Subgroups — C₂₃×#21615 #19169241

Order 5888.

868 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#475	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₄²×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1988	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1947	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1920	C₂	1	2	yes	maximal
7	C₂₃×#1863	C₂	1	2	yes	maximal
8	C₂₃×#1803	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1776	C₂	1	2	yes	maximal
10	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic
11	C₂₃×#1763	C₂	1	2	yes	maximal
12	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
14	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#475	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×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₂₃×#103	C₂²	1	2	yes	—
23	C₂×C₂₃×#103	C₂²	1	2	yes	—
24	C₂×C₂₃×#97	C₂²	1	2	yes	—
25	C₂₃×#391	—	2	1	no	—
26	C₂₃×#389	—	2	1	no	—
27	C₂₃×#389	C₂²	1	2	yes	—
28	C₂₃×#384	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
30	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
31	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
32	C₄×C₄⋊C₄×C₂₃	—	2	1	no	—
33	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
37	C₄³×C₂₃	—	2	1	no	—
38	C₄³×C₂₃	C₂²	1	1	yes	characteristic
39	C₂₃×#340	C₂²	1	2	yes	—
40	C₂₃×#340	C₂²	1	2	yes	—
41	C₂₃×#340	C₂²	1	2	yes	—
42	C₂₃×#340	C₂²	1	2	yes	—
43	C₂₃×#336	C₂²	1	2	yes	—
44	C₂²×Q₈×C₂₃	—	2	1	no	—
45	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂₃×C₄⋊Q₈	—	2	2	no	—
49	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
50	C₄×Q₈×C₂₃	—	2	2	no	—
51	C₄×Q₈×C₂₃	—	2	2	no	—
52	C₄×Q₈×C₂₃	—	2	2	no	—
53	C₄×Q₈×C₂₃	—	2	2	no	—
54	C₄×Q₈×C₂₃	—	4	4	no	—
55	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
56	C₄×Q₈×C₂₃	Q₈	1	4	yes	complemented
57	C₂×C₄⋊C₄×C₂₃	—	2	1	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	—	2	1	no	—
63	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
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×C₄²×C₂₃	—	2	1	no	—
70	C₂×C₄²×C₂₃	—	2	1	no	—
71	C₂×C₄²×C₂₃	—	2	1	no	—
72	C₂×C₄²×C₂₃	—	2	1	no	—
73	C₂×C₄²×C₂₃	—	2	1	no	—
74	C₂×C₄²×C₂₃	—	2	1	no	—
75	C₂×C₄²×C₂₃	—	2	2	no	—
76	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
83	C₂₃×#107	—	2	4	no	—
84	C₂₃×#106	C₂×C₄	1	8	yes	—
85	C₂₃×C₄⋊C₈	—	2	2	no	—
86	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
87	C₂₃×#103	—	2	4	no	—
88	C₂₃×#103	C₂×C₄	1	8	yes	—
89	C₂₃×#97	—	2	4	no	—
90	C₂₃×#95	—	2	1	no	—
91	C₂₃×#95	—	2	1	no	—
92	C₂₃×#95	—	2	1	no	—
93	C₂₃×#95	—	2	2	no	—
94	C₂₃×#95	—	2	2	no	—
95	C₂×Q₈×C₂₃	—	2	4	no	—
96	C₂×Q₈×C₂₃	—	2	1	no	—
97	C₂×Q₈×C₂₃	—	2	1	no	—
98	C₂×Q₈×C₂₃	—	2	1	no	—
99	C₂×Q₈×C₂₃	—	4	2	no	—
100	C₂×Q₈×C₂₃	—	4	2	no	—
101	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
102	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
103	C₂×Q₈×C₂₃	#41	1	2	yes	—
104	C₂²×C₄×C₂₃	—	2	1	no	—
105	C₂²×C₄×C₂₃	—	2	1	no	—
106	C₂²×C₄×C₂₃	—	2	1	no	—
107	C₂²×C₄×C₂₃	—	2	1	no	—
108	C₂²×C₄×C₂₃	—	2	2	no	—
109	C₂²×C₄×C₂₃	—	2	2	no	—
110	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
117	C₂×C₈×C₂₃	—	2	2	no	—
118	C₂×C₈×C₂₃	—	2	2	no	—
119	C₂×C₈×C₂₃	—	2	2	no	—
120	C₂×C₈×C₂₃	—	2	2	no	—
121	C₂×C₈×C₂₃	—	2	2	no	—
122	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
123	C₄⋊C₄×C₂₃	—	2	4	no	—
124	C₄⋊C₄×C₂₃	—	2	4	no	—
125	C₄⋊C₄×C₂₃	—	2	4	no	—
126	C₄⋊C₄×C₂₃	—	2	4	no	—
127	C₄⋊C₄×C₂₃	—	2	1	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	2	no	—
132	C₄⋊C₄×C₂₃	—	2	2	no	—
133	C₄⋊C₄×C₂₃	—	4	2	no	—
134	C₄⋊C₄×C₂₃	—	4	4	no	—
135	C₄⋊C₄×C₂₃	—	4	2	no	—
136	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
137	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
138	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
139	C₄⋊C₄×C₂₃	#41	1	2	yes	—
140	C₄²×C₂₃	—	2	4	no	—
141	C₄²×C₂₃	—	2	4	no	—
142	C₄²×C₂₃	—	2	4	no	—
143	C₄²×C₂₃	—	2	2	no	—
144	C₄²×C₂₃	—	2	2	no	—
145	C₄²×C₂₃	—	2	2	no	—
146	C₄²×C₂₃	—	2	1	no	—
147	C₄²×C₂₃	—	2	1	no	—
148	C₄²×C₂₃	—	4	1	no	—
149	C₄²×C₂₃	—	4	2	no	—
150	C₄²×C₂₃	—	4	4	no	—
151	C₄²×C₂₃	—	4	2	no	—
152	C₄²×C₂₃	—	4	2	no	—
153	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
154	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
155	C₄²×C₂₃	#41	1	2	yes	—
156	C₄²×C₂₃	#41	1	2	yes	—
157	#21615	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
158	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₂₃	—	2	2	no	—
160	Q₈×C₂₃	—	2	2	no	—
161	Q₈×C₂₃	—	2	2	no	—
162	Q₈×C₂₃	—	4	8	no	—
163	Q₈×C₂₃	C₄×Q₈	1	4	yes	complemented
164	C₂×C₄×C₂₃	—	2	2	no	—
165	C₂×C₄×C₂₃	—	2	2	no	—
166	C₂×C₄×C₂₃	—	2	1	no	—
167	C₂×C₄×C₂₃	—	2	2	no	—
168	C₂×C₄×C₂₃	—	2	1	no	—
169	C₂×C₄×C₂₃	—	2	1	no	—
170	C₂×C₄×C₂₃	—	2	1	no	—
171	C₂×C₄×C₂₃	—	2	1	no	—
172	C₂×C₄×C₂₃	—	2	1	no	—
173	C₂×C₄×C₂₃	—	2	1	no	—
174	C₂×C₄×C₂₃	—	2	1	no	—
175	C₂×C₄×C₂₃	—	2	2	no	—
176	C₂×C₄×C₂₃	—	2	4	no	—
177	C₂×C₄×C₂₃	—	2	2	no	—
178	C₂×C₄×C₂₃	—	2	1	no	—
179	C₂×C₄×C₂₃	—	2	2	no	—
180	C₂×C₄×C₂₃	—	2	1	no	—
181	C₂×C₄×C₂₃	—	2	1	no	—
182	C₂×C₄×C₂₃	—	4	2	no	—
183	C₂×C₄×C₂₃	—	4	1	no	—
184	C₂×C₄×C₂₃	—	4	1	no	—
185	C₂×C₄×C₂₃	—	4	1	no	—
186	C₂×C₄×C₂₃	—	4	1	no	—
187	C₂×C₄×C₂₃	—	4	2	no	—
188	C₂×C₄×C₂₃	—	4	1	no	—
189	C₂×C₄×C₂₃	—	4	1	no	—
190	C₂×C₄×C₂₃	—	4	2	no	—
191	C₂×C₄×C₂₃	—	4	2	no	—
192	C₂×C₄×C₂₃	—	4	2	no	—
193	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
194	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
195	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
196	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
197	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
198	C₂×C₄×C₂₃	#137	1	2	yes	—
199	C₂×C₄×C₂₃	#137	1	2	yes	—
200	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
202	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
203	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
204	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
206	C₈×C₂₃	—	2	4	no	—
207	C₈×C₂₃	—	4	2	no	—
208	C₂×#475	C₂×C₂₃	1	1	yes	characteristic
209	C₄×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic
210	C₄²×Q₈	C₂×C₂₃	1	1	yes	characteristic
211	#1988	C₂×C₂₃	1	1	yes	characteristic
212	#1947	C₂×C₂₃	1	1	yes	characteristic
213	#1920	C₂×C₂₃	1	2	yes	—
214	#1863	C₂×C₂₃	1	2	yes	—
215	#1803	C₂×C₂₃	1	1	yes	characteristic
216	#1776	C₂×C₂₃	1	2	yes	—
217	#1767	C₂×C₂₃	1	1	yes	characteristic
218	#1763	C₂×C₂₃	1	2	yes	—
219	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
220	C₂²×C₂₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
221	C₂²×C₂₃	#525	1	1	yes	characteristic
222	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
224	C₂²×C₂₃	C₂×#137	1	1	yes	characteristic
225	C₂²×C₂₃	#575	1	1	yes	characteristic
226	C₄×C₂₃	—	2	2	no	—
227	C₄×C₂₃	—	2	2	no	—
228	C₄×C₂₃	—	2	2	no	—
229	C₄×C₂₃	—	2	2	no	—
230	C₄×C₂₃	—	2	4	no	—
231	C₄×C₂₃	—	4	2	no	—
232	C₄×C₂₃	—	4	1	no	—
233	C₄×C₂₃	—	4	2	no	—
234	C₄×C₂₃	—	4	4	no	—
235	C₄×C₂₃	—	4	2	no	—
236	C₄×C₂₃	—	8	2	no	—
237	C₄×C₂₃	#440	1	4	yes	—
238	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
239	C₄×C₂₃	C₄×#41	1	1	yes	characteristic
240	C₄×C₂₃	#529	1	1	yes	characteristic
241	C₄×C₂₃	#533	1	1	yes	characteristic
242	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
243	C₂×C₄×Q₈	—	2	1	no	—
244	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
245	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
246	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
247	#475	C₄×C₂₃	1	8	yes	complemented
248	C₂×#107	C₂²×C₂₃	1	2	yes	—
249	C₂×#106	C₂²×C₂₃	1	2	yes	—
250	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
251	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
252	C₂×#103	C₂²×C₂₃	1	2	yes	—
253	C₂×#103	C₂²×C₂₃	1	2	yes	—
254	C₂×#97	C₂²×C₂₃	1	2	yes	—
255	#391	—	2	1	no	—
256	#389	—	2	1	no	—
257	#389	C₂²×C₂₃	1	2	yes	—
258	#384	C₂²×C₂₃	1	1	yes	characteristic
259	C₄×C₄⋊C₄	—	2	1	no	—
260	C₄×C₄⋊C₄	—	2	1	no	—
261	C₄×C₄⋊C₄	—	2	1	no	—
262	C₄×C₄⋊C₄	—	2	1	no	—
263	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
264	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
265	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
266	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
267	C₄³	—	2	1	no	—
268	C₄³	C₂²×C₂₃	1	1	yes	characteristic
269	#340	C₂²×C₂₃	1	2	yes	—
270	#340	C₂²×C₂₃	1	2	yes	—
271	#340	C₂²×C₂₃	1	2	yes	—
272	#340	C₂²×C₂₃	1	2	yes	—
273	#336	C₂²×C₂₃	1	2	yes	—
274	C₂×C₂₃	C₄×C₂²⋊Q₈	1	1	yes	characteristic
275	C₂×C₂₃	C₂×#440	1	1	yes	characteristic
276	C₂×C₂₃	C₄×#137	1	1	yes	characteristic
277	C₂×C₂₃	#3078	1	1	yes	characteristic
278	C₂×C₂₃	#3079	1	1	yes	characteristic
279	C₂×C₂₃	#3113	1	1	yes	characteristic
280	C₂×C₂₃	#3115	1	1	yes	characteristic
281	C₂²×Q₈	—	2	1	no	—
282	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
283	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
284	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
285	C₄⋊Q₈	—	2	2	no	—
286	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
287	C₄×Q₈	—	2	2	no	—
288	C₄×Q₈	—	2	2	no	—
289	C₄×Q₈	—	2	2	no	—
290	C₄×Q₈	—	2	2	no	—
291	C₄×Q₈	—	4	4	no	—
292	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
293	C₄×Q₈	Q₈×C₂₃	1	4	yes	complemented
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	2	2	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
302	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
306	C₂×C₄²	—	2	1	no	—
307	C₂×C₄²	—	2	1	no	—
308	C₂×C₄²	—	2	1	no	—
309	C₂×C₄²	—	2	2	no	—
310	C₂×C₄²	—	2	1	no	—
311	C₂×C₄²	—	2	1	no	—
312	C₂×C₄²	—	2	1	no	—
313	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
314	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
315	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
316	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
317	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
318	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
320	#107	—	2	4	no	—
321	#106	C₂×C₄×C₂₃	1	8	yes	—
322	C₄⋊C₈	—	2	2	no	—
323	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
324	#103	—	2	4	no	—
325	#103	C₂×C₄×C₂₃	1	8	yes	—
326	#97	—	2	4	no	—
327	#95	—	2	1	no	—
328	#95	—	2	1	no	—
329	#95	—	2	1	no	—
330	#95	—	2	2	no	—
331	#95	—	2	2	no	—
332	C₂₃	#21615	1	1	yes	characteristic, direct product factor, Sylow
333	C₂×Q₈	—	2	4	no	—
334	C₂×Q₈	—	2	1	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	—	2	1	no	—
337	C₂×Q₈	—	4	2	no	—
338	C₂×Q₈	—	4	2	no	—
339	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
340	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
341	C₂×Q₈	#41×C₂₃	1	2	yes	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
349	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
353	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
354	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
361	C₄⋊C₄	—	2	4	no	—
362	C₄⋊C₄	—	2	4	no	—
363	C₄⋊C₄	—	2	4	no	—
364	C₄⋊C₄	—	2	4	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	1	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	2	2	no	—
371	C₄⋊C₄	—	4	4	no	—
372	C₄⋊C₄	—	4	2	no	—
373	C₄⋊C₄	—	4	2	no	—
374	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
375	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
376	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
377	C₄⋊C₄	#41×C₂₃	1	2	yes	—
378	C₄²	—	2	4	no	—
379	C₄²	—	2	4	no	—
380	C₄²	—	2	4	no	—
381	C₄²	—	2	2	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	2	1	no	—
385	C₄²	—	2	1	no	—
386	C₄²	—	4	1	no	—
387	C₄²	—	4	4	no	—
388	C₄²	—	4	2	no	—
389	C₄²	—	4	2	no	—
390	C₄²	—	4	2	no	—
391	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
392	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
393	C₄²	#41×C₂₃	1	2	yes	—
394	C₄²	#41×C₂₃	1	2	yes	—
395	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
396	Q₈	—	2	2	no	—
397	Q₈	—	2	2	no	—
398	Q₈	—	2	2	no	—
399	Q₈	—	4	8	no	—
400	Q₈	C₄×Q₈×C₂₃	1	4	yes	complemented
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
431	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
432	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
433	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
434	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
435	C₂×C₄	C₂₃×#137	1	2	yes	—
436	C₂×C₄	C₂₃×#137	1	2	yes	—
437	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
438	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
439	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
440	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
441	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
442	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
443	C₈	—	2	4	no	—
444	C₈	—	4	2	no	—
445	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
446	C₂²	C₂×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
447	C₂²	C₂₃×#525	1	1	yes	characteristic
448	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
449	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
450	C₂²	C₂×C₂₃×#137	1	1	yes	characteristic
451	C₂²	C₂₃×#575	1	1	yes	characteristic
452	C₄	—	2	2	no	—
453	C₄	—	2	4	no	—
454	C₄	—	2	2	no	—
455	C₄	—	2	2	no	—
456	C₄	—	2	2	no	—
457	C₄	—	4	4	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	1	no	—
461	C₄	—	4	2	no	—
462	C₄	—	8	2	no	—
463	C₄	C₂₃×#440	1	4	yes	—
464	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
465	C₄	C₄×#41×C₂₃	1	1	yes	characteristic
466	C₄	C₂₃×#529	1	1	yes	characteristic
467	C₄	C₂₃×#533	1	1	yes	characteristic
468	C₂	C₄×C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
469	C₂	C₂×C₂₃×#440	1	1	yes	characteristic
470	C₂	C₄×C₂₃×#137	1	1	yes	characteristic
471	C₂	C₂₃×#3078	1	1	yes	characteristic
472	C₂	C₂₃×#3079	1	1	yes	characteristic
473	C₂	C₂₃×#3113	1	1	yes	characteristic
474	C₂	C₂₃×#3115	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×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
17. C₂₃×#475	C₄	semidirect
56. C₄×Q₈×C₂₃	Q₈	semidirect
157. #21615	C₂₃	direct (N × H)
163. Q₈×C₂₃	C₄×Q₈	semidirect
247. #475	C₄×C₂₃	semidirect
293. C₄×Q₈	Q₈×C₂₃	semidirect
332. C₂₃	#21615	direct (N × H)
400. Q₈	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×#475	2
C₄×C₂₃×C₄⋊Q₈	2
C₄²×Q₈×C₂₃	2
C₂₃×#1988	2
C₂₃×#1947	2
C₂₃×#1920	2
C₂₃×#1863	2
C₂₃×#1803	2
C₂₃×#1776	2
C₂₃×#1767	2
C₂₃×#1763	2
#21615	23

Subgroups — C23×#21615 #19169241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21615 #19169241