Subgroups — C₂₃×#34211 #19181837

Order 5888.

3102 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#518	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2367	C₂	1	8	yes	maximal, complemented
5	C₂₃×#2342	C₂	1	8	yes	maximal, complemented
6	C₂₃×#2326	C₂	1	4	yes	maximal, complemented
7	C₂₃×#2310	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#2309	C₂	1	1	yes	maximal, characteristic
9	C₄×C₂₃×#124	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#124	C₄	1	16	yes	complemented
15	C₂×C₂₃×#124	C₂²	1	2	yes	—
16	C₂×C₂₃×#124	C₂²	1	2	yes	—
17	C₂×C₂₃×#124	C₂²	1	4	yes	—
18	C₂₃×#519	—	2	8	no	—
19	C₂₃×#518	—	2	8	no	—
20	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
21	C₂×C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
22	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
23	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
24	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#117	C₂²	1	4	yes	—
27	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
30	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
31	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#391	C₂²	1	4	yes	complemented
33	C₂₃×#391	C₂²	1	4	yes	complemented
34	C₂₃×#390	C₂²	1	4	yes	—
35	C₂₃×#390	C₂²	1	4	yes	—
36	C₂₃×#388	C₂²	1	16	yes	complemented
37	C₂₃×#388	C₂²	1	16	yes	complemented
38	C₂₃×#385	C₂²	1	16	yes	complemented
39	C₂₃×#384	C₂²	1	2	yes	complemented
40	C₂₃×#382	C₂²	1	2	yes	complemented
41	C₂₃×#381	C₂²	1	4	yes	—
42	C₂₃×#380	C₂²	1	4	yes	—
43	C₂₃×#379	C₂²	1	4	yes	—
44	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
45	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
46	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
47	C₄×C₂²⋊C₄×C₂₃	C₂²	1	16	yes	complemented
48	C₂₃×#376	C₂²	1	2	yes	complemented
49	C₄³×C₂₃	C₂²	1	2	yes	complemented
50	C₂⁵×C₂₃	C₂³	1	2	yes	—
51	C₂²×Q₈×C₂₃	—	2	4	no	—
52	C₂²×Q₈×C₂₃	—	2	4	no	—
53	C₂²×Q₈×C₂₃	C₂×C₄	1	8	yes	complemented
54	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
55	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
56	C₂²×D₄×C₂₃	—	2	4	no	—
57	C₂²×D₄×C₂₃	—	2	4	no	—
58	C₂²×D₄×C₂₃	C₂×C₄	1	8	yes	—
59	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
60	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
61	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
62	C₂³×C₄×C₂₃	C₂³	1	8	yes	—
63	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
64	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
65	C₂₃×#124	C₂×C₄	1	16	yes	complemented
66	C₂₃×#124	C₂×C₄	1	16	yes	complemented
67	C₂₃×#124	C₂×C₄	1	32	yes	complemented
68	C₄×Q₈×C₂₃	—	2	16	no	—
69	C₄×Q₈×C₂₃	—	2	16	no	—
70	C₄×D₄×C₂₃	—	2	16	no	—
71	C₄×D₄×C₂₃	—	2	16	no	—
72	C₂₃×#117	—	2	4	no	—
73	C₂₃×#117	—	2	4	no	—
74	C₂₃×#117	—	2	4	no	—
75	C₂₃×#117	—	2	16	no	—
76	C₂₃×#117	D₄	1	8	yes	—
77	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₂₃	—	2	16	no	—
84	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	32	yes	complemented
85	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
87	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
88	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
89	C₂×C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
90	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
93	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	complemented
94	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
95	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
96	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
97	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
98	C₂×C₄²×C₂₃	C₂³	1	8	yes	—
99	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
102	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
103	C₂₃×#95	C₂³	1	8	yes	—
104	C₂₃×#95	C₂³	1	8	yes	—
105	C₂₃×#95	C₂³	1	16	yes	—
106	C₂⁴×C₂₃	—	2	4	no	—
107	C₂⁴×C₂₃	—	2	4	no	—
108	C₂⁴×C₂₃	—	4	8	no	—
109	C₂⁴×C₂₃	C₂²×C₄	1	8	yes	—
110	C₂⁴×C₂₃	C₂⁴	1	4	yes	—
111	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
112	C₂×Q₈×C₂₃	—	2	32	no	—
113	C₂×Q₈×C₂₃	—	2	4	no	—
114	C₂×Q₈×C₂₃	—	2	4	no	—
115	C₂×Q₈×C₂₃	—	2	4	no	—
116	C₂×Q₈×C₂₃	—	2	4	no	—
117	C₂×Q₈×C₂₃	C₂²×C₄	1	8	yes	—
118	C₂×Q₈×C₂₃	C₂²×C₄	1	8	yes	—
119	C₂×D₄×C₂₃	—	2	32	no	—
120	C₂×D₄×C₂₃	—	2	4	no	—
121	C₂×D₄×C₂₃	—	2	4	no	—
122	C₂×D₄×C₂₃	—	2	4	no	—
123	C₂×D₄×C₂₃	—	2	4	no	—
124	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
125	C₂×D₄×C₂₃	C₂²×C₄	1	8	yes	—
126	C₂²×C₄×C₂₃	—	2	4	no	—
127	C₂²×C₄×C₂₃	—	2	4	no	—
128	C₂²×C₄×C₂₃	—	2	16	no	—
129	C₂²×C₄×C₂₃	—	4	4	no	—
130	C₂²×C₄×C₂₃	—	4	16	no	—
131	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
132	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
133	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
134	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
135	C₂²×C₄×C₂₃	#41	1	4	yes	—
136	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
137	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
138	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
139	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
140	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
144	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
145	C₄⋊C₄×C₂₃	—	2	4	no	—
146	C₄⋊C₄×C₂₃	—	2	16	no	—
147	C₄⋊C₄×C₂₃	—	2	4	no	—
148	C₄⋊C₄×C₂₃	—	2	4	no	—
149	C₄⋊C₄×C₂₃	—	4	8	no	—
150	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
151	C₂²⋊C₄×C₂₃	—	2	4	no	—
152	C₂²⋊C₄×C₂₃	—	2	4	no	—
153	C₂²⋊C₄×C₂₃	—	2	4	no	—
154	C₂²⋊C₄×C₂₃	—	2	16	no	—
155	C₂²⋊C₄×C₂₃	—	4	16	no	—
156	C₂²⋊C₄×C₂₃	C₂²×C₄	1	32	yes	—
157	C₂²⋊C₄×C₂₃	C₂²×C₄	1	32	yes	—
158	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
159	C₄²×C₂₃	—	2	4	no	—
160	C₄²×C₂₃	—	2	16	no	—
161	C₄²×C₂₃	—	2	4	no	—
162	C₄²×C₂₃	—	2	4	no	—
163	C₄²×C₂₃	—	4	8	no	—
164	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
165	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
166	C₄²×C₂₃	C₂²×C₄	1	8	yes	—
167	C₄²×C₂₃	C₂×D₄	1	8	yes	—
168	#34211	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂³×C₂₃	—	2	4	no	—
170	C₂³×C₂₃	—	2	4	no	—
171	C₂³×C₂₃	—	2	4	no	—
172	C₂³×C₂₃	—	2	16	no	—
173	C₂³×C₂₃	—	2	16	no	—
174	C₂³×C₂₃	—	4	16	no	—
175	C₂³×C₂₃	—	8	8	no	—
176	C₂³×C₂₃	—	8	8	no	—
177	C₂³×C₂₃	C₂³×C₄	1	8	yes	—
178	C₂³×C₂₃	C₂³×C₄	1	2	yes	—
179	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
180	C₂³×C₂₃	C₂×#41	1	2	yes	—
181	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
182	Q₈×C₂₃	—	2	16	no	—
183	Q₈×C₂₃	—	2	16	no	—
184	D₄×C₂₃	—	2	16	no	—
185	D₄×C₂₃	—	2	16	no	—
186	C₂×C₄×C₂₃	—	2	16	no	—
187	C₂×C₄×C₂₃	—	2	4	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	2	4	no	—
190	C₂×C₄×C₂₃	—	2	8	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	8	no	—
196	C₂×C₄×C₂₃	—	2	8	no	—
197	C₂×C₄×C₂₃	—	2	8	no	—
198	C₂×C₄×C₂₃	—	4	8	no	—
199	C₂×C₄×C₂₃	—	8	16	no	—
200	C₂×C₄×C₂₃	C₄×D₄	1	16	yes	—
201	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
202	C₂×C₄×C₂₃	C₂³×C₄	1	4	yes	—
203	C₂×C₄×C₂₃	C₂³×C₄	1	8	yes	—
204	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
205	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
206	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
207	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
208	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
209	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
210	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
211	C₂²×#124	C₂×C₂₃	1	1	yes	characteristic
212	C₂×#519	C₂×C₂₃	1	1	yes	characteristic, complemented
213	C₂×#518	C₂×C₂₃	1	1	yes	characteristic
214	#2367	C₂×C₂₃	1	8	yes	complemented
215	#2342	C₂×C₂₃	1	8	yes	complemented
216	#2326	C₂×C₂₃	1	4	yes	complemented
217	#2310	C₂×C₂₃	1	1	yes	characteristic, complemented
218	#2309	C₂×C₂₃	1	1	yes	characteristic
219	C₄×#124	C₂×C₂₃	1	4	yes	complemented
220	#2285	C₂×C₂₃	1	1	yes	characteristic, complemented
221	C₄×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
222	C₂²×C₂₃	—	2	4	no	—
223	C₂²×C₂₃	—	2	4	no	—
224	C₂²×C₂₃	—	2	4	no	—
225	C₂²×C₂₃	—	2	8	no	—
226	C₂²×C₂₃	—	2	16	no	—
227	C₂²×C₂₃	—	4	8	no	—
228	C₂²×C₂₃	—	4	8	no	—
229	C₂²×C₂₃	—	8	16	no	—
230	C₂²×C₂₃	C₂×C₄×D₄	1	4	yes	—
231	C₂²×C₂₃	C₂⁴×C₄	1	1	yes	characteristic
232	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
233	C₂²×C₂₃	C₂²×#41	1	1	yes	characteristic
234	C₂²×C₂₃	#585	1	4	yes	—
235	C₄×C₂₃	—	2	8	no	—
236	C₄×C₂₃	—	2	4	no	—
237	C₄×C₂₃	—	2	4	no	—
238	C₄×C₂₃	—	2	8	no	—
239	C₄×C₂₃	—	4	4	no	—
240	C₄×C₂₃	—	4	4	no	—
241	C₄×C₂₃	—	4	4	no	—
242	C₄×C₂₃	C₂×C₄×D₄	1	4	yes	—
243	C₄×C₂₃	C₂×C₄×D₄	1	4	yes	—
244	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
245	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
246	C₂×#124	C₄×C₂₃	1	16	yes	complemented
247	C₂×#124	C₂²×C₂₃	1	2	yes	—
248	C₂×#124	C₂²×C₂₃	1	4	yes	—
249	C₂×#124	C₂²×C₂₃	1	2	yes	—
250	#519	—	2	8	no	—
251	#518	—	2	8	no	—
252	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
253	C₂×C₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
254	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
255	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
256	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
257	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
258	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×#117	C₂²×C₂₃	1	4	yes	—
260	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
261	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
262	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
263	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
264	#391	C₂²×C₂₃	1	4	yes	complemented
265	#391	C₂²×C₂₃	1	4	yes	complemented
266	#390	C₂²×C₂₃	1	4	yes	—
267	#390	C₂²×C₂₃	1	4	yes	—
268	#388	C₂²×C₂₃	1	16	yes	complemented
269	#388	C₂²×C₂₃	1	16	yes	complemented
270	#385	C₂²×C₂₃	1	16	yes	complemented
271	#384	C₂²×C₂₃	1	2	yes	complemented
272	#382	C₂²×C₂₃	1	2	yes	complemented
273	#381	C₂²×C₂₃	1	4	yes	—
274	#380	C₂²×C₂₃	1	4	yes	—
275	#379	C₂²×C₂₃	1	4	yes	—
276	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
277	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
278	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
279	C₄×C₂²⋊C₄	C₂²×C₂₃	1	16	yes	complemented
280	#376	C₂²×C₂₃	1	2	yes	complemented
281	C₄³	C₂²×C₂₃	1	2	yes	complemented
282	C₂×C₂₃	—	2	4	no	—
283	C₂×C₂₃	—	4	8	no	—
284	C₂×C₂₃	C₂²×C₄×D₄	1	1	yes	characteristic
285	C₂×C₂₃	#3428	1	2	yes	—
286	C₂×C₂₃	#3500	1	2	yes	—
287	C₂×C₂₃	#3510	1	2	yes	—
288	C₂⁵	C₂³×C₂₃	1	2	yes	—
289	C₂²×Q₈	—	2	4	no	—
290	C₂²×Q₈	—	2	4	no	—
291	C₂²×Q₈	C₂×C₄×C₂₃	1	8	yes	complemented
292	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
293	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
294	C₂²×D₄	—	2	4	no	—
295	C₂²×D₄	—	2	4	no	—
296	C₂²×D₄	C₂×C₄×C₂₃	1	8	yes	—
297	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
298	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
299	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
300	C₂³×C₄	C₂³×C₂₃	1	8	yes	—
301	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
302	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
303	#124	C₂×C₄×C₂₃	1	16	yes	complemented
304	#124	C₂×C₄×C₂₃	1	32	yes	complemented
305	#124	C₂×C₄×C₂₃	1	16	yes	complemented
306	C₄×Q₈	—	2	16	no	—
307	C₄×Q₈	—	2	16	no	—
308	C₄×D₄	—	2	16	no	—
309	C₄×D₄	—	2	16	no	—
310	#117	—	2	4	no	—
311	#117	—	2	4	no	—
312	#117	—	2	4	no	—
313	#117	—	2	16	no	—
314	#117	D₄×C₂₃	1	8	yes	—
315	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
316	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
318	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
319	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
320	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
321	C₂×C₂²⋊C₄	—	2	16	no	—
322	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	32	yes	complemented
323	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
324	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
325	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
326	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
327	C₂×C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
328	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
329	C₂×C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
330	C₂×C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
331	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	complemented
332	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
333	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
334	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
335	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
336	C₂×C₄²	C₂³×C₂₃	1	8	yes	—
337	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
338	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
339	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
340	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
341	#95	C₂³×C₂₃	1	8	yes	—
342	#95	C₂³×C₂₃	1	8	yes	—
343	#95	C₂³×C₂₃	1	16	yes	—
344	C₂₃	#34211	1	1	yes	characteristic, direct product factor, Sylow
345	C₂⁴	—	2	4	no	—
346	C₂⁴	—	2	4	no	—
347	C₂⁴	—	4	8	no	—
348	C₂⁴	C₂²×C₄×C₂₃	1	8	yes	—
349	C₂⁴	C₂⁴×C₂₃	1	4	yes	—
350	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
351	C₂×Q₈	—	2	32	no	—
352	C₂×Q₈	—	2	4	no	—
353	C₂×Q₈	—	2	4	no	—
354	C₂×Q₈	—	2	4	no	—
355	C₂×Q₈	—	2	4	no	—
356	C₂×Q₈	C₂²×C₄×C₂₃	1	8	yes	—
357	C₂×Q₈	C₂²×C₄×C₂₃	1	8	yes	—
358	C₂×D₄	—	2	32	no	—
359	C₂×D₄	—	2	4	no	—
360	C₂×D₄	—	2	4	no	—
361	C₂×D₄	—	2	4	no	—
362	C₂×D₄	—	2	4	no	—
363	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
364	C₂×D₄	C₂²×C₄×C₂₃	1	8	yes	—
365	C₂²×C₄	—	2	4	no	—
366	C₂²×C₄	—	2	4	no	—
367	C₂²×C₄	—	2	16	no	—
368	C₂²×C₄	—	4	4	no	—
369	C₂²×C₄	—	4	16	no	—
370	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
371	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
372	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
373	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
374	C₂²×C₄	#41×C₂₃	1	4	yes	—
375	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
376	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
377	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
378	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
379	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
380	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
381	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
382	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
383	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	16	no	—
386	C₄⋊C₄	—	2	4	no	—
387	C₄⋊C₄	—	2	4	no	—
388	C₄⋊C₄	—	4	8	no	—
389	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
390	C₂²⋊C₄	—	2	4	no	—
391	C₂²⋊C₄	—	2	4	no	—
392	C₂²⋊C₄	—	2	4	no	—
393	C₂²⋊C₄	—	2	16	no	—
394	C₂²⋊C₄	—	4	16	no	—
395	C₂²⋊C₄	C₂²×C₄×C₂₃	1	32	yes	—
396	C₂²⋊C₄	C₂²×C₄×C₂₃	1	32	yes	—
397	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
398	C₄²	—	2	4	no	—
399	C₄²	—	2	16	no	—
400	C₄²	—	2	4	no	—
401	C₄²	—	2	4	no	—
402	C₄²	—	4	8	no	—
403	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
404	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
405	C₄²	C₂²×C₄×C₂₃	1	8	yes	—
406	C₄²	C₂×D₄×C₂₃	1	8	yes	—
407	C₂³	—	2	4	no	—
408	C₂³	—	2	4	no	—
409	C₂³	—	2	4	no	—
410	C₂³	—	2	16	no	—
411	C₂³	—	2	16	no	—
412	C₂³	—	4	16	no	—
413	C₂³	—	8	8	no	—
414	C₂³	—	8	8	no	—
415	C₂³	C₂³×C₄×C₂₃	1	8	yes	—
416	C₂³	C₂³×C₄×C₂₃	1	2	yes	—
417	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
418	C₂³	C₂×#41×C₂₃	1	2	yes	—
419	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
420	Q₈	—	2	16	no	—
421	Q₈	—	2	16	no	—
422	D₄	—	2	16	no	—
423	D₄	—	2	16	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	8	no	—
428	C₂×C₄	—	2	4	no	—
429	C₂×C₄	—	2	16	no	—
430	C₂×C₄	—	2	4	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	2	8	no	—
434	C₂×C₄	—	2	8	no	—
435	C₂×C₄	—	2	8	no	—
436	C₂×C₄	—	4	8	no	—
437	C₂×C₄	—	8	16	no	—
438	C₂×C₄	C₄×D₄×C₂₃	1	16	yes	—
439	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
440	C₂×C₄	C₂³×C₄×C₂₃	1	4	yes	—
441	C₂×C₄	C₂³×C₄×C₂₃	1	8	yes	—
442	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
443	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
444	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
445	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
446	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
447	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
448	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
449	C₂²	—	2	4	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	2	4	no	—
452	C₂²	—	2	8	no	—
453	C₂²	—	2	16	no	—
454	C₂²	—	4	8	no	—
455	C₂²	—	4	8	no	—
456	C₂²	—	8	16	no	—
457	C₂²	C₂×C₄×D₄×C₂₃	1	4	yes	—
458	C₂²	C₂⁴×C₄×C₂₃	1	1	yes	characteristic
459	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
460	C₂²	C₂²×#41×C₂₃	1	1	yes	characteristic
461	C₂²	C₂₃×#585	1	4	yes	—
462	C₄	—	2	8	no	—
463	C₄	—	2	4	no	—
464	C₄	—	2	4	no	—
465	C₄	—	2	8	no	—
466	C₄	—	4	4	no	—
467	C₄	—	4	4	no	—
468	C₄	—	4	4	no	—
469	C₄	C₂×C₄×D₄×C₂₃	1	4	yes	—
470	C₄	C₂×C₄×D₄×C₂₃	1	4	yes	—
471	C₂	—	2	4	no	—
472	C₂	—	4	8	no	—
473	C₂	C₂²×C₄×D₄×C₂₃	1	1	yes	characteristic
474	C₂	C₂₃×#3428	1	2	yes	—
475	C₂	C₂₃×#3500	1	2	yes	—
476	C₂	C₂₃×#3510	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₂₃
Frattini, cosocle	C₂³	C₂⁵×C₂₃
prime radical	C₂³×D₄×C₂₃	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₂₃×#519	C₂	semidirect
4. C₂₃×#2367	C₂	semidirect
5. C₂₃×#2342	C₂	semidirect
6. C₂₃×#2326	C₂	semidirect
7. C₂₃×#2310	C₂	semidirect
9. C₄×C₂₃×#124	C₂	semidirect
10. C₂₃×#2285	C₂	semidirect
11. C₄×C₂₃×#117	C₂	semidirect
14. C₂×C₂₃×#124	C₄	semidirect
20. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
21. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
32. C₂₃×#391	C₂²	semidirect
33. C₂₃×#391	C₂²	semidirect
36. C₂₃×#388	C₂²	semidirect
37. C₂₃×#388	C₂²	semidirect
38. C₂₃×#385	C₂²	semidirect
39. C₂₃×#384	C₂²	semidirect
40. C₂₃×#382	C₂²	semidirect
44. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
45. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
47. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
48. C₂₃×#376	C₂²	semidirect
49. C₄³×C₂₃	C₂²	semidirect
53. C₂²×Q₈×C₂₃	C₂×C₄	semidirect
65. C₂₃×#124	C₂×C₄	semidirect
66. C₂₃×#124	C₂×C₄	semidirect
67. C₂₃×#124	C₂×C₄	semidirect
84. C₂×C₂²⋊C₄×C₂₃	C₂×C₄	semidirect
93. C₂×C₄²×C₂₃	C₂×C₄	semidirect
168. #34211	C₂₃	direct (N × H)
212. C₂×#519	C₂×C₂₃	semidirect
214. #2367	C₂×C₂₃	semidirect
215. #2342	C₂×C₂₃	semidirect
216. #2326	C₂×C₂₃	semidirect
217. #2310	C₂×C₂₃	semidirect
219. C₄×#124	C₂×C₂₃	semidirect
220. #2285	C₂×C₂₃	semidirect
221. C₄×#117	C₂×C₂₃	semidirect
246. C₂×#124	C₄×C₂₃	semidirect
252. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
253. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
264. #391	C₂²×C₂₃	semidirect
265. #391	C₂²×C₂₃	semidirect
268. #388	C₂²×C₂₃	semidirect
269. #388	C₂²×C₂₃	semidirect
270. #385	C₂²×C₂₃	semidirect
271. #384	C₂²×C₂₃	semidirect
272. #382	C₂²×C₂₃	semidirect
276. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
277. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
279. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
280. #376	C₂²×C₂₃	semidirect
281. C₄³	C₂²×C₂₃	semidirect
291. C₂²×Q₈	C₂×C₄×C₂₃	semidirect
303. #124	C₂×C₄×C₂₃	semidirect
304. #124	C₂×C₄×C₂₃	semidirect
305. #124	C₂×C₄×C₂₃	semidirect
322. C₂×C₂²⋊C₄	C₂×C₄×C₂₃	semidirect
331. C₂×C₄²	C₂×C₄×C₂₃	semidirect
344. C₂₃	#34211	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#124	2
C₂×C₂₃×#519	2
C₂×C₂₃×#518	2
C₂₃×#2367	2
C₂₃×#2342	2
C₂₃×#2326	2
C₂₃×#2310	2
C₂₃×#2309	2
C₄×C₂₃×#124	2
C₂₃×#2285	2
C₄×C₂₃×#117	2
#34211	23

Subgroups — C23×#34211 #19181837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#34211 #19181837