Subgroups — C₂₃×#21058 #19168684

Order 5888.

1154 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#427	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#416	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#416	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2297	C₂	1	1	yes	maximal, characteristic
5	C₄×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1928	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1927	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1926	C₂	1	2	yes	maximal, complemented
9	C₂₃×#1925	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1735	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	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#427	—	2	4	no	—
20	C₂×C₂₃×#107	C₂²	1	2	yes	—
21	C₂×C₂₃×#106	C₂²	1	2	yes	—
22	C₂₃×#416	—	2	4	no	—
23	C₂₃×#416	C₄	1	8	yes	complemented
24	C₂×C₂₃×#103	C₂²	1	2	yes	—
25	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#102	C₂²	1	2	yes	—
28	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
30	C₂₃×#391	C₂²	1	2	yes	—
31	C₂₃×#390	C₂²	1	2	yes	—
32	C₂₃×#386	—	2	2	no	—
33	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
35	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
36	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
37	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
38	C₂₃×#376	—	2	1	no	—
39	C₄³×C₂₃	—	2	1	no	—
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂₃×#340	C₂²	1	2	yes	—
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	—	2	2	no	—
49	C₂×#41×C₂₃	C₂×C₄	1	4	yes	—
50	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×D₄×C₂₃	—	2	1	no	—
52	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₂₃	—	2	1	no	—
54	C₂³×C₄×C₂₃	D₄	1	2	yes	—
55	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₂₃	—	2	2	no	—
57	C₂²×C₈×C₂₃	—	2	2	no	—
58	C₂²×C₈×C₂₃	—	2	2	no	—
59	C₂²×C₈×C₂₃	—	2	2	no	—
60	C₂²×C₈×C₂₃	—	2	2	no	—
61	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
62	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
63	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂₃×#117	—	2	2	no	—
66	C₂₃×#117	—	2	2	no	—
67	C₂₃×#117	—	2	2	no	—
68	C₂₃×#117	—	2	2	no	—
69	C₂₃×#117	C₂×C₄	1	4	yes	—
70	C₂₃×#117	D₄	1	4	yes	—
71	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
75	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
78	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
79	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
80	C₂×C₄²×C₂₃	—	2	1	no	—
81	C₂×C₄²×C₂₃	—	2	1	no	—
82	C₂×C₄²×C₂₃	—	2	2	no	—
83	C₂×C₄²×C₂₃	—	2	2	no	—
84	C₂×C₄²×C₂₃	—	2	2	no	—
85	C₂×C₄²×C₂₃	—	2	2	no	—
86	C₂×C₄²×C₂₃	—	2	1	no	—
87	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
88	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
89	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
90	C₂₃×#107	—	2	4	no	—
91	C₂₃×#106	—	2	4	no	—
92	C₂₃×#103	—	2	4	no	—
93	C₂₃×#103	C₂×C₄	1	4	yes	complemented
94	C₂₃×#103	C₂×C₄	1	4	yes	complemented
95	C₂₃×#102	—	2	4	no	—
96	C₂₃×#102	C₂×C₄	1	4	yes	complemented
97	C₂₃×#102	C₂×C₄	1	4	yes	complemented
98	C₂₃×#95	—	2	2	no	—
99	C₂₃×#95	—	2	2	no	—
100	C₂₃×#95	—	2	2	no	—
101	C₂₃×#95	—	2	2	no	—
102	C₂⁴×C₂₃	—	2	1	no	—
103	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
104	#41×C₂₃	—	2	8	no	—
105	#41×C₂₃	—	2	8	no	—
106	#41×C₂₃	—	4	4	no	—
107	#41×C₂₃	—	4	4	no	—
108	C₂×Q₈×C₂₃	—	2	1	no	—
109	C₂×Q₈×C₂₃	—	2	1	no	—
110	C₂×Q₈×C₂₃	—	2	1	no	—
111	C₂×Q₈×C₂₃	—	2	1	no	—
112	C₂×Q₈×C₂₃	—	2	1	no	—
113	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
114	C₂×D₄×C₂₃	—	2	4	no	—
115	C₂×D₄×C₂₃	—	2	1	no	—
116	C₂×D₄×C₂₃	—	2	1	no	—
117	C₂×D₄×C₂₃	—	2	1	no	—
118	C₂×D₄×C₂₃	—	2	1	no	—
119	C₂×D₄×C₂₃	—	2	1	no	—
120	C₂×D₄×C₂₃	—	4	2	no	—
121	C₂×D₄×C₂₃	—	4	2	no	—
122	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
123	C₂²×C₄×C₂₃	—	2	4	no	—
124	C₂²×C₄×C₂₃	—	2	1	no	—
125	C₂²×C₄×C₂₃	—	2	1	no	—
126	C₂²×C₄×C₂₃	—	2	2	no	—
127	C₂²×C₄×C₂₃	—	2	1	no	—
128	C₂²×C₄×C₂₃	—	2	4	no	—
129	C₂²×C₄×C₂₃	—	2	4	no	—
130	C₂²×C₄×C₂₃	—	2	2	no	—
131	C₂²×C₄×C₂₃	—	4	2	no	—
132	C₂²×C₄×C₂₃	—	4	2	no	—
133	C₂²×C₄×C₂₃	—	4	2	no	—
134	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
135	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₂₃	#41	1	2	yes	—
140	C₂²×C₄×C₂₃	#41	1	2	yes	—
141	C₂²×C₄×C₂₃	#41	1	2	yes	—
142	C₂²×C₄×C₂₃	#41	1	2	yes	—
143	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
144	C₂×C₈×C₂₃	—	2	8	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	1	no	—
149	C₂×C₈×C₂₃	—	2	2	no	—
150	C₂×C₈×C₂₃	—	2	2	no	—
151	C₂×C₈×C₂₃	—	2	2	no	—
152	C₂×C₈×C₂₃	—	4	4	no	—
153	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
154	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
155	C₄⋊C₄×C₂₃	—	2	1	no	—
156	C₄⋊C₄×C₂₃	—	2	1	no	—
157	C₄⋊C₄×C₂₃	—	2	2	no	—
158	C₄⋊C₄×C₂₃	—	2	2	no	—
159	C₄⋊C₄×C₂₃	—	2	2	no	—
160	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
161	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
162	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
163	C₂²⋊C₄×C₂₃	—	2	4	no	—
164	C₂²⋊C₄×C₂₃	—	2	4	no	—
165	C₂²⋊C₄×C₂₃	—	4	4	no	—
166	C₂²⋊C₄×C₂₃	—	4	2	no	—
167	C₂²⋊C₄×C₂₃	—	4	4	no	—
168	C₄²×C₂₃	—	2	4	no	—
169	C₄²×C₂₃	—	2	4	no	—
170	C₄²×C₂₃	—	2	2	no	—
171	C₄²×C₂₃	—	2	2	no	—
172	C₄²×C₂₃	—	2	2	no	—
173	C₄²×C₂₃	—	2	2	no	—
174	C₄²×C₂₃	—	4	4	no	—
175	C₄²×C₂₃	—	4	2	no	—
176	C₄²×C₂₃	—	4	4	no	—
177	#21058	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂³×C₂₃	—	2	2	no	—
179	C₂³×C₂₃	—	2	2	no	—
180	C₂³×C₂₃	—	2	2	no	—
181	C₂³×C₂₃	—	4	1	no	—
182	C₂³×C₂₃	—	4	1	no	—
183	C₂³×C₂₃	—	4	2	no	—
184	C₂³×C₂₃	—	4	2	no	—
185	C₂³×C₂₃	C₄×D₄	1	2	yes	—
186	C₂³×C₂₃	#121	1	2	yes	—
187	C₂³×C₂₃	#123	1	2	yes	—
188	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
189	Q₈×C₂₃	—	2	2	no	—
190	Q₈×C₂₃	—	2	2	no	—
191	Q₈×C₂₃	—	4	1	no	—
192	Q₈×C₂₃	—	4	1	no	—
193	D₄×C₂₃	—	2	2	no	—
194	D₄×C₂₃	—	2	2	no	—
195	D₄×C₂₃	—	4	8	no	—
196	D₄×C₂₃	—	4	1	no	—
197	D₄×C₂₃	—	4	1	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	2	no	—
200	C₂×C₄×C₂₃	—	2	2	no	—
201	C₂×C₄×C₂₃	—	2	2	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	4	no	—
204	C₂×C₄×C₂₃	—	2	2	no	—
205	C₂×C₄×C₂₃	—	2	2	no	—
206	C₂×C₄×C₂₃	—	2	2	no	—
207	C₂×C₄×C₂₃	—	4	8	no	—
208	C₂×C₄×C₂₃	—	4	1	no	—
209	C₂×C₄×C₂₃	—	4	1	no	—
210	C₂×C₄×C₂₃	—	4	2	no	—
211	C₂×C₄×C₂₃	—	4	2	no	—
212	C₂×C₄×C₂₃	—	4	2	no	—
213	C₂×C₄×C₂₃	—	4	2	no	—
214	C₂×C₄×C₂₃	—	4	8	no	—
215	C₂×C₄×C₂₃	—	4	2	no	—
216	C₂×C₄×C₂₃	—	4	2	no	—
217	C₂×C₄×C₂₃	—	4	2	no	—
218	C₂×C₄×C₂₃	#117	1	4	yes	—
219	C₂×C₄×C₂₃	C₄×D₄	1	4	yes	—
220	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
221	C₂×C₄×C₂₃	#121	1	2	yes	—
222	C₂×C₄×C₂₃	#123	1	2	yes	—
223	C₂×C₄×C₂₃	#124	1	4	yes	—
224	C₂×C₄×C₂₃	#126	1	4	yes	—
225	C₂×C₄×C₂₃	#135	1	4	yes	—
226	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
227	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
229	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
230	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
231	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
232	C₈×C₂₃	—	2	2	no	—
233	C₈×C₂₃	—	2	2	no	—
234	C₈×C₂₃	—	4	2	no	—
235	C₂×#427	C₂×C₂₃	1	1	yes	characteristic, complemented
236	C₂×#416	C₂×C₂₃	1	1	yes	characteristic
237	C₂×#416	C₂×C₂₃	1	1	yes	characteristic
238	#2297	C₂×C₂₃	1	1	yes	characteristic
239	C₄×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
240	#1928	C₂×C₂₃	1	2	yes	complemented
241	#1927	C₂×C₂₃	1	2	yes	complemented
242	#1926	C₂×C₂₃	1	2	yes	complemented
243	#1925	C₂×C₂₃	1	2	yes	complemented
244	#1735	C₂×C₂₃	1	1	yes	characteristic, complemented
245	C₂×#336	C₂×C₂₃	1	1	yes	characteristic, complemented
246	C₂²×C₂₃	—	2	4	no	—
247	C₂²×C₂₃	—	2	2	no	—
248	C₂²×C₂₃	—	2	2	no	—
249	C₂²×C₂₃	—	2	4	no	—
250	C₂²×C₂₃	—	4	2	no	—
251	C₂²×C₂₃	—	4	2	no	—
252	C₂²×C₂₃	—	4	2	no	—
253	C₂²×C₂₃	—	8	2	no	—
254	C₂²×C₂₃	—	8	2	no	—
255	C₂²×C₂₃	#388	1	4	yes	—
256	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
257	C₂²×C₂₃	C₂×#121	1	1	yes	characteristic
258	C₂²×C₂₃	C₂×#123	1	1	yes	characteristic
259	C₂²×C₂₃	C₂×#135	1	2	yes	—
260	C₂²×C₂₃	#575	1	2	yes	—
261	C₄×C₂₃	—	2	4	no	—
262	C₄×C₂₃	—	4	2	no	—
263	C₄×C₂₃	—	4	2	no	—
264	C₄×C₂₃	—	4	2	no	—
265	C₄×C₂₃	—	4	4	no	—
266	C₄×C₂₃	—	4	2	no	—
267	C₄×C₂₃	—	8	2	no	—
268	C₄×C₂₃	#388	1	4	yes	—
269	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
270	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
271	C₄×C₂₃	C₂×#124	1	1	yes	characteristic
272	C₄×C₂₃	C₂×#126	1	1	yes	characteristic
273	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
274	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
275	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
276	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
277	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
278	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
279	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
280	#427	—	2	4	no	—
281	C₂×#107	C₂²×C₂₃	1	2	yes	—
282	C₂×#106	C₂²×C₂₃	1	2	yes	—
283	#416	—	2	4	no	—
284	#416	C₄×C₂₃	1	8	yes	complemented
285	C₂×#103	C₂²×C₂₃	1	2	yes	—
286	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
287	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
288	C₂×#102	C₂²×C₂₃	1	2	yes	—
289	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
290	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
291	#391	C₂²×C₂₃	1	2	yes	—
292	#390	C₂²×C₂₃	1	2	yes	—
293	#386	—	2	2	no	—
294	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
295	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
296	C₄×C₂²⋊C₄	—	2	2	no	—
297	C₄×C₂²⋊C₄	—	2	1	no	—
298	C₄×C₂²⋊C₄	—	2	1	no	—
299	#376	—	2	1	no	—
300	C₄³	—	2	1	no	—
301	#340	C₂²×C₂₃	1	2	yes	—
302	#340	C₂²×C₂₃	1	2	yes	—
303	#336	C₂²×C₂₃	1	2	yes	—
304	#336	C₂²×C₂₃	1	2	yes	—
305	C₂×C₂₃	—	2	4	no	—
306	C₂×C₂₃	—	8	2	no	—
307	C₂×C₂₃	C₂×#388	1	1	yes	characteristic
308	C₂×C₂₃	C₄×#135	1	1	yes	characteristic
309	C₂×C₂₃	#2943	1	1	yes	characteristic
310	C₂×C₂₃	#3035	1	2	yes	—
311	C₂×C₂₃	#3091	1	2	yes	—
312	C₂×#41	—	2	2	no	—
313	C₂×#41	—	2	2	no	—
314	C₂×#41	—	2	2	no	—
315	C₂×#41	—	2	2	no	—
316	C₂×#41	—	2	2	no	—
317	C₂×#41	C₂×C₄×C₂₃	1	4	yes	—
318	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
319	C₂²×D₄	—	2	1	no	—
320	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
321	C₂³×C₄	—	2	1	no	—
322	C₂³×C₄	D₄×C₂₃	1	2	yes	—
323	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
324	C₂²×C₈	—	2	2	no	—
325	C₂²×C₈	—	2	2	no	—
326	C₂²×C₈	—	2	2	no	—
327	C₂²×C₈	—	2	2	no	—
328	C₂²×C₈	—	2	2	no	—
329	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
330	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
331	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
332	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
333	#117	—	2	2	no	—
334	#117	—	2	2	no	—
335	#117	—	2	2	no	—
336	#117	—	2	2	no	—
337	#117	C₂×C₄×C₂₃	1	4	yes	—
338	#117	D₄×C₂₃	1	4	yes	—
339	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
340	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
342	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
343	C₂×C₂²⋊C₄	—	2	1	no	—
344	C₂×C₂²⋊C₄	—	2	1	no	—
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₂²⋊C₄	—	2	2	no	—
347	C₂×C₂²⋊C₄	—	2	2	no	—
348	C₂×C₄²	—	2	1	no	—
349	C₂×C₄²	—	2	1	no	—
350	C₂×C₄²	—	2	2	no	—
351	C₂×C₄²	—	2	2	no	—
352	C₂×C₄²	—	2	1	no	—
353	C₂×C₄²	—	2	2	no	—
354	C₂×C₄²	—	2	2	no	—
355	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
356	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
357	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
358	#107	—	2	4	no	—
359	#106	—	2	4	no	—
360	#103	—	2	4	no	—
361	#103	C₂×C₄×C₂₃	1	4	yes	complemented
362	#103	C₂×C₄×C₂₃	1	4	yes	complemented
363	#102	—	2	4	no	—
364	#102	C₂×C₄×C₂₃	1	4	yes	complemented
365	#102	C₂×C₄×C₂₃	1	4	yes	complemented
366	#95	—	2	2	no	—
367	#95	—	2	2	no	—
368	#95	—	2	2	no	—
369	#95	—	2	2	no	—
370	C₂₃	#21058	1	1	yes	characteristic, direct product factor, Sylow
371	C₂⁴	—	2	1	no	—
372	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
373	#41	—	2	8	no	—
374	#41	—	2	8	no	—
375	#41	—	4	4	no	—
376	#41	—	4	4	no	—
377	C₂×Q₈	—	2	1	no	—
378	C₂×Q₈	—	2	1	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×Q₈	—	2	1	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
383	C₂×D₄	—	2	4	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	4	2	no	—
390	C₂×D₄	—	4	2	no	—
391	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	4	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	4	2	no	—
401	C₂²×C₄	—	4	2	no	—
402	C₂²×C₄	—	4	2	no	—
403	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
404	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
405	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
406	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
408	C₂²×C₄	#41×C₂₃	1	2	yes	—
409	C₂²×C₄	#41×C₂₃	1	2	yes	—
410	C₂²×C₄	#41×C₂₃	1	2	yes	—
411	C₂²×C₄	#41×C₂₃	1	2	yes	—
412	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
413	C₂×C₈	—	2	8	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	4	4	no	—
422	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
423	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	2	no	—
428	C₄⋊C₄	—	2	2	no	—
429	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
430	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
431	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
432	C₂²⋊C₄	—	2	4	no	—
433	C₂²⋊C₄	—	2	4	no	—
434	C₂²⋊C₄	—	4	4	no	—
435	C₂²⋊C₄	—	4	2	no	—
436	C₂²⋊C₄	—	4	4	no	—
437	C₄²	—	2	4	no	—
438	C₄²	—	2	4	no	—
439	C₄²	—	2	2	no	—
440	C₄²	—	2	2	no	—
441	C₄²	—	2	2	no	—
442	C₄²	—	2	2	no	—
443	C₄²	—	4	4	no	—
444	C₄²	—	4	2	no	—
445	C₄²	—	4	4	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	4	2	no	—
451	C₂³	—	4	1	no	—
452	C₂³	—	4	1	no	—
453	C₂³	C₄×D₄×C₂₃	1	2	yes	—
454	C₂³	C₂₃×#121	1	2	yes	—
455	C₂³	C₂₃×#123	1	2	yes	—
456	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
457	Q₈	—	2	2	no	—
458	Q₈	—	2	2	no	—
459	Q₈	—	4	1	no	—
460	Q₈	—	4	1	no	—
461	D₄	—	2	2	no	—
462	D₄	—	2	2	no	—
463	D₄	—	4	8	no	—
464	D₄	—	4	1	no	—
465	D₄	—	4	1	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	4	8	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	8	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	C₂₃×#117	1	4	yes	—
487	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
488	C₂×C₄	C₄×D₄×C₂₃	1	4	yes	—
489	C₂×C₄	C₂₃×#121	1	2	yes	—
490	C₂×C₄	C₂₃×#123	1	2	yes	—
491	C₂×C₄	C₂₃×#124	1	4	yes	—
492	C₂×C₄	C₂₃×#126	1	4	yes	—
493	C₂×C₄	C₂₃×#135	1	4	yes	—
494	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
495	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
496	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
497	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
498	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
499	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
500	C₈	—	2	2	no	—
501	C₈	—	2	2	no	—
502	C₈	—	4	2	no	—
503	C₂²	—	2	2	no	—
504	C₂²	—	2	2	no	—
505	C₂²	—	2	4	no	—
506	C₂²	—	2	4	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	8	2	no	—
511	C₂²	—	8	2	no	—
512	C₂²	C₂₃×#388	1	4	yes	—
513	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
514	C₂²	C₂×C₂₃×#121	1	1	yes	characteristic
515	C₂²	C₂×C₂₃×#123	1	1	yes	characteristic
516	C₂²	C₂×C₂₃×#135	1	2	yes	—
517	C₂²	C₂₃×#575	1	2	yes	—
518	C₄	—	2	4	no	—
519	C₄	—	4	2	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	4	no	—
522	C₄	—	4	2	no	—
523	C₄	—	4	2	no	—
524	C₄	—	8	2	no	—
525	C₄	C₂₃×#388	1	4	yes	—
526	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
527	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
528	C₄	C₂×C₂₃×#124	1	1	yes	characteristic
529	C₄	C₂×C₂₃×#126	1	1	yes	characteristic
530	C₂	—	2	4	no	—
531	C₂	—	8	2	no	—
532	C₂	C₂×C₂₃×#388	1	1	yes	characteristic
533	C₂	C₄×C₂₃×#135	1	1	yes	characteristic
534	C₂	C₂₃×#2943	1	1	yes	characteristic
535	C₂	C₂₃×#3035	1	2	yes	—
536	C₂	C₂₃×#3091	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂²×D₄
prime radical	C₂²×#41×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#427	C₂	semidirect
5. C₄×C₂₃×#117	C₂	semidirect
6. C₂₃×#1928	C₂	semidirect
7. C₂₃×#1927	C₂	semidirect
8. C₂₃×#1926	C₂	semidirect
9. C₂₃×#1925	C₂	semidirect
10. C₂₃×#1735	C₂	semidirect
11. C₂×C₂₃×#336	C₂	semidirect
23. C₂₃×#416	C₄	semidirect
93. C₂₃×#103	C₂×C₄	semidirect
94. C₂₃×#103	C₂×C₄	semidirect
96. C₂₃×#102	C₂×C₄	semidirect
97. C₂₃×#102	C₂×C₄	semidirect
177. #21058	C₂₃	direct (N × H)
235. C₂×#427	C₂×C₂₃	semidirect
239. C₄×#117	C₂×C₂₃	semidirect
240. #1928	C₂×C₂₃	semidirect
241. #1927	C₂×C₂₃	semidirect
242. #1926	C₂×C₂₃	semidirect
243. #1925	C₂×C₂₃	semidirect
244. #1735	C₂×C₂₃	semidirect
245. C₂×#336	C₂×C₂₃	semidirect
284. #416	C₄×C₂₃	semidirect
361. #103	C₂×C₄×C₂₃	semidirect
362. #103	C₂×C₄×C₂₃	semidirect
364. #102	C₂×C₄×C₂₃	semidirect
365. #102	C₂×C₄×C₂₃	semidirect
370. C₂₃	#21058	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×#427	2
C₂×C₂₃×#416	2
C₂₃×#2297	2
C₄×C₂₃×#117	2
C₂₃×#1928	2
C₂₃×#1927	2
C₂₃×#1926	2
C₂₃×#1925	2
C₂₃×#1735	2
C₂×C₂₃×#336	2
#21058	23

Subgroups — C23×#21058 #19168684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21058 #19168684