Subgroups — C₂₃×#61891 #19209517

Order 5888.

2034 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3497	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3399	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3373	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3213	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3205	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3109	C₂	1	4	yes	maximal, complemented
8	C₂₃×#3103	C₂	1	2	yes	maximal, complemented
9	C₂₃×#2983	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×2₋¹⁺⁴	—	2	1	no	—
11	C₂₃×#559	—	2	4	no	—
12	C₂₃×#559	C₂²	1	8	yes	—
13	C₂₃×#550	C₂²	1	4	yes	complemented
14	D₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
15	C₂₃×#548	—	2	4	no	—
16	C₂₃×#537	—	2	1	no	—
17	C₂₃×#537	—	2	2	no	—
18	C₂₃×#532	C₂²	1	2	yes	—
19	C₂₃×#532	C₂²	1	4	yes	—
20	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#124	—	2	1	no	—
22	C₂×C₂₃×#124	C₂²	1	2	yes	—
23	C₂×C₂₃×#123	—	2	2	no	—
24	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₂₃	—	2	1	no	—
27	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
28	C₂₃×#496	C₂²	1	8	yes	complemented
29	C₂₃×#493	C₂²	1	4	yes	—
30	C₂₃×#489	C₂²	1	8	yes	complemented
31	C₂₃×#486	C₂²	1	4	yes	—
32	C₂₃×#481	C₂²	1	4	yes	—
33	C₂₃×#476	C₂²	1	8	yes	complemented
34	C₂₃×#475	C₂²	1	4	yes	complemented
35	C₂₃×#463	C₂²	1	16	yes	complemented
36	C₂₃×#460	C₂²	1	4	yes	complemented
37	C₂₃×#459	C₂²	1	8	yes	—
38	C₂₃×#459	C₂²	1	8	yes	complemented
39	C₂₃×#448	C₂²	1	4	yes	—
40	C₂₃×#447	C₂²	1	8	yes	—
41	C₂₃×#446	C₂²	1	8	yes	complemented
42	C₂₃×#442	C₂²	1	16	yes	complemented
43	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
44	C₂₃×#432	C₂²	1	4	yes	—
45	C₂₃×#431	C₂²	1	2	yes	—
46	C₂₃×2₋¹⁺⁴	—	4	8	no	—
47	C₂×#41×C₂₃	—	2	4	no	—
48	C₂×#41×C₂₃	—	2	4	no	—
49	C₂×#41×C₂₃	—	2	2	no	—
50	C₂²×Q₈×C₂₃	—	2	4	no	—
51	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
52	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₂₃	—	2	1	no	—
54	C₂²×D₄×C₂₃	—	2	1	no	—
55	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
56	C₂³×C₄×C₂₃	—	2	2	no	—
57	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
58	C₂×D₈×C₂₃	C₂³	1	16	yes	—
59	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
60	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
61	C₂₃×#127	—	2	2	no	—
62	C₂₃×#127	C₂³	1	4	yes	—
63	C₂₃×#127	C₂³	1	2	yes	—
64	C₂₃×#127	C₂³	1	4	yes	—
65	C₂₃×#124	—	2	8	no	—
66	C₂₃×#124	—	2	4	no	—
67	C₂₃×#124	—	2	2	no	—
68	C₂₃×#124	—	2	4	no	—
69	C₂₃×#124	C₂³	1	8	yes	—
70	C₂₃×#123	—	2	16	no	—
71	C₂₃×C₂²⋊Q₈	—	2	4	no	—
72	C₂₃×C₂²⋊Q₈	—	2	8	no	—
73	C₂₃×#121	—	2	4	no	—
74	C₂₃×#121	—	2	2	no	—
75	C₂₃×#121	—	2	4	no	—
76	C₂₃×#121	—	2	8	no	—
77	C₂₃×#121	C₂³	1	4	yes	—
78	C₄×Q₈×C₂₃	—	2	2	no	—
79	C₄×Q₈×C₂₃	D₄	1	4	yes	—
80	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
81	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
82	C₄×D₄×C₂₃	—	2	8	no	—
83	C₄×D₄×C₂₃	—	2	4	no	—
84	C₄×D₄×C₂₃	—	2	2	no	—
85	C₄×D₄×C₂₃	—	2	4	no	—
86	C₄×D₄×C₂₃	C₂³	1	4	yes	—
87	C₄×D₄×C₂₃	C₂³	1	8	yes	—
88	C₄×D₄×C₂₃	C₂³	1	8	yes	—
89	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
90	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
92	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
96	C₂×C₄²×C₂₃	—	2	1	no	—
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
99	C₂₃×#106	C₂³	1	8	yes	—
100	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
101	C₂₃×C₄⋊C₈	C₂³	1	8	yes	—
102	C₂₃×#103	C₂³	1	8	yes	—
103	C₂₃×#102	C₂³	1	8	yes	—
104	C₂₃×#102	C₂³	1	16	yes	—
105	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
106	C₂₃×#97	C₂³	1	8	yes	—
107	C₄×C₈×C₂₃	C₂³	1	4	yes	—
108	C₂⁴×C₂₃	—	2	2	no	—
109	#41×C₂₃	—	4	16	no	—
110	#41×C₂₃	—	4	16	no	—
111	#41×C₂₃	—	4	8	no	—
112	C₂×Q₈×C₂₃	—	2	4	no	—
113	C₂×Q₈×C₂₃	—	2	1	no	—
114	C₂×Q₈×C₂₃	—	2	4	no	—
115	C₂×Q₈×C₂₃	—	2	2	no	—
116	C₂×Q₈×C₂₃	—	4	2	no	—
117	C₂×Q₈×C₂₃	—	4	16	no	—
118	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
119	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
120	C₂×D₄×C₂₃	—	2	4	no	—
121	C₂×D₄×C₂₃	—	2	2	no	—
122	C₂×D₄×C₂₃	—	2	4	no	—
123	C₂×D₄×C₂₃	—	2	4	no	—
124	C₂×D₄×C₂₃	—	2	2	no	—
125	C₂×D₄×C₂₃	—	2	4	no	—
126	C₂×D₄×C₂₃	—	2	2	no	—
127	C₂×D₄×C₂₃	—	2	4	no	—
128	C₂×D₄×C₂₃	—	4	4	no	—
129	C₂×D₄×C₂₃	—	4	4	no	—
130	C₂×D₄×C₂₃	—	8	2	no	—
131	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
132	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₂₃	—	2	1	no	—
134	C₂²×C₄×C₂₃	—	2	1	no	—
135	C₂²×C₄×C₂₃	—	2	8	no	—
136	C₂²×C₄×C₂₃	—	2	4	no	—
137	C₂²×C₄×C₂₃	—	2	8	no	—
138	C₂²×C₄×C₂₃	—	8	4	no	—
139	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
141	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
144	SD₁₆×C₂₃	—	4	8	no	—
145	D₈×C₂₃	—	4	16	no	—
146	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
147	C₄⋊C₄×C₂₃	—	2	4	no	—
148	C₄⋊C₄×C₂₃	—	2	1	no	—
149	C₄⋊C₄×C₂₃	—	2	4	no	—
150	C₄⋊C₄×C₂₃	—	2	2	no	—
151	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
152	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
153	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
154	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
155	C₂²⋊C₄×C₂₃	—	2	8	no	—
156	C₂²⋊C₄×C₂₃	—	2	8	no	—
157	C₂²⋊C₄×C₂₃	—	2	8	no	—
158	C₂²⋊C₄×C₂₃	—	2	4	no	—
159	C₂²⋊C₄×C₂₃	—	2	8	no	—
160	C₂²⋊C₄×C₂₃	—	2	4	no	—
161	C₄²×C₂₃	—	2	4	no	—
162	C₄²×C₂₃	C₂×D₄	1	4	yes	—
163	C₄²×C₂₃	C₂×D₄	1	2	yes	—
164	C₄²×C₂₃	C₂⁴	1	4	yes	—
165	C₄²×C₂₃	C₂⁴	1	2	yes	—
166	#61891	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂³×C₂₃	—	2	4	no	—
168	C₂³×C₂₃	—	2	4	no	—
169	C₂³×C₂₃	—	4	4	no	—
170	C₂³×C₂₃	—	4	4	no	—
171	C₂³×C₂₃	—	4	4	no	—
172	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
173	Q₈×C₂₃	—	2	8	no	—
174	Q₈×C₂₃	—	2	4	no	—
175	Q₈×C₂₃	—	4	2	no	—
176	Q₈×C₂₃	—	4	8	no	—
177	Q₈×C₂₃	—	4	4	no	—
178	D₄×C₂₃	—	2	8	no	—
179	D₄×C₂₃	—	2	8	no	—
180	D₄×C₂₃	—	4	4	no	—
181	D₄×C₂₃	—	4	8	no	—
182	D₄×C₂₃	—	4	8	no	—
183	D₄×C₂₃	—	4	8	no	—
184	D₄×C₂₃	—	4	4	no	—
185	D₄×C₂₃	—	8	4	no	—
186	D₄×C₂₃	—	8	2	no	—
187	D₄×C₂₃	—	8	4	no	—
188	C₂×C₄×C₂₃	—	2	4	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	—	4	2	no	—
194	C₂×C₄×C₂₃	—	4	1	no	—
195	C₂×C₄×C₂₃	—	4	16	no	—
196	C₂×C₄×C₂₃	—	4	2	no	—
197	C₂×C₄×C₂₃	—	4	2	no	—
198	C₂×C₄×C₂₃	—	4	8	no	—
199	C₂×C₄×C₂₃	—	8	8	no	—
200	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
201	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
203	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
204	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
205	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
206	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
207	C₈×C₂₃	—	2	8	no	—
208	#3497	C₂×C₂₃	1	1	yes	characteristic, complemented
209	C₂×#559	C₂×C₂₃	1	1	yes	characteristic, complemented
210	#3399	C₂×C₂₃	1	8	yes	complemented
211	#3373	C₂×C₂₃	1	8	yes	complemented
212	#3213	C₂×C₂₃	1	4	yes	complemented
213	#3205	C₂×C₂₃	1	2	yes	complemented
214	#3109	C₂×C₂₃	1	4	yes	complemented
215	#3103	C₂×C₂₃	1	2	yes	complemented
216	#2983	C₂×C₂₃	1	1	yes	characteristic, complemented
217	C₂²×C₂₃	—	2	1	no	—
218	C₂²×C₂₃	—	2	1	no	—
219	C₂²×C₂₃	—	4	4	no	—
220	C₂²×C₂₃	—	4	4	no	—
221	C₂²×C₂₃	—	4	4	no	—
222	C₂²×C₂₃	—	4	4	no	—
223	C₂²×C₂₃	—	4	4	no	—
224	C₂²×C₂₃	—	4	4	no	—
225	C₂²×C₂₃	—	8	8	no	—
226	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
227	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
228	C₄×C₂₃	—	2	4	no	—
229	C₄×C₂₃	—	2	2	no	—
230	C₄×C₂₃	—	2	1	no	—
231	C₄×C₂₃	—	2	4	no	—
232	C₄×C₂₃	—	4	2	no	—
233	C₄×C₂₃	—	4	2	no	—
234	C₄×C₂₃	—	4	4	no	—
235	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
236	C₄×C₂₃	#585	1	2	yes	—
237	C₂×2₋¹⁺⁴	—	2	1	no	—
238	#559	—	2	4	no	—
239	#559	C₂²×C₂₃	1	8	yes	—
240	#550	C₂²×C₂₃	1	4	yes	complemented
241	D₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
242	#548	—	2	4	no	—
243	#537	—	2	1	no	—
244	#537	—	2	2	no	—
245	#532	C₂²×C₂₃	1	2	yes	—
246	#532	C₂²×C₂₃	1	4	yes	—
247	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
248	C₂×#124	—	2	1	no	—
249	C₂×#124	C₂²×C₂₃	1	2	yes	—
250	C₂×#123	—	2	2	no	—
251	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
252	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
253	C₂×C₄×D₄	—	2	1	no	—
254	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
255	#496	C₂²×C₂₃	1	8	yes	complemented
256	#493	C₂²×C₂₃	1	4	yes	—
257	#489	C₂²×C₂₃	1	8	yes	complemented
258	#486	C₂²×C₂₃	1	4	yes	—
259	#481	C₂²×C₂₃	1	4	yes	—
260	#476	C₂²×C₂₃	1	8	yes	complemented
261	#475	C₂²×C₂₃	1	4	yes	complemented
262	#463	C₂²×C₂₃	1	16	yes	complemented
263	#460	C₂²×C₂₃	1	4	yes	complemented
264	#459	C₂²×C₂₃	1	8	yes	—
265	#459	C₂²×C₂₃	1	8	yes	complemented
266	#448	C₂²×C₂₃	1	4	yes	—
267	#447	C₂²×C₂₃	1	8	yes	—
268	#446	C₂²×C₂₃	1	8	yes	complemented
269	#442	C₂²×C₂₃	1	16	yes	complemented
270	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
271	#432	C₂²×C₂₃	1	4	yes	—
272	#431	C₂²×C₂₃	1	2	yes	—
273	C₂×C₂₃	—	2	2	no	—
274	C₂×C₂₃	—	8	4	no	—
275	C₂×C₂₃	—	8	4	no	—
276	C₂×C₂₃	#3487	1	1	yes	characteristic
277	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
278	C₂×C₂₃	#3585	1	1	yes	characteristic
279	2₋¹⁺⁴	—	4	8	no	—
280	C₂×#41	—	2	4	no	—
281	C₂×#41	—	2	4	no	—
282	C₂×#41	—	2	2	no	—
283	C₂²×Q₈	—	2	4	no	—
284	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
285	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
286	C₂²×D₄	—	2	1	no	—
287	C₂²×D₄	—	2	1	no	—
288	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
289	C₂³×C₄	—	2	2	no	—
290	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
291	C₂×D₈	C₂³×C₂₃	1	16	yes	—
292	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
293	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
294	#127	—	2	2	no	—
295	#127	C₂³×C₂₃	1	4	yes	—
296	#127	C₂³×C₂₃	1	4	yes	—
297	#127	C₂³×C₂₃	1	2	yes	—
298	#124	—	2	8	no	—
299	#124	—	2	4	no	—
300	#124	—	2	4	no	—
301	#124	—	2	2	no	—
302	#124	C₂³×C₂₃	1	8	yes	—
303	#123	—	2	16	no	—
304	C₂²⋊Q₈	—	2	4	no	—
305	C₂²⋊Q₈	—	2	8	no	—
306	#121	—	2	4	no	—
307	#121	—	2	2	no	—
308	#121	—	2	4	no	—
309	#121	—	2	8	no	—
310	#121	C₂³×C₂₃	1	4	yes	—
311	C₄×Q₈	—	2	2	no	—
312	C₄×Q₈	D₄×C₂₃	1	4	yes	—
313	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
314	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
315	C₄×D₄	—	2	8	no	—
316	C₄×D₄	—	2	4	no	—
317	C₄×D₄	—	2	4	no	—
318	C₄×D₄	—	2	2	no	—
319	C₄×D₄	C₂³×C₂₃	1	8	yes	—
320	C₄×D₄	C₂³×C₂₃	1	4	yes	—
321	C₄×D₄	C₂³×C₂₃	1	8	yes	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
324	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
326	C₂×C₂²⋊C₄	—	2	2	no	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₄²	—	2	1	no	—
330	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
331	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
332	#106	C₂³×C₂₃	1	8	yes	—
333	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
334	C₄⋊C₈	C₂³×C₂₃	1	8	yes	—
335	#103	C₂³×C₂₃	1	8	yes	—
336	#102	C₂³×C₂₃	1	8	yes	—
337	#102	C₂³×C₂₃	1	16	yes	—
338	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
339	#97	C₂³×C₂₃	1	8	yes	—
340	C₄×C₈	C₂³×C₂₃	1	4	yes	—
341	C₂₃	#61891	1	1	yes	characteristic, direct product factor, Sylow
342	C₂⁴	—	2	2	no	—
343	#41	—	4	16	no	—
344	#41	—	4	16	no	—
345	#41	—	4	8	no	—
346	C₂×Q₈	—	2	4	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	4	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	—	4	2	no	—
351	C₂×Q₈	—	4	16	no	—
352	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
353	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	4	no	—
357	C₂×D₄	—	2	4	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	4	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	4	no	—
362	C₂×D₄	—	4	4	no	—
363	C₂×D₄	—	4	4	no	—
364	C₂×D₄	—	8	2	no	—
365	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
366	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	8	no	—
370	C₂²×C₄	—	2	4	no	—
371	C₂²×C₄	—	2	8	no	—
372	C₂²×C₄	—	8	4	no	—
373	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
374	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
375	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
377	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
378	SD₁₆	—	4	8	no	—
379	D₈	—	4	16	no	—
380	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
381	C₄⋊C₄	—	2	4	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	4	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
386	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
387	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
388	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
389	C₂²⋊C₄	—	2	8	no	—
390	C₂²⋊C₄	—	2	8	no	—
391	C₂²⋊C₄	—	2	8	no	—
392	C₂²⋊C₄	—	2	4	no	—
393	C₂²⋊C₄	—	2	8	no	—
394	C₂²⋊C₄	—	2	4	no	—
395	C₄²	—	2	4	no	—
396	C₄²	C₂×D₄×C₂₃	1	4	yes	—
397	C₄²	C₂×D₄×C₂₃	1	2	yes	—
398	C₄²	C₂⁴×C₂₃	1	2	yes	—
399	C₄²	C₂⁴×C₂₃	1	4	yes	—
400	C₂³	—	2	4	no	—
401	C₂³	—	2	4	no	—
402	C₂³	—	4	4	no	—
403	C₂³	—	4	4	no	—
404	C₂³	—	4	4	no	—
405	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
406	Q₈	—	2	8	no	—
407	Q₈	—	2	4	no	—
408	Q₈	—	4	8	no	—
409	Q₈	—	4	2	no	—
410	Q₈	—	4	4	no	—
411	D₄	—	2	8	no	—
412	D₄	—	2	8	no	—
413	D₄	—	4	4	no	—
414	D₄	—	4	8	no	—
415	D₄	—	4	8	no	—
416	D₄	—	4	8	no	—
417	D₄	—	4	4	no	—
418	D₄	—	8	4	no	—
419	D₄	—	8	4	no	—
420	D₄	—	8	2	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	16	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	8	no	—
432	C₂×C₄	—	8	8	no	—
433	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
434	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
435	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
436	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
437	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
438	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
439	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
440	C₈	—	2	8	no	—
441	C₂²	—	2	1	no	—
442	C₂²	—	2	1	no	—
443	C₂²	—	4	4	no	—
444	C₂²	—	4	4	no	—
445	C₂²	—	4	4	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	4	4	no	—
448	C₂²	—	4	4	no	—
449	C₂²	—	8	8	no	—
450	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
451	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
452	C₄	—	2	1	no	—
453	C₄	—	2	4	no	—
454	C₄	—	2	4	no	—
455	C₄	—	2	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	4	4	no	—
458	C₄	—	4	2	no	—
459	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	4	yes	—
460	C₄	C₂₃×#585	1	2	yes	—
461	C₂	—	2	2	no	—
462	C₂	—	8	4	no	—
463	C₂	—	8	4	no	—
464	C₂	C₂₃×#3487	1	1	yes	characteristic
465	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
466	C₂	C₂₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3497	C₂	semidirect
2. C₂×C₂₃×#559	C₂	semidirect
3. C₂₃×#3399	C₂	semidirect
4. C₂₃×#3373	C₂	semidirect
5. C₂₃×#3213	C₂	semidirect
6. C₂₃×#3205	C₂	semidirect
7. C₂₃×#3109	C₂	semidirect
8. C₂₃×#3103	C₂	semidirect
9. C₂₃×#2983	C₂	semidirect
13. C₂₃×#550	C₂²	semidirect
14. D₄×Q₈×C₂₃	C₂²	semidirect
28. C₂₃×#496	C₂²	semidirect
30. C₂₃×#489	C₂²	semidirect
33. C₂₃×#476	C₂²	semidirect
34. C₂₃×#475	C₂²	semidirect
35. C₂₃×#463	C₂²	semidirect
36. C₂₃×#460	C₂²	semidirect
38. C₂₃×#459	C₂²	semidirect
41. C₂₃×#446	C₂²	semidirect
42. C₂₃×#442	C₂²	semidirect
43. C₄×SD₁₆×C₂₃	C₂²	semidirect
166. #61891	C₂₃	direct (N × H)
208. #3497	C₂×C₂₃	semidirect
209. C₂×#559	C₂×C₂₃	semidirect
210. #3399	C₂×C₂₃	semidirect
211. #3373	C₂×C₂₃	semidirect
212. #3213	C₂×C₂₃	semidirect
213. #3205	C₂×C₂₃	semidirect
214. #3109	C₂×C₂₃	semidirect
215. #3103	C₂×C₂₃	semidirect
216. #2983	C₂×C₂₃	semidirect
240. #550	C₂²×C₂₃	semidirect
241. D₄×Q₈	C₂²×C₂₃	semidirect
255. #496	C₂²×C₂₃	semidirect
257. #489	C₂²×C₂₃	semidirect
260. #476	C₂²×C₂₃	semidirect
261. #475	C₂²×C₂₃	semidirect
262. #463	C₂²×C₂₃	semidirect
263. #460	C₂²×C₂₃	semidirect
265. #459	C₂²×C₂₃	semidirect
268. #446	C₂²×C₂₃	semidirect
269. #442	C₂²×C₂₃	semidirect
270. C₄×SD₁₆	C₂²×C₂₃	semidirect
341. C₂₃	#61891	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×#3497	2
C₂×C₂₃×#559	2
C₂₃×#3399	2
C₂₃×#3373	2
C₂₃×#3213	2
C₂₃×#3205	2
C₂₃×#3109	2
C₂₃×#3103	2
C₂₃×#2983	2
#61891	23

Subgroups — C23×#61891 #19209517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61891 #19209517