Subgroups — C₂₃×#22574 #19170200

Order 5888.

1190 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#102	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2345	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1934	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1926	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1864	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1824	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1786	C₂	1	1	yes	maximal, characteristic
10	C₄×C₂₃×#102	C₂	1	2	yes	maximal, complemented
11	C₈×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊C₄×C₂₃	—	2	1	no	—
17	C₂×C₂₃×#106	C₂²	1	2	yes	complemented
18	C₂×C₂₃×#102	C₄	1	8	yes	complemented
19	C₂×C₂₃×#102	C₂²	1	2	yes	—
20	C₂×C₂₃×#102	C₂²	1	2	yes	—
21	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
23	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
25	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
27	C₂₃×#390	C₂²	1	2	yes	—
28	C₂₃×#389	C₂²	1	2	yes	complemented
29	C₂₃×#387	—	2	2	no	—
30	C₂₃×#386	—	2	2	no	—
31	C₂₃×#380	—	2	1	no	—
32	C₂₃×#380	C₂²	1	1	yes	characteristic
33	C₂₃×#380	C₂²	1	1	yes	characteristic
34	C₂₃×#379	—	2	1	no	—
35	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
36	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
37	C₄×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#340	C₂²	1	2	yes	complemented
39	C₂₃×#340	C₂²	1	2	yes	complemented
40	C₂₃×#340	C₂²	1	2	yes	complemented
41	C₂₃×#336	C₂²	1	2	yes	complemented
42	C₂⁵×C₂₃	—	2	1	no	—
43	C₂²×D₄×C₂₃	—	2	4	no	—
44	C₂²×D₄×C₂₃	—	2	1	no	—
45	C₂²×D₄×C₂₃	—	2	2	no	—
46	C₂²×D₄×C₂₃	—	4	2	no	—
47	C₂²×D₄×C₂₃	C₂×C₄	1	4	yes	—
48	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₂₃	—	2	1	no	—
51	C₂³×C₄×C₂₃	—	2	2	no	—
52	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₂₃	—	4	2	no	—
54	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
55	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
56	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₄×D₄×C₂₃	—	2	4	no	—
59	C₄×D₄×C₂₃	—	4	2	no	—
60	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	complemented
64	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₂₃	—	4	4	no	—
74	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
75	C₂×C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₂₃	—	2	2	no	—
77	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
78	C₂₃×#106	—	4	2	no	—
79	C₂₃×#102	—	2	4	no	—
80	C₂₃×#102	—	4	2	no	—
81	C₂₃×#102	C₂×C₄	1	4	yes	—
82	C₂₃×#102	C₂×C₄	1	4	yes	—
83	C₂₃×#102	C₂×C₄	1	4	yes	complemented
84	C₂₃×#102	C₂×C₄	1	4	yes	complemented
85	C₂₃×C₂²⋊C₈	—	2	2	no	—
86	C₄×C₈×C₂₃	—	4	2	no	—
87	C₂₃×#95	—	2	2	no	—
88	C₂₃×#95	—	2	2	no	—
89	C₂₃×#95	—	2	1	no	—
90	C₂₃×#95	—	2	1	no	—
91	C₂₃×#95	D₄	1	2	yes	—
92	C₂⁴×C₂₃	—	2	1	no	—
93	C₂⁴×C₂₃	—	2	1	no	—
94	C₂⁴×C₂₃	—	2	4	no	—
95	C₂⁴×C₂₃	—	4	4	no	—
96	C₂⁴×C₂₃	—	4	2	no	—
97	C₂⁴×C₂₃	—	4	2	no	—
98	C₂⁴×C₂₃	—	8	2	no	—
99	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂×D₄×C₂₃	—	2	8	no	—
101	C₂×D₄×C₂₃	—	2	4	no	—
102	C₂×D₄×C₂₃	—	2	1	no	—
103	C₂×D₄×C₂₃	—	2	2	no	—
104	C₂×D₄×C₂₃	—	2	1	no	—
105	C₂×D₄×C₂₃	—	2	2	no	—
106	C₂×D₄×C₂₃	—	2	2	no	—
107	C₂×D₄×C₂₃	—	4	4	no	—
108	C₂×D₄×C₂₃	—	4	8	no	—
109	C₂×D₄×C₂₃	—	4	4	no	—
110	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
111	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
112	C₂×D₄×C₂₃	#41	1	4	yes	—
113	C₂²×C₄×C₂₃	—	2	2	no	—
114	C₂²×C₄×C₂₃	—	2	2	no	—
115	C₂²×C₄×C₂₃	—	2	1	no	—
116	C₂²×C₄×C₂₃	—	2	1	no	—
117	C₂²×C₄×C₂₃	—	2	4	no	—
118	C₂²×C₄×C₂₃	—	2	2	no	—
119	C₂²×C₄×C₂₃	—	2	2	no	—
120	C₂²×C₄×C₂₃	—	4	4	no	—
121	C₂²×C₄×C₂₃	—	4	4	no	—
122	C₂²×C₄×C₂₃	—	8	2	no	—
123	C₂²×C₄×C₂₃	—	8	2	no	—
124	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
125	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₂₃	#41	1	2	yes	—
129	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
130	C₂×C₈×C₂₃	—	2	2	no	—
131	C₂×C₈×C₂₃	—	2	2	no	—
132	C₂×C₈×C₂₃	—	2	2	no	—
133	C₂×C₈×C₂₃	—	2	2	no	—
134	C₂×C₈×C₂₃	—	2	2	no	—
135	C₂×C₈×C₂₃	—	4	4	no	—
136	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
137	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
138	C₄⋊C₄×C₂₃	—	2	4	no	—
139	C₄⋊C₄×C₂₃	—	2	1	no	—
140	C₄⋊C₄×C₂₃	—	2	1	no	—
141	C₄⋊C₄×C₂₃	—	2	2	no	—
142	C₄⋊C₄×C₂₃	—	2	2	no	—
143	C₄⋊C₄×C₂₃	—	2	2	no	—
144	C₄⋊C₄×C₂₃	—	2	2	no	—
145	C₄⋊C₄×C₂₃	—	8	2	no	—
146	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
147	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
148	C₂²⋊C₄×C₂₃	—	2	2	no	—
149	C₂²⋊C₄×C₂₃	—	4	4	no	—
150	C₂²⋊C₄×C₂₃	—	4	2	no	—
151	C₂²⋊C₄×C₂₃	—	4	2	no	—
152	C₂²⋊C₄×C₂₃	—	4	4	no	—
153	C₂²⋊C₄×C₂₃	—	8	2	no	—
154	C₂²⋊C₄×C₂₃	SD₁₆	1	4	yes	—
155	C₄²×C₂₃	—	2	2	no	—
156	C₄²×C₂₃	—	2	2	no	—
157	C₄²×C₂₃	—	8	2	no	—
158	#22574	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₂₃	—	2	2	no	—
160	C₂³×C₂₃	—	2	2	no	—
161	C₂³×C₂₃	—	2	2	no	—
162	C₂³×C₂₃	—	2	4	no	—
163	C₂³×C₂₃	—	2	2	no	—
164	C₂³×C₂₃	—	4	8	no	—
165	C₂³×C₂₃	—	4	1	no	—
166	C₂³×C₂₃	—	4	1	no	—
167	C₂³×C₂₃	—	4	2	no	—
168	C₂³×C₂₃	—	4	2	no	—
169	C₂³×C₂₃	—	4	1	no	—
170	C₂³×C₂₃	—	4	1	no	—
171	C₂³×C₂₃	—	4	2	no	—
172	C₂³×C₂₃	—	4	8	no	—
173	C₂³×C₂₃	—	8	4	no	—
174	C₂³×C₂₃	—	8	8	no	—
175	C₂³×C₂₃	—	8	8	no	—
176	C₂³×C₂₃	C₄×D₄	1	2	yes	—
177	C₂³×C₂₃	C₂×SD₁₆	1	2	yes	—
178	C₂³×C₂₃	#135	1	2	yes	—
179	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
180	D₄×C₂₃	—	2	2	no	—
181	D₄×C₂₃	—	2	2	no	—
182	D₄×C₂₃	—	2	2	no	—
183	D₄×C₂₃	—	2	4	no	—
184	D₄×C₂₃	—	4	8	no	—
185	D₄×C₂₃	—	4	2	no	—
186	D₄×C₂₃	#117	1	4	yes	—
187	C₂×C₄×C₂₃	—	2	2	no	—
188	C₂×C₄×C₂₃	—	2	2	no	—
189	C₂×C₄×C₂₃	—	2	2	no	—
190	C₂×C₄×C₂₃	—	2	2	no	—
191	C₂×C₄×C₂₃	—	2	2	no	—
192	C₂×C₄×C₂₃	—	2	4	no	—
193	C₂×C₄×C₂₃	—	2	2	no	—
194	C₂×C₄×C₂₃	—	2	2	no	—
195	C₂×C₄×C₂₃	—	2	2	no	—
196	C₂×C₄×C₂₃	—	2	2	no	—
197	C₂×C₄×C₂₃	—	2	2	no	—
198	C₂×C₄×C₂₃	—	4	2	no	—
199	C₂×C₄×C₂₃	—	4	2	no	—
200	C₂×C₄×C₂₃	—	4	2	no	—
201	C₂×C₄×C₂₃	—	4	2	no	—
202	C₂×C₄×C₂₃	—	4	2	no	—
203	C₂×C₄×C₂₃	—	4	2	no	—
204	C₂×C₄×C₂₃	—	4	2	no	—
205	C₂×C₄×C₂₃	—	4	2	no	—
206	C₂×C₄×C₂₃	—	8	8	no	—
207	C₂×C₄×C₂₃	—	8	4	no	—
208	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
209	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
210	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
211	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
212	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
214	C₈×C₂₃	—	2	2	no	—
215	C₈×C₂₃	—	2	2	no	—
216	C₈×C₂₃	—	4	2	no	—
217	C₂²×#102	C₂×C₂₃	1	1	yes	characteristic
218	#2345	C₂×C₂₃	1	1	yes	characteristic, complemented
219	D₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
220	#1934	C₂×C₂₃	1	2	yes	complemented
221	#1926	C₂×C₂₃	1	2	yes	complemented
222	#1864	C₂×C₂₃	1	2	yes	complemented
223	#1824	C₂×C₂₃	1	1	yes	characteristic, complemented
224	#1807	C₂×C₂₃	1	1	yes	characteristic, complemented
225	#1786	C₂×C₂₃	1	1	yes	characteristic
226	C₄×#102	C₂×C₂₃	1	2	yes	complemented
227	C₈×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
228	C₂²×C₂₃	—	2	2	no	—
229	C₂²×C₂₃	—	2	2	no	—
230	C₂²×C₂₃	—	2	4	no	—
231	C₂²×C₂₃	—	2	4	no	—
232	C₂²×C₂₃	—	2	4	no	—
233	C₂²×C₂₃	—	4	2	no	—
234	C₂²×C₂₃	—	4	2	no	—
235	C₂²×C₂₃	—	4	4	no	—
236	C₂²×C₂₃	—	4	4	no	—
237	C₂²×C₂₃	—	4	2	no	—
238	C₂²×C₂₃	—	4	2	no	—
239	C₂²×C₂₃	—	4	2	no	—
240	C₂²×C₂₃	—	8	16	no	—
241	C₂²×C₂₃	—	8	2	no	—
242	C₂²×C₂₃	—	8	2	no	—
243	C₂²×C₂₃	C₄×SD₁₆	1	4	yes	—
244	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
245	C₂²×C₂₃	#546	1	2	yes	—
246	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
247	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
248	C₂²×C₂₃	#576	1	2	yes	—
249	C₄×C₂₃	—	2	2	no	—
250	C₄×C₂₃	—	2	4	no	—
251	C₄×C₂₃	—	4	2	no	—
252	C₄×C₂₃	—	4	2	no	—
253	C₄×C₂₃	—	4	2	no	—
254	C₄×C₂₃	—	8	2	no	—
255	C₄×C₂₃	—	8	2	no	—
256	C₄×C₂₃	C₂×#117	1	1	yes	characteristic
257	C₄×C₂₃	#518	1	1	yes	characteristic
258	C₄×C₂₃	#551	1	2	yes	—
259	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
260	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
261	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
262	C₂²×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
263	C₂²×C₂²⋊C₄	—	2	1	no	—
264	C₂×#106	C₂²×C₂₃	1	2	yes	complemented
265	C₂×#102	C₄×C₂₃	1	8	yes	complemented
266	C₂×#102	C₂²×C₂₃	1	2	yes	—
267	C₂×#102	C₂²×C₂₃	1	2	yes	—
268	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
269	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
270	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
271	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
272	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
273	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
274	#390	C₂²×C₂₃	1	2	yes	—
275	#389	C₂²×C₂₃	1	2	yes	complemented
276	#387	—	2	2	no	—
277	#386	—	2	2	no	—
278	#380	—	2	1	no	—
279	#380	C₂²×C₂₃	1	1	yes	characteristic
280	#380	C₂²×C₂₃	1	1	yes	characteristic
281	#379	—	2	1	no	—
282	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
283	C₄×C₂²⋊C₄	—	2	1	no	—
284	C₄×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
285	#340	C₂²×C₂₃	1	2	yes	complemented
286	#340	C₂²×C₂₃	1	2	yes	complemented
287	#340	C₂²×C₂₃	1	2	yes	complemented
288	#336	C₂²×C₂₃	1	2	yes	complemented
289	C₂×C₂₃	—	2	4	no	—
290	C₂×C₂₃	—	4	4	no	—
291	C₂×C₂₃	—	8	2	no	—
292	C₂×C₂₃	#2342	1	1	yes	characteristic
293	C₂×C₂₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
294	C₂×C₂₃	#2948	1	1	yes	characteristic
295	C₂×C₂₃	#3295	1	2	yes	—
296	C₂×C₂₃	#3297	1	2	yes	—
297	C₂⁵	—	2	1	no	—
298	C₂²×D₄	—	2	4	no	—
299	C₂²×D₄	—	2	1	no	—
300	C₂²×D₄	—	2	2	no	—
301	C₂²×D₄	—	4	2	no	—
302	C₂²×D₄	C₂×C₄×C₂₃	1	4	yes	—
303	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
304	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
305	C₂³×C₄	—	2	1	no	—
306	C₂³×C₄	—	2	2	no	—
307	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
308	C₂²×C₈	—	4	2	no	—
309	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	complemented
310	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
311	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
313	C₄×D₄	—	2	4	no	—
314	C₄×D₄	—	4	2	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	complemented
319	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
320	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
321	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
322	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
323	C₂×C₂²⋊C₄	—	2	1	no	—
324	C₂×C₂²⋊C₄	—	2	1	no	—
325	C₂×C₂²⋊C₄	—	2	2	no	—
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₂²⋊C₄	—	4	4	no	—
329	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
330	C₂×C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
331	C₂×C₄²	—	2	2	no	—
332	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
333	#106	—	4	2	no	—
334	#102	—	2	4	no	—
335	#102	—	4	2	no	—
336	#102	C₂×C₄×C₂₃	1	4	yes	—
337	#102	C₂×C₄×C₂₃	1	4	yes	—
338	#102	C₂×C₄×C₂₃	1	4	yes	complemented
339	#102	C₂×C₄×C₂₃	1	4	yes	complemented
340	C₂²⋊C₈	—	2	2	no	—
341	C₄×C₈	—	4	2	no	—
342	#95	—	2	2	no	—
343	#95	—	2	2	no	—
344	#95	—	2	1	no	—
345	#95	—	2	1	no	—
346	#95	D₄×C₂₃	1	2	yes	—
347	C₂₃	#22574	1	1	yes	characteristic, direct product factor, Sylow
348	C₂⁴	—	2	1	no	—
349	C₂⁴	—	2	1	no	—
350	C₂⁴	—	2	4	no	—
351	C₂⁴	—	4	4	no	—
352	C₂⁴	—	4	2	no	—
353	C₂⁴	—	4	2	no	—
354	C₂⁴	—	8	2	no	—
355	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
356	C₂×D₄	—	2	8	no	—
357	C₂×D₄	—	2	4	no	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	1	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	4	4	no	—
364	C₂×D₄	—	4	8	no	—
365	C₂×D₄	—	4	4	no	—
366	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
367	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
368	C₂×D₄	#41×C₂₃	1	4	yes	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	4	4	no	—
377	C₂²×C₄	—	4	4	no	—
378	C₂²×C₄	—	8	2	no	—
379	C₂²×C₄	—	8	2	no	—
380	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
381	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
382	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
384	C₂²×C₄	#41×C₂₃	1	2	yes	—
385	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	—	4	4	no	—
392	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
393	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
394	C₄⋊C₄	—	2	4	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	—	8	2	no	—
402	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
403	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
404	C₂²⋊C₄	—	2	2	no	—
405	C₂²⋊C₄	—	4	4	no	—
406	C₂²⋊C₄	—	4	2	no	—
407	C₂²⋊C₄	—	4	2	no	—
408	C₂²⋊C₄	—	4	4	no	—
409	C₂²⋊C₄	—	8	2	no	—
410	C₂²⋊C₄	SD₁₆×C₂₃	1	4	yes	—
411	C₄²	—	2	2	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	8	2	no	—
414	C₂³	—	2	2	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	2	2	no	—
417	C₂³	—	2	4	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	4	8	no	—
420	C₂³	—	4	1	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	1	no	—
424	C₂³	—	4	1	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	4	8	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	8	4	no	—
429	C₂³	—	8	8	no	—
430	C₂³	—	8	8	no	—
431	C₂³	C₄×D₄×C₂₃	1	2	yes	—
432	C₂³	C₂×SD₁₆×C₂₃	1	2	yes	—
433	C₂³	C₂₃×#135	1	2	yes	—
434	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
435	D₄	—	2	4	no	—
436	D₄	—	2	2	no	—
437	D₄	—	2	2	no	—
438	D₄	—	2	2	no	—
439	D₄	—	4	8	no	—
440	D₄	—	4	2	no	—
441	D₄	C₂₃×#117	1	4	yes	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	4	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	2	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	8	8	no	—
462	C₂×C₄	—	8	4	no	—
463	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
464	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
465	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
466	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
467	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
468	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
469	C₈	—	2	2	no	—
470	C₈	—	2	2	no	—
471	C₈	—	4	2	no	—
472	C₂²	—	2	2	no	—
473	C₂²	—	2	4	no	—
474	C₂²	—	2	4	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	4	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	4	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	8	16	no	—
485	C₂²	—	8	2	no	—
486	C₂²	—	8	2	no	—
487	C₂²	C₄×SD₁₆×C₂₃	1	4	yes	—
488	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
489	C₂²	C₂₃×#546	1	2	yes	—
490	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
491	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
492	C₂²	C₂₃×#576	1	2	yes	—
493	C₄	—	2	4	no	—
494	C₄	—	2	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	8	2	no	—
499	C₄	—	8	2	no	—
500	C₄	C₂×C₂₃×#117	1	1	yes	characteristic
501	C₄	C₂₃×#518	1	1	yes	characteristic
502	C₄	C₂₃×#551	1	2	yes	—
503	C₂	—	2	4	no	—
504	C₂	—	4	4	no	—
505	C₂	—	8	2	no	—
506	C₂	C₂₃×#2342	1	1	yes	characteristic
507	C₂	C₂×C₄×SD₁₆×C₂₃	1	1	yes	characteristic
508	C₂	C₂₃×#2948	1	1	yes	characteristic
509	C₂	C₂₃×#3295	1	2	yes	—
510	C₂	C₂₃×#3297	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂³×D₄×C₂₃	C₂²
LCS	G ▷ C₂×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
2. C₂₃×#2345	C₂	semidirect
4. C₂₃×#1934	C₂	semidirect
5. C₂₃×#1926	C₂	semidirect
6. C₂₃×#1864	C₂	semidirect
7. C₂₃×#1824	C₂	semidirect
8. C₂₃×#1807	C₂	semidirect
10. C₄×C₂₃×#102	C₂	semidirect
11. C₈×C₂²⋊C₄×C₂₃	C₂	semidirect
17. C₂×C₂₃×#106	C₂²	semidirect
18. C₂×C₂₃×#102	C₄	semidirect
26. C₂×C₄×C₈×C₂₃	C₂²	semidirect
28. C₂₃×#389	C₂²	semidirect
35. C₄×C₄⋊C₄×C₂₃	C₂²	semidirect
38. C₂₃×#340	C₂²	semidirect
39. C₂₃×#340	C₂²	semidirect
40. C₂₃×#340	C₂²	semidirect
41. C₂₃×#336	C₂²	semidirect
54. C₂²×C₈×C₂₃	C₂×C₄	semidirect
63. C₂×C₄⋊C₄×C₂₃	C₂×C₄	semidirect
83. C₂₃×#102	C₂×C₄	semidirect
84. C₂₃×#102	C₂×C₄	semidirect
158. #22574	C₂₃	direct (N × H)
218. #2345	C₂×C₂₃	semidirect
220. #1934	C₂×C₂₃	semidirect
221. #1926	C₂×C₂₃	semidirect
222. #1864	C₂×C₂₃	semidirect
223. #1824	C₂×C₂₃	semidirect
224. #1807	C₂×C₂₃	semidirect
226. C₄×#102	C₂×C₂₃	semidirect
227. C₈×C₂²⋊C₄	C₂×C₂₃	semidirect
264. C₂×#106	C₂²×C₂₃	semidirect
265. C₂×#102	C₄×C₂₃	semidirect
273. C₂×C₄×C₈	C₂²×C₂₃	semidirect
275. #389	C₂²×C₂₃	semidirect
282. C₄×C₄⋊C₄	C₂²×C₂₃	semidirect
285. #340	C₂²×C₂₃	semidirect
286. #340	C₂²×C₂₃	semidirect
287. #340	C₂²×C₂₃	semidirect
288. #336	C₂²×C₂₃	semidirect
309. C₂²×C₈	C₂×C₄×C₂₃	semidirect
318. C₂×C₄⋊C₄	C₂×C₄×C₂₃	semidirect
338. #102	C₂×C₄×C₂₃	semidirect
339. #102	C₂×C₄×C₂₃	semidirect
347. C₂₃	#22574	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×#102	2
C₂₃×#2345	2
D₄×C₂²⋊C₄×C₂₃	2
C₂₃×#1934	2
C₂₃×#1926	2
C₂₃×#1864	2
C₂₃×#1824	2
C₂₃×#1807	2
C₂₃×#1786	2
C₄×C₂₃×#102	2
C₈×C₂²⋊C₄×C₂₃	2
#22574	23

Subgroups — C23×#22574 #19170200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22574 #19170200