Subgroups — C₂₃×#60799 #19208425

Order 5888.

2262 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3444	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3215	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3213	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3211	C₂	1	8	yes	maximal, complemented
6	C₂₃×#3153	C₂	1	2	yes	maximal, complemented
7	C₂₃×#3140	C₂	1	2	yes	maximal, complemented
8	C₂₃×#3040	C₂	1	4	yes	maximal, complemented
9	C₂₃×#3004	C₂	1	2	yes	maximal, complemented
10	C₂₃×#3003	C₂	1	2	yes	maximal, complemented
11	C₂₃×#2920	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#137	C₂²	1	4	yes	complemented
15	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	4	yes	—
16	C₂₃×#533	C₂²	1	2	yes	complemented
17	C₂₃×#533	C₂²	1	1	yes	characteristic, complemented
18	C₂₃×#532	C₂²	1	1	yes	characteristic, complemented
19	C₂₃×#532	C₂²	1	2	yes	complemented
20	C₂₃×#529	—	2	2	no	—
21	C₂₃×#529	—	2	2	no	—
22	C₂₃×#529	C₂²	1	4	yes	complemented
23	C₂×C₂₃×#124	—	2	2	no	—
24	C₂×C₂₃×#124	—	2	2	no	—
25	C₂×C₂₃×#124	—	2	2	no	—
26	C₂×C₂₃×#124	C₂²	1	4	yes	—
27	C₂×C₂₃×#124	C₂²	1	4	yes	—
28	C₂×C₂₃×#124	C₂²	1	2	yes	—
29	C₂×C₂₃×#124	C₂²	1	2	yes	—
30	C₂₃×#525	—	2	2	no	—
31	C₂₃×#525	C₂²	1	2	yes	—
32	C₄×#41×C₂₃	—	2	2	no	—
33	C₄×#41×C₂₃	C₂²	1	2	yes	complemented
34	C₂²×C₂²⋊C₄×C₂₃	—	2	2	no	—
35	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
36	C₂₃×#497	C₂²	1	4	yes	complemented
37	C₂₃×#496	C₂²	1	4	yes	complemented
38	C₂₃×#495	C₂²	1	8	yes	complemented
39	C₂₃×#489	C₂²	1	8	yes	complemented
40	C₂₃×#488	C₂²	1	8	yes	complemented
41	C₂₃×#464	C₂²	1	16	yes	complemented
42	C₂₃×#463	C₂²	1	16	yes	complemented
43	C₂₃×#451	C₂²	1	8	yes	complemented
44	C₂₃×#450	C₂²	1	8	yes	—
45	C₂₃×#448	C₂²	1	8	yes	complemented
46	C₂₃×#447	C₂²	1	8	yes	—
47	C₂₃×#432	C₂²	1	4	yes	complemented
48	C₂₃×#431	C₂²	1	4	yes	complemented
49	C₂₃×#423	C₂²	1	2	yes	complemented
50	C₂₃×#419	C₂²	1	4	yes	complemented
51	C₂₃×#418	C₂²	1	4	yes	—
52	C₂₃×#407	C₂²	1	2	yes	—
53	C₄×M₁₆×C₂₃	C₂²	1	2	yes	complemented
54	C₂⁵×C₂₃	—	2	1	no	—
55	C₂×#41×C₂₃	—	2	2	no	—
56	C₂×#41×C₂₃	C₂³	1	2	yes	—
57	C₂²×Q₈×C₂₃	—	2	2	no	—
58	C₂²×Q₈×C₂₃	—	2	1	no	—
59	C₂²×Q₈×C₂₃	—	4	4	no	—
60	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
61	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
62	C₂²×D₄×C₂₃	—	2	2	no	—
63	C₂²×D₄×C₂₃	—	2	1	no	—
64	C₂²×D₄×C₂₃	—	4	4	no	—
65	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
66	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
67	C₂³×C₄×C₂₃	—	2	1	no	—
68	C₂³×C₄×C₂₃	D₄	1	2	yes	—
69	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂₃×#137	—	4	8	no	—
71	C₂₃×AGL(1,ℤ/8)	—	4	8	no	—
72	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	complemented
73	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
74	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	complemented
75	C₂×D₈×C₂₃	C₂³	1	8	yes	—
76	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
77	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic, complemented
78	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	complemented
79	C₂₃×#127	C₂³	1	1	yes	characteristic, complemented
80	C₂₃×#126	—	2	4	no	—
81	C₂₃×#125	—	2	2	no	—
82	C₂₃×#125	C₂³	1	2	yes	complemented
83	C₂₃×#124	—	2	2	no	—
84	C₂₃×#124	—	2	4	no	—
85	C₂₃×#124	—	2	2	no	—
86	C₂₃×#124	—	2	4	no	—
87	C₂₃×#124	—	2	4	no	—
88	C₂₃×#124	—	2	4	no	—
89	C₂₃×#124	—	2	4	no	—
90	C₂₃×#124	—	2	2	no	—
91	C₂₃×#124	C₂³	1	8	yes	—
92	C₂₃×#124	C₂³	1	4	yes	—
93	C₂₃×#124	C₂³	1	4	yes	—
94	C₂₃×#124	C₂³	1	2	yes	complemented
95	C₂₃×#123	—	2	4	no	—
96	C₂₃×#123	—	2	2	no	—
97	C₂₃×C₂²⋊Q₈	—	2	2	no	—
98	C₂₃×C₂²⋊Q₈	—	2	2	no	—
99	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
100	C₂₃×#121	—	2	2	no	—
101	C₂₃×#121	—	2	2	no	—
102	C₂₃×#121	C₂³	1	4	yes	—
103	C₂₃×#120	—	2	2	no	—
104	C₄×Q₈×C₂₃	—	2	2	no	—
105	C₄×Q₈×C₂₃	—	2	2	no	—
106	C₄×Q₈×C₂₃	C₂³	1	4	yes	complemented
107	C₄×D₄×C₂₃	—	2	4	no	—
108	C₄×D₄×C₂₃	—	2	4	no	—
109	C₄×D₄×C₂₃	—	2	4	no	—
110	C₄×D₄×C₂₃	—	2	2	no	—
111	C₄×D₄×C₂₃	—	2	2	no	—
112	C₄×D₄×C₂₃	C₂³	1	4	yes	complemented
113	C₂₃×#117	—	2	2	no	—
114	C₂₃×#117	—	2	4	no	—
115	C₂₃×#117	—	2	2	no	—
116	C₂₃×#117	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
118	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
119	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
120	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
121	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
122	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
123	C₂×C₄²×C₂₃	—	2	2	no	—
124	C₂×C₄²×C₂₃	D₄	1	2	yes	—
125	C₂×C₄²×C₂₃	D₄	1	2	yes	—
126	C₂×C₄²×C₂₃	D₄	1	2	yes	—
127	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
129	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
130	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
131	C₂₃×C₄⋊C₈	C₂³	1	8	yes	complemented
132	C₂₃×#103	C₂³	1	16	yes	complemented
133	C₂₃×#102	C₂³	1	16	yes	—
134	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
135	C₂₃×#97	C₂³	1	4	yes	complemented
136	C₄×C₈×C₂₃	C₂³	1	4	yes	complemented
137	C₂⁴×C₂₃	—	2	4	no	—
138	C₂⁴×C₂₃	—	2	2	no	—
139	C₂⁴×C₂₃	—	4	4	no	—
140	C₂⁴×C₂₃	—	4	8	no	—
141	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
142	#41×C₂₃	—	2	8	no	—
143	#41×C₂₃	—	4	8	no	—
144	C₂×Q₈×C₂₃	—	2	2	no	—
145	C₂×Q₈×C₂₃	—	2	4	no	—
146	C₂×Q₈×C₂₃	—	2	2	no	—
147	C₂×Q₈×C₂₃	—	2	8	no	—
148	C₂×Q₈×C₂₃	—	4	8	no	—
149	C₂×Q₈×C₂₃	—	4	4	no	—
150	C₂×Q₈×C₂₃	—	4	8	no	—
151	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
152	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
153	C₂×D₄×C₂₃	—	2	2	no	—
154	C₂×D₄×C₂₃	—	2	4	no	—
155	C₂×D₄×C₂₃	—	2	2	no	—
156	C₂×D₄×C₂₃	—	2	4	no	—
157	C₂×D₄×C₂₃	—	2	2	no	—
158	C₂×D₄×C₂₃	—	2	8	no	—
159	C₂×D₄×C₂₃	—	4	8	no	—
160	C₂×D₄×C₂₃	—	4	4	no	—
161	C₂×D₄×C₂₃	—	4	8	no	—
162	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
163	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₂₃	—	2	4	no	—
165	C₂²×C₄×C₂₃	—	2	4	no	—
166	C₂²×C₄×C₂₃	—	2	2	no	—
167	C₂²×C₄×C₂₃	—	2	2	no	—
168	C₂²×C₄×C₂₃	—	2	4	no	—
169	C₂²×C₄×C₂₃	—	4	4	no	—
170	C₂²×C₄×C₂₃	—	4	4	no	—
171	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
176	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
177	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
179	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
181	Q₁₆×C₂₃	—	4	8	no	—
182	SD₁₆×C₂₃	—	4	8	no	—
183	SD₁₆×C₂₃	—	4	8	no	—
184	D₈×C₂₃	—	4	8	no	—
185	M₁₆×C₂₃	—	2	8	no	—
186	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
187	C₄⋊C₄×C₂₃	—	2	2	no	—
188	C₄⋊C₄×C₂₃	—	2	2	no	—
189	C₄⋊C₄×C₂₃	—	2	4	no	—
190	C₄⋊C₄×C₂₃	—	2	4	no	—
191	C₄⋊C₄×C₂₃	—	2	4	no	—
192	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
193	C₂²⋊C₄×C₂₃	—	2	4	no	—
194	C₂²⋊C₄×C₂₃	—	2	2	no	—
195	C₂²⋊C₄×C₂₃	—	2	8	no	—
196	C₂²⋊C₄×C₂₃	—	2	8	no	—
197	C₂²⋊C₄×C₂₃	—	2	8	no	—
198	C₂²⋊C₄×C₂₃	—	2	8	no	—
199	C₂²⋊C₄×C₂₃	—	2	4	no	—
200	C₂²⋊C₄×C₂₃	—	2	4	no	—
201	C₂²⋊C₄×C₂₃	—	2	4	no	—
202	C₄²×C₂₃	—	2	4	no	—
203	C₄²×C₂₃	—	2	2	no	—
204	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
205	C₄²×C₂₃	C₂×D₄	1	2	yes	—
206	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₂₃	C₂×D₄	1	2	yes	—
208	C₄²×C₂₃	C₂×D₄	1	2	yes	—
209	C₄²×C₂₃	C₂×D₄	1	2	yes	—
210	C₄²×C₂₃	C₂×D₄	1	2	yes	—
211	C₄²×C₂₃	C₂⁴	1	2	yes	—
212	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
213	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
214	#60799	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂³×C₂₃	—	2	2	no	—
216	C₂³×C₂₃	—	2	4	no	—
217	C₂³×C₂₃	—	2	4	no	—
218	C₂³×C₂₃	—	2	2	no	—
219	C₂³×C₂₃	—	4	8	no	—
220	C₂³×C₂₃	—	4	8	no	—
221	C₂³×C₂₃	—	4	8	no	—
222	C₂³×C₂₃	—	4	4	no	—
223	C₂³×C₂₃	—	4	8	no	—
224	C₂³×C₂₃	—	8	16	no	—
225	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
226	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
227	Q₈×C₂₃	—	2	4	no	—
228	Q₈×C₂₃	—	2	8	no	—
229	Q₈×C₂₃	—	4	8	no	—
230	Q₈×C₂₃	—	4	4	no	—
231	Q₈×C₂₃	—	4	4	no	—
232	D₄×C₂₃	—	2	4	no	—
233	D₄×C₂₃	—	2	8	no	—
234	D₄×C₂₃	—	4	8	no	—
235	D₄×C₂₃	—	4	4	no	—
236	D₄×C₂₃	—	4	4	no	—
237	D₄×C₂₃	—	4	8	no	—
238	D₄×C₂₃	—	4	4	no	—
239	C₂×C₄×C₂₃	—	2	2	no	—
240	C₂×C₄×C₂₃	—	2	4	no	—
241	C₂×C₄×C₂₃	—	2	4	no	—
242	C₂×C₄×C₂₃	—	2	4	no	—
243	C₂×C₄×C₂₃	—	2	4	no	—
244	C₂×C₄×C₂₃	—	2	2	no	—
245	C₂×C₄×C₂₃	—	2	4	no	—
246	C₂×C₄×C₂₃	—	2	4	no	—
247	C₂×C₄×C₂₃	—	4	2	no	—
248	C₂×C₄×C₂₃	—	4	2	no	—
249	C₂×C₄×C₂₃	—	4	2	no	—
250	C₂×C₄×C₂₃	—	4	2	no	—
251	C₂×C₄×C₂₃	—	4	8	no	—
252	C₂×C₄×C₂₃	—	4	8	no	—
253	C₂×C₄×C₂₃	—	4	4	no	—
254	C₂×C₄×C₂₃	—	4	4	no	—
255	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
262	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
263	C₈×C₂₃	—	2	8	no	—
264	#3444	C₂×C₂₃	1	1	yes	characteristic, complemented
265	C₂²×#124	C₂×C₂₃	1	1	yes	characteristic, complemented
266	#3215	C₂×C₂₃	1	4	yes	complemented
267	#3213	C₂×C₂₃	1	4	yes	complemented
268	#3211	C₂×C₂₃	1	8	yes	complemented
269	#3153	C₂×C₂₃	1	2	yes	complemented
270	#3140	C₂×C₂₃	1	2	yes	complemented
271	#3040	C₂×C₂₃	1	4	yes	complemented
272	#3004	C₂×C₂₃	1	2	yes	complemented
273	#3003	C₂×C₂₃	1	2	yes	complemented
274	#2920	C₂×C₂₃	1	1	yes	characteristic, complemented
275	C₂²×C₂₃	—	2	8	no	—
276	C₂²×C₂₃	—	2	2	no	—
277	C₂²×C₂₃	—	2	2	no	—
278	C₂²×C₂₃	—	2	2	no	—
279	C₂²×C₂₃	—	4	2	no	—
280	C₂²×C₂₃	—	4	4	no	—
281	C₂²×C₂₃	—	4	4	no	—
282	C₂²×C₂₃	—	4	8	no	—
283	C₂²×C₂₃	—	8	8	no	—
284	C₂²×C₂₃	—	8	16	no	—
285	C₂²×C₂₃	#535	1	2	yes	—
286	C₂²×C₂₃	#575	1	4	yes	—
287	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
288	C₄×C₂₃	—	2	2	no	—
289	C₄×C₂₃	—	2	2	no	—
290	C₄×C₂₃	—	2	2	no	—
291	C₄×C₂₃	—	2	2	no	—
292	C₄×C₂₃	—	2	2	no	—
293	C₄×C₂₃	—	4	4	no	—
294	C₄×C₂₃	—	4	2	no	—
295	C₄×C₂₃	—	4	4	no	—
296	C₄×C₂₃	#535	1	2	yes	—
297	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
298	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
299	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
300	C₂×#137	C₂²×C₂₃	1	4	yes	complemented
301	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	4	yes	—
302	#533	C₂²×C₂₃	1	2	yes	complemented
303	#533	C₂²×C₂₃	1	1	yes	characteristic, complemented
304	#532	C₂²×C₂₃	1	1	yes	characteristic, complemented
305	#532	C₂²×C₂₃	1	2	yes	complemented
306	#529	—	2	2	no	—
307	#529	—	2	2	no	—
308	#529	C₂²×C₂₃	1	4	yes	complemented
309	C₂×#124	—	2	2	no	—
310	C₂×#124	—	2	2	no	—
311	C₂×#124	—	2	2	no	—
312	C₂×#124	C₂²×C₂₃	1	2	yes	—
313	C₂×#124	C₂²×C₂₃	1	4	yes	—
314	C₂×#124	C₂²×C₂₃	1	4	yes	—
315	C₂×#124	C₂²×C₂₃	1	2	yes	—
316	#525	—	2	2	no	—
317	#525	C₂²×C₂₃	1	2	yes	—
318	C₄×#41	—	2	2	no	—
319	C₄×#41	C₂²×C₂₃	1	2	yes	complemented
320	C₂²×C₂²⋊C₄	—	2	2	no	—
321	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
322	#497	C₂²×C₂₃	1	4	yes	complemented
323	#496	C₂²×C₂₃	1	4	yes	complemented
324	#495	C₂²×C₂₃	1	8	yes	complemented
325	#489	C₂²×C₂₃	1	8	yes	complemented
326	#488	C₂²×C₂₃	1	8	yes	complemented
327	#464	C₂²×C₂₃	1	16	yes	complemented
328	#463	C₂²×C₂₃	1	16	yes	complemented
329	#451	C₂²×C₂₃	1	8	yes	complemented
330	#450	C₂²×C₂₃	1	8	yes	—
331	#448	C₂²×C₂₃	1	8	yes	complemented
332	#447	C₂²×C₂₃	1	8	yes	—
333	#432	C₂²×C₂₃	1	4	yes	complemented
334	#431	C₂²×C₂₃	1	4	yes	complemented
335	#423	C₂²×C₂₃	1	2	yes	complemented
336	#419	C₂²×C₂₃	1	4	yes	complemented
337	#418	C₂²×C₂₃	1	4	yes	—
338	#407	C₂²×C₂₃	1	2	yes	—
339	C₄×M₁₆	C₂²×C₂₃	1	2	yes	complemented
340	C₂×C₂₃	—	2	2	no	—
341	C₂×C₂₃	—	2	4	no	—
342	C₂×C₂₃	—	8	2	no	—
343	C₂×C₂₃	—	8	4	no	—
344	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
345	C₂×C₂₃	C₂×#575	1	2	yes	—
346	C₂⁵	—	2	1	no	—
347	C₂×#41	—	2	2	no	—
348	C₂×#41	C₂³×C₂₃	1	2	yes	—
349	C₂²×Q₈	—	2	2	no	—
350	C₂²×Q₈	—	2	1	no	—
351	C₂²×Q₈	—	4	4	no	—
352	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
353	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
354	C₂²×D₄	—	2	2	no	—
355	C₂²×D₄	—	2	1	no	—
356	C₂²×D₄	—	4	4	no	—
357	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
358	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
359	C₂³×C₄	—	2	1	no	—
360	C₂³×C₄	D₄×C₂₃	1	2	yes	—
361	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
362	#137	—	4	8	no	—
363	AGL(1,ℤ/8)	—	4	8	no	—
364	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	complemented
365	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
366	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	complemented
367	C₂×D₈	C₂³×C₂₃	1	8	yes	—
368	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
369	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic, complemented
370	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	complemented
371	#127	C₂³×C₂₃	1	1	yes	characteristic, complemented
372	#126	—	2	4	no	—
373	#125	—	2	2	no	—
374	#125	C₂³×C₂₃	1	2	yes	complemented
375	#124	—	2	2	no	—
376	#124	—	2	4	no	—
377	#124	—	2	2	no	—
378	#124	—	2	4	no	—
379	#124	—	2	4	no	—
380	#124	—	2	4	no	—
381	#124	—	2	4	no	—
382	#124	—	2	2	no	—
383	#124	C₂³×C₂₃	1	8	yes	—
384	#124	C₂³×C₂₃	1	4	yes	—
385	#124	C₂³×C₂₃	1	4	yes	—
386	#124	C₂³×C₂₃	1	2	yes	complemented
387	#123	—	2	4	no	—
388	#123	—	2	2	no	—
389	C₂²⋊Q₈	—	2	2	no	—
390	C₂²⋊Q₈	—	2	2	no	—
391	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
392	#121	—	2	2	no	—
393	#121	—	2	2	no	—
394	#121	C₂³×C₂₃	1	4	yes	—
395	#120	—	2	2	no	—
396	C₄×Q₈	—	2	2	no	—
397	C₄×Q₈	—	2	2	no	—
398	C₄×Q₈	C₂³×C₂₃	1	4	yes	complemented
399	C₄×D₄	—	2	4	no	—
400	C₄×D₄	—	2	4	no	—
401	C₄×D₄	—	2	4	no	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	C₂³×C₂₃	1	4	yes	complemented
405	#117	—	2	2	no	—
406	#117	—	2	4	no	—
407	#117	—	2	2	no	—
408	#117	C₂³×C₂₃	1	2	yes	—
409	C₂×C₂²⋊C₄	—	2	4	no	—
410	C₂×C₂²⋊C₄	—	2	4	no	—
411	C₂×C₂²⋊C₄	—	2	4	no	—
412	C₂×C₂²⋊C₄	—	2	4	no	—
413	C₂×C₂²⋊C₄	—	2	4	no	—
414	C₂×C₂²⋊C₄	—	2	4	no	—
415	C₂×C₄²	—	2	2	no	—
416	C₂×C₄²	D₄×C₂₃	1	2	yes	—
417	C₂×C₄²	D₄×C₂₃	1	2	yes	—
418	C₂×C₄²	D₄×C₂₃	1	2	yes	—
419	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
420	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
421	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
423	C₄⋊C₈	C₂³×C₂₃	1	8	yes	complemented
424	#103	C₂³×C₂₃	1	16	yes	complemented
425	#102	C₂³×C₂₃	1	16	yes	—
426	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
427	#97	C₂³×C₂₃	1	4	yes	complemented
428	C₄×C₈	C₂³×C₂₃	1	4	yes	complemented
429	C₂₃	#60799	1	1	yes	characteristic, direct product factor, Sylow
430	C₂⁴	—	2	4	no	—
431	C₂⁴	—	2	2	no	—
432	C₂⁴	—	4	4	no	—
433	C₂⁴	—	4	8	no	—
434	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
435	#41	—	2	8	no	—
436	#41	—	4	8	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	—	2	4	no	—
439	C₂×Q₈	—	2	2	no	—
440	C₂×Q₈	—	2	8	no	—
441	C₂×Q₈	—	4	8	no	—
442	C₂×Q₈	—	4	4	no	—
443	C₂×Q₈	—	4	8	no	—
444	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
445	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	2	4	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	4	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	8	no	—
452	C₂×D₄	—	4	8	no	—
453	C₂×D₄	—	4	4	no	—
454	C₂×D₄	—	4	8	no	—
455	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
456	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
457	C₂²×C₄	—	2	4	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	4	4	no	—
463	C₂²×C₄	—	4	4	no	—
464	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
465	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
468	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
469	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
470	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
471	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
472	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
473	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
474	Q₁₆	—	4	8	no	—
475	SD₁₆	—	4	8	no	—
476	SD₁₆	—	4	8	no	—
477	D₈	—	4	8	no	—
478	M₁₆	—	2	8	no	—
479	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
480	C₄⋊C₄	—	2	2	no	—
481	C₄⋊C₄	—	2	2	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	2	no	—
488	C₂²⋊C₄	—	2	8	no	—
489	C₂²⋊C₄	—	2	8	no	—
490	C₂²⋊C₄	—	2	8	no	—
491	C₂²⋊C₄	—	2	8	no	—
492	C₂²⋊C₄	—	2	4	no	—
493	C₂²⋊C₄	—	2	4	no	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₄²	—	2	4	no	—
496	C₄²	—	2	2	no	—
497	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
498	C₄²	C₂×D₄×C₂₃	1	2	yes	—
499	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
500	C₄²	C₂×D₄×C₂₃	1	2	yes	—
501	C₄²	C₂×D₄×C₂₃	1	2	yes	—
502	C₄²	C₂×D₄×C₂₃	1	2	yes	—
503	C₄²	C₂×D₄×C₂₃	1	2	yes	—
504	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
505	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
506	C₄²	C₂⁴×C₂₃	1	2	yes	—
507	C₂³	—	2	4	no	—
508	C₂³	—	2	2	no	—
509	C₂³	—	2	4	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	4	8	no	—
512	C₂³	—	4	8	no	—
513	C₂³	—	4	8	no	—
514	C₂³	—	4	8	no	—
515	C₂³	—	4	4	no	—
516	C₂³	—	8	16	no	—
517	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
518	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
519	Q₈	—	2	4	no	—
520	Q₈	—	2	8	no	—
521	Q₈	—	4	8	no	—
522	Q₈	—	4	4	no	—
523	Q₈	—	4	4	no	—
524	D₄	—	2	4	no	—
525	D₄	—	2	8	no	—
526	D₄	—	4	8	no	—
527	D₄	—	4	4	no	—
528	D₄	—	4	8	no	—
529	D₄	—	4	4	no	—
530	D₄	—	4	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	2	4	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	2	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	2	no	—
543	C₂×C₄	—	4	8	no	—
544	C₂×C₄	—	4	8	no	—
545	C₂×C₄	—	4	4	no	—
546	C₂×C₄	—	4	4	no	—
547	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
548	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
549	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
550	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
551	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
552	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
553	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
554	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
555	C₈	—	2	8	no	—
556	C₂²	—	2	8	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	4	2	no	—
561	C₂²	—	4	4	no	—
562	C₂²	—	4	4	no	—
563	C₂²	—	4	8	no	—
564	C₂²	—	8	16	no	—
565	C₂²	—	8	8	no	—
566	C₂²	C₂₃×#535	1	2	yes	—
567	C₂²	C₂₃×#575	1	4	yes	—
568	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
569	C₄	—	2	2	no	—
570	C₄	—	2	2	no	—
571	C₄	—	2	2	no	—
572	C₄	—	2	2	no	—
573	C₄	—	2	2	no	—
574	C₄	—	4	4	no	—
575	C₄	—	4	2	no	—
576	C₄	—	4	4	no	—
577	C₄	C₂₃×#535	1	2	yes	—
578	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
579	C₂	—	2	4	no	—
580	C₂	—	2	2	no	—
581	C₂	—	8	4	no	—
582	C₂	—	8	2	no	—
583	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
584	C₂	C₂×C₂₃×#575	1	2	yes	—

	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 ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3444	C₂	semidirect
2. C₂²×C₂₃×#124	C₂	semidirect
3. C₂₃×#3215	C₂	semidirect
4. C₂₃×#3213	C₂	semidirect
5. C₂₃×#3211	C₂	semidirect
6. C₂₃×#3153	C₂	semidirect
7. C₂₃×#3140	C₂	semidirect
8. C₂₃×#3040	C₂	semidirect
9. C₂₃×#3004	C₂	semidirect
10. C₂₃×#3003	C₂	semidirect
11. C₂₃×#2920	C₂	semidirect
14. C₂×C₂₃×#137	C₂²	semidirect
16. C₂₃×#533	C₂²	semidirect
17. C₂₃×#533	C₂²	semidirect
18. C₂₃×#532	C₂²	semidirect
19. C₂₃×#532	C₂²	semidirect
22. C₂₃×#529	C₂²	semidirect
33. C₄×#41×C₂₃	C₂²	semidirect
36. C₂₃×#497	C₂²	semidirect
37. C₂₃×#496	C₂²	semidirect
38. C₂₃×#495	C₂²	semidirect
39. C₂₃×#489	C₂²	semidirect
40. C₂₃×#488	C₂²	semidirect
41. C₂₃×#464	C₂²	semidirect
42. C₂₃×#463	C₂²	semidirect
43. C₂₃×#451	C₂²	semidirect
45. C₂₃×#448	C₂²	semidirect
47. C₂₃×#432	C₂²	semidirect
48. C₂₃×#431	C₂²	semidirect
49. C₂₃×#423	C₂²	semidirect
50. C₂₃×#419	C₂²	semidirect
53. C₄×M₁₆×C₂₃	C₂²	semidirect
72. C₂×Q₁₆×C₂₃	C₂³	semidirect
74. C₂×SD₁₆×C₂₃	C₂³	semidirect
77. C₂₃×C₄⋊Q₈	C₂³	semidirect
78. C₂₃×C₄⋊Q₈	C₂³	semidirect
79. C₂₃×#127	C₂³	semidirect
82. C₂₃×#125	C₂³	semidirect
94. C₂₃×#124	C₂³	semidirect
106. C₄×Q₈×C₂₃	C₂³	semidirect
112. C₄×D₄×C₂₃	C₂³	semidirect
131. C₂₃×C₄⋊C₈	C₂³	semidirect
132. C₂₃×#103	C₂³	semidirect
135. C₂₃×#97	C₂³	semidirect
136. C₄×C₈×C₂₃	C₂³	semidirect
214. #60799	C₂₃	direct (N × H)
264. #3444	C₂×C₂₃	semidirect
265. C₂²×#124	C₂×C₂₃	semidirect
266. #3215	C₂×C₂₃	semidirect
267. #3213	C₂×C₂₃	semidirect
268. #3211	C₂×C₂₃	semidirect
269. #3153	C₂×C₂₃	semidirect
270. #3140	C₂×C₂₃	semidirect
271. #3040	C₂×C₂₃	semidirect
272. #3004	C₂×C₂₃	semidirect
273. #3003	C₂×C₂₃	semidirect
274. #2920	C₂×C₂₃	semidirect
300. C₂×#137	C₂²×C₂₃	semidirect
302. #533	C₂²×C₂₃	semidirect
303. #533	C₂²×C₂₃	semidirect
304. #532	C₂²×C₂₃	semidirect
305. #532	C₂²×C₂₃	semidirect
308. #529	C₂²×C₂₃	semidirect
319. C₄×#41	C₂²×C₂₃	semidirect
322. #497	C₂²×C₂₃	semidirect
323. #496	C₂²×C₂₃	semidirect
324. #495	C₂²×C₂₃	semidirect
325. #489	C₂²×C₂₃	semidirect
326. #488	C₂²×C₂₃	semidirect
327. #464	C₂²×C₂₃	semidirect
328. #463	C₂²×C₂₃	semidirect
329. #451	C₂²×C₂₃	semidirect
331. #448	C₂²×C₂₃	semidirect
333. #432	C₂²×C₂₃	semidirect
334. #431	C₂²×C₂₃	semidirect
335. #423	C₂²×C₂₃	semidirect
336. #419	C₂²×C₂₃	semidirect
339. C₄×M₁₆	C₂²×C₂₃	semidirect
364. C₂×Q₁₆	C₂³×C₂₃	semidirect
366. C₂×SD₁₆	C₂³×C₂₃	semidirect
369. C₄⋊Q₈	C₂³×C₂₃	semidirect
370. C₄⋊Q₈	C₂³×C₂₃	semidirect
371. #127	C₂³×C₂₃	semidirect
374. #125	C₂³×C₂₃	semidirect
386. #124	C₂³×C₂₃	semidirect
398. C₄×Q₈	C₂³×C₂₃	semidirect
404. C₄×D₄	C₂³×C₂₃	semidirect
423. C₄⋊C₈	C₂³×C₂₃	semidirect
424. #103	C₂³×C₂₃	semidirect
427. #97	C₂³×C₂₃	semidirect
428. C₄×C₈	C₂³×C₂₃	semidirect
429. C₂₃	#60799	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×#3444	2
C₂²×C₂₃×#124	2
C₂₃×#3215	2
C₂₃×#3213	2
C₂₃×#3211	2
C₂₃×#3153	2
C₂₃×#3140	2
C₂₃×#3040	2
C₂₃×#3004	2
C₂₃×#3003	2
C₂₃×#2920	2
#60799	23

Subgroups — C23×#60799 #19208425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60799 #19208425