Subgroups — C₂₃×#22968 #19170594

Order 5888.

914 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#474	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2365	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#1948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#1943	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1921	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1864	C₂	1	2	yes	maximal
9	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#1762	C₂	1	2	yes	maximal
11	C₄×C₂₃×#102	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
14	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
17	C₂₃×#474	C₄	1	8	yes	complemented
18	C₂×C₂₃×#107	C₂²	1	2	yes	—
19	C₂×C₂₃×#106	C₂²	1	2	yes	—
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#102	C₂²	1	2	yes	—
23	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
24	C₂×C₂₃×#97	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#391	—	2	1	no	—
27	C₂₃×#389	C₂²	1	2	yes	—
28	C₂₃×#389	C₂²	1	2	yes	—
29	C₂₃×#387	—	2	2	no	—
30	C₂₃×#385	—	2	2	no	—
31	C₂₃×#384	C₂²	1	1	yes	characteristic
32	C₂₃×#380	—	2	1	no	—
33	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
36	C₂₃×#340	C₂²	1	1	yes	characteristic
37	C₂₃×#340	C₂²	1	2	yes	—
38	C₂₃×#340	C₂²	1	2	yes	—
39	C₂₃×#340	C₂²	1	1	yes	characteristic
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂₃×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₂₃	—	2	1	no	—
43	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₂₃	—	2	1	no	—
45	C₂³×C₄×C₂₃	—	2	2	no	—
46	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂₃×C₄⋊Q₈	—	2	2	no	—
50	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
51	C₄×Q₈×C₂₃	—	4	4	no	—
52	C₄×D₄×C₂₃	—	2	4	no	—
53	C₄×D₄×C₂₃	—	2	2	no	—
54	C₄×D₄×C₂₃	—	4	2	no	—
55	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
56	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₂₃	—	4	4	no	—
62	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₂×C₄²×C₂₃	—	2	2	no	—
72	C₂×C₄²×C₂₃	—	2	1	no	—
73	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
74	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
75	C₂₃×#107	C₂×C₄	1	8	yes	complemented
76	C₂₃×#106	—	4	2	no	—
77	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
78	C₂₃×#102	—	2	2	no	—
79	C₂₃×#102	C₂×C₄	1	8	yes	—
80	C₂₃×#102	Q₈	1	4	yes	complemented
81	C₂₃×#97	—	4	2	no	—
82	C₄×C₈×C₂₃	—	2	2	no	—
83	C₂₃×#95	—	2	1	no	—
84	C₂₃×#95	—	2	2	no	—
85	C₂₃×#95	—	2	2	no	—
86	C₂₃×#95	—	2	2	no	—
87	C₂₃×#95	D₄	1	2	yes	—
88	C₂⁴×C₂₃	—	2	1	no	—
89	C₂×Q₈×C₂₃	—	2	4	no	—
90	C₂×Q₈×C₂₃	—	4	2	no	—
91	C₂×Q₈×C₂₃	—	4	2	no	—
92	C₂×D₄×C₂₃	—	2	1	no	—
93	C₂×D₄×C₂₃	—	2	1	no	—
94	C₂×D₄×C₂₃	—	2	1	no	—
95	C₂×D₄×C₂₃	C₂²×C₄	1	2	yes	—
96	C₂×D₄×C₂₃	C₂×Q₈	1	2	yes	—
97	C₂×D₄×C₂₃	#41	1	2	yes	—
98	C₂²×C₄×C₂₃	—	2	1	no	—
99	C₂²×C₄×C₂₃	—	2	2	no	—
100	C₂²×C₄×C₂₃	—	2	1	no	—
101	C₂²×C₄×C₂₃	—	2	4	no	—
102	C₂²×C₄×C₂₃	—	2	1	no	—
103	C₂²×C₄×C₂₃	—	2	2	no	—
104	C₂²×C₄×C₂₃	—	4	4	no	—
105	C₂²×C₄×C₂₃	—	4	4	no	—
106	C₂²×C₄×C₂₃	—	4	2	no	—
107	C₂²×C₄×C₂₃	—	4	2	no	—
108	C₂²×C₄×C₂₃	—	8	2	no	—
109	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
114	C₂×C₈×C₂₃	—	2	1	no	—
115	C₂×C₈×C₂₃	—	2	2	no	—
116	C₂×C₈×C₂₃	—	2	1	no	—
117	C₂×C₈×C₂₃	—	2	2	no	—
118	C₂×C₈×C₂₃	—	2	1	no	—
119	C₂×C₈×C₂₃	—	2	1	no	—
120	C₂×C₈×C₂₃	—	2	1	no	—
121	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
122	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
123	C₄⋊C₄×C₂₃	—	2	4	no	—
124	C₄⋊C₄×C₂₃	—	2	2	no	—
125	C₄⋊C₄×C₂₃	—	2	2	no	—
126	C₄⋊C₄×C₂₃	—	2	1	no	—
127	C₄⋊C₄×C₂₃	—	2	2	no	—
128	C₄⋊C₄×C₂₃	—	2	1	no	—
129	C₄⋊C₄×C₂₃	—	2	1	no	—
130	C₄⋊C₄×C₂₃	—	2	2	no	—
131	C₄⋊C₄×C₂₃	—	2	1	no	—
132	C₄⋊C₄×C₂₃	—	2	2	no	—
133	C₄⋊C₄×C₂₃	—	4	4	no	—
134	C₄⋊C₄×C₂₃	—	4	4	no	—
135	C₄⋊C₄×C₂₃	—	4	2	no	—
136	C₄⋊C₄×C₂₃	—	8	1	no	—
137	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
138	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
139	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
140	C₂²⋊C₄×C₂₃	—	2	4	no	—
141	C₂²⋊C₄×C₂₃	—	4	4	no	—
142	C₄²×C₂₃	—	2	2	no	—
143	C₄²×C₂₃	—	2	2	no	—
144	C₄²×C₂₃	—	4	4	no	—
145	C₄²×C₂₃	—	4	2	no	—
146	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
147	C₄²×C₂₃	#41	1	2	yes	—
148	#22968	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂³×C₂₃	—	2	2	no	—
150	C₂³×C₂₃	—	2	2	no	—
151	C₂³×C₂₃	—	2	2	no	—
152	C₂³×C₂₃	—	4	1	no	—
153	C₂³×C₂₃	—	4	1	no	—
154	C₂³×C₂₃	—	4	1	no	—
155	C₂³×C₂₃	—	4	1	no	—
156	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
157	Q₈×C₂₃	—	4	8	no	—
158	D₄×C₂₃	—	2	2	no	—
159	D₄×C₂₃	—	2	2	no	—
160	D₄×C₂₃	—	2	2	no	—
161	D₄×C₂₃	C₄×Q₈	1	4	yes	complemented
162	C₂×C₄×C₂₃	—	2	1	no	—
163	C₂×C₄×C₂₃	—	2	1	no	—
164	C₂×C₄×C₂₃	—	2	2	no	—
165	C₂×C₄×C₂₃	—	2	4	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	2	2	no	—
168	C₂×C₄×C₂₃	—	2	2	no	—
169	C₂×C₄×C₂₃	—	2	2	no	—
170	C₂×C₄×C₂₃	—	2	2	no	—
171	C₂×C₄×C₂₃	—	2	2	no	—
172	C₂×C₄×C₂₃	—	2	2	no	—
173	C₂×C₄×C₂₃	—	4	1	no	—
174	C₂×C₄×C₂₃	—	4	2	no	—
175	C₂×C₄×C₂₃	—	4	1	no	—
176	C₂×C₄×C₂₃	—	4	1	no	—
177	C₂×C₄×C₂₃	—	4	2	no	—
178	C₂×C₄×C₂₃	—	4	1	no	—
179	C₂×C₄×C₂₃	—	4	8	no	—
180	C₂×C₄×C₂₃	—	4	2	no	—
181	C₂×C₄×C₂₃	—	4	1	no	—
182	C₂×C₄×C₂₃	—	4	1	no	—
183	C₂×C₄×C₂₃	—	4	2	no	—
184	C₂×C₄×C₂₃	—	4	2	no	—
185	C₂×C₄×C₂₃	—	4	1	no	—
186	C₂×C₄×C₂₃	—	8	8	no	—
187	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
188	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
189	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
190	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
191	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
193	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
196	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
197	C₈×C₂₃	—	2	4	no	—
198	C₈×C₂₃	—	2	2	no	—
199	C₈×C₂₃	—	4	1	no	—
200	C₂×#474	C₂×C₂₃	1	1	yes	characteristic
201	#2365	C₂×C₂₃	1	1	yes	characteristic, complemented
202	D₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
203	#1990	C₂×C₂₃	1	1	yes	characteristic, complemented
204	#1948	C₂×C₂₃	1	1	yes	characteristic, complemented
205	#1943	C₂×C₂₃	1	2	yes	complemented
206	#1921	C₂×C₂₃	1	2	yes	complemented
207	#1864	C₂×C₂₃	1	2	yes	—
208	#1770	C₂×C₂₃	1	1	yes	characteristic, complemented
209	#1762	C₂×C₂₃	1	2	yes	—
210	C₄×#102	C₂×C₂₃	1	1	yes	characteristic
211	C₂²×C₂₃	—	2	4	no	—
212	C₂²×C₂₃	—	4	2	no	—
213	C₂²×C₂₃	—	4	2	no	—
214	C₂²×C₂₃	—	4	2	no	—
215	C₂²×C₂₃	—	4	2	no	—
216	C₂²×C₂₃	—	4	2	no	—
217	C₂²×C₂₃	—	4	2	no	—
218	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
219	C₂²×C₂₃	#546	1	1	yes	characteristic
220	C₂²×C₂₃	D₄×Q₈	1	1	yes	characteristic
221	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂²×C₂₃	#577	1	1	yes	characteristic
224	C₂²×C₂₃	#578	1	1	yes	characteristic
225	C₄×C₂₃	—	2	2	no	—
226	C₄×C₂₃	—	4	2	no	—
227	C₄×C₂₃	—	4	2	no	—
228	C₄×C₂₃	—	4	2	no	—
229	C₄×C₂₃	—	4	4	no	—
230	C₄×C₂₃	—	4	2	no	—
231	C₄×C₂₃	—	8	1	no	—
232	C₄×C₂₃	—	8	2	no	—
233	C₄×C₂₃	#442	1	4	yes	—
234	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
235	C₄×C₂₃	#520	1	1	yes	characteristic
236	C₄×C₂₃	#552	1	1	yes	characteristic
237	C₄×C₂₃	#554	1	1	yes	characteristic
238	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
239	C₂×C₄×Q₈	—	2	1	no	—
240	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
241	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
242	C₂²×C₄⋊C₄	—	2	1	no	—
243	#474	C₄×C₂₃	1	8	yes	complemented
244	C₂×#107	C₂²×C₂₃	1	2	yes	—
245	C₂×#106	C₂²×C₂₃	1	2	yes	—
246	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
247	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
248	C₂×#102	C₂²×C₂₃	1	2	yes	—
249	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
250	C₂×#97	C₂²×C₂₃	1	2	yes	—
251	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
252	#391	—	2	1	no	—
253	#389	C₂²×C₂₃	1	2	yes	—
254	#389	C₂²×C₂₃	1	2	yes	—
255	#387	—	2	2	no	—
256	#385	—	2	2	no	—
257	#384	C₂²×C₂₃	1	1	yes	characteristic
258	#380	—	2	1	no	—
259	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
260	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
261	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
262	#340	C₂²×C₂₃	1	2	yes	—
263	#340	C₂²×C₂₃	1	1	yes	characteristic
264	#340	C₂²×C₂₃	1	2	yes	—
265	#340	C₂²×C₂₃	1	1	yes	characteristic
266	#340	C₂²×C₂₃	1	2	yes	—
267	#336	C₂²×C₂₃	1	2	yes	—
268	C₂×C₂₃	—	4	4	no	—
269	C₂×C₂₃	#2360	1	1	yes	characteristic
270	C₂×C₂₃	C₂×#442	1	1	yes	characteristic
271	C₂×C₂₃	#2947	1	1	yes	characteristic
272	C₂×C₂₃	#3324	1	1	yes	characteristic
273	C₂×C₂₃	#3330	1	1	yes	characteristic
274	C₂×C₂₃	#3384	1	1	yes	characteristic
275	C₂×C₂₃	#3386	1	1	yes	characteristic
276	C₂²×Q₈	—	2	1	no	—
277	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
278	C₂³×C₄	—	2	1	no	—
279	C₂³×C₄	—	2	2	no	—
280	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
282	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
283	C₄⋊Q₈	—	2	2	no	—
284	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	complemented
285	C₄×Q₈	—	4	4	no	—
286	C₄×D₄	—	2	4	no	—
287	C₄×D₄	—	2	2	no	—
288	C₄×D₄	—	4	2	no	—
289	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	—	2	1	no	—
292	C₂×C₄⋊C₄	—	2	2	no	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	4	4	no	—
296	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
297	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
299	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
302	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
303	C₂×C₂²⋊C₄	—	2	1	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₄²	—	2	2	no	—
306	C₂×C₄²	—	2	1	no	—
307	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
308	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
309	#107	C₂×C₄×C₂₃	1	8	yes	complemented
310	#106	—	4	2	no	—
311	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	complemented
312	#102	—	2	2	no	—
313	#102	C₂×C₄×C₂₃	1	8	yes	—
314	#102	Q₈×C₂₃	1	4	yes	complemented
315	#97	—	4	2	no	—
316	C₄×C₈	—	2	2	no	—
317	#95	—	2	1	no	—
318	#95	—	2	2	no	—
319	#95	—	2	2	no	—
320	#95	—	2	2	no	—
321	#95	D₄×C₂₃	1	2	yes	—
322	C₂₃	#22968	1	1	yes	characteristic, direct product factor, Sylow
323	C₂⁴	—	2	1	no	—
324	C₂×Q₈	—	2	4	no	—
325	C₂×Q₈	—	4	2	no	—
326	C₂×Q₈	—	4	2	no	—
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	—	2	1	no	—
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	C₂²×C₄×C₂₃	1	2	yes	—
331	C₂×D₄	C₂×Q₈×C₂₃	1	2	yes	—
332	C₂×D₄	#41×C₂₃	1	2	yes	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	4	4	no	—
340	C₂²×C₄	—	4	4	no	—
341	C₂²×C₄	—	4	2	no	—
342	C₂²×C₄	—	4	2	no	—
343	C₂²×C₄	—	8	2	no	—
344	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
345	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
346	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
347	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
348	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
357	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
358	C₄⋊C₄	—	2	4	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	4	4	no	—
369	C₄⋊C₄	—	4	4	no	—
370	C₄⋊C₄	—	4	2	no	—
371	C₄⋊C₄	—	8	1	no	—
372	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
373	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
374	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
375	C₂²⋊C₄	—	2	4	no	—
376	C₂²⋊C₄	—	4	4	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	4	4	no	—
380	C₄²	—	4	2	no	—
381	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
382	C₄²	#41×C₂₃	1	2	yes	—
383	C₂³	—	2	2	no	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	4	1	no	—
387	C₂³	—	4	1	no	—
388	C₂³	—	4	1	no	—
389	C₂³	—	4	1	no	—
390	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
391	Q₈	—	4	8	no	—
392	D₄	—	2	2	no	—
393	D₄	—	2	2	no	—
394	D₄	—	2	2	no	—
395	D₄	C₄×Q₈×C₂₃	1	4	yes	complemented
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	4	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	8	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	8	8	no	—
421	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
422	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
423	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
424	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
425	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
426	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
429	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
430	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	2	4	no	—
433	C₈	—	4	1	no	—
434	C₂²	—	2	4	no	—
435	C₂²	—	4	2	no	—
436	C₂²	—	4	2	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
442	C₂²	C₂₃×#546	1	1	yes	characteristic
443	C₂²	D₄×Q₈×C₂₃	1	1	yes	characteristic
444	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
445	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
446	C₂²	C₂₃×#577	1	1	yes	characteristic
447	C₂²	C₂₃×#578	1	1	yes	characteristic
448	C₄	—	2	2	no	—
449	C₄	—	4	2	no	—
450	C₄	—	4	4	no	—
451	C₄	—	4	2	no	—
452	C₄	—	4	2	no	—
453	C₄	—	4	2	no	—
454	C₄	—	8	2	no	—
455	C₄	—	8	1	no	—
456	C₄	C₂₃×#442	1	4	yes	—
457	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
458	C₄	C₂₃×#520	1	1	yes	characteristic
459	C₄	C₂₃×#552	1	1	yes	characteristic
460	C₄	C₂₃×#554	1	1	yes	characteristic
461	C₂	—	4	4	no	—
462	C₂	C₂₃×#2360	1	1	yes	characteristic
463	C₂	C₂×C₂₃×#442	1	1	yes	characteristic
464	C₂	C₂₃×#2947	1	1	yes	characteristic
465	C₂	C₂₃×#3324	1	1	yes	characteristic
466	C₂	C₂₃×#3330	1	1	yes	characteristic
467	C₂	C₂₃×#3384	1	1	yes	characteristic
468	C₂	C₂₃×#3386	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2365	C₂	semidirect
4. C₂₃×#1990	C₂	semidirect
5. C₂₃×#1948	C₂	semidirect
6. C₂₃×#1943	C₂	semidirect
7. C₂₃×#1921	C₂	semidirect
9. C₂₃×#1770	C₂	semidirect
17. C₂₃×#474	C₄	semidirect
50. C₂₃×C₄⋊Q₈	C₂×C₄	semidirect
75. C₂₃×#107	C₂×C₄	semidirect
77. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
80. C₂₃×#102	Q₈	semidirect
148. #22968	C₂₃	direct (N × H)
161. D₄×C₂₃	C₄×Q₈	semidirect
201. #2365	C₂×C₂₃	semidirect
203. #1990	C₂×C₂₃	semidirect
204. #1948	C₂×C₂₃	semidirect
205. #1943	C₂×C₂₃	semidirect
206. #1921	C₂×C₂₃	semidirect
208. #1770	C₂×C₂₃	semidirect
243. #474	C₄×C₂₃	semidirect
284. C₄⋊Q₈	C₂×C₄×C₂₃	semidirect
309. #107	C₂×C₄×C₂₃	semidirect
311. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
314. #102	Q₈×C₂₃	semidirect
322. C₂₃	#22968	direct (N × H)
395. D₄	C₄×Q₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#474	2
C₂₃×#2365	2
D₄×C₄⋊C₄×C₂₃	2
C₂₃×#1990	2
C₂₃×#1948	2
C₂₃×#1943	2
C₂₃×#1921	2
C₂₃×#1864	2
C₂₃×#1770	2
C₂₃×#1762	2
C₄×C₂₃×#102	2
#22968	23

Subgroups — C23×#22968 #19170594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22968 #19170594