Subgroups — C₂₃×#61662 #19209288

Order 5888.

2144 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3550	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3498	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3324	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3304	C₂	1	4	yes	maximal, complemented
5	D₄×SD₁₆×C₂₃	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3278	C₂	1	4	yes	maximal, complemented
7	C₂₃×#3208	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3200	C₂	1	4	yes	maximal, complemented
9	C₂₃×#3107	C₂	1	2	yes	maximal, complemented
10	C₂₃×#3009	C₂	1	4	yes	maximal, complemented
11	C₂₃×#2979	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×H(5,𝔽₂)	C₂²	1	2	yes	—
13	C₂×C₂₃×#137	C₂²	1	4	yes	complemented
14	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
15	C₂₃×#563	—	2	2	no	—
16	C₂₃×#559	—	2	1	no	—
17	Q₈²×C₂₃	—	2	1	no	—
18	C₂₃×#556	C₂²	1	4	yes	complemented
19	C₂₃×#550	C₂²	1	4	yes	—
20	D₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
21	D₄²×C₂₃	—	2	4	no	—
22	D₄²×C₂₃	C₂²	1	4	yes	—
23	C₂₃×#542	—	2	4	no	—
24	C₂₃×#536	C₂²	1	4	yes	complemented
25	C₂₃×#535	—	2	2	no	—
26	C₂₃×#535	C₂²	1	2	yes	—
27	C₂₃×#533	C₂²	1	2	yes	complemented
28	C₂₃×#532	—	2	1	no	—
29	C₂₃×#532	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#127	—	2	1	no	—
31	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
32	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#492	C₂²	1	4	yes	complemented
34	C₂₃×#486	C₂²	1	4	yes	complemented
35	C₂₃×#485	C₂²	1	4	yes	complemented
36	C₂₃×#482	C₂²	1	8	yes	complemented
37	C₂₃×#475	C₂²	1	2	yes	complemented
38	C₂₃×#474	C₂²	1	2	yes	complemented
39	C₂₃×#468	C₂²	1	8	yes	complemented
40	C₂₃×#465	C₂²	1	8	yes	complemented
41	C₂₃×#462	C₂²	1	2	yes	complemented
42	C₂₃×#461	C₂²	1	2	yes	complemented
43	C₂₃×#460	C₂²	1	4	yes	complemented
44	C₂₃×#460	C₂²	1	4	yes	complemented
45	C₂₃×#452	C₂²	1	4	yes	—
46	C₂₃×#452	C₂²	1	8	yes	complemented
47	C₂₃×#450	C₂²	1	8	yes	complemented
48	C₂₃×#450	C₂²	1	8	yes	complemented
49	C₂₃×#448	C₂²	1	8	yes	complemented
50	C₄×Q₁₆×C₂₃	C₂²	1	4	yes	complemented
51	C₄×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
52	C₂₃×#436	C₂²	1	4	yes	complemented
53	C₈×D₄×C₂₃	C₂²	1	4	yes	complemented
54	C₂₃×#431	C₂²	1	2	yes	complemented
55	C₂₃×#418	C₂²	1	4	yes	complemented
56	C₂₃×#416	C₂²	1	4	yes	complemented
57	C₂₃×#408	C₂²	1	4	yes	complemented
58	C₂₃×H(5,𝔽₂)	—	4	4	no	—
59	C₂₃×H(5,𝔽₂)	—	4	4	no	—
60	C₂×#41×C₂₃	—	2	2	no	—
61	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
62	C₂×#41×C₂₃	C₂³	1	4	yes	—
63	C₂×#41×C₂₃	C₂³	1	2	yes	—
64	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
65	C₂²×D₄×C₂₃	—	2	2	no	—
66	C₂²×D₄×C₂₃	—	2	4	no	—
67	C₂²×D₄×C₂₃	—	2	4	no	—
68	C₂²×D₄×C₂₃	—	2	2	no	—
69	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
70	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
71	C₂₃×#137	—	4	8	no	—
72	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
73	C₂×SD₁₆×C₂₃	—	4	8	no	—
74	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
75	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
76	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
77	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
78	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
79	C₂₃×C₄⋊Q₈	—	2	4	no	—
80	C₂₃×C₄⋊Q₈	—	2	4	no	—
81	C₂₃×C₄⋊Q₈	D₄	1	1	yes	characteristic
82	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
83	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
84	C₂₃×#127	—	2	4	no	—
85	C₂₃×#127	—	2	2	no	—
86	C₂₃×#127	—	2	1	no	—
87	C₂₃×#127	—	2	1	no	—
88	C₂₃×#127	D₄	1	1	yes	characteristic
89	C₂₃×#127	C₂³	1	4	yes	—
90	C₂₃×#126	—	2	8	no	—
91	C₂₃×#125	—	2	2	no	—
92	C₂₃×#124	—	2	2	no	—
93	C₂₃×#124	—	2	4	no	—
94	C₂₃×#124	—	2	4	no	—
95	C₂₃×#124	D₄	1	2	yes	—
96	C₂₃×#123	—	2	4	no	—
97	C₂₃×#123	—	2	4	no	—
98	C₂₃×C₂²⋊Q₈	—	2	2	no	—
99	C₂₃×C₂²⋊Q₈	—	2	8	no	—
100	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
101	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
102	C₂₃×#121	—	2	4	no	—
103	C₂₃×#121	—	2	8	no	—
104	C₂₃×#121	D₄	1	4	yes	—
105	C₂₃×#121	C₂³	1	8	yes	—
106	C₂₃×#120	—	2	4	no	—
107	C₂₃×#120	—	2	8	no	—
108	C₄×Q₈×C₂₃	—	2	4	no	—
109	C₄×Q₈×C₂₃	—	2	2	no	—
110	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
111	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
112	C₄×D₄×C₂₃	—	2	4	no	—
113	C₄×D₄×C₂₃	—	2	2	no	—
114	C₄×D₄×C₂₃	—	2	4	no	—
115	C₄×D₄×C₂₃	C₂³	1	2	yes	—
116	C₄×D₄×C₂₃	C₂³	1	4	yes	—
117	C₂₃×#117	—	2	2	no	—
118	C₂₃×#117	—	2	1	no	—
119	C₂₃×#117	C₂³	1	4	yes	—
120	C₂₃×#117	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₂₃	—	2	1	no	—
122	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
123	C₂₃×#107	C₂³	1	4	yes	—
124	C₂₃×#106	C₂³	1	4	yes	—
125	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
126	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
127	C₂₃×#103	C₂³	1	4	yes	—
128	C₂₃×#103	C₂³	1	4	yes	—
129	C₂₃×#103	C₂³	1	4	yes	—
130	C₂₃×#102	C₂³	1	8	yes	—
131	C₂₃×#102	C₂³	1	4	yes	—
132	C₂₃×#102	C₂³	1	8	yes	—
133	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
134	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
135	C₄×C₈×C₂₃	C₂³	1	4	yes	—
136	C₂⁴×C₂₃	—	2	4	no	—
137	C₂⁴×C₂₃	—	2	2	no	—
138	#41×C₂₃	—	2	2	no	—
139	#41×C₂₃	—	2	2	no	—
140	#41×C₂₃	—	2	8	no	—
141	#41×C₂₃	—	4	2	no	—
142	#41×C₂₃	—	4	2	no	—
143	#41×C₂₃	—	4	4	no	—
144	#41×C₂₃	—	4	8	no	—
145	C₂×Q₈×C₂₃	—	2	2	no	—
146	C₂×Q₈×C₂₃	—	2	4	no	—
147	C₂×Q₈×C₂₃	—	4	4	no	—
148	C₂×Q₈×C₂₃	—	4	4	no	—
149	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
150	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
151	C₂×Q₈×C₂₃	C₂⁴	1	1	yes	characteristic
152	C₂×D₄×C₂₃	—	2	1	no	—
153	C₂×D₄×C₂₃	—	2	8	no	—
154	C₂×D₄×C₂₃	—	2	4	no	—
155	C₂×D₄×C₂₃	—	2	8	no	—
156	C₂×D₄×C₂₃	—	2	4	no	—
157	C₂×D₄×C₂₃	—	2	4	no	—
158	C₂×D₄×C₂₃	—	2	2	no	—
159	C₂×D₄×C₂₃	—	2	4	no	—
160	C₂×D₄×C₂₃	—	4	4	no	—
161	C₂×D₄×C₂₃	—	4	2	no	—
162	C₂×D₄×C₂₃	—	4	2	no	—
163	C₂×D₄×C₂₃	—	4	8	no	—
164	C₂×D₄×C₂₃	—	4	16	no	—
165	C₂×D₄×C₂₃	—	8	4	no	—
166	C₂×D₄×C₂₃	—	8	8	no	—
167	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
168	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
169	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
170	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
171	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₂₃	—	2	1	no	—
173	C₂²×C₄×C₂₃	—	2	4	no	—
174	C₂²×C₄×C₂₃	—	2	2	no	—
175	C₂²×C₄×C₂₃	—	2	4	no	—
176	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
179	Q₁₆×C₂₃	—	2	8	no	—
180	SD₁₆×C₂₃	—	2	8	no	—
181	SD₁₆×C₂₃	—	4	8	no	—
182	SD₁₆×C₂₃	—	4	8	no	—
183	M₁₆×C₂₃	—	4	4	no	—
184	C₂×C₈×C₂₃	—	4	4	no	—
185	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
186	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
187	C₄⋊C₄×C₂₃	—	2	2	no	—
188	C₄⋊C₄×C₂₃	—	2	2	no	—
189	C₄⋊C₄×C₂₃	—	2	2	no	—
190	C₄⋊C₄×C₂₃	—	2	2	no	—
191	C₄⋊C₄×C₂₃	—	2	8	no	—
192	C₄⋊C₄×C₂₃	—	2	4	no	—
193	C₄⋊C₄×C₂₃	—	2	4	no	—
194	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
195	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
197	C₂²⋊C₄×C₂₃	—	2	4	no	—
198	C₂²⋊C₄×C₂₃	—	2	1	no	—
199	C₂²⋊C₄×C₂₃	—	2	4	no	—
200	C₂²⋊C₄×C₂₃	—	2	8	no	—
201	C₂²⋊C₄×C₂₃	—	2	2	no	—
202	C₂²⋊C₄×C₂₃	—	2	8	no	—
203	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
204	C₄²×C₂₃	—	2	1	no	—
205	C₄²×C₂₃	—	2	2	no	—
206	C₄²×C₂₃	—	2	4	no	—
207	C₄²×C₂₃	—	2	4	no	—
208	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
209	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
210	C₄²×C₂₃	C₂⁴	1	2	yes	—
211	#61662	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂³×C₂₃	—	2	1	no	—
213	C₂³×C₂₃	—	2	4	no	—
214	C₂³×C₂₃	—	4	8	no	—
215	C₂³×C₂₃	—	4	4	no	—
216	C₂³×C₂₃	—	8	4	no	—
217	C₂³×C₂₃	—	8	8	no	—
218	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
219	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
220	Q₈×C₂₃	—	2	1	no	—
221	Q₈×C₂₃	—	2	4	no	—
222	Q₈×C₂₃	—	2	4	no	—
223	Q₈×C₂₃	—	4	2	no	—
224	Q₈×C₂₃	—	4	4	no	—
225	Q₈×C₂₃	—	8	4	no	—
226	Q₈×C₂₃	H(5,𝔽₂)	1	2	yes	—
227	D₄×C₂₃	—	2	4	no	—
228	D₄×C₂₃	—	2	8	no	—
229	D₄×C₂₃	—	2	1	no	—
230	D₄×C₂₃	—	2	4	no	—
231	D₄×C₂₃	—	2	4	no	—
232	D₄×C₂₃	—	4	2	no	—
233	D₄×C₂₃	—	4	4	no	—
234	D₄×C₂₃	—	4	8	no	—
235	D₄×C₂₃	—	4	8	no	—
236	D₄×C₂₃	—	4	4	no	—
237	D₄×C₂₃	—	4	8	no	—
238	D₄×C₂₃	—	8	4	no	—
239	D₄×C₂₃	—	8	8	no	—
240	D₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
241	C₂×C₄×C₂₃	—	2	4	no	—
242	C₂×C₄×C₂₃	—	2	1	no	—
243	C₂×C₄×C₂₃	—	2	4	no	—
244	C₂×C₄×C₂₃	—	2	4	no	—
245	C₂×C₄×C₂₃	—	2	4	no	—
246	C₂×C₄×C₂₃	—	2	1	no	—
247	C₂×C₄×C₂₃	—	2	1	no	—
248	C₂×C₄×C₂₃	—	2	4	no	—
249	C₂×C₄×C₂₃	—	2	2	no	—
250	C₂×C₄×C₂₃	—	2	4	no	—
251	C₂×C₄×C₂₃	—	4	8	no	—
252	C₂×C₄×C₂₃	—	4	4	no	—
253	C₂×C₄×C₂₃	—	8	4	no	—
254	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
256	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
259	C₈×C₂₃	—	2	4	no	—
260	C₈×C₂₃	—	2	4	no	—
261	#3550	C₂×C₂₃	1	1	yes	characteristic, complemented
262	#3498	C₂×C₂₃	1	1	yes	characteristic
263	#3324	C₂×C₂₃	1	4	yes	complemented
264	#3304	C₂×C₂₃	1	4	yes	complemented
265	D₄×SD₁₆	C₂×C₂₃	1	4	yes	complemented
266	#3278	C₂×C₂₃	1	4	yes	complemented
267	#3208	C₂×C₂₃	1	2	yes	complemented
268	#3200	C₂×C₂₃	1	4	yes	complemented
269	#3107	C₂×C₂₃	1	2	yes	complemented
270	#3009	C₂×C₂₃	1	4	yes	complemented
271	#2979	C₂×C₂₃	1	1	yes	characteristic, complemented
272	C₂²×C₂₃	—	2	4	no	—
273	C₂²×C₂₃	—	2	1	no	—
274	C₂²×C₂₃	—	2	2	no	—
275	C₂²×C₂₃	—	2	2	no	—
276	C₂²×C₂₃	—	2	8	no	—
277	C₂²×C₂₃	—	4	2	no	—
278	C₂²×C₂₃	—	4	4	no	—
279	C₂²×C₂₃	—	4	8	no	—
280	C₂²×C₂₃	—	16	8	no	—
281	C₂²×C₂₃	—	16	4	no	—
282	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
283	C₄×C₂₃	—	2	1	no	—
284	C₄×C₂₃	—	2	2	no	—
285	C₄×C₂₃	—	2	4	no	—
286	C₄×C₂₃	—	2	2	no	—
287	C₄×C₂₃	—	4	4	no	—
288	C₄×C₂₃	—	4	2	no	—
289	C₄×C₂₃	—	4	4	no	—
290	C₄×C₂₃	#577	1	4	yes	—
291	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
292	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
293	C₂×H(5,𝔽₂)	C₂²×C₂₃	1	2	yes	—
294	C₂×#137	C₂²×C₂₃	1	4	yes	complemented
295	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
296	#563	—	2	2	no	—
297	#559	—	2	1	no	—
298	Q₈²	—	2	1	no	—
299	#556	C₂²×C₂₃	1	4	yes	complemented
300	#550	C₂²×C₂₃	1	4	yes	—
301	D₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
302	D₄²	—	2	4	no	—
303	D₄²	C₂²×C₂₃	1	4	yes	—
304	#542	—	2	4	no	—
305	#536	C₂²×C₂₃	1	4	yes	complemented
306	#535	—	2	2	no	—
307	#535	C₂²×C₂₃	1	2	yes	—
308	#533	C₂²×C₂₃	1	2	yes	complemented
309	#532	—	2	1	no	—
310	#532	C₂²×C₂₃	1	1	yes	characteristic
311	C₂×#127	—	2	1	no	—
312	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
313	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
314	#492	C₂²×C₂₃	1	4	yes	complemented
315	#486	C₂²×C₂₃	1	4	yes	complemented
316	#485	C₂²×C₂₃	1	4	yes	complemented
317	#482	C₂²×C₂₃	1	8	yes	complemented
318	#475	C₂²×C₂₃	1	2	yes	complemented
319	#474	C₂²×C₂₃	1	2	yes	complemented
320	#468	C₂²×C₂₃	1	8	yes	complemented
321	#465	C₂²×C₂₃	1	8	yes	complemented
322	#462	C₂²×C₂₃	1	2	yes	complemented
323	#461	C₂²×C₂₃	1	2	yes	complemented
324	#460	C₂²×C₂₃	1	4	yes	complemented
325	#460	C₂²×C₂₃	1	4	yes	complemented
326	#452	C₂²×C₂₃	1	4	yes	—
327	#452	C₂²×C₂₃	1	8	yes	complemented
328	#450	C₂²×C₂₃	1	8	yes	complemented
329	#450	C₂²×C₂₃	1	8	yes	complemented
330	#448	C₂²×C₂₃	1	8	yes	complemented
331	C₄×Q₁₆	C₂²×C₂₃	1	4	yes	complemented
332	C₄×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
333	#436	C₂²×C₂₃	1	4	yes	complemented
334	C₈×D₄	C₂²×C₂₃	1	4	yes	complemented
335	#431	C₂²×C₂₃	1	2	yes	complemented
336	#418	C₂²×C₂₃	1	4	yes	complemented
337	#416	C₂²×C₂₃	1	4	yes	complemented
338	#408	C₂²×C₂₃	1	4	yes	complemented
339	C₂×C₂₃	—	4	4	no	—
340	C₂×C₂₃	—	4	1	no	—
341	C₂×C₂₃	—	4	2	no	—
342	C₂×C₂₃	—	8	4	no	—
343	C₂×C₂₃	#3484	1	1	yes	characteristic
344	C₂×C₂₃	C₂×#577	1	2	yes	—
345	H(5,𝔽₂)	—	4	4	no	—
346	H(5,𝔽₂)	—	4	4	no	—
347	C₂×#41	—	2	2	no	—
348	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
349	C₂×#41	C₂³×C₂₃	1	4	yes	—
350	C₂×#41	C₂³×C₂₃	1	2	yes	—
351	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
352	C₂²×D₄	—	2	2	no	—
353	C₂²×D₄	—	2	4	no	—
354	C₂²×D₄	—	2	4	no	—
355	C₂²×D₄	—	2	2	no	—
356	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
357	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
358	#137	—	4	8	no	—
359	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
360	C₂×SD₁₆	—	4	8	no	—
361	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
362	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
363	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
364	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
365	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
366	C₄⋊Q₈	—	2	4	no	—
367	C₄⋊Q₈	—	2	4	no	—
368	C₄⋊Q₈	D₄×C₂₃	1	1	yes	characteristic
369	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
370	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
371	#127	—	2	2	no	—
372	#127	—	2	4	no	—
373	#127	—	2	1	no	—
374	#127	—	2	1	no	—
375	#127	D₄×C₂₃	1	1	yes	characteristic
376	#127	C₂³×C₂₃	1	4	yes	—
377	#126	—	2	8	no	—
378	#125	—	2	2	no	—
379	#124	—	2	2	no	—
380	#124	—	2	4	no	—
381	#124	—	2	4	no	—
382	#124	D₄×C₂₃	1	2	yes	—
383	#123	—	2	4	no	—
384	#123	—	2	4	no	—
385	C₂²⋊Q₈	—	2	2	no	—
386	C₂²⋊Q₈	—	2	8	no	—
387	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
388	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
389	#121	—	2	4	no	—
390	#121	—	2	8	no	—
391	#121	D₄×C₂₃	1	4	yes	—
392	#121	C₂³×C₂₃	1	8	yes	—
393	#120	—	2	4	no	—
394	#120	—	2	8	no	—
395	C₄×Q₈	—	2	4	no	—
396	C₄×Q₈	—	2	2	no	—
397	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
398	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
399	C₄×D₄	—	2	4	no	—
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	—	2	4	no	—
402	C₄×D₄	C₂³×C₂₃	1	2	yes	—
403	C₄×D₄	C₂³×C₂₃	1	4	yes	—
404	#117	—	2	2	no	—
405	#117	—	2	1	no	—
406	#117	C₂³×C₂₃	1	1	yes	characteristic
407	#117	C₂³×C₂₃	1	4	yes	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
410	#107	C₂³×C₂₃	1	4	yes	—
411	#106	C₂³×C₂₃	1	4	yes	—
412	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
413	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
414	#103	C₂³×C₂₃	1	4	yes	—
415	#103	C₂³×C₂₃	1	4	yes	—
416	#103	C₂³×C₂₃	1	4	yes	—
417	#102	C₂³×C₂₃	1	8	yes	—
418	#102	C₂³×C₂₃	1	4	yes	—
419	#102	C₂³×C₂₃	1	8	yes	—
420	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
421	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
422	C₄×C₈	C₂³×C₂₃	1	4	yes	—
423	C₂₃	#61662	1	1	yes	characteristic, direct product factor, Sylow
424	C₂⁴	—	2	4	no	—
425	C₂⁴	—	2	2	no	—
426	#41	—	2	2	no	—
427	#41	—	2	2	no	—
428	#41	—	2	8	no	—
429	#41	—	4	2	no	—
430	#41	—	4	2	no	—
431	#41	—	4	4	no	—
432	#41	—	4	8	no	—
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	—	2	4	no	—
435	C₂×Q₈	—	4	4	no	—
436	C₂×Q₈	—	4	4	no	—
437	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
438	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
439	C₂×Q₈	C₂⁴×C₂₃	1	1	yes	characteristic
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	—	2	8	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	8	no	—
444	C₂×D₄	—	2	4	no	—
445	C₂×D₄	—	2	4	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	4	4	no	—
449	C₂×D₄	—	4	2	no	—
450	C₂×D₄	—	4	2	no	—
451	C₂×D₄	—	4	8	no	—
452	C₂×D₄	—	4	16	no	—
453	C₂×D₄	—	8	4	no	—
454	C₂×D₄	—	8	8	no	—
455	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
456	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
457	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
458	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
459	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
465	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
466	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
467	Q₁₆	—	2	8	no	—
468	SD₁₆	—	2	8	no	—
469	SD₁₆	—	4	8	no	—
470	SD₁₆	—	4	8	no	—
471	M₁₆	—	4	4	no	—
472	C₂×C₈	—	4	4	no	—
473	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
474	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	—	2	2	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	2	8	no	—
480	C₄⋊C₄	—	2	4	no	—
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
483	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
484	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	1	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	8	no	—
489	C₂²⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
492	C₄²	—	2	1	no	—
493	C₄²	—	2	2	no	—
494	C₄²	—	2	4	no	—
495	C₄²	—	2	4	no	—
496	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
497	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
498	C₄²	C₂⁴×C₂₃	1	2	yes	—
499	C₂³	—	2	1	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	4	8	no	—
502	C₂³	—	4	4	no	—
503	C₂³	—	8	4	no	—
504	C₂³	—	8	8	no	—
505	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
506	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
507	Q₈	—	2	1	no	—
508	Q₈	—	2	4	no	—
509	Q₈	—	2	4	no	—
510	Q₈	—	4	2	no	—
511	Q₈	—	4	4	no	—
512	Q₈	—	8	4	no	—
513	Q₈	C₂₃×H(5,𝔽₂)	1	2	yes	—
514	D₄	—	2	4	no	—
515	D₄	—	2	8	no	—
516	D₄	—	2	1	no	—
517	D₄	—	2	4	no	—
518	D₄	—	2	4	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	4	no	—
521	D₄	—	4	8	no	—
522	D₄	—	4	8	no	—
523	D₄	—	4	4	no	—
524	D₄	—	4	8	no	—
525	D₄	—	8	4	no	—
526	D₄	—	8	8	no	—
527	D₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	1	no	—
534	C₂×C₄	—	2	1	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	4	8	no	—
539	C₂×C₄	—	4	4	no	—
540	C₂×C₄	—	8	4	no	—
541	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
543	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
544	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
545	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
546	C₈	—	2	4	no	—
547	C₈	—	2	4	no	—
548	C₂²	—	2	4	no	—
549	C₂²	—	2	1	no	—
550	C₂²	—	2	2	no	—
551	C₂²	—	2	2	no	—
552	C₂²	—	2	8	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	4	no	—
555	C₂²	—	4	8	no	—
556	C₂²	—	16	8	no	—
557	C₂²	—	16	4	no	—
558	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
559	C₄	—	2	1	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	4	no	—
562	C₄	—	2	2	no	—
563	C₄	—	4	4	no	—
564	C₄	—	4	2	no	—
565	C₄	—	4	4	no	—
566	C₄	C₂₃×#577	1	4	yes	—
567	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
568	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
569	C₂	—	4	1	no	—
570	C₂	—	4	4	no	—
571	C₂	—	4	2	no	—
572	C₂	—	8	4	no	—
573	C₂	C₂₃×#3484	1	1	yes	characteristic
574	C₂	C₂×C₂₃×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂₃×#3498	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#532 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3550	C₂	semidirect
3. C₂₃×#3324	C₂	semidirect
4. C₂₃×#3304	C₂	semidirect
5. D₄×SD₁₆×C₂₃	C₂	semidirect
6. C₂₃×#3278	C₂	semidirect
7. C₂₃×#3208	C₂	semidirect
8. C₂₃×#3200	C₂	semidirect
9. C₂₃×#3107	C₂	semidirect
10. C₂₃×#3009	C₂	semidirect
11. C₂₃×#2979	C₂	semidirect
13. C₂×C₂₃×#137	C₂²	semidirect
14. C₂²×SD₁₆×C₂₃	C₂²	semidirect
18. C₂₃×#556	C₂²	semidirect
20. D₄×Q₈×C₂₃	C₂²	semidirect
24. C₂₃×#536	C₂²	semidirect
27. C₂₃×#533	C₂²	semidirect
33. C₂₃×#492	C₂²	semidirect
34. C₂₃×#486	C₂²	semidirect
35. C₂₃×#485	C₂²	semidirect
36. C₂₃×#482	C₂²	semidirect
37. C₂₃×#475	C₂²	semidirect
38. C₂₃×#474	C₂²	semidirect
39. C₂₃×#468	C₂²	semidirect
40. C₂₃×#465	C₂²	semidirect
41. C₂₃×#462	C₂²	semidirect
42. C₂₃×#461	C₂²	semidirect
43. C₂₃×#460	C₂²	semidirect
44. C₂₃×#460	C₂²	semidirect
46. C₂₃×#452	C₂²	semidirect
47. C₂₃×#450	C₂²	semidirect
48. C₂₃×#450	C₂²	semidirect
49. C₂₃×#448	C₂²	semidirect
50. C₄×Q₁₆×C₂₃	C₂²	semidirect
51. C₄×SD₁₆×C₂₃	C₂²	semidirect
52. C₂₃×#436	C₂²	semidirect
53. C₈×D₄×C₂₃	C₂²	semidirect
54. C₂₃×#431	C₂²	semidirect
55. C₂₃×#418	C₂²	semidirect
56. C₂₃×#416	C₂²	semidirect
57. C₂₃×#408	C₂²	semidirect
211. #61662	C₂₃	direct (N × H)
261. #3550	C₂×C₂₃	semidirect
263. #3324	C₂×C₂₃	semidirect
264. #3304	C₂×C₂₃	semidirect
265. D₄×SD₁₆	C₂×C₂₃	semidirect
266. #3278	C₂×C₂₃	semidirect
267. #3208	C₂×C₂₃	semidirect
268. #3200	C₂×C₂₃	semidirect
269. #3107	C₂×C₂₃	semidirect
270. #3009	C₂×C₂₃	semidirect
271. #2979	C₂×C₂₃	semidirect
294. C₂×#137	C₂²×C₂₃	semidirect
295. C₂²×SD₁₆	C₂²×C₂₃	semidirect
299. #556	C₂²×C₂₃	semidirect
301. D₄×Q₈	C₂²×C₂₃	semidirect
305. #536	C₂²×C₂₃	semidirect
308. #533	C₂²×C₂₃	semidirect
314. #492	C₂²×C₂₃	semidirect
315. #486	C₂²×C₂₃	semidirect
316. #485	C₂²×C₂₃	semidirect
317. #482	C₂²×C₂₃	semidirect
318. #475	C₂²×C₂₃	semidirect
319. #474	C₂²×C₂₃	semidirect
320. #468	C₂²×C₂₃	semidirect
321. #465	C₂²×C₂₃	semidirect
322. #462	C₂²×C₂₃	semidirect
323. #461	C₂²×C₂₃	semidirect
324. #460	C₂²×C₂₃	semidirect
325. #460	C₂²×C₂₃	semidirect
327. #452	C₂²×C₂₃	semidirect
328. #450	C₂²×C₂₃	semidirect
329. #450	C₂²×C₂₃	semidirect
330. #448	C₂²×C₂₃	semidirect
331. C₄×Q₁₆	C₂²×C₂₃	semidirect
332. C₄×SD₁₆	C₂²×C₂₃	semidirect
333. #436	C₂²×C₂₃	semidirect
334. C₈×D₄	C₂²×C₂₃	semidirect
335. #431	C₂²×C₂₃	semidirect
336. #418	C₂²×C₂₃	semidirect
337. #416	C₂²×C₂₃	semidirect
338. #408	C₂²×C₂₃	semidirect
423. C₂₃	#61662	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×#3550	2
C₂₃×#3498	2
C₂₃×#3324	2
C₂₃×#3304	2
D₄×SD₁₆×C₂₃	2
C₂₃×#3278	2
C₂₃×#3208	2
C₂₃×#3200	2
C₂₃×#3107	2
C₂₃×#3009	2
C₂₃×#2979	2
#61662	23

Subgroups — C23×#61662 #19209288

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61662 #19209288