Subgroups — C₂₃×#27030 #19174656

Order 5888.

812 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2496	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2361	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2071	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2067	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#2033	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#2033	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1942	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1923	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1918	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#1918	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#1862	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂₃×#1852	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂₃×#1823	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₂₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₂₃×#126	—	2	1	no	—
17	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₂₃	—	2	1	no	—
19	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
31	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#400	—	2	1	no	—
34	C₂₃×#397	—	2	1	no	—
35	C₂₃×#390	C₂²	1	1	yes	characteristic
36	C₂₃×#389	C₂²	1	1	yes	characteristic
37	C₂₃×#389	C₂²	1	1	yes	characteristic
38	C₂₃×#389	C₂²	1	1	yes	characteristic
39	C₂₃×#388	—	2	1	no	—
40	C₂₃×#387	—	2	1	no	—
41	C₂₃×#386	—	2	1	no	—
42	C₂₃×#385	—	2	1	no	—
43	C₂₃×#384	C₂²	1	1	yes	characteristic
44	C₂₃×#380	C₂²	1	1	yes	characteristic
45	C₂₃×#380	C₂²	1	1	yes	characteristic
46	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
47	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
48	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
49	C₂₃×#340	C₂²	1	1	yes	characteristic
50	C₂₃×#340	C₂²	1	1	yes	characteristic
51	C₂₃×#340	C₂²	1	1	yes	characteristic
52	C₂₃×#340	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×D₄×C₂₃	—	2	1	no	—
60	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₂₃	—	2	1	no	—
62	C₂³×C₄×C₂₃	—	2	1	no	—
63	C₂³×C₄×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	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂₃×#126	—	4	4	no	—
69	C₂₃×#121	—	2	2	no	—
70	C₂₃×#121	—	2	2	no	—
71	C₄×D₄×C₂₃	—	2	2	no	—
72	C₄×D₄×C₂₃	—	4	1	no	—
73	C₄×D₄×C₂₃	—	4	4	no	—
74	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
76	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
77	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
78	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
79	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
80	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
81	C₂×C₄⋊C₄×C₂₃	—	4	1	no	—
82	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
95	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₂₃	—	2	1	no	—
98	C₂×C₄²×C₂₃	—	2	1	no	—
99	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂₃×#107	—	2	2	no	—
102	C₂₃×#107	—	2	2	no	—
103	C₂₃×#107	—	2	2	no	—
104	C₂₃×#107	—	4	1	no	—
105	C₂₃×C₄⋊C₈	—	2	2	no	—
106	C₂₃×#102	—	2	2	no	—
107	C₂₃×#102	—	2	2	no	—
108	C₂₃×#102	—	2	2	no	—
109	C₂₃×#102	—	4	1	no	—
110	C₂₃×C₂²⋊C₈	—	2	2	no	—
111	C₂₃×C₂²⋊C₈	—	4	1	no	—
112	C₄×C₈×C₂₃	—	4	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	D₄	1	1	yes	characteristic
121	C₂₃×#95	D₄	1	1	yes	characteristic
122	C₂⁴×C₂₃	—	2	1	no	—
123	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
124	C₂×D₄×C₂₃	—	2	4	no	—
125	C₂×D₄×C₂₃	—	2	1	no	—
126	C₂×D₄×C₂₃	—	2	1	no	—
127	C₂×D₄×C₂₃	—	2	1	no	—
128	C₂×D₄×C₂₃	—	2	1	no	—
129	C₂×D₄×C₂₃	—	2	1	no	—
130	C₂×D₄×C₂₃	—	4	2	no	—
131	C₂×D₄×C₂₃	—	4	2	no	—
132	C₂×D₄×C₂₃	#41	1	2	yes	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	2	1	no	—
136	C₂²×C₄×C₂₃	—	2	1	no	—
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	—	2	1	no	—
145	C₂²×C₄×C₂₃	—	2	1	no	—
146	C₂²×C₄×C₂₃	—	4	2	no	—
147	C₂²×C₄×C₂₃	—	4	2	no	—
148	C₂²×C₄×C₂₃	—	4	2	no	—
149	C₂²×C₄×C₂₃	—	4	2	no	—
150	C₂²×C₄×C₂₃	—	8	1	no	—
151	C₂²×C₄×C₂₃	—	8	1	no	—
152	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
165	C₂×C₈×C₂₃	—	2	1	no	—
166	C₂×C₈×C₂₃	—	2	1	no	—
167	C₂×C₈×C₂₃	—	2	1	no	—
168	C₂×C₈×C₂₃	—	2	1	no	—
169	C₂×C₈×C₂₃	#41	1	2	yes	—
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	—	2	1	no	—
179	C₄⋊C₄×C₂₃	—	2	1	no	—
180	C₄⋊C₄×C₂₃	—	2	1	no	—
181	C₄⋊C₄×C₂₃	—	2	1	no	—
182	C₄⋊C₄×C₂₃	—	4	2	no	—
183	C₄⋊C₄×C₂₃	—	4	2	no	—
184	C₄⋊C₄×C₂₃	—	4	2	no	—
185	C₄⋊C₄×C₂₃	—	4	2	no	—
186	C₄⋊C₄×C₂₃	—	4	2	no	—
187	C₄⋊C₄×C₂₃	—	4	2	no	—
188	C₄⋊C₄×C₂₃	#41	1	2	yes	—
189	C₄⋊C₄×C₂₃	#41	1	2	yes	—
190	C₂²⋊C₄×C₂₃	—	2	4	no	—
191	C₂²⋊C₄×C₂₃	—	4	1	no	—
192	C₂²⋊C₄×C₂₃	—	4	2	no	—
193	C₂²⋊C₄×C₂₃	—	4	2	no	—
194	C₂²⋊C₄×C₂₃	—	4	2	no	—
195	C₂²⋊C₄×C₂₃	—	4	2	no	—
196	C₂²⋊C₄×C₂₃	—	8	1	no	—
197	C₂²⋊C₄×C₂₃	D₈	1	4	yes	—
198	C₄²×C₂₃	—	2	1	no	—
199	C₄²×C₂₃	—	2	1	no	—
200	C₄²×C₂₃	—	2	1	no	—
201	C₄²×C₂₃	—	2	1	no	—
202	C₄²×C₂₃	—	4	2	no	—
203	C₄²×C₂₃	—	4	2	no	—
204	#27030	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂³×C₂₃	—	2	1	no	—
206	C₂³×C₂₃	—	2	1	no	—
207	C₂³×C₂₃	—	2	1	no	—
208	C₂³×C₂₃	—	2	1	no	—
209	C₂³×C₂₃	—	2	1	no	—
210	C₂³×C₂₃	—	2	1	no	—
211	C₂³×C₂₃	—	2	2	no	—
212	C₂³×C₂₃	—	4	1	no	—
213	C₂³×C₂₃	—	4	1	no	—
214	C₂³×C₂₃	—	4	1	no	—
215	C₂³×C₂₃	—	4	1	no	—
216	C₂³×C₂₃	—	4	1	no	—
217	C₂³×C₂₃	—	4	1	no	—
218	C₂³×C₂₃	C₂×D₈	1	2	yes	—
219	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
220	D₄×C₂₃	—	2	2	no	—
221	D₄×C₂₃	—	2	2	no	—
222	D₄×C₂₃	—	4	8	no	—
223	D₄×C₂₃	—	4	1	no	—
224	D₄×C₂₃	—	4	1	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	2	no	—
231	C₂×C₄×C₂₃	—	2	1	no	—
232	C₂×C₄×C₂₃	—	2	2	no	—
233	C₂×C₄×C₂₃	—	2	1	no	—
234	C₂×C₄×C₂₃	—	2	1	no	—
235	C₂×C₄×C₂₃	—	2	1	no	—
236	C₂×C₄×C₂₃	—	2	1	no	—
237	C₂×C₄×C₂₃	—	2	1	no	—
238	C₂×C₄×C₂₃	—	2	1	no	—
239	C₂×C₄×C₂₃	—	2	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	—	4	1	no	—
252	C₂×C₄×C₂₃	—	4	1	no	—
253	C₂×C₄×C₂₃	—	4	1	no	—
254	C₂×C₄×C₂₃	—	4	1	no	—
255	C₂×C₄×C₂₃	—	4	1	no	—
256	C₂×C₄×C₂₃	—	4	1	no	—
257	C₂×C₄×C₂₃	—	4	1	no	—
258	C₂×C₄×C₂₃	—	4	1	no	—
259	C₂×C₄×C₂₃	—	4	1	no	—
260	C₂×C₄×C₂₃	—	4	1	no	—
261	C₂×C₄×C₂₃	—	4	1	no	—
262	C₂×C₄×C₂₃	—	4	1	no	—
263	C₂×C₄×C₂₃	—	8	4	no	—
264	C₂×C₄×C₂₃	—	8	4	no	—
265	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
266	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
267	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
268	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
269	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
270	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
271	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
272	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
273	C₈×C₂₃	—	2	2	no	—
274	C₈×C₂₃	—	4	1	no	—
275	C₈×C₂₃	—	4	1	no	—
276	C₈×C₂₃	—	4	1	no	—
277	#2496	C₂×C₂₃	1	1	yes	characteristic, complemented
278	#2361	C₂×C₂₃	1	1	yes	characteristic
279	#2071	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#2067	C₂×C₂₃	1	1	yes	characteristic, complemented
281	#2033	C₂×C₂₃	1	1	yes	characteristic
282	#2033	C₂×C₂₃	1	1	yes	characteristic
283	#1947	C₂×C₂₃	1	1	yes	characteristic, complemented
284	#1942	C₂×C₂₃	1	1	yes	characteristic, complemented
285	#1923	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#1918	C₂×C₂₃	1	1	yes	characteristic, complemented
287	#1918	C₂×C₂₃	1	1	yes	characteristic, complemented
288	#1862	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#1852	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#1823	C₂×C₂₃	1	1	yes	characteristic, complemented
291	C₄×#102	C₂×C₂₃	1	1	yes	characteristic, complemented
292	C₂²×C₂₃	—	2	2	no	—
293	C₂²×C₂₃	—	2	2	no	—
294	C₂²×C₂₃	—	2	2	no	—
295	C₂²×C₂₃	—	4	1	no	—
296	C₂²×C₂₃	—	4	1	no	—
297	C₂²×C₂₃	—	4	1	no	—
298	C₂²×C₂₃	—	4	1	no	—
299	C₂²×C₂₃	—	4	2	no	—
300	C₂²×C₂₃	—	4	2	no	—
301	C₂²×C₂₃	—	4	2	no	—
302	C₂²×C₂₃	—	8	1	no	—
303	C₂²×C₂₃	—	8	1	no	—
304	C₂²×C₂₃	—	8	1	no	—
305	C₂²×C₂₃	—	8	1	no	—
306	C₂²×C₂₃	#525	1	1	yes	characteristic
307	C₂²×C₂₃	#546	1	1	yes	characteristic
308	C₂²×C₂₃	#547	1	1	yes	characteristic
309	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
310	C₂²×C₂₃	#575	1	1	yes	characteristic
311	C₂²×C₂₃	#577	1	1	yes	characteristic
312	C₂²×C₂₃	#578	1	1	yes	characteristic
313	C₄×C₂₃	—	2	2	no	—
314	C₄×C₂₃	—	2	2	no	—
315	C₄×C₂₃	—	4	1	no	—
316	C₄×C₂₃	—	4	2	no	—
317	C₄×C₂₃	—	4	1	no	—
318	C₄×C₂₃	—	4	2	no	—
319	C₄×C₂₃	—	4	1	no	—
320	C₄×C₂₃	—	8	1	no	—
321	C₄×C₂₃	—	8	1	no	—
322	C₄×C₂₃	—	8	1	no	—
323	C₄×C₂₃	—	8	1	no	—
324	C₄×C₂₃	—	8	1	no	—
325	C₄×C₂₃	#529	1	1	yes	characteristic
326	C₄×C₂₃	#539	1	1	yes	characteristic
327	C₄×C₂₃	#553	1	1	yes	characteristic
328	C₄×C₂₃	#556	1	1	yes	characteristic
329	C₂×#126	—	2	1	no	—
330	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
331	C₂×C₄×D₄	—	2	1	no	—
332	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
333	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
334	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
335	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
336	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
337	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
339	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
341	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
342	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
345	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
346	#400	—	2	1	no	—
347	#397	—	2	1	no	—
348	#390	C₂²×C₂₃	1	1	yes	characteristic
349	#389	C₂²×C₂₃	1	1	yes	characteristic
350	#389	C₂²×C₂₃	1	1	yes	characteristic
351	#389	C₂²×C₂₃	1	1	yes	characteristic
352	#388	—	2	1	no	—
353	#387	—	2	1	no	—
354	#386	—	2	1	no	—
355	#385	—	2	1	no	—
356	#384	C₂²×C₂₃	1	1	yes	characteristic
357	#380	C₂²×C₂₃	1	1	yes	characteristic
358	#380	C₂²×C₂₃	1	1	yes	characteristic
359	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
360	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
361	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
362	#340	C₂²×C₂₃	1	1	yes	characteristic
363	#340	C₂²×C₂₃	1	1	yes	characteristic
364	#340	C₂²×C₂₃	1	1	yes	characteristic
365	#340	C₂²×C₂₃	1	1	yes	characteristic
366	#340	C₂²×C₂₃	1	1	yes	characteristic
367	#340	C₂²×C₂₃	1	1	yes	characteristic
368	#340	C₂²×C₂₃	1	1	yes	characteristic
369	#340	C₂²×C₂₃	1	1	yes	characteristic
370	#336	C₂²×C₂₃	1	1	yes	characteristic
371	#336	C₂²×C₂₃	1	1	yes	characteristic
372	C₂×C₂₃	—	4	2	no	—
373	C₂×C₂₃	—	8	2	no	—
374	C₂×C₂₃	#2466	1	1	yes	characteristic
375	C₂×C₂₃	#3100	1	1	yes	characteristic
376	C₂×C₂₃	#3120	1	1	yes	characteristic
377	C₂×C₂₃	#3294	1	1	yes	characteristic
378	C₂×C₂₃	#3313	1	1	yes	characteristic
379	C₂×C₂₃	#3334	1	1	yes	characteristic
380	C₂×C₂₃	#3351	1	1	yes	characteristic
381	C₂²×D₄	—	2	1	no	—
382	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
383	C₂³×C₄	—	2	1	no	—
384	C₂³×C₄	—	2	1	no	—
385	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
386	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
387	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
388	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
389	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
390	#126	—	4	4	no	—
391	#121	—	2	2	no	—
392	#121	—	2	2	no	—
393	C₄×D₄	—	2	2	no	—
394	C₄×D₄	—	4	1	no	—
395	C₄×D₄	—	4	4	no	—
396	C₂×C₄⋊C₄	—	2	2	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	—	2	1	no	—
399	C₂×C₄⋊C₄	—	2	1	no	—
400	C₂×C₄⋊C₄	—	2	1	no	—
401	C₂×C₄⋊C₄	—	2	1	no	—
402	C₂×C₄⋊C₄	—	2	1	no	—
403	C₂×C₄⋊C₄	—	4	1	no	—
404	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
405	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
406	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
410	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
411	C₂×C₂²⋊C₄	—	2	1	no	—
412	C₂×C₂²⋊C₄	—	2	1	no	—
413	C₂×C₂²⋊C₄	—	2	1	no	—
414	C₂×C₂²⋊C₄	—	2	1	no	—
415	C₂×C₂²⋊C₄	—	2	1	no	—
416	C₂×C₂²⋊C₄	—	2	1	no	—
417	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
418	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
419	C₂×C₄²	—	2	1	no	—
420	C₂×C₄²	—	2	1	no	—
421	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
423	#107	—	2	2	no	—
424	#107	—	2	2	no	—
425	#107	—	2	2	no	—
426	#107	—	4	1	no	—
427	C₄⋊C₈	—	2	2	no	—
428	#102	—	2	2	no	—
429	#102	—	2	2	no	—
430	#102	—	2	2	no	—
431	#102	—	4	1	no	—
432	C₂²⋊C₈	—	2	2	no	—
433	C₂²⋊C₈	—	4	1	no	—
434	C₄×C₈	—	4	1	no	—
435	#95	—	2	1	no	—
436	#95	—	2	1	no	—
437	#95	—	2	1	no	—
438	#95	—	2	1	no	—
439	#95	—	2	1	no	—
440	#95	—	2	1	no	—
441	#95	—	2	1	no	—
442	#95	D₄×C₂₃	1	1	yes	characteristic
443	#95	D₄×C₂₃	1	1	yes	characteristic
444	C₂₃	#27030	1	1	yes	characteristic, direct product factor, Sylow
445	C₂⁴	—	2	1	no	—
446	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	1	no	—
449	C₂×D₄	—	2	1	no	—
450	C₂×D₄	—	2	1	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	4	2	no	—
454	C₂×D₄	—	4	2	no	—
455	C₂×D₄	#41×C₂₃	1	2	yes	—
456	C₂²×C₄	—	2	1	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	—	4	2	no	—
472	C₂²×C₄	—	4	2	no	—
473	C₂²×C₄	—	8	1	no	—
474	C₂²×C₄	—	8	1	no	—
475	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
480	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	#41×C₂₃	1	2	yes	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	4	2	no	—
506	C₄⋊C₄	—	4	2	no	—
507	C₄⋊C₄	—	4	2	no	—
508	C₄⋊C₄	—	4	2	no	—
509	C₄⋊C₄	—	4	2	no	—
510	C₄⋊C₄	—	4	2	no	—
511	C₄⋊C₄	#41×C₂₃	1	2	yes	—
512	C₄⋊C₄	#41×C₂₃	1	2	yes	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₂²⋊C₄	—	4	1	no	—
515	C₂²⋊C₄	—	4	2	no	—
516	C₂²⋊C₄	—	4	2	no	—
517	C₂²⋊C₄	—	4	2	no	—
518	C₂²⋊C₄	—	4	2	no	—
519	C₂²⋊C₄	—	8	1	no	—
520	C₂²⋊C₄	D₈×C₂₃	1	4	yes	—
521	C₄²	—	2	1	no	—
522	C₄²	—	2	1	no	—
523	C₄²	—	2	1	no	—
524	C₄²	—	2	1	no	—
525	C₄²	—	4	2	no	—
526	C₄²	—	4	2	no	—
527	C₂³	—	2	1	no	—
528	C₂³	—	2	1	no	—
529	C₂³	—	2	1	no	—
530	C₂³	—	2	1	no	—
531	C₂³	—	2	1	no	—
532	C₂³	—	2	1	no	—
533	C₂³	—	2	2	no	—
534	C₂³	—	4	1	no	—
535	C₂³	—	4	1	no	—
536	C₂³	—	4	1	no	—
537	C₂³	—	4	1	no	—
538	C₂³	—	4	1	no	—
539	C₂³	—	4	1	no	—
540	C₂³	C₂×D₈×C₂₃	1	2	yes	—
541	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
542	D₄	—	2	2	no	—
543	D₄	—	2	2	no	—
544	D₄	—	4	8	no	—
545	D₄	—	4	1	no	—
546	D₄	—	4	1	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	2	1	no	—
553	C₂×C₄	—	2	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	8	4	no	—
586	C₂×C₄	—	8	4	no	—
587	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
588	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
589	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
590	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
591	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
592	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
593	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
594	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
595	C₈	—	2	2	no	—
596	C₈	—	4	1	no	—
597	C₈	—	4	1	no	—
598	C₈	—	4	1	no	—
599	C₂²	—	2	2	no	—
600	C₂²	—	2	2	no	—
601	C₂²	—	2	2	no	—
602	C₂²	—	4	1	no	—
603	C₂²	—	4	1	no	—
604	C₂²	—	4	1	no	—
605	C₂²	—	4	1	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	8	1	no	—
610	C₂²	—	8	1	no	—
611	C₂²	—	8	1	no	—
612	C₂²	—	8	1	no	—
613	C₂²	C₂₃×#525	1	1	yes	characteristic
614	C₂²	C₂₃×#546	1	1	yes	characteristic
615	C₂²	C₂₃×#547	1	1	yes	characteristic
616	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
617	C₂²	C₂₃×#575	1	1	yes	characteristic
618	C₂²	C₂₃×#577	1	1	yes	characteristic
619	C₂²	C₂₃×#578	1	1	yes	characteristic
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	4	1	no	—
623	C₄	—	4	2	no	—
624	C₄	—	4	1	no	—
625	C₄	—	4	1	no	—
626	C₄	—	4	2	no	—
627	C₄	—	8	1	no	—
628	C₄	—	8	1	no	—
629	C₄	—	8	1	no	—
630	C₄	—	8	1	no	—
631	C₄	—	8	1	no	—
632	C₄	C₂₃×#529	1	1	yes	characteristic
633	C₄	C₂₃×#539	1	1	yes	characteristic
634	C₄	C₂₃×#553	1	1	yes	characteristic
635	C₄	C₂₃×#556	1	1	yes	characteristic
636	C₂	—	4	2	no	—
637	C₂	—	8	2	no	—
638	C₂	C₂₃×#2466	1	1	yes	characteristic
639	C₂	C₂₃×#3100	1	1	yes	characteristic
640	C₂	C₂₃×#3120	1	1	yes	characteristic
641	C₂	C₂₃×#3294	1	1	yes	characteristic
642	C₂	C₂₃×#3313	1	1	yes	characteristic
643	C₂	C₂₃×#3334	1	1	yes	characteristic
644	C₂	C₂₃×#3351	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₂²⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2496	C₂	semidirect
3. C₂₃×#2071	C₂	semidirect
4. C₂₃×#2067	C₂	semidirect
7. C₂₃×#1947	C₂	semidirect
8. C₂₃×#1942	C₂	semidirect
9. C₂₃×#1923	C₂	semidirect
10. C₂₃×#1918	C₂	semidirect
11. C₂₃×#1918	C₂	semidirect
12. C₂₃×#1862	C₂	semidirect
13. C₂₃×#1852	C₂	semidirect
14. C₂₃×#1823	C₂	semidirect
15. C₄×C₂₃×#102	C₂	semidirect
204. #27030	C₂₃	direct (N × H)
277. #2496	C₂×C₂₃	semidirect
279. #2071	C₂×C₂₃	semidirect
280. #2067	C₂×C₂₃	semidirect
283. #1947	C₂×C₂₃	semidirect
284. #1942	C₂×C₂₃	semidirect
285. #1923	C₂×C₂₃	semidirect
286. #1918	C₂×C₂₃	semidirect
287. #1918	C₂×C₂₃	semidirect
288. #1862	C₂×C₂₃	semidirect
289. #1852	C₂×C₂₃	semidirect
290. #1823	C₂×C₂₃	semidirect
291. C₄×#102	C₂×C₂₃	semidirect
444. C₂₃	#27030	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×#2496	2
C₂₃×#2361	2
C₂₃×#2071	2
C₂₃×#2067	2
C₂₃×#2033	2
C₂₃×#1947	2
C₂₃×#1942	2
C₂₃×#1923	2
C₂₃×#1918	2
C₂₃×#1862	2
C₂₃×#1852	2
C₂₃×#1823	2
C₄×C₂₃×#102	2
#27030	23

Subgroups — C23×#27030 #19174656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#27030 #19174656