Subgroups — C₂₃×#29194 #19176820

Order 5888.

734 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2864	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2068	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2067	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#2066	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#2065	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1958	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1958	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1865	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1864	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1863	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1862	C₂	1	1	yes	maximal, characteristic
13	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1835	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#391	C₂²	1	1	yes	characteristic
36	C₂₃×#390	C₂²	1	1	yes	characteristic
37	C₂₃×#389	C₂²	1	1	yes	characteristic
38	C₂₃×#389	C₂²	1	1	yes	characteristic
39	C₂₃×#389	C₂²	1	1	yes	characteristic
40	C₂₃×#389	C₂²	1	1	yes	characteristic
41	C₂₃×#388	—	2	1	no	—
42	C₂₃×#388	—	2	1	no	—
43	C₂₃×#387	—	2	1	no	—
44	C₂₃×#387	—	2	1	no	—
45	C₂₃×#387	—	2	1	no	—
46	C₂₃×#387	—	2	1	no	—
47	C₂₃×#382	C₂²	1	1	yes	characteristic
48	C₂₃×#382	C₂²	1	1	yes	characteristic
49	C₂₃×#382	C₂²	1	1	yes	characteristic
50	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
51	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
52	C₄³×C₂₃	C₂²	1	1	yes	characteristic
53	C₂₃×#340	C₂²	1	1	yes	characteristic
54	C₂₃×#340	C₂²	1	1	yes	characteristic
55	C₂₃×#340	C₂²	1	1	yes	characteristic
56	C₂₃×#340	C₂²	1	1	yes	characteristic
57	C₂₃×#336	C₂²	1	1	yes	characteristic
58	C₂₃×#336	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₂₃	—	2	1	no	—
62	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂₃×#124	—	2	2	no	—
67	C₂₃×#124	Q₈	1	4	yes	—
68	C₄×Q₈×C₂₃	—	4	1	no	—
69	C₄×Q₈×C₂₃	—	4	1	no	—
70	C₄×D₄×C₂₃	—	4	1	no	—
71	C₄×D₄×C₂₃	—	4	1	no	—
72	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
73	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
88	C₂×C₄²×C₂₃	—	2	1	no	—
89	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
90	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
91	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
92	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
93	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
96	C₂₃×#107	—	4	1	no	—
97	C₂₃×#107	—	4	1	no	—
98	C₂₃×#106	—	4	1	no	—
99	C₂₃×#106	—	4	1	no	—
100	C₂₃×C₄⋊C₈	—	2	2	no	—
101	C₂₃×C₄⋊C₈	—	2	2	no	—
102	C₂₃×#103	—	2	2	no	—
103	C₂₃×#103	—	2	2	no	—
104	C₂₃×#103	—	4	1	no	—
105	C₂₃×#103	—	4	1	no	—
106	C₂₃×#102	—	2	2	no	—
107	C₂₃×#102	—	2	2	no	—
108	C₂₃×#102	—	4	1	no	—
109	C₂₃×#102	—	4	1	no	—
110	C₄×C₈×C₂₃	—	2	2	no	—
111	C₄×C₈×C₂₃	—	2	2	no	—
112	C₂₃×#95	—	2	1	no	—
113	C₂₃×#95	—	2	1	no	—
114	C₂₃×#95	—	2	1	no	—
115	C₂₃×#95	—	2	1	no	—
116	C₂₃×#95	—	2	1	no	—
117	C₂₃×#95	—	2	1	no	—
118	C₂₃×#95	—	2	1	no	—
119	C₂₃×#95	—	2	1	no	—
120	C₂₃×#95	—	2	1	no	—
121	C₂₃×#95	—	2	1	no	—
122	C₂₃×#95	—	2	1	no	—
123	C₂₃×#95	—	2	1	no	—
124	C₂⁴×C₂₃	—	2	1	no	—
125	C₂×Q₈×C₂₃	—	2	1	no	—
126	C₂×Q₈×C₂₃	—	2	1	no	—
127	C₂×Q₈×C₂₃	—	2	1	no	—
128	C₂×Q₈×C₂₃	—	2	1	no	—
129	C₂×Q₈×C₂₃	—	2	1	no	—
130	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
131	C₂×D₄×C₂₃	—	2	1	no	—
132	C₂×D₄×C₂₃	—	2	1	no	—
133	C₂×D₄×C₂₃	—	2	1	no	—
134	C₂×D₄×C₂₃	—	2	1	no	—
135	C₂×D₄×C₂₃	—	2	1	no	—
136	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
137	C₂²×C₄×C₂₃	—	2	1	no	—
138	C₂²×C₄×C₂₃	—	2	1	no	—
139	C₂²×C₄×C₂₃	—	2	1	no	—
140	C₂²×C₄×C₂₃	—	2	1	no	—
141	C₂²×C₄×C₂₃	—	2	1	no	—
142	C₂²×C₄×C₂₃	—	2	1	no	—
143	C₂²×C₄×C₂₃	—	2	1	no	—
144	C₂²×C₄×C₂₃	—	4	2	no	—
145	C₂²×C₄×C₂₃	—	4	2	no	—
146	C₂²×C₄×C₂₃	—	4	2	no	—
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
152	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
154	C₂×C₈×C₂₃	—	2	1	no	—
155	C₂×C₈×C₂₃	—	2	1	no	—
156	C₂×C₈×C₂₃	—	2	1	no	—
157	C₂×C₈×C₂₃	—	2	1	no	—
158	C₂×C₈×C₂₃	—	2	1	no	—
159	C₂×C₈×C₂₃	—	2	1	no	—
160	C₂×C₈×C₂₃	—	2	1	no	—
161	C₂×C₈×C₂₃	—	2	1	no	—
162	C₂×C₈×C₂₃	—	2	1	no	—
163	C₂×C₈×C₂₃	—	2	1	no	—
164	C₂×C₈×C₂₃	#41	1	2	yes	—
165	C₂×C₈×C₂₃	#41	1	2	yes	—
166	C₄⋊C₄×C₂₃	—	2	1	no	—
167	C₄⋊C₄×C₂₃	—	2	1	no	—
168	C₄⋊C₄×C₂₃	—	2	1	no	—
169	C₄⋊C₄×C₂₃	—	2	1	no	—
170	C₄⋊C₄×C₂₃	—	2	1	no	—
171	C₄⋊C₄×C₂₃	—	2	1	no	—
172	C₄⋊C₄×C₂₃	—	2	1	no	—
173	C₄⋊C₄×C₂₃	—	2	1	no	—
174	C₄⋊C₄×C₂₃	—	2	1	no	—
175	C₄⋊C₄×C₂₃	—	2	1	no	—
176	C₄⋊C₄×C₂₃	—	2	1	no	—
177	C₄⋊C₄×C₂₃	—	2	1	no	—
178	C₄⋊C₄×C₂₃	—	4	2	no	—
179	C₄⋊C₄×C₂₃	—	4	2	no	—
180	C₄⋊C₄×C₂₃	—	4	2	no	—
181	C₄⋊C₄×C₂₃	—	4	2	no	—
182	C₄⋊C₄×C₂₃	—	4	2	no	—
183	C₂²⋊C₄×C₂₃	—	2	4	no	—
184	C₂²⋊C₄×C₂₃	—	2	4	no	—
185	C₂²⋊C₄×C₂₃	—	4	2	no	—
186	C₂²⋊C₄×C₂₃	—	4	2	no	—
187	C₂²⋊C₄×C₂₃	—	4	2	no	—
188	C₄²×C₂₃	—	2	2	no	—
189	C₄²×C₂₃	—	2	2	no	—
190	C₄²×C₂₃	—	2	1	no	—
191	C₄²×C₂₃	—	2	1	no	—
192	C₄²×C₂₃	—	2	1	no	—
193	C₄²×C₂₃	—	2	1	no	—
194	C₄²×C₂₃	—	4	1	no	—
195	C₄²×C₂₃	—	4	1	no	—
196	C₄²×C₂₃	—	4	2	no	—
197	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
198	C₄²×C₂₃	#41	1	2	yes	—
199	#29194	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂³×C₂₃	—	2	2	no	—
201	C₂³×C₂₃	—	4	1	no	—
202	C₂³×C₂₃	—	4	1	no	—
203	C₂³×C₂₃	—	4	1	no	—
204	C₂³×C₂₃	—	4	1	no	—
205	C₂³×C₂₃	—	4	1	no	—
206	C₂³×C₂₃	—	4	1	no	—
207	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₂₃	—	2	2	no	—
209	Q₈×C₂₃	—	2	2	no	—
210	Q₈×C₂₃	—	4	1	no	—
211	Q₈×C₂₃	—	4	1	no	—
212	D₄×C₂₃	—	2	2	no	—
213	D₄×C₂₃	—	2	2	no	—
214	D₄×C₂₃	—	4	1	no	—
215	D₄×C₂₃	—	4	1	no	—
216	C₂×C₄×C₂₃	—	2	1	no	—
217	C₂×C₄×C₂₃	—	2	1	no	—
218	C₂×C₄×C₂₃	—	2	1	no	—
219	C₂×C₄×C₂₃	—	2	1	no	—
220	C₂×C₄×C₂₃	—	2	1	no	—
221	C₂×C₄×C₂₃	—	2	1	no	—
222	C₂×C₄×C₂₃	—	2	1	no	—
223	C₂×C₄×C₂₃	—	2	1	no	—
224	C₂×C₄×C₂₃	—	2	2	no	—
225	C₂×C₄×C₂₃	—	2	1	no	—
226	C₂×C₄×C₂₃	—	2	1	no	—
227	C₂×C₄×C₂₃	—	2	1	no	—
228	C₂×C₄×C₂₃	—	2	1	no	—
229	C₂×C₄×C₂₃	—	2	1	no	—
230	C₂×C₄×C₂₃	—	2	1	no	—
231	C₂×C₄×C₂₃	—	4	1	no	—
232	C₂×C₄×C₂₃	—	4	1	no	—
233	C₂×C₄×C₂₃	—	4	1	no	—
234	C₂×C₄×C₂₃	—	4	1	no	—
235	C₂×C₄×C₂₃	—	4	1	no	—
236	C₂×C₄×C₂₃	—	4	1	no	—
237	C₂×C₄×C₂₃	—	4	1	no	—
238	C₂×C₄×C₂₃	—	4	1	no	—
239	C₂×C₄×C₂₃	—	4	1	no	—
240	C₂×C₄×C₂₃	—	4	1	no	—
241	C₂×C₄×C₂₃	—	4	1	no	—
242	C₂×C₄×C₂₃	—	4	1	no	—
243	C₂×C₄×C₂₃	—	4	1	no	—
244	C₂×C₄×C₂₃	—	4	1	no	—
245	C₂×C₄×C₂₃	—	4	1	no	—
246	C₂×C₄×C₂₃	—	4	1	no	—
247	C₂×C₄×C₂₃	—	4	1	no	—
248	C₂×C₄×C₂₃	—	4	1	no	—
249	C₂×C₄×C₂₃	—	4	1	no	—
250	C₂×C₄×C₂₃	—	4	1	no	—
251	C₂×C₄×C₂₃	—	8	4	no	—
252	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
253	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
254	C₂×C₄×C₂₃	#135	1	2	yes	—
255	C₂×C₄×C₂₃	#135	1	2	yes	—
256	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
257	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
258	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
259	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
260	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
261	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
262	C₈×C₂₃	—	2	2	no	—
263	C₈×C₂₃	—	2	2	no	—
264	C₈×C₂₃	—	4	1	no	—
265	C₈×C₂₃	—	4	1	no	—
266	#2864	C₂×C₂₃	1	1	yes	characteristic, complemented
267	C₄×#124	C₂×C₂₃	1	1	yes	characteristic
268	#2068	C₂×C₂₃	1	1	yes	characteristic, complemented
269	#2067	C₂×C₂₃	1	1	yes	characteristic
270	#2066	C₂×C₂₃	1	1	yes	characteristic, complemented
271	#2065	C₂×C₂₃	1	1	yes	characteristic
272	#1958	C₂×C₂₃	1	1	yes	characteristic
273	#1958	C₂×C₂₃	1	1	yes	characteristic
274	#1865	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#1864	C₂×C₂₃	1	1	yes	characteristic
276	#1863	C₂×C₂₃	1	1	yes	characteristic, complemented
277	#1862	C₂×C₂₃	1	1	yes	characteristic
278	#1846	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#1835	C₂×C₂₃	1	1	yes	characteristic, complemented
280	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
281	C₂²×C₂₃	—	4	2	no	—
282	C₂²×C₂₃	—	4	2	no	—
283	C₂²×C₂₃	—	4	2	no	—
284	C₂²×C₂₃	—	8	1	no	—
285	C₂²×C₂₃	—	8	1	no	—
286	C₂²×C₂₃	—	8	1	no	—
287	C₂²×C₂₃	—	8	1	no	—
288	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
289	C₂²×C₂₃	#532	1	1	yes	characteristic
290	C₂²×C₂₃	#535	1	1	yes	characteristic
291	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
292	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
293	C₂²×C₂₃	#575	1	1	yes	characteristic
294	C₂²×C₂₃	#575	1	1	yes	characteristic
295	C₄×C₂₃	—	2	2	no	—
296	C₄×C₂₃	—	2	2	no	—
297	C₄×C₂₃	—	2	2	no	—
298	C₄×C₂₃	—	2	2	no	—
299	C₄×C₂₃	—	4	2	no	—
300	C₄×C₂₃	—	4	1	no	—
301	C₄×C₂₃	—	4	1	no	—
302	C₄×C₂₃	—	8	1	no	—
303	C₄×C₂₃	—	8	1	no	—
304	C₄×C₂₃	—	8	1	no	—
305	C₄×C₂₃	—	8	1	no	—
306	C₄×C₂₃	—	8	1	no	—
307	C₄×C₂₃	—	8	1	no	—
308	C₄×C₂₃	#553	1	1	yes	characteristic
309	C₄×C₂₃	#553	1	1	yes	characteristic
310	C₄×C₂₃	#554	1	1	yes	characteristic
311	C₄×C₂₃	#554	1	1	yes	characteristic
312	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic
313	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
314	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
315	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
316	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
317	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
318	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
319	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
320	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
321	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
322	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
323	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
324	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
325	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
326	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
327	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
328	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
329	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
330	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
331	#391	C₂²×C₂₃	1	1	yes	characteristic
332	#390	C₂²×C₂₃	1	1	yes	characteristic
333	#389	C₂²×C₂₃	1	1	yes	characteristic
334	#389	C₂²×C₂₃	1	1	yes	characteristic
335	#389	C₂²×C₂₃	1	1	yes	characteristic
336	#389	C₂²×C₂₃	1	1	yes	characteristic
337	#388	—	2	1	no	—
338	#388	—	2	1	no	—
339	#387	—	2	1	no	—
340	#387	—	2	1	no	—
341	#387	—	2	1	no	—
342	#387	—	2	1	no	—
343	#382	C₂²×C₂₃	1	1	yes	characteristic
344	#382	C₂²×C₂₃	1	1	yes	characteristic
345	#382	C₂²×C₂₃	1	1	yes	characteristic
346	C₄×C₂²⋊C₄	—	2	1	no	—
347	C₄×C₂²⋊C₄	—	2	1	no	—
348	C₄³	C₂²×C₂₃	1	1	yes	characteristic
349	#340	C₂²×C₂₃	1	1	yes	characteristic
350	#340	C₂²×C₂₃	1	1	yes	characteristic
351	#340	C₂²×C₂₃	1	1	yes	characteristic
352	#340	C₂²×C₂₃	1	1	yes	characteristic
353	#336	C₂²×C₂₃	1	1	yes	characteristic
354	#336	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×C₂₃	—	8	2	no	—
356	C₂×C₂₃	#2566	1	1	yes	characteristic
357	C₂×C₂₃	#3076	1	1	yes	characteristic
358	C₂×C₂₃	#3076	1	1	yes	characteristic
359	C₂×C₂₃	#3168	1	1	yes	characteristic
360	C₂×C₂₃	#3183	1	1	yes	characteristic
361	C₂×C₂₃	#3211	1	1	yes	characteristic
362	C₂×C₂₃	#3212	1	1	yes	characteristic
363	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
364	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
365	C₂³×C₄	—	2	1	no	—
366	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
367	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
368	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
369	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
370	#124	—	2	2	no	—
371	#124	Q₈×C₂₃	1	4	yes	—
372	C₄×Q₈	—	4	1	no	—
373	C₄×Q₈	—	4	1	no	—
374	C₄×D₄	—	4	1	no	—
375	C₄×D₄	—	4	1	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	—	2	1	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
384	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
385	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
386	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₂²⋊C₄	—	2	1	no	—
390	C₂×C₂²⋊C₄	—	2	1	no	—
391	C₂×C₂²⋊C₄	—	2	1	no	—
392	C₂×C₄²	—	2	1	no	—
393	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
394	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
395	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
396	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
397	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
399	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
400	#107	—	4	1	no	—
401	#107	—	4	1	no	—
402	#106	—	4	1	no	—
403	#106	—	4	1	no	—
404	C₄⋊C₈	—	2	2	no	—
405	C₄⋊C₈	—	2	2	no	—
406	#103	—	2	2	no	—
407	#103	—	2	2	no	—
408	#103	—	4	1	no	—
409	#103	—	4	1	no	—
410	#102	—	2	2	no	—
411	#102	—	2	2	no	—
412	#102	—	4	1	no	—
413	#102	—	4	1	no	—
414	C₄×C₈	—	2	2	no	—
415	C₄×C₈	—	2	2	no	—
416	#95	—	2	1	no	—
417	#95	—	2	1	no	—
418	#95	—	2	1	no	—
419	#95	—	2	1	no	—
420	#95	—	2	1	no	—
421	#95	—	2	1	no	—
422	#95	—	2	1	no	—
423	#95	—	2	1	no	—
424	#95	—	2	1	no	—
425	#95	—	2	1	no	—
426	#95	—	2	1	no	—
427	#95	—	2	1	no	—
428	C₂₃	#29194	1	1	yes	characteristic, direct product factor, Sylow
429	C₂⁴	—	2	1	no	—
430	C₂×Q₈	—	2	1	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	4	2	no	—
450	C₂²×C₄	—	4	2	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
457	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
458	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
459	C₂×C₈	—	2	1	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	1	no	—
469	C₂×C₈	#41×C₂₃	1	2	yes	—
470	C₂×C₈	#41×C₂₃	1	2	yes	—
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	4	2	no	—
484	C₄⋊C₄	—	4	2	no	—
485	C₄⋊C₄	—	4	2	no	—
486	C₄⋊C₄	—	4	2	no	—
487	C₄⋊C₄	—	4	2	no	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	4	2	no	—
491	C₂²⋊C₄	—	4	2	no	—
492	C₂²⋊C₄	—	4	2	no	—
493	C₄²	—	2	2	no	—
494	C₄²	—	2	2	no	—
495	C₄²	—	2	1	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	2	1	no	—
499	C₄²	—	4	1	no	—
500	C₄²	—	4	1	no	—
501	C₄²	—	4	2	no	—
502	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
503	C₄²	#41×C₂₃	1	2	yes	—
504	C₂³	—	2	2	no	—
505	C₂³	—	4	1	no	—
506	C₂³	—	4	1	no	—
507	C₂³	—	4	1	no	—
508	C₂³	—	4	1	no	—
509	C₂³	—	4	1	no	—
510	C₂³	—	4	1	no	—
511	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
512	Q₈	—	2	2	no	—
513	Q₈	—	2	2	no	—
514	Q₈	—	4	1	no	—
515	Q₈	—	4	1	no	—
516	D₄	—	2	2	no	—
517	D₄	—	2	2	no	—
518	D₄	—	4	1	no	—
519	D₄	—	4	1	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	1	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	1	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	1	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	1	no	—
542	C₂×C₄	—	4	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	8	4	no	—
556	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
557	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
558	C₂×C₄	C₂₃×#135	1	2	yes	—
559	C₂×C₄	C₂₃×#135	1	2	yes	—
560	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
561	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
562	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
563	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
564	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
565	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
566	C₈	—	2	2	no	—
567	C₈	—	2	2	no	—
568	C₈	—	4	1	no	—
569	C₈	—	4	1	no	—
570	C₂²	—	4	2	no	—
571	C₂²	—	4	2	no	—
572	C₂²	—	4	2	no	—
573	C₂²	—	8	1	no	—
574	C₂²	—	8	1	no	—
575	C₂²	—	8	1	no	—
576	C₂²	—	8	1	no	—
577	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
578	C₂²	C₂₃×#532	1	1	yes	characteristic
579	C₂²	C₂₃×#535	1	1	yes	characteristic
580	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
581	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
582	C₂²	C₂₃×#575	1	1	yes	characteristic
583	C₂²	C₂₃×#575	1	1	yes	characteristic
584	C₄	—	2	2	no	—
585	C₄	—	2	2	no	—
586	C₄	—	2	2	no	—
587	C₄	—	2	2	no	—
588	C₄	—	4	1	no	—
589	C₄	—	4	1	no	—
590	C₄	—	4	2	no	—
591	C₄	—	8	1	no	—
592	C₄	—	8	1	no	—
593	C₄	—	8	1	no	—
594	C₄	—	8	1	no	—
595	C₄	—	8	1	no	—
596	C₄	—	8	1	no	—
597	C₄	C₂₃×#553	1	1	yes	characteristic
598	C₄	C₂₃×#553	1	1	yes	characteristic
599	C₄	C₂₃×#554	1	1	yes	characteristic
600	C₄	C₂₃×#554	1	1	yes	characteristic
601	C₂	—	8	2	no	—
602	C₂	C₂₃×#2566	1	1	yes	characteristic
603	C₂	C₂₃×#3076	1	1	yes	characteristic
604	C₂	C₂₃×#3076	1	1	yes	characteristic
605	C₂	C₂₃×#3168	1	1	yes	characteristic
606	C₂	C₂₃×#3183	1	1	yes	characteristic
607	C₂	C₂₃×#3211	1	1	yes	characteristic
608	C₂	C₂₃×#3212	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2864	C₂	semidirect
3. C₂₃×#2068	C₂	semidirect
5. C₂₃×#2066	C₂	semidirect
9. C₂₃×#1865	C₂	semidirect
11. C₂₃×#1863	C₂	semidirect
13. C₂₃×#1846	C₂	semidirect
14. C₂₃×#1835	C₂	semidirect
15. C₄×C₂₃×C₄⋊C₈	C₂	semidirect
199. #29194	C₂₃	direct (N × H)
266. #2864	C₂×C₂₃	semidirect
268. #2068	C₂×C₂₃	semidirect
270. #2066	C₂×C₂₃	semidirect
274. #1865	C₂×C₂₃	semidirect
276. #1863	C₂×C₂₃	semidirect
278. #1846	C₂×C₂₃	semidirect
279. #1835	C₂×C₂₃	semidirect
280. C₄×C₄⋊C₈	C₂×C₂₃	semidirect
428. C₂₃	#29194	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2864	2
C₄×C₂₃×#124	2
C₂₃×#2068	2
C₂₃×#2067	2
C₂₃×#2066	2
C₂₃×#2065	2
C₂₃×#1958	2
C₂₃×#1865	2
C₂₃×#1864	2
C₂₃×#1863	2
C₂₃×#1862	2
C₂₃×#1846	2
C₂₃×#1835	2
C₄×C₂₃×C₄⋊C₈	2
#29194	23

Subgroups — C23×#29194 #19176820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29194 #19176820