristic
395	C₂²	C₂₃×#519	1	1	yes	characteristic
396	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
397	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
398	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
399	C₂²	C₂₃×#568	1	1	yes	characteristic
400	C₄	—	2	2	no	—
401	C₄	—	2	4	no	—
402	C₄	—	2	2	no	—
403	C₄	—	2	4	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	1	no	—
406	C₄	—	4	2	no	—
407	C₄	—	4	1	no	—
408	C₄	—	4	1	no	—
409	C₄	C₂₃×#424	1	4	yes	—
410	C₄	C₂₃×#548	1	1	yes	characteristic
411	C₄	C₂₃×#548	1	1	yes	characteristic
412	C₄	C₂₃×#557	1	1	yes	characteristic
413	C₄	C₂₃×#557	1	1	yes	characteristic
414	C₂	C₂₃×#2351	1	1	yes	characteristic
415	C₂	C₂×C₂₃×#424	1	1	yes	characteristic
416	C₂	C₂₃×#2906	1	1	yes	characteristic
417	C₂	C₂₃×#2966	1	1	yes	characteristic
418	C₂	C₂₃×#2969	1	1	yes	characteristic
419	C₂	C₂₃×#2985	1	1	yes	characteristic
420	C₂	C₂₃×#2986	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle, prime radical	C₂³×C₂₃	C₂³×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
138. #22735	C₂₃	direct (N × H)
296. C₂₃	#22735	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#446	2
C₂×C₈×Q₈×C₂₃	2
C₂₃×#2295	2
C₂₃×#1940	2
C₂₃×#1939	2
C₂₃×#1832	2
C₂₃×#1831	2
C₂₃×#1768	2
C₄×C₂₃×C₄⋊C₈	2
#22735	23

Subgroups — C23×#22735 #19170361

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22735 #19170361