Subgroups — C₂₃×#24968 #19172594

Order 5888.

854 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	C₂₃×#2537	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2354	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2076	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2033	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1954	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1875	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#1825	C₂	1	2	yes	maximal, complemented
10	C₂₃×#1802	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#121	—	2	1	no	—
13	C₂×C₂₃×#121	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#117	C₂²	1	2	yes	—
17	C₂×C₂₃×#107	C₂²	1	2	yes	—
18	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#102	C₂²	1	2	yes	—
22	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂₃×#394	—	2	2	no	—
26	C₂₃×#390	—	2	1	no	—
27	C₂₃×#388	—	2	2	no	—
28	C₂₃×#384	C₂²	1	1	yes	characteristic
29	C₂₃×#384	C₂²	1	1	yes	characteristic
30	C₂₃×#380	—	2	1	no	—
31	C₂₃×#380	C₂²	1	2	yes	—
32	C₂₃×#380	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
35	C₂₃×#340	C₂²	1	2	yes	—
36	C₂₃×#340	C₂²	1	1	yes	characteristic
37	C₂₃×#340	C₂²	1	1	yes	characteristic
38	C₂₃×#340	C₂²	1	2	yes	—
39	C₂₃×#340	C₂²	1	2	yes	—
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂²×D₄×C₂₃	—	2	2	no	—
42	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
43	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₂₃	—	2	1	no	—
45	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
49	C₂₃×#121	—	2	4	no	—
50	C₂₃×#121	—	4	2	no	—
51	C₂₃×#121	—	4	4	no	—
52	C₄×D₄×C₂₃	—	2	2	no	—
53	C₄×D₄×C₂₃	—	2	2	no	—
54	C₄×D₄×C₂₃	—	2	2	no	—
55	C₄×D₄×C₂₃	—	2	2	no	—
56	C₂₃×#117	—	2	4	no	—
57	C₂₃×#117	—	4	2	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
59	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
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	2	no	—
69	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
74	C₂×C₄²×C₂₃	—	2	1	no	—
75	C₂×C₄²×C₂₃	—	2	2	no	—
76	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
77	C₂₃×#107	—	4	2	no	—
78	C₂₃×C₄⋊C₈	—	2	2	no	—
79	C₂₃×#102	—	2	2	no	—
80	C₂₃×#102	—	2	2	no	—
81	C₂₃×#102	—	4	2	no	—
82	C₂₃×C₂²⋊C₈	—	4	2	no	—
83	C₂₃×C₂²⋊C₈	—	4	2	no	—
84	C₄×C₈×C₂₃	—	2	2	no	—
85	C₂₃×#95	—	2	2	no	—
86	C₂₃×#95	—	2	1	no	—
87	C₂₃×#95	—	2	2	no	—
88	C₂⁴×C₂₃	—	2	1	no	—
89	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
90	C₂×D₄×C₂₃	—	2	2	no	—
91	C₂×D₄×C₂₃	—	2	2	no	—
92	C₂×D₄×C₂₃	—	2	2	no	—
93	C₂×D₄×C₂₃	—	2	1	no	—
94	C₂×D₄×C₂₃	—	2	1	no	—
95	C₂×D₄×C₂₃	—	2	1	no	—
96	C₂×D₄×C₂₃	—	2	1	no	—
97	C₂×D₄×C₂₃	—	2	1	no	—
98	C₂×D₄×C₂₃	—	4	4	no	—
99	C₂×D₄×C₂₃	—	4	4	no	—
100	C₂×D₄×C₂₃	—	8	2	no	—
101	C₂×D₄×C₂₃	#41	1	2	yes	—
102	C₂²×C₄×C₂₃	—	2	1	no	—
103	C₂²×C₄×C₂₃	—	2	2	no	—
104	C₂²×C₄×C₂₃	—	2	2	no	—
105	C₂²×C₄×C₂₃	—	2	2	no	—
106	C₂²×C₄×C₂₃	—	2	2	no	—
107	C₂²×C₄×C₂₃	—	2	2	no	—
108	C₂²×C₄×C₂₃	—	2	2	no	—
109	C₂²×C₄×C₂₃	—	2	2	no	—
110	C₂²×C₄×C₂₃	—	2	1	no	—
111	C₂²×C₄×C₂₃	—	2	1	no	—
112	C₂²×C₄×C₂₃	—	2	2	no	—
113	C₂²×C₄×C₂₃	—	4	2	no	—
114	C₂²×C₄×C₂₃	—	4	2	no	—
115	C₂²×C₄×C₂₃	—	4	2	no	—
116	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₈×C₂₃	—	2	1	no	—
121	C₂×C₈×C₂₃	—	2	1	no	—
122	C₂×C₈×C₂₃	—	2	2	no	—
123	C₂×C₈×C₂₃	—	2	1	no	—
124	C₂×C₈×C₂₃	—	2	1	no	—
125	C₂×C₈×C₂₃	—	2	2	no	—
126	C₂×C₈×C₂₃	—	2	2	no	—
127	C₂×C₈×C₂₃	—	2	1	no	—
128	C₂×C₈×C₂₃	—	2	1	no	—
129	C₄⋊C₄×C₂₃	—	2	2	no	—
130	C₄⋊C₄×C₂₃	—	2	1	no	—
131	C₄⋊C₄×C₂₃	—	2	1	no	—
132	C₄⋊C₄×C₂₃	—	2	2	no	—
133	C₄⋊C₄×C₂₃	—	2	1	no	—
134	C₄⋊C₄×C₂₃	—	2	1	no	—
135	C₄⋊C₄×C₂₃	—	2	2	no	—
136	C₄⋊C₄×C₂₃	—	2	2	no	—
137	C₄⋊C₄×C₂₃	—	2	1	no	—
138	C₄⋊C₄×C₂₃	—	2	2	no	—
139	C₄⋊C₄×C₂₃	—	4	2	no	—
140	C₄⋊C₄×C₂₃	—	4	2	no	—
141	C₄⋊C₄×C₂₃	—	8	1	no	—
142	C₄⋊C₄×C₂₃	—	8	1	no	—
143	C₄⋊C₄×C₂₃	#41	1	2	yes	—
144	C₂²⋊C₄×C₂₃	—	2	4	no	—
145	C₂²⋊C₄×C₂₃	—	4	4	no	—
146	C₂²⋊C₄×C₂₃	—	4	4	no	—
147	C₂²⋊C₄×C₂₃	—	4	4	no	—
148	C₂²⋊C₄×C₂₃	—	4	2	no	—
149	C₂²⋊C₄×C₂₃	—	8	2	no	—
150	C₄²×C₂₃	—	4	4	no	—
151	C₄²×C₂₃	—	4	2	no	—
152	C₄²×C₂₃	#41	1	2	yes	—
153	C₄²×C₂₃	#41	1	2	yes	—
154	#24968	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₂₃	—	2	2	no	—
156	C₂³×C₂₃	—	2	2	no	—
157	C₂³×C₂₃	—	2	2	no	—
158	C₂³×C₂₃	—	2	2	no	—
159	C₂³×C₂₃	—	2	2	no	—
160	C₂³×C₂₃	—	2	2	no	—
161	C₂³×C₂₃	—	2	2	no	—
162	C₂³×C₂₃	—	2	2	no	—
163	C₂³×C₂₃	—	4	1	no	—
164	C₂³×C₂₃	—	4	1	no	—
165	C₂³×C₂₃	—	4	1	no	—
166	C₂³×C₂₃	—	4	1	no	—
167	C₂³×C₂₃	—	4	1	no	—
168	C₂³×C₂₃	—	4	1	no	—
169	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
170	D₄×C₂₃	—	2	2	no	—
171	D₄×C₂₃	—	2	2	no	—
172	D₄×C₂₃	—	4	1	no	—
173	D₄×C₂₃	—	4	1	no	—
174	D₄×C₂₃	—	4	4	no	—
175	D₄×C₂₃	—	8	8	no	—
176	C₂×C₄×C₂₃	—	2	1	no	—
177	C₂×C₄×C₂₃	—	2	1	no	—
178	C₂×C₄×C₂₃	—	2	2	no	—
179	C₂×C₄×C₂₃	—	2	2	no	—
180	C₂×C₄×C₂₃	—	2	2	no	—
181	C₂×C₄×C₂₃	—	2	2	no	—
182	C₂×C₄×C₂₃	—	4	1	no	—
183	C₂×C₄×C₂₃	—	4	1	no	—
184	C₂×C₄×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	—	4	2	no	—
186	C₂×C₄×C₂₃	—	4	2	no	—
187	C₂×C₄×C₂₃	—	4	2	no	—
188	C₂×C₄×C₂₃	—	4	1	no	—
189	C₂×C₄×C₂₃	—	4	2	no	—
190	C₂×C₄×C₂₃	—	4	2	no	—
191	C₂×C₄×C₂₃	—	4	1	no	—
192	C₂×C₄×C₂₃	—	4	2	no	—
193	C₂×C₄×C₂₃	—	4	1	no	—
194	C₂×C₄×C₂₃	—	4	1	no	—
195	C₂×C₄×C₂₃	—	4	1	no	—
196	C₂×C₄×C₂₃	—	4	2	no	—
197	C₂×C₄×C₂₃	—	4	1	no	—
198	C₂×C₄×C₂₃	—	4	2	no	—
199	C₂×C₄×C₂₃	—	4	2	no	—
200	C₂×C₄×C₂₃	—	8	4	no	—
201	C₂×C₄×C₂₃	#123	1	2	yes	—
202	C₂×C₄×C₂₃	#123	1	2	yes	—
203	C₂×C₄×C₂₃	#135	1	2	yes	—
204	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
205	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
211	C₈×C₂₃	—	4	2	no	—
212	C₈×C₂₃	—	4	1	no	—
213	C₈×C₂₃	—	4	1	no	—
214	#2537	C₂×C₂₃	1	1	yes	characteristic
215	#2354	C₂×C₂₃	1	1	yes	characteristic, complemented
216	#2076	C₂×C₂₃	1	2	yes	complemented
217	#2033	C₂×C₂₃	1	2	yes	complemented
218	#1954	C₂×C₂₃	1	1	yes	characteristic, complemented
219	#1947	C₂×C₂₃	1	1	yes	characteristic, complemented
220	#1875	C₂×C₂₃	1	2	yes	complemented
221	#1834	C₂×C₂₃	1	1	yes	characteristic, complemented
222	#1825	C₂×C₂₃	1	2	yes	complemented
223	#1802	C₂×C₂₃	1	1	yes	characteristic, complemented
224	C₄×#102	C₂×C₂₃	1	1	yes	characteristic, complemented
225	C₂²×C₂₃	—	4	2	no	—
226	C₂²×C₂₃	—	4	2	no	—
227	C₂²×C₂₃	—	4	2	no	—
228	C₂²×C₂₃	—	4	2	no	—
229	C₂²×C₂₃	—	4	2	no	—
230	C₂²×C₂₃	—	4	2	no	—
231	C₂²×C₂₃	—	4	2	no	—
232	C₂²×C₂₃	—	4	2	no	—
233	C₂²×C₂₃	—	4	2	no	—
234	C₂²×C₂₃	—	4	2	no	—
235	C₂²×C₂₃	—	8	1	no	—
236	C₂²×C₂₃	—	8	1	no	—
237	C₂²×C₂₃	—	8	1	no	—
238	C₂²×C₂₃	—	8	1	no	—
239	C₂²×C₂₃	C₂×#123	1	1	yes	characteristic
240	C₂²×C₂₃	#546	1	1	yes	characteristic
241	C₂²×C₂₃	#546	1	1	yes	characteristic
242	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
243	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂²×C₂₃	#576	1	1	yes	characteristic
245	C₂²×C₂₃	#578	1	1	yes	characteristic
246	C₄×C₂₃	—	2	2	no	—
247	C₄×C₂₃	—	4	2	no	—
248	C₄×C₂₃	—	4	2	no	—
249	C₄×C₂₃	—	8	1	no	—
250	C₄×C₂₃	—	8	2	no	—
251	C₄×C₂₃	—	8	1	no	—
252	C₄×C₂₃	—	8	2	no	—
253	C₄×C₂₃	#482	1	4	yes	—
254	C₄×C₂₃	#529	1	1	yes	characteristic
255	C₄×C₂₃	#540	1	1	yes	characteristic
256	C₄×C₂₃	#553	1	1	yes	characteristic
257	C₄×C₂₃	#555	1	1	yes	characteristic
258	C₂×#121	—	2	1	no	—
259	C₂×#121	C₂²×C₂₃	1	2	yes	—
260	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
261	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
262	C₂×#117	C₂²×C₂₃	1	2	yes	—
263	C₂×#107	C₂²×C₂₃	1	2	yes	—
264	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
265	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
266	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
267	C₂×#102	C₂²×C₂₃	1	2	yes	—
268	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
269	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
270	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
271	#394	—	2	2	no	—
272	#390	—	2	1	no	—
273	#388	—	2	2	no	—
274	#384	C₂²×C₂₃	1	1	yes	characteristic
275	#384	C₂²×C₂₃	1	1	yes	characteristic
276	#380	—	2	1	no	—
277	#380	C₂²×C₂₃	1	2	yes	—
278	#380	C₂²×C₂₃	1	2	yes	—
279	C₄×C₄⋊C₄	—	2	1	no	—
280	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
281	#340	C₂²×C₂₃	1	2	yes	—
282	#340	C₂²×C₂₃	1	1	yes	characteristic
283	#340	C₂²×C₂₃	1	2	yes	—
284	#340	C₂²×C₂₃	1	1	yes	characteristic
285	#340	C₂²×C₂₃	1	2	yes	—
286	#340	C₂²×C₂₃	1	2	yes	—
287	C₂×C₂₃	—	8	2	no	—
288	C₂×C₂₃	—	8	2	no	—
289	C₂×C₂₃	#2420	1	1	yes	characteristic
290	C₂×C₂₃	C₂×#482	1	1	yes	characteristic
291	C₂×C₂₃	#3098	1	1	yes	characteristic
292	C₂×C₂₃	#3302	1	1	yes	characteristic
293	C₂×C₂₃	#3305	1	1	yes	characteristic
294	C₂×C₂₃	#3325	1	1	yes	characteristic
295	C₂×C₂₃	#3330	1	1	yes	characteristic
296	C₂²×D₄	—	2	2	no	—
297	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
298	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
299	C₂³×C₄	—	2	1	no	—
300	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
301	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
303	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
304	#121	—	2	4	no	—
305	#121	—	4	2	no	—
306	#121	—	4	4	no	—
307	C₄×D₄	—	2	2	no	—
308	C₄×D₄	—	2	2	no	—
309	C₄×D₄	—	2	2	no	—
310	C₄×D₄	—	2	2	no	—
311	#117	—	2	4	no	—
312	#117	—	4	2	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
318	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
319	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
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	2	no	—
324	C₂×C₂²⋊C₄	—	2	2	no	—
325	C₂×C₂²⋊C₄	—	2	1	no	—
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
329	C₂×C₄²	—	2	2	no	—
330	C₂×C₄²	—	2	1	no	—
331	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
332	#107	—	4	2	no	—
333	C₄⋊C₈	—	2	2	no	—
334	#102	—	2	2	no	—
335	#102	—	2	2	no	—
336	#102	—	4	2	no	—
337	C₂²⋊C₈	—	4	2	no	—
338	C₂²⋊C₈	—	4	2	no	—
339	C₄×C₈	—	2	2	no	—
340	#95	—	2	2	no	—
341	#95	—	2	1	no	—
342	#95	—	2	2	no	—
343	C₂₃	#24968	1	1	yes	characteristic, direct product factor, Sylow
344	C₂⁴	—	2	1	no	—
345	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	4	4	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂×D₄	—	8	2	no	—
357	C₂×D₄	#41×C₂₃	1	2	yes	—
358	C₂²×C₄	—	2	1	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	2	no	—
363	C₂²×C₄	—	2	2	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	2	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	4	2	no	—
370	C₂²×C₄	—	4	2	no	—
371	C₂²×C₄	—	4	2	no	—
372	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
373	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
374	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
375	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	2	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	4	2	no	—
396	C₄⋊C₄	—	4	2	no	—
397	C₄⋊C₄	—	8	1	no	—
398	C₄⋊C₄	—	8	1	no	—
399	C₄⋊C₄	#41×C₂₃	1	2	yes	—
400	C₂²⋊C₄	—	2	4	no	—
401	C₂²⋊C₄	—	4	4	no	—
402	C₂²⋊C₄	—	4	4	no	—
403	C₂²⋊C₄	—	4	4	no	—
404	C₂²⋊C₄	—	4	2	no	—
405	C₂²⋊C₄	—	8	2	no	—
406	C₄²	—	4	4	no	—
407	C₄²	—	4	2	no	—
408	C₄²	#41×C₂₃	1	2	yes	—
409	C₄²	#41×C₂₃	1	2	yes	—
410	C₂³	—	2	2	no	—
411	C₂³	—	2	2	no	—
412	C₂³	—	2	2	no	—
413	C₂³	—	2	2	no	—
414	C₂³	—	2	2	no	—
415	C₂³	—	2	2	no	—
416	C₂³	—	2	2	no	—
417	C₂³	—	2	2	no	—
418	C₂³	—	4	1	no	—
419	C₂³	—	4	1	no	—
420	C₂³	—	4	1	no	—
421	C₂³	—	4	1	no	—
422	C₂³	—	4	1	no	—
423	C₂³	—	4	1	no	—
424	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
425	D₄	—	2	2	no	—
426	D₄	—	2	2	no	—
427	D₄	—	4	1	no	—
428	D₄	—	4	4	no	—
429	D₄	—	4	1	no	—
430	D₄	—	8	8	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	2	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	1	no	—
455	C₂×C₄	—	8	4	no	—
456	C₂×C₄	C₂₃×#123	1	2	yes	—
457	C₂×C₄	C₂₃×#123	1	2	yes	—
458	C₂×C₄	C₂₃×#135	1	2	yes	—
459	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
460	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
462	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
463	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
464	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
465	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
466	C₈	—	4	1	no	—
467	C₈	—	4	1	no	—
468	C₈	—	4	2	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	2	no	—
473	C₂²	—	4	2	no	—
474	C₂²	—	4	2	no	—
475	C₂²	—	4	2	no	—
476	C₂²	—	4	2	no	—
477	C₂²	—	4	2	no	—
478	C₂²	—	4	2	no	—
479	C₂²	—	8	1	no	—
480	C₂²	—	8	1	no	—
481	C₂²	—	8	1	no	—
482	C₂²	—	8	1	no	—
483	C₂²	C₂×C₂₃×#123	1	1	yes	characteristic
484	C₂²	C₂₃×#546	1	1	yes	characteristic
485	C₂²	C₂₃×#546	1	1	yes	characteristic
486	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
487	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
488	C₂²	C₂₃×#576	1	1	yes	characteristic
489	C₂²	C₂₃×#578	1	1	yes	characteristic
490	C₄	—	2	2	no	—
491	C₄	—	4	2	no	—
492	C₄	—	4	2	no	—
493	C₄	—	8	2	no	—
494	C₄	—	8	2	no	—
495	C₄	—	8	1	no	—
496	C₄	—	8	1	no	—
497	C₄	C₂₃×#482	1	4	yes	—
498	C₄	C₂₃×#529	1	1	yes	characteristic
499	C₄	C₂₃×#540	1	1	yes	characteristic
500	C₄	C₂₃×#553	1	1	yes	characteristic
501	C₄	C₂₃×#555	1	1	yes	characteristic
502	C₂	—	8	2	no	—
503	C₂	—	8	2	no	—
504	C₂	C₂₃×#2420	1	1	yes	characteristic
505	C₂	C₂×C₂₃×#482	1	1	yes	characteristic
506	C₂	C₂₃×#3098	1	1	yes	characteristic
507	C₂	C₂₃×#3302	1	1	yes	characteristic
508	C₂	C₂₃×#3305	1	1	yes	characteristic
509	C₂	C₂₃×#3325	1	1	yes	characteristic
510	C₂	C₂₃×#3330	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#2537	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2354	C₂	semidirect
3. C₂₃×#2076	C₂	semidirect
4. C₂₃×#2033	C₂	semidirect
5. C₂₃×#1954	C₂	semidirect
6. C₂₃×#1947	C₂	semidirect
7. C₂₃×#1875	C₂	semidirect
8. C₂₃×#1834	C₂	semidirect
9. C₂₃×#1825	C₂	semidirect
10. C₂₃×#1802	C₂	semidirect
11. C₄×C₂₃×#102	C₂	semidirect
154. #24968	C₂₃	direct (N × H)
215. #2354	C₂×C₂₃	semidirect
216. #2076	C₂×C₂₃	semidirect
217. #2033	C₂×C₂₃	semidirect
218. #1954	C₂×C₂₃	semidirect
219. #1947	C₂×C₂₃	semidirect
220. #1875	C₂×C₂₃	semidirect
221. #1834	C₂×C₂₃	semidirect
222. #1825	C₂×C₂₃	semidirect
223. #1802	C₂×C₂₃	semidirect
224. C₄×#102	C₂×C₂₃	semidirect
343. C₂₃	#24968	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×#2537	2
C₂₃×#2354	2
C₂₃×#2076	2
C₂₃×#2033	2
C₂₃×#1954	2
C₂₃×#1947	2
C₂₃×#1875	2
C₂₃×#1834	2
C₂₃×#1825	2
C₂₃×#1802	2
C₄×C₂₃×#102	2
#24968	23

Subgroups — C23×#24968 #19172594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#24968 #19172594