Subgroups — C₂₃×#61717 #19209343

Order 5888.

2014 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3506	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3354	C₂	1	4	yes	maximal, complemented
4	C₂₃×#3353	C₂	1	4	yes	maximal, complemented
5	C₂₃×#3332	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3330	C₂	1	4	yes	maximal, complemented
7	C₂₃×#3229	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3226	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#3194	C₂	1	4	yes	maximal, complemented
10	C₂₃×#3101	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#3100	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂₃×#3092	C₂	1	4	yes	maximal, complemented
13	C₂₃×#2975	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₂₃	—	2	1	no	—
15	C₂₃×#559	—	2	2	no	—
16	C₂₃×#556	C₂²	1	2	yes	complemented
17	C₂₃×#555	C₂²	1	2	yes	complemented
18	C₂₃×#553	—	2	2	no	—
19	C₂₃×#550	C₂²	1	2	yes	complemented
20	C₂₃×#547	—	2	2	no	—
21	C₂₃×#547	—	2	2	no	—
22	C₂₃×#547	C₂²	1	4	yes	—
23	C₂₃×#547	C₂²	1	4	yes	—
24	C₂₃×#547	C₂²	1	2	yes	complemented
25	C₂₃×#538	—	2	2	no	—
26	C₂₃×#535	—	2	1	no	—
27	C₂₃×#533	C₂²	1	1	yes	characteristic
28	C₂₃×#532	C₂²	1	1	yes	characteristic
29	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
30	C₂₃×#525	C₂²	1	4	yes	—
31	C₂×C₂₃×#123	—	2	2	no	—
32	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
33	C₂×C₂₃×#121	C₂²	1	2	yes	—
34	C₂₃×#520	—	2	1	no	—
35	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
37	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
38	C₂₃×#500	C₂²	1	2	yes	complemented
39	C₂₃×#498	C₂²	1	2	yes	complemented
40	C₂₃×#487	C₂²	1	2	yes	complemented
41	C₂₃×#486	C₂²	1	2	yes	complemented
42	C₂₃×#484	C₂²	1	8	yes	complemented
43	C₂₃×#483	C₂²	1	4	yes	—
44	C₂₃×#483	C₂²	1	2	yes	—
45	C₂₃×#482	C₂²	1	8	yes	complemented
46	C₂₃×#481	C₂²	1	8	yes	complemented
47	C₂₃×#480	C₂²	1	4	yes	—
48	C₂₃×#480	C₂²	1	2	yes	—
49	C₂₃×#476	C₂²	1	2	yes	complemented
50	C₂₃×#475	C₂²	1	4	yes	complemented
51	C₂₃×#474	C₂²	1	4	yes	complemented
52	C₂₃×#474	C₂²	1	2	yes	complemented
53	C₂₃×#469	C₂²	1	8	yes	complemented
54	C₂₃×#468	C₂²	1	8	yes	complemented
55	C₂₃×#461	C₂²	1	2	yes	complemented
56	C₂₃×#459	C₂²	1	2	yes	complemented
57	C₂₃×#450	C₂²	1	2	yes	—
58	C₂₃×#449	C₂²	1	8	yes	complemented
59	C₂₃×#447	C₂²	1	2	yes	—
60	C₄×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
61	C₄×D₈×C₂₃	C₂²	1	4	yes	complemented
62	C₂₃×#436	C₂²	1	4	yes	complemented
63	C₂₃×#436	C₂²	1	4	yes	complemented
64	C₂₃×#431	C₂²	1	1	yes	characteristic
65	C₂₃×#431	C₂²	1	1	yes	characteristic
66	C₂₃×#428	C₂²	1	8	yes	complemented
67	C₂₃×#417	C₂²	1	8	yes	complemented
68	C₂₃×#407	C₂²	1	4	yes	—
69	C₂×#41×C₂₃	—	2	4	no	—
70	C₂×#41×C₂₃	—	2	4	no	—
71	C₂×#41×C₂₃	—	2	1	no	—
72	C₂×#41×C₂₃	—	4	8	no	—
73	C₂×#41×C₂₃	C₂³	1	4	yes	—
74	C₂²×Q₈×C₂₃	—	2	1	no	—
75	C₂²×D₄×C₂₃	—	2	2	no	—
76	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
77	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₂₃	—	2	2	no	—
79	C₂³×C₄×C₂₃	—	2	1	no	—
80	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
81	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
82	C₂×D₈×C₂₃	C₂³	1	4	yes	—
83	C₂×M₁₆×C₂₃	C₂³	1	8	yes	—
84	C₂₃×C₄⋊Q₈	—	2	1	no	—
85	C₂₃×C₄⋊Q₈	—	2	1	no	—
86	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
88	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
89	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
90	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
91	C₂₃×#127	—	2	2	no	—
92	C₂₃×#127	C₂³	1	1	yes	characteristic
93	C₂₃×#126	—	2	4	no	—
94	C₂₃×#125	—	2	1	no	—
95	C₂₃×#124	—	2	2	no	—
96	C₂₃×#124	—	2	4	no	—
97	C₂₃×#124	—	2	1	no	—
98	C₂₃×#123	—	2	4	no	—
99	C₂₃×#123	—	2	8	no	—
100	C₂₃×#123	—	2	8	no	—
101	C₂₃×#123	—	2	4	no	—
102	C₂₃×C₂²⋊Q₈	—	2	1	no	—
103	C₂₃×C₂²⋊Q₈	—	2	4	no	—
104	C₂₃×C₂²⋊Q₈	—	2	4	no	—
105	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
106	C₂₃×C₂²⋊Q₈	C₂³	1	2	yes	—
107	C₂₃×#121	—	2	2	no	—
108	C₂₃×#121	—	2	2	no	—
109	C₂₃×#121	—	2	4	no	—
110	C₂₃×#121	—	2	1	no	—
111	C₂₃×#121	—	2	4	no	—
112	C₂₃×#121	C₂³	1	8	yes	—
113	C₂₃×#121	C₂³	1	4	yes	—
114	C₂₃×#121	C₂³	1	2	yes	—
115	C₂₃×#120	—	2	4	no	—
116	C₄×Q₈×C₂₃	—	2	2	no	—
117	C₄×Q₈×C₂₃	—	2	1	no	—
118	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
119	C₄×D₄×C₂₃	—	2	4	no	—
120	C₄×D₄×C₂₃	—	2	4	no	—
121	C₄×D₄×C₂₃	—	2	2	no	—
122	C₄×D₄×C₂₃	—	2	1	no	—
123	C₄×D₄×C₂₃	—	2	1	no	—
124	C₄×D₄×C₂₃	—	2	1	no	—
125	C₄×D₄×C₂₃	D₄	1	2	yes	—
126	C₄×D₄×C₂₃	D₄	1	2	yes	—
127	C₄×D₄×C₂₃	C₂³	1	2	yes	—
128	C₄×D₄×C₂₃	C₂³	1	2	yes	—
129	C₄×D₄×C₂₃	C₂³	1	2	yes	—
130	C₄×D₄×C₂₃	C₂³	1	2	yes	—
131	C₄×D₄×C₂₃	C₂³	1	2	yes	—
132	C₂₃×#117	—	2	1	no	—
133	C₂₃×#117	—	2	2	no	—
134	C₂₃×#117	C₂³	1	4	yes	—
135	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
136	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
137	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
138	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
139	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
140	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
141	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
142	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
143	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
144	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
145	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
146	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
147	C₂₃×#107	C₂³	1	4	yes	—
148	C₂₃×#107	C₂³	1	8	yes	—
149	C₂₃×#106	C₂³	1	4	yes	—
150	C₂₃×#106	C₂³	1	8	yes	—
151	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
152	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
153	C₂₃×#103	C₂³	1	4	yes	—
154	C₂₃×#103	C₂³	1	8	yes	—
155	C₂₃×#102	C₂³	1	4	yes	—
156	C₂₃×#102	C₂³	1	8	yes	—
157	C₂₃×#102	C₂³	1	4	yes	—
158	C₂₃×#102	C₂³	1	4	yes	—
159	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
160	C₂₃×C₂²⋊C₈	C₂³	1	2	yes	—
161	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
162	C₄×C₈×C₂₃	C₂³	1	2	yes	—
163	C₄×C₈×C₂₃	C₂³	1	2	yes	—
164	C₂⁴×C₂₃	—	2	1	no	—
165	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
166	#41×C₂₃	—	4	8	no	—
167	#41×C₂₃	—	4	16	no	—
168	#41×C₂₃	—	4	16	no	—
169	#41×C₂₃	—	4	4	no	—
170	C₂×Q₈×C₂₃	—	2	1	no	—
171	C₂×Q₈×C₂₃	—	2	2	no	—
172	C₂×Q₈×C₂₃	—	2	2	no	—
173	C₂×Q₈×C₂₃	—	2	1	no	—
174	C₂×Q₈×C₂₃	—	4	4	no	—
175	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
176	C₂×D₄×C₂₃	—	2	2	no	—
177	C₂×D₄×C₂₃	—	2	8	no	—
178	C₂×D₄×C₂₃	—	2	4	no	—
179	C₂×D₄×C₂₃	—	2	4	no	—
180	C₂×D₄×C₂₃	—	2	4	no	—
181	C₂×D₄×C₂₃	—	2	1	no	—
182	C₂×D₄×C₂₃	—	2	1	no	—
183	C₂×D₄×C₂₃	—	2	1	no	—
184	C₂×D₄×C₂₃	—	4	2	no	—
185	C₂×D₄×C₂₃	—	4	8	no	—
186	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
187	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
188	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
189	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₂₃	—	2	1	no	—
191	C₂²×C₄×C₂₃	—	2	2	no	—
192	C₂²×C₄×C₂₃	—	2	8	no	—
193	C₂²×C₄×C₂₃	—	2	2	no	—
194	C₂²×C₄×C₂₃	—	2	2	no	—
195	C₂²×C₄×C₂₃	—	2	2	no	—
196	C₂²×C₄×C₂₃	—	2	4	no	—
197	C₂²×C₄×C₂₃	—	4	4	no	—
198	C₂²×C₄×C₂₃	—	4	8	no	—
199	C₂²×C₄×C₂₃	—	4	4	no	—
200	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
201	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
202	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
203	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
204	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
205	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
206	SD₁₆×C₂₃	—	4	4	no	—
207	D₈×C₂₃	—	4	4	no	—
208	M₁₆×C₂₃	—	4	8	no	—
209	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
210	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
211	C₄⋊C₄×C₂₃	—	2	1	no	—
212	C₄⋊C₄×C₂₃	—	2	2	no	—
213	C₄⋊C₄×C₂₃	—	2	8	no	—
214	C₄⋊C₄×C₂₃	—	2	1	no	—
215	C₄⋊C₄×C₂₃	—	2	4	no	—
216	C₄⋊C₄×C₂₃	—	2	2	no	—
217	C₄⋊C₄×C₂₃	—	2	2	no	—
218	C₄⋊C₄×C₂₃	—	2	1	no	—
219	C₄⋊C₄×C₂₃	—	2	2	no	—
220	C₄⋊C₄×C₂₃	—	2	1	no	—
221	C₄⋊C₄×C₂₃	—	2	1	no	—
222	C₄⋊C₄×C₂₃	—	2	2	no	—
223	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
224	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
225	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
226	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
227	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
228	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
229	C₂²⋊C₄×C₂₃	—	2	2	no	—
230	C₂²⋊C₄×C₂₃	—	2	4	no	—
231	C₂²⋊C₄×C₂₃	—	2	4	no	—
232	C₂²⋊C₄×C₂₃	—	2	2	no	—
233	C₂²⋊C₄×C₂₃	—	2	2	no	—
234	C₂²⋊C₄×C₂₃	—	2	8	no	—
235	C₂²⋊C₄×C₂₃	—	2	4	no	—
236	C₂²⋊C₄×C₂₃	—	2	2	no	—
237	C₂²⋊C₄×C₂₃	—	2	2	no	—
238	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
239	C₂²⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
240	C₄²×C₂₃	—	2	4	no	—
241	C₄²×C₂₃	—	2	2	no	—
242	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
243	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
244	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
245	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
246	#61717	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
247	C₂³×C₂₃	—	2	4	no	—
248	C₂³×C₂₃	—	2	2	no	—
249	C₂³×C₂₃	—	2	2	no	—
250	C₂³×C₂₃	—	4	2	no	—
251	C₂³×C₂₃	—	4	2	no	—
252	C₂³×C₂₃	—	4	2	no	—
253	C₂³×C₂₃	—	4	2	no	—
254	C₂³×C₂₃	—	4	2	no	—
255	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
256	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
257	Q₈×C₂₃	—	2	4	no	—
258	Q₈×C₂₃	—	4	4	no	—
259	Q₈×C₂₃	—	4	4	no	—
260	Q₈×C₂₃	—	4	2	no	—
261	Q₈×C₂₃	—	8	1	no	—
262	D₄×C₂₃	—	2	4	no	—
263	D₄×C₂₃	—	2	4	no	—
264	D₄×C₂₃	—	2	2	no	—
265	D₄×C₂₃	—	2	4	no	—
266	D₄×C₂₃	—	2	2	no	—
267	D₄×C₂₃	—	4	2	no	—
268	D₄×C₂₃	—	4	16	no	—
269	D₄×C₂₃	—	4	4	no	—
270	D₄×C₂₃	—	4	2	no	—
271	D₄×C₂₃	—	8	1	no	—
272	D₄×C₂₃	—	8	4	no	—
273	D₄×C₂₃	—	8	4	no	—
274	C₂×C₄×C₂₃	—	2	8	no	—
275	C₂×C₄×C₂₃	—	2	2	no	—
276	C₂×C₄×C₂₃	—	2	2	no	—
277	C₂×C₄×C₂₃	—	2	2	no	—
278	C₂×C₄×C₂₃	—	2	2	no	—
279	C₂×C₄×C₂₃	—	2	2	no	—
280	C₂×C₄×C₂₃	—	2	4	no	—
281	C₂×C₄×C₂₃	—	4	4	no	—
282	C₂×C₄×C₂₃	—	4	2	no	—
283	C₂×C₄×C₂₃	—	4	1	no	—
284	C₂×C₄×C₂₃	—	4	16	no	—
285	C₂×C₄×C₂₃	—	4	2	no	—
286	C₂×C₄×C₂₃	—	4	4	no	—
287	C₂×C₄×C₂₃	—	4	4	no	—
288	C₂×C₄×C₂₃	—	4	4	no	—
289	C₂×C₄×C₂₃	—	4	4	no	—
290	C₂×C₄×C₂₃	—	8	4	no	—
291	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
292	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
293	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
294	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
295	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
297	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
298	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
299	C₈×C₂₃	—	2	4	no	—
300	C₈×C₂₃	—	2	4	no	—
301	#3506	C₂×C₂₃	1	1	yes	characteristic, complemented
302	C₂×#547	C₂×C₂₃	1	1	yes	characteristic, complemented
303	#3354	C₂×C₂₃	1	4	yes	complemented
304	#3353	C₂×C₂₃	1	4	yes	complemented
305	#3332	C₂×C₂₃	1	4	yes	complemented
306	#3330	C₂×C₂₃	1	4	yes	complemented
307	#3229	C₂×C₂₃	1	1	yes	characteristic, complemented
308	#3226	C₂×C₂₃	1	1	yes	characteristic, complemented
309	#3194	C₂×C₂₃	1	4	yes	complemented
310	#3101	C₂×C₂₃	1	1	yes	characteristic, complemented
311	#3100	C₂×C₂₃	1	1	yes	characteristic, complemented
312	#3092	C₂×C₂₃	1	4	yes	complemented
313	#2975	C₂×C₂₃	1	1	yes	characteristic, complemented
314	C₂²×C₂₃	—	2	1	no	—
315	C₂²×C₂₃	—	2	1	no	—
316	C₂²×C₂₃	—	2	4	no	—
317	C₂²×C₂₃	—	2	4	no	—
318	C₂²×C₂₃	—	2	4	no	—
319	C₂²×C₂₃	—	4	8	no	—
320	C₂²×C₂₃	—	4	2	no	—
321	C₂²×C₂₃	—	4	2	no	—
322	C₂²×C₂₃	—	4	2	no	—
323	C₂²×C₂₃	—	4	2	no	—
324	C₂²×C₂₃	—	4	2	no	—
325	C₂²×C₂₃	—	4	2	no	—
326	C₂²×C₂₃	—	8	4	no	—
327	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
328	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
329	C₄×C₂₃	—	2	1	no	—
330	C₄×C₂₃	—	2	1	no	—
331	C₄×C₂₃	—	2	1	no	—
332	C₄×C₂₃	—	2	4	no	—
333	C₄×C₂₃	—	4	2	no	—
334	C₄×C₂₃	—	4	2	no	—
335	C₄×C₂₃	—	4	2	no	—
336	C₄×C₂₃	—	4	4	no	—
337	C₄×C₂₃	—	4	2	no	—
338	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
339	C₄×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
340	C₄×C₂₃	#585	1	2	yes	—
341	C₂²×#41	—	2	1	no	—
342	#559	—	2	2	no	—
343	#556	C₂²×C₂₃	1	2	yes	complemented
344	#555	C₂²×C₂₃	1	2	yes	complemented
345	#553	—	2	2	no	—
346	#550	C₂²×C₂₃	1	2	yes	complemented
347	#547	—	2	2	no	—
348	#547	—	2	2	no	—
349	#547	C₂²×C₂₃	1	4	yes	—
350	#547	C₂²×C₂₃	1	4	yes	—
351	#547	C₂²×C₂₃	1	2	yes	complemented
352	#538	—	2	2	no	—
353	#535	—	2	1	no	—
354	#533	C₂²×C₂₃	1	1	yes	characteristic
355	#532	C₂²×C₂₃	1	1	yes	characteristic
356	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
357	#525	C₂²×C₂₃	1	4	yes	—
358	C₂×#123	—	2	2	no	—
359	C₂×C₂²⋊Q₈	—	2	1	no	—
360	C₂×#121	C₂²×C₂₃	1	2	yes	—
361	#520	—	2	1	no	—
362	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
363	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
364	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
365	#500	C₂²×C₂₃	1	2	yes	complemented
366	#498	C₂²×C₂₃	1	2	yes	complemented
367	#487	C₂²×C₂₃	1	2	yes	complemented
368	#486	C₂²×C₂₃	1	2	yes	complemented
369	#484	C₂²×C₂₃	1	8	yes	complemented
370	#483	C₂²×C₂₃	1	2	yes	—
371	#483	C₂²×C₂₃	1	4	yes	—
372	#482	C₂²×C₂₃	1	8	yes	complemented
373	#481	C₂²×C₂₃	1	8	yes	complemented
374	#480	C₂²×C₂₃	1	2	yes	—
375	#480	C₂²×C₂₃	1	4	yes	—
376	#476	C₂²×C₂₃	1	2	yes	complemented
377	#475	C₂²×C₂₃	1	4	yes	complemented
378	#474	C₂²×C₂₃	1	4	yes	complemented
379	#474	C₂²×C₂₃	1	2	yes	complemented
380	#469	C₂²×C₂₃	1	8	yes	complemented
381	#468	C₂²×C₂₃	1	8	yes	complemented
382	#461	C₂²×C₂₃	1	2	yes	complemented
383	#459	C₂²×C₂₃	1	2	yes	complemented
384	#450	C₂²×C₂₃	1	2	yes	—
385	#449	C₂²×C₂₃	1	8	yes	complemented
386	#447	C₂²×C₂₃	1	2	yes	—
387	C₄×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
388	C₄×D₈	C₂²×C₂₃	1	4	yes	complemented
389	#436	C₂²×C₂₃	1	4	yes	complemented
390	#436	C₂²×C₂₃	1	4	yes	complemented
391	#431	C₂²×C₂₃	1	1	yes	characteristic
392	#431	C₂²×C₂₃	1	1	yes	characteristic
393	#428	C₂²×C₂₃	1	8	yes	complemented
394	#417	C₂²×C₂₃	1	8	yes	complemented
395	#407	C₂²×C₂₃	1	4	yes	—
396	C₂×C₂₃	—	2	2	no	—
397	C₂×C₂₃	—	4	4	no	—
398	C₂×C₂₃	—	8	2	no	—
399	C₂×C₂₃	—	8	2	no	—
400	C₂×C₂₃	#3485	1	1	yes	characteristic
401	C₂×C₂₃	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
402	C₂×C₂₃	#3586	1	1	yes	characteristic
403	C₂×#41	—	2	4	no	—
404	C₂×#41	—	2	4	no	—
405	C₂×#41	—	2	1	no	—
406	C₂×#41	—	4	8	no	—
407	C₂×#41	C₂³×C₂₃	1	4	yes	—
408	C₂²×Q₈	—	2	1	no	—
409	C₂²×D₄	—	2	2	no	—
410	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
411	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
412	C₂³×C₄	—	2	2	no	—
413	C₂³×C₄	—	2	1	no	—
414	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
415	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
416	C₂×D₈	C₂³×C₂₃	1	4	yes	—
417	C₂×M₁₆	C₂³×C₂₃	1	8	yes	—
418	C₄⋊Q₈	—	2	1	no	—
419	C₄⋊Q₈	—	2	1	no	—
420	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
421	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
422	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
423	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
424	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
425	#127	—	2	2	no	—
426	#127	C₂³×C₂₃	1	1	yes	characteristic
427	#126	—	2	4	no	—
428	#125	—	2	1	no	—
429	#124	—	2	2	no	—
430	#124	—	2	4	no	—
431	#124	—	2	1	no	—
432	#123	—	2	4	no	—
433	#123	—	2	8	no	—
434	#123	—	2	8	no	—
435	#123	—	2	4	no	—
436	C₂²⋊Q₈	—	2	1	no	—
437	C₂²⋊Q₈	—	2	4	no	—
438	C₂²⋊Q₈	—	2	4	no	—
439	C₂²⋊Q₈	C₂³×C₂₃	1	2	yes	—
440	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
441	#121	—	2	2	no	—
442	#121	—	2	2	no	—
443	#121	—	2	4	no	—
444	#121	—	2	1	no	—
445	#121	—	2	4	no	—
446	#121	C₂³×C₂₃	1	8	yes	—
447	#121	C₂³×C₂₃	1	2	yes	—
448	#121	C₂³×C₂₃	1	4	yes	—
449	#120	—	2	4	no	—
450	C₄×Q₈	—	2	2	no	—
451	C₄×Q₈	—	2	1	no	—
452	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
453	C₄×D₄	—	2	4	no	—
454	C₄×D₄	—	2	4	no	—
455	C₄×D₄	—	2	2	no	—
456	C₄×D₄	—	2	1	no	—
457	C₄×D₄	—	2	1	no	—
458	C₄×D₄	—	2	1	no	—
459	C₄×D₄	D₄×C₂₃	1	2	yes	—
460	C₄×D₄	D₄×C₂₃	1	2	yes	—
461	C₄×D₄	C₂³×C₂₃	1	2	yes	—
462	C₄×D₄	C₂³×C₂₃	1	2	yes	—
463	C₄×D₄	C₂³×C₂₃	1	2	yes	—
464	C₄×D₄	C₂³×C₂₃	1	2	yes	—
465	C₄×D₄	C₂³×C₂₃	1	2	yes	—
466	#117	—	2	1	no	—
467	#117	—	2	2	no	—
468	#117	C₂³×C₂₃	1	4	yes	—
469	C₂×C₄⋊C₄	—	2	2	no	—
470	C₂×C₄⋊C₄	—	2	4	no	—
471	C₂×C₄⋊C₄	—	2	1	no	—
472	C₂×C₄⋊C₄	—	2	2	no	—
473	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
474	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
475	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
476	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
477	C₂×C₂²⋊C₄	—	2	1	no	—
478	C₂×C₂²⋊C₄	—	2	2	no	—
479	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
480	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
481	#107	C₂³×C₂₃	1	4	yes	—
482	#107	C₂³×C₂₃	1	8	yes	—
483	#106	C₂³×C₂₃	1	4	yes	—
484	#106	C₂³×C₂₃	1	8	yes	—
485	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
486	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
487	#103	C₂³×C₂₃	1	4	yes	—
488	#103	C₂³×C₂₃	1	8	yes	—
489	#102	C₂³×C₂₃	1	4	yes	—
490	#102	C₂³×C₂₃	1	8	yes	—
491	#102	C₂³×C₂₃	1	4	yes	—
492	#102	C₂³×C₂₃	1	4	yes	—
493	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
494	C₂²⋊C₈	C₂³×C₂₃	1	2	yes	—
495	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
496	C₄×C₈	C₂³×C₂₃	1	2	yes	—
497	C₄×C₈	C₂³×C₂₃	1	2	yes	—
498	C₂₃	#61717	1	1	yes	characteristic, direct product factor, Sylow
499	C₂⁴	—	2	1	no	—
500	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
501	#41	—	4	8	no	—
502	#41	—	4	16	no	—
503	#41	—	4	16	no	—
504	#41	—	4	4	no	—
505	C₂×Q₈	—	2	1	no	—
506	C₂×Q₈	—	2	2	no	—
507	C₂×Q₈	—	2	2	no	—
508	C₂×Q₈	—	2	1	no	—
509	C₂×Q₈	—	4	4	no	—
510	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
511	C₂×D₄	—	2	2	no	—
512	C₂×D₄	—	2	8	no	—
513	C₂×D₄	—	2	4	no	—
514	C₂×D₄	—	2	4	no	—
515	C₂×D₄	—	2	4	no	—
516	C₂×D₄	—	2	1	no	—
517	C₂×D₄	—	2	1	no	—
518	C₂×D₄	—	2	1	no	—
519	C₂×D₄	—	4	2	no	—
520	C₂×D₄	—	4	8	no	—
521	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
522	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
523	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
524	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	8	no	—
528	C₂²×C₄	—	2	2	no	—
529	C₂²×C₄	—	2	2	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	—	2	4	no	—
532	C₂²×C₄	—	4	4	no	—
533	C₂²×C₄	—	4	8	no	—
534	C₂²×C₄	—	4	4	no	—
535	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
536	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
537	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
538	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
539	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
540	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
541	SD₁₆	—	4	4	no	—
542	D₈	—	4	4	no	—
543	M₁₆	—	4	8	no	—
544	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
545	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	2	no	—
548	C₄⋊C₄	—	2	8	no	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	2	4	no	—
551	C₄⋊C₄	—	2	2	no	—
552	C₄⋊C₄	—	2	2	no	—
553	C₄⋊C₄	—	2	1	no	—
554	C₄⋊C₄	—	2	2	no	—
555	C₄⋊C₄	—	2	1	no	—
556	C₄⋊C₄	—	2	1	no	—
557	C₄⋊C₄	—	2	2	no	—
558	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
559	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
560	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
561	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
562	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
563	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
564	C₂²⋊C₄	—	2	2	no	—
565	C₂²⋊C₄	—	2	4	no	—
566	C₂²⋊C₄	—	2	4	no	—
567	C₂²⋊C₄	—	2	2	no	—
568	C₂²⋊C₄	—	2	2	no	—
569	C₂²⋊C₄	—	2	8	no	—
570	C₂²⋊C₄	—	2	4	no	—
571	C₂²⋊C₄	—	2	2	no	—
572	C₂²⋊C₄	—	2	2	no	—
573	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
574	C₂²⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
575	C₄²	—	2	4	no	—
576	C₄²	—	2	2	no	—
577	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
578	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
579	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
580	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
581	C₂³	—	2	4	no	—
582	C₂³	—	2	2	no	—
583	C₂³	—	2	2	no	—
584	C₂³	—	4	2	no	—
585	C₂³	—	4	2	no	—
586	C₂³	—	4	2	no	—
587	C₂³	—	4	2	no	—
588	C₂³	—	4	2	no	—
589	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
590	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
591	Q₈	—	2	4	no	—
592	Q₈	—	4	4	no	—
593	Q₈	—	4	4	no	—
594	Q₈	—	4	2	no	—
595	Q₈	—	8	1	no	—
596	D₄	—	2	4	no	—
597	D₄	—	2	4	no	—
598	D₄	—	2	2	no	—
599	D₄	—	2	4	no	—
600	D₄	—	2	2	no	—
601	D₄	—	4	2	no	—
602	D₄	—	4	16	no	—
603	D₄	—	4	4	no	—
604	D₄	—	4	2	no	—
605	D₄	—	8	1	no	—
606	D₄	—	8	4	no	—
607	D₄	—	8	4	no	—
608	C₂×C₄	—	2	8	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	4	4	no	—
616	C₂×C₄	—	4	2	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	16	no	—
619	C₂×C₄	—	4	2	no	—
620	C₂×C₄	—	4	4	no	—
621	C₂×C₄	—	4	4	no	—
622	C₂×C₄	—	4	4	no	—
623	C₂×C₄	—	4	4	no	—
624	C₂×C₄	—	8	4	no	—
625	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
626	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
627	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
628	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
629	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
630	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
631	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
632	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
633	C₈	—	2	4	no	—
634	C₈	—	2	4	no	—
635	C₂²	—	2	1	no	—
636	C₂²	—	2	1	no	—
637	C₂²	—	2	4	no	—
638	C₂²	—	2	4	no	—
639	C₂²	—	2	4	no	—
640	C₂²	—	4	8	no	—
641	C₂²	—	4	2	no	—
642	C₂²	—	4	2	no	—
643	C₂²	—	4	2	no	—
644	C₂²	—	4	2	no	—
645	C₂²	—	4	2	no	—
646	C₂²	—	4	2	no	—
647	C₂²	—	8	4	no	—
648	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
649	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
650	C₄	—	2	1	no	—
651	C₄	—	2	4	no	—
652	C₄	—	2	1	no	—
653	C₄	—	2	1	no	—
654	C₄	—	4	2	no	—
655	C₄	—	4	2	no	—
656	C₄	—	4	4	no	—
657	C₄	—	4	2	no	—
658	C₄	—	4	2	no	—
659	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
660	C₄	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
661	C₄	C₂₃×#585	1	2	yes	—
662	C₂	—	2	2	no	—
663	C₂	—	4	4	no	—
664	C₂	—	8	2	no	—
665	C₂	—	8	2	no	—
666	C₂	C₂₃×#3485	1	1	yes	characteristic
667	C₂	C₂²×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
668	C₂	C₂₃×#3586	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×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3506	C₂	semidirect
2. C₂×C₂₃×#547	C₂	semidirect
3. C₂₃×#3354	C₂	semidirect
4. C₂₃×#3353	C₂	semidirect
5. C₂₃×#3332	C₂	semidirect
6. C₂₃×#3330	C₂	semidirect
7. C₂₃×#3229	C₂	semidirect
8. C₂₃×#3226	C₂	semidirect
9. C₂₃×#3194	C₂	semidirect
10. C₂₃×#3101	C₂	semidirect
11. C₂₃×#3100	C₂	semidirect
12. C₂₃×#3092	C₂	semidirect
13. C₂₃×#2975	C₂	semidirect
16. C₂₃×#556	C₂²	semidirect
17. C₂₃×#555	C₂²	semidirect
19. C₂₃×#550	C₂²	semidirect
24. C₂₃×#547	C₂²	semidirect
38. C₂₃×#500	C₂²	semidirect
39. C₂₃×#498	C₂²	semidirect
40. C₂₃×#487	C₂²	semidirect
41. C₂₃×#486	C₂²	semidirect
42. C₂₃×#484	C₂²	semidirect
45. C₂₃×#482	C₂²	semidirect
46. C₂₃×#481	C₂²	semidirect
49. C₂₃×#476	C₂²	semidirect
50. C₂₃×#475	C₂²	semidirect
51. C₂₃×#474	C₂²	semidirect
52. C₂₃×#474	C₂²	semidirect
53. C₂₃×#469	C₂²	semidirect
54. C₂₃×#468	C₂²	semidirect
55. C₂₃×#461	C₂²	semidirect
56. C₂₃×#459	C₂²	semidirect
58. C₂₃×#449	C₂²	semidirect
60. C₄×SD₁₆×C₂₃	C₂²	semidirect
61. C₄×D₈×C₂₃	C₂²	semidirect
62. C₂₃×#436	C₂²	semidirect
63. C₂₃×#436	C₂²	semidirect
66. C₂₃×#428	C₂²	semidirect
67. C₂₃×#417	C₂²	semidirect
246. #61717	C₂₃	direct (N × H)
301. #3506	C₂×C₂₃	semidirect
302. C₂×#547	C₂×C₂₃	semidirect
303. #3354	C₂×C₂₃	semidirect
304. #3353	C₂×C₂₃	semidirect
305. #3332	C₂×C₂₃	semidirect
306. #3330	C₂×C₂₃	semidirect
307. #3229	C₂×C₂₃	semidirect
308. #3226	C₂×C₂₃	semidirect
309. #3194	C₂×C₂₃	semidirect
310. #3101	C₂×C₂₃	semidirect
311. #3100	C₂×C₂₃	semidirect
312. #3092	C₂×C₂₃	semidirect
313. #2975	C₂×C₂₃	semidirect
343. #556	C₂²×C₂₃	semidirect
344. #555	C₂²×C₂₃	semidirect
346. #550	C₂²×C₂₃	semidirect
351. #547	C₂²×C₂₃	semidirect
365. #500	C₂²×C₂₃	semidirect
366. #498	C₂²×C₂₃	semidirect
367. #487	C₂²×C₂₃	semidirect
368. #486	C₂²×C₂₃	semidirect
369. #484	C₂²×C₂₃	semidirect
372. #482	C₂²×C₂₃	semidirect
373. #481	C₂²×C₂₃	semidirect
376. #476	C₂²×C₂₃	semidirect
377. #475	C₂²×C₂₃	semidirect
378. #474	C₂²×C₂₃	semidirect
379. #474	C₂²×C₂₃	semidirect
380. #469	C₂²×C₂₃	semidirect
381. #468	C₂²×C₂₃	semidirect
382. #461	C₂²×C₂₃	semidirect
383. #459	C₂²×C₂₃	semidirect
385. #449	C₂²×C₂₃	semidirect
387. C₄×SD₁₆	C₂²×C₂₃	semidirect
388. C₄×D₈	C₂²×C₂₃	semidirect
389. #436	C₂²×C₂₃	semidirect
390. #436	C₂²×C₂₃	semidirect
393. #428	C₂²×C₂₃	semidirect
394. #417	C₂²×C₂₃	semidirect
498. C₂₃	#61717	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×#3506	2
C₂×C₂₃×#547	2
C₂₃×#3354	2
C₂₃×#3353	2
C₂₃×#3332	2
C₂₃×#3330	2
C₂₃×#3229	2
C₂₃×#3226	2
C₂₃×#3194	2
C₂₃×#3101	2
C₂₃×#3100	2
C₂₃×#3092	2
C₂₃×#2975	2
#61717	23

Subgroups — C23×#61717 #19209343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61717 #19209343