Subgroups — C₂₃×#22964 #19170590

Order 5888.

822 conjugacy classes of subgroups.

The subgroup lattice has 434 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	C₂₃×#2365	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1990	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1948	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1941	C₂	1	2	yes	maximal
7	C₂₃×#1920	C₂	1	2	yes	maximal
8	C₂₃×#1865	C₂	1	2	yes	maximal
9	C₈×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
10	C₄×C₂₃×#103	C₂	1	1	yes	maximal, characteristic
11	C₄×C₂₃×#103	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	2	yes	—
15	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#475	C₄	1	8	yes	complemented
17	C₂×C₂₃×#106	C₂²	1	2	yes	—
18	C₂×C₂₃×#106	C₂²	1	2	yes	—
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#103	C₂²	1	2	yes	—
22	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
25	C₂₃×#391	—	2	1	no	—
26	C₂₃×#389	—	2	2	no	—
27	C₂₃×#389	C₂²	1	2	yes	—
28	C₂₃×#389	C₂²	1	2	yes	—
29	C₂₃×#387	—	2	2	no	—
30	C₂₃×#384	—	2	1	no	—
31	C₂₃×#384	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₂₃	—	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×Q₈×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
46	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂₃×C₄⋊Q₈	—	2	2	no	—
48	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
49	C₄×Q₈×C₂₃	—	2	2	no	—
50	C₄×Q₈×C₂₃	—	2	4	no	—
51	C₄×Q₈×C₂₃	—	4	2	no	—
52	C₄×Q₈×C₂₃	—	4	4	no	—
53	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
54	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
57	C₂×C₄⋊C₄×C₂₃	—	2	2	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	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
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×C₄²×C₂₃	—	2	1	no	—
69	C₂×C₄²×C₂₃	—	2	2	no	—
70	C₂×C₄²×C₂₃	—	2	2	no	—
71	C₂×C₄²×C₂₃	—	2	1	no	—
72	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
73	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
74	C₂₃×#106	—	4	2	no	—
75	C₂₃×#106	C₂×C₄	1	8	yes	—
76	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
77	C₂₃×#103	—	2	2	no	—
78	C₂₃×#103	C₂×C₄	1	8	yes	—
79	C₂₃×#103	Q₈	1	4	yes	complemented
80	C₄×C₈×C₂₃	—	2	2	no	—
81	C₄×C₈×C₂₃	—	4	2	no	—
82	C₂₃×#95	—	2	2	no	—
83	C₂₃×#95	—	2	1	no	—
84	C₂₃×#95	—	2	2	no	—
85	C₂₃×#95	—	2	2	no	—
86	C₂₃×#95	D₄	1	2	yes	—
87	C₂×Q₈×C₂₃	—	2	4	no	—
88	C₂×Q₈×C₂₃	—	2	1	no	—
89	C₂×Q₈×C₂₃	—	2	1	no	—
90	C₂×Q₈×C₂₃	—	2	1	no	—
91	C₂×Q₈×C₂₃	—	4	2	no	—
92	C₂×Q₈×C₂₃	—	4	2	no	—
93	C₂×Q₈×C₂₃	C₂²×C₄	1	2	yes	—
94	C₂×Q₈×C₂₃	C₂×Q₈	1	2	yes	—
95	C₂×Q₈×C₂₃	#41	1	2	yes	—
96	C₂²×C₄×C₂₃	—	2	1	no	—
97	C₂²×C₄×C₂₃	—	2	1	no	—
98	C₂²×C₄×C₂₃	—	2	1	no	—
99	C₂²×C₄×C₂₃	—	2	2	no	—
100	C₂²×C₄×C₂₃	—	2	2	no	—
101	C₂²×C₄×C₂₃	—	2	1	no	—
102	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₂₃	—	2	1	no	—
108	C₂×C₈×C₂₃	—	2	2	no	—
109	C₂×C₈×C₂₃	—	2	1	no	—
110	C₂×C₈×C₂₃	—	2	2	no	—
111	C₂×C₈×C₂₃	—	2	1	no	—
112	C₂×C₈×C₂₃	—	2	1	no	—
113	C₂×C₈×C₂₃	—	2	1	no	—
114	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
115	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
116	C₄⋊C₄×C₂₃	—	2	4	no	—
117	C₄⋊C₄×C₂₃	—	2	4	no	—
118	C₄⋊C₄×C₂₃	—	2	2	no	—
119	C₄⋊C₄×C₂₃	—	2	1	no	—
120	C₄⋊C₄×C₂₃	—	2	2	no	—
121	C₄⋊C₄×C₂₃	—	2	1	no	—
122	C₄⋊C₄×C₂₃	—	2	1	no	—
123	C₄⋊C₄×C₂₃	—	2	2	no	—
124	C₄⋊C₄×C₂₃	—	2	1	no	—
125	C₄⋊C₄×C₂₃	—	2	2	no	—
126	C₄⋊C₄×C₂₃	—	4	4	no	—
127	C₄⋊C₄×C₂₃	—	4	4	no	—
128	C₄⋊C₄×C₂₃	—	4	2	no	—
129	C₄⋊C₄×C₂₃	—	8	1	no	—
130	C₄⋊C₄×C₂₃	—	8	2	no	—
131	C₄⋊C₄×C₂₃	SD₁₆	1	4	yes	—
132	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
133	C₄⋊C₄×C₂₃	C₂²×C₄	1	2	yes	—
134	C₄⋊C₄×C₂₃	C₂×Q₈	1	2	yes	—
135	C₄²×C₂₃	—	2	4	no	—
136	C₄²×C₂₃	—	2	2	no	—
137	C₄²×C₂₃	—	2	2	no	—
138	C₄²×C₂₃	—	4	4	no	—
139	C₄²×C₂₃	—	4	4	no	—
140	C₄²×C₂₃	—	4	2	no	—
141	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
142	C₄²×C₂₃	#41	1	2	yes	—
143	#22964	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
145	Q₈×C₂₃	—	2	2	no	—
146	Q₈×C₂₃	—	2	2	no	—
147	Q₈×C₂₃	—	2	2	no	—
148	Q₈×C₂₃	—	4	8	no	—
149	Q₈×C₂₃	C₄×Q₈	1	4	yes	complemented
150	C₂×C₄×C₂₃	—	2	2	no	—
151	C₂×C₄×C₂₃	—	2	2	no	—
152	C₂×C₄×C₂₃	—	2	2	no	—
153	C₂×C₄×C₂₃	—	2	2	no	—
154	C₂×C₄×C₂₃	—	2	1	no	—
155	C₂×C₄×C₂₃	—	2	1	no	—
156	C₂×C₄×C₂₃	—	2	2	no	—
157	C₂×C₄×C₂₃	—	2	2	no	—
158	C₂×C₄×C₂₃	—	2	4	no	—
159	C₂×C₄×C₂₃	—	2	2	no	—
160	C₂×C₄×C₂₃	—	2	2	no	—
161	C₂×C₄×C₂₃	—	2	2	no	—
162	C₂×C₄×C₂₃	—	2	2	no	—
163	C₂×C₄×C₂₃	—	4	1	no	—
164	C₂×C₄×C₂₃	—	4	2	no	—
165	C₂×C₄×C₂₃	—	4	2	no	—
166	C₂×C₄×C₂₃	—	4	1	no	—
167	C₂×C₄×C₂₃	—	4	1	no	—
168	C₂×C₄×C₂₃	—	4	1	no	—
169	C₂×C₄×C₂₃	—	4	2	no	—
170	C₂×C₄×C₂₃	—	4	1	no	—
171	C₂×C₄×C₂₃	—	4	1	no	—
172	C₂×C₄×C₂₃	—	4	2	no	—
173	C₂×C₄×C₂₃	—	4	2	no	—
174	C₂×C₄×C₂₃	—	4	1	no	—
175	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
176	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
177	C₂×C₄×C₂₃	C₂×SD₁₆	1	4	yes	—
178	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
179	C₂×C₄×C₂₃	#135	1	2	yes	—
180	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
181	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
182	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
186	C₈×C₂₃	—	2	4	no	—
187	C₈×C₂₃	—	2	2	no	—
188	C₈×C₂₃	—	4	1	no	—
189	C₂×#475	C₂×C₂₃	1	1	yes	characteristic
190	#2365	C₂×C₂₃	1	1	yes	characteristic
191	Q₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
192	#1990	C₂×C₂₃	1	1	yes	characteristic
193	#1948	C₂×C₂₃	1	1	yes	characteristic
194	#1941	C₂×C₂₃	1	2	yes	—
195	#1920	C₂×C₂₃	1	2	yes	—
196	#1865	C₂×C₂₃	1	2	yes	—
197	C₈×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
198	C₄×#103	C₂×C₂₃	1	2	yes	—
199	C₄×#103	C₂×C₂₃	1	1	yes	characteristic
200	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
201	C₂²×C₂₃	#546	1	1	yes	characteristic
202	C₂²×C₂₃	D₄×Q₈	1	1	yes	characteristic
203	C₂²×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
204	C₂²×C₂₃	C₂×#135	1	1	yes	characteristic
205	C₂²×C₂₃	#577	1	1	yes	characteristic
206	C₂²×C₂₃	#578	1	1	yes	characteristic
207	C₄×C₂₃	—	2	4	no	—
208	C₄×C₂₃	—	2	2	no	—
209	C₄×C₂₃	—	2	4	no	—
210	C₄×C₂₃	—	4	2	no	—
211	C₄×C₂₃	—	4	4	no	—
212	C₄×C₂₃	—	4	2	no	—
213	C₄×C₂₃	—	4	2	no	—
214	C₄×C₂₃	—	8	1	no	—
215	C₄×C₂₃	—	8	2	no	—
216	C₄×C₂₃	C₄×SD₁₆	1	4	yes	—
217	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
218	C₄×C₂₃	#520	1	1	yes	characteristic
219	C₄×C₂₃	#552	1	1	yes	characteristic
220	C₄×C₂₃	#554	1	1	yes	characteristic
221	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
222	C₂×C₄×Q₈	—	2	1	no	—
223	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
224	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
225	#475	C₄×C₂₃	1	8	yes	complemented
226	C₂×#106	C₂²×C₂₃	1	2	yes	—
227	C₂×#106	C₂²×C₂₃	1	2	yes	—
228	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
229	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
230	C₂×#103	C₂²×C₂₃	1	2	yes	—
231	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
232	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
233	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
234	#391	—	2	1	no	—
235	#389	—	2	2	no	—
236	#389	C₂²×C₂₃	1	2	yes	—
237	#389	C₂²×C₂₃	1	2	yes	—
238	#387	—	2	2	no	—
239	#384	—	2	1	no	—
240	#384	C₂²×C₂₃	1	1	yes	characteristic
241	C₄×C₄⋊C₄	—	2	1	no	—
242	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
243	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
244	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
245	#340	C₂²×C₂₃	1	2	yes	—
246	#340	C₂²×C₂₃	1	1	yes	characteristic
247	#340	C₂²×C₂₃	1	2	yes	—
248	#340	C₂²×C₂₃	1	1	yes	characteristic
249	#340	C₂²×C₂₃	1	2	yes	—
250	#336	C₂²×C₂₃	1	2	yes	—
251	C₂×C₂₃	#2360	1	1	yes	characteristic
252	C₂×C₂₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
253	C₂×C₂₃	#2947	1	1	yes	characteristic
254	C₂×C₂₃	#3298	1	1	yes	characteristic
255	C₂×C₂₃	#3304	1	1	yes	characteristic
256	C₂×C₂₃	Q₈×SD₁₆	1	1	yes	characteristic
257	C₂×C₂₃	#3383	1	1	yes	characteristic
258	C₂²×Q₈	—	2	1	no	—
259	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
262	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
263	C₄⋊Q₈	—	2	2	no	—
264	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
265	C₄×Q₈	—	2	2	no	—
266	C₄×Q₈	—	2	4	no	—
267	C₄×Q₈	—	4	2	no	—
268	C₄×Q₈	—	4	4	no	—
269	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	—	2	2	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	1	no	—
277	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
278	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
279	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
281	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
282	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
283	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
284	C₂×C₄²	—	2	1	no	—
285	C₂×C₄²	—	2	2	no	—
286	C₂×C₄²	—	2	1	no	—
287	C₂×C₄²	—	2	2	no	—
288	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
289	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
290	#106	—	4	2	no	—
291	#106	C₂×C₄×C₂₃	1	8	yes	—
292	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
293	#103	—	2	2	no	—
294	#103	C₂×C₄×C₂₃	1	8	yes	—
295	#103	Q₈×C₂₃	1	4	yes	complemented
296	C₄×C₈	—	2	2	no	—
297	C₄×C₈	—	4	2	no	—
298	#95	—	2	2	no	—
299	#95	—	2	1	no	—
300	#95	—	2	2	no	—
301	#95	—	2	2	no	—
302	#95	D₄×C₂₃	1	2	yes	—
303	C₂₃	#22964	1	1	yes	characteristic, direct product factor, Sylow
304	C₂×Q₈	—	2	4	no	—
305	C₂×Q₈	—	2	1	no	—
306	C₂×Q₈	—	2	1	no	—
307	C₂×Q₈	—	2	1	no	—
308	C₂×Q₈	—	4	2	no	—
309	C₂×Q₈	—	4	2	no	—
310	C₂×Q₈	C₂²×C₄×C₂₃	1	2	yes	—
311	C₂×Q₈	C₂×Q₈×C₂₃	1	2	yes	—
312	C₂×Q₈	#41×C₂₃	1	2	yes	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
320	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
323	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
332	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	4	4	no	—
344	C₄⋊C₄	—	4	4	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	—	8	1	no	—
347	C₄⋊C₄	—	8	2	no	—
348	C₄⋊C₄	SD₁₆×C₂₃	1	4	yes	—
349	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
350	C₄⋊C₄	C₂²×C₄×C₂₃	1	2	yes	—
351	C₄⋊C₄	C₂×Q₈×C₂₃	1	2	yes	—
352	C₄²	—	2	4	no	—
353	C₄²	—	2	2	no	—
354	C₄²	—	2	2	no	—
355	C₄²	—	4	4	no	—
356	C₄²	—	4	4	no	—
357	C₄²	—	4	2	no	—
358	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
359	C₄²	#41×C₂₃	1	2	yes	—
360	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
361	Q₈	—	2	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	2	2	no	—
364	Q₈	—	4	8	no	—
365	Q₈	C₄×Q₈×C₂₃	1	4	yes	complemented
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	4	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
392	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
393	C₂×C₄	C₂×SD₁₆×C₂₃	1	4	yes	—
394	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
395	C₂×C₄	C₂₃×#135	1	2	yes	—
396	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
397	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
398	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
399	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
400	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
401	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
402	C₈	—	2	2	no	—
403	C₈	—	2	4	no	—
404	C₈	—	4	1	no	—
405	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
406	C₂²	C₂₃×#546	1	1	yes	characteristic
407	C₂²	D₄×Q₈×C₂₃	1	1	yes	characteristic
408	C₂²	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
409	C₂²	C₂×C₂₃×#135	1	1	yes	characteristic
410	C₂²	C₂₃×#577	1	1	yes	characteristic
411	C₂²	C₂₃×#578	1	1	yes	characteristic
412	C₄	—	2	2	no	—
413	C₄	—	2	4	no	—
414	C₄	—	2	4	no	—
415	C₄	—	4	4	no	—
416	C₄	—	4	2	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	2	no	—
419	C₄	—	8	1	no	—
420	C₄	—	8	2	no	—
421	C₄	C₄×SD₁₆×C₂₃	1	4	yes	—
422	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
423	C₄	C₂₃×#520	1	1	yes	characteristic
424	C₄	C₂₃×#552	1	1	yes	characteristic
425	C₄	C₂₃×#554	1	1	yes	characteristic
426	C₂	C₂₃×#2360	1	1	yes	characteristic
427	C₂	C₂×C₄×SD₁₆×C₂₃	1	1	yes	characteristic
428	C₂	C₂₃×#2947	1	1	yes	characteristic
429	C₂	C₂₃×#3298	1	1	yes	characteristic
430	C₂	C₂₃×#3304	1	1	yes	characteristic
431	C₂	Q₈×SD₁₆×C₂₃	1	1	yes	characteristic
432	C₂	C₂₃×#3383	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 ◁ C₄×C₄⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
16. C₂₃×#475	C₄	semidirect
79. C₂₃×#103	Q₈	semidirect
143. #22964	C₂₃	direct (N × H)
149. Q₈×C₂₃	C₄×Q₈	semidirect
225. #475	C₄×C₂₃	semidirect
295. #103	Q₈×C₂₃	semidirect
303. C₂₃	#22964	direct (N × H)
365. 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
C₂₃×#2365	2
Q₈×C₄⋊C₄×C₂₃	2
C₂₃×#1990	2
C₂₃×#1948	2
C₂₃×#1941	2
C₂₃×#1920	2
C₂₃×#1865	2
C₈×C₄⋊C₄×C₂₃	2
C₄×C₂₃×#103	2
#22964	23

Subgroups — C23×#22964 #19170590

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22964 #19170590