Subgroups — C₂₃×#61303 #19208929

Order 5888.

1840 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3480	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3365	C₂	1	3	yes	maximal, complemented
4	C₂₃×#3364	C₂	1	3	yes	maximal, complemented
5	C₂₃×#3363	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#3362	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#3361	C₂	1	3	yes	maximal, complemented
8	C₂₃×#3360	C₂	1	3	yes	maximal, complemented
9	C₂₃×#3359	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#3358	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#3131	C₂	1	3	yes	maximal, complemented
12	C₂₃×#3076	C₂	1	3	yes	maximal, complemented
13	C₂₃×#3035	C₂	1	3	yes	maximal, complemented
14	C₂×C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₂₃×#135	C₂	1	3	yes	maximal, complemented
16	C₂×C₂₃×#135	C₂²	1	3	yes	—
17	C₂₃×#559	C₂²	1	3	yes	—
18	Q₈²×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#557	C₂²	1	1	yes	characteristic
20	C₂₃×#557	C₂²	1	3	yes	—
21	C₂₃×#556	C₂²	1	3	yes	—
22	C₂₃×#554	—	2	3	no	—
23	C₂₃×#554	C₂²	1	3	yes	—
24	C₂₃×#553	—	2	3	no	—
25	C₂₃×#550	C₂²	1	1	yes	characteristic
26	D₄×Q₈×C₂₃	—	2	1	no	—
27	D₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#548	—	2	1	no	—
29	C₂₃×#533	—	2	3	no	—
30	C₂₃×#533	C₂²	1	3	yes	—
31	C₂₃×#532	C₂²	1	3	yes	—
32	C₂₃×#529	—	2	3	no	—
33	C₄×#41×C₂₃	—	2	1	no	—
34	C₄×#41×C₂₃	C₂²	1	3	yes	—
35	C₄×#41×C₂₃	C₂²	1	3	yes	—
36	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
37	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
38	C₂₃×#488	C₂²	1	6	yes	—
39	C₂₃×#488	C₂²	1	6	yes	—
40	C₂₃×#487	C₂²	1	3	yes	—
41	C₂₃×#487	C₂²	1	3	yes	—
42	C₂₃×#486	C₂²	1	3	yes	—
43	C₂₃×#486	C₂²	1	3	yes	—
44	C₂₃×#479	C₂²	1	6	yes	—
45	C₂₃×#478	C₂²	1	6	yes	—
46	C₂₃×#477	C₂²	1	3	yes	—
47	C₂₃×#476	C₂²	1	3	yes	—
48	C₂₃×#475	C₂²	1	3	yes	—
49	C₂₃×#474	C₂²	1	3	yes	—
50	C₂₃×#464	C₂²	1	6	yes	—
51	C₂₃×#463	C₂²	1	6	yes	—
52	C₂₃×#462	C₂²	1	3	yes	—
53	C₂₃×#461	C₂²	1	3	yes	—
54	C₂₃×#460	C₂²	1	3	yes	—
55	C₂₃×#459	C₂²	1	3	yes	—
56	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
57	C₈×Q₈×C₂₃	C₂²	1	1	yes	characteristic
58	C₈×Q₈×C₂₃	C₂²	1	3	yes	—
59	C₈×Q₈×C₂₃	C₂²	1	3	yes	—
60	C₄×Q₁₆×C₂₃	C₂²	1	3	yes	—
61	C₄×Q₁₆×C₂₃	C₂²	1	3	yes	—
62	C₄×SD₁₆×C₂₃	C₂²	1	3	yes	—
63	C₄×SD₁₆×C₂₃	C₂²	1	3	yes	—
64	C₄×SD₁₆×C₂₃	C₂²	1	3	yes	—
65	C₄×SD₁₆×C₂₃	C₂²	1	3	yes	—
66	C₄×D₈×C₂₃	C₂²	1	3	yes	—
67	C₄×D₈×C₂₃	C₂²	1	3	yes	—
68	C₂₃×#427	C₂²	1	3	yes	—
69	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	—
70	C₂₃×#416	C₂²	1	1	yes	characteristic
71	C₂₃×#416	C₂²	1	3	yes	—
72	C₂₃×#416	C₂²	1	6	yes	—
73	C₂×C₄×C₈×C₂₃	C₂²	1	3	yes	—
74	C₂×#41×C₂₃	—	2	1	no	—
75	C₂×#41×C₂₃	C₂³	1	3	yes	—
76	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
77	C₂²×Q₈×C₂₃	—	2	1	no	—
78	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
79	C₂₃×#135	—	2	12	no	—
80	C₂×Q₁₆×C₂₃	C₂³	1	3	yes	—
81	C₂×SD₁₆×C₂₃	C₂³	1	3	yes	—
82	C₂×SD₁₆×C₂₃	C₂³	1	3	yes	—
83	C₂×D₈×C₂₃	C₂³	1	3	yes	—
84	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
85	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
86	C₂₃×C₄⋊Q₈	—	2	3	no	—
87	C₂₃×C₄⋊Q₈	—	2	3	no	—
88	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
89	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
90	C₂₃×C₄⋊Q₈	C₂³	1	3	yes	—
91	C₂₃×#127	C₂³	1	3	yes	—
92	C₂₃×#126	—	2	6	no	—
93	C₂₃×#125	—	2	3	no	—
94	C₂₃×#125	—	2	6	no	—
95	C₂₃×#125	C₂³	1	6	yes	—
96	C₂₃×#124	—	2	3	no	—
97	C₂₃×#124	C₂³	1	6	yes	—
98	C₂₃×#123	—	2	6	no	—
99	C₂₃×C₂²⋊Q₈	—	2	3	no	—
100	C₂₃×C₂²⋊Q₈	—	2	3	no	—
101	C₂₃×C₂²⋊Q₈	—	2	3	no	—
102	C₂₃×C₂²⋊Q₈	—	2	3	no	—
103	C₂₃×C₂²⋊Q₈	—	2	3	no	—
104	C₂₃×C₂²⋊Q₈	—	2	6	no	—
105	C₂₃×#121	—	2	3	no	—
106	C₂₃×#121	—	2	3	no	—
107	C₂₃×#121	—	2	3	no	—
108	C₄×Q₈×C₂₃	—	2	1	no	—
109	C₄×Q₈×C₂₃	—	2	1	no	—
110	C₄×Q₈×C₂₃	—	2	3	no	—
111	C₄×Q₈×C₂₃	—	2	3	no	—
112	C₄×Q₈×C₂₃	—	2	3	no	—
113	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
114	C₄×Q₈×C₂₃	D₄	1	3	yes	—
115	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
116	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
117	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
118	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
119	C₄×Q₈×C₂₃	C₂³	1	1	yes	characteristic
120	C₄×Q₈×C₂₃	C₂³	1	3	yes	—
121	C₄×D₄×C₂₃	—	2	3	no	—
122	C₄×D₄×C₂₃	—	2	3	no	—
123	C₄×D₄×C₂₃	—	2	3	no	—
124	C₄×D₄×C₂₃	—	2	3	no	—
125	C₄×D₄×C₂₃	—	2	3	no	—
126	C₄×D₄×C₂₃	—	2	3	no	—
127	C₄×D₄×C₂₃	—	2	3	no	—
128	C₄×D₄×C₂₃	C₂³	1	3	yes	—
129	C₄×D₄×C₂₃	C₂³	1	3	yes	—
130	C₄×D₄×C₂₃	C₂³	1	3	yes	—
131	C₄×D₄×C₂₃	C₂³	1	3	yes	—
132	C₂₃×#117	—	2	3	no	—
133	C₂₃×#117	—	2	3	no	—
134	C₂₃×#117	—	2	3	no	—
135	C₂₃×#117	C₂³	1	3	yes	—
136	C₂₃×#117	C₂³	1	1	yes	characteristic
137	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
138	C₂×C₄⋊C₄×C₂₃	D₄	1	3	yes	—
139	C₂×C₄²×C₂₃	—	2	3	no	—
140	C₂×C₄²×C₂₃	—	2	3	no	—
141	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
142	C₂₃×#107	C₂³	1	3	yes	—
143	C₂₃×#107	C₂³	1	3	yes	—
144	C₂₃×#106	C₂³	1	3	yes	—
145	C₂₃×#106	C₂³	1	3	yes	—
146	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
147	C₂₃×C₄⋊C₈	C₂³	1	3	yes	—
148	C₂₃×C₄⋊C₈	C₂³	1	6	yes	—
149	C₂₃×#103	C₂³	1	1	yes	characteristic
150	C₂₃×#103	C₂³	1	1	yes	characteristic
151	C₂₃×#103	C₂³	1	3	yes	—
152	C₂₃×#103	C₂³	1	3	yes	—
153	C₂₃×#103	C₂³	1	6	yes	—
154	C₂₃×#103	C₂³	1	6	yes	—
155	C₂₃×#102	C₂³	1	1	yes	characteristic
156	C₂₃×#102	C₂³	1	1	yes	characteristic
157	C₂₃×#102	C₂³	1	3	yes	—
158	C₂₃×#102	C₂³	1	3	yes	—
159	C₂₃×#102	C₂³	1	6	yes	—
160	C₂₃×#102	C₂³	1	6	yes	—
161	C₄×C₈×C₂₃	C₂³	1	3	yes	—
162	C₄×C₈×C₂₃	C₂³	1	3	yes	—
163	C₄×C₈×C₂₃	C₂³	1	3	yes	—
164	C₄×C₈×C₂₃	C₂³	1	3	yes	—
165	#41×C₂₃	—	2	2	no	—
166	#41×C₂₃	—	2	6	no	—
167	#41×C₂₃	—	2	6	no	—
168	#41×C₂₃	—	4	4	no	—
169	#41×C₂₃	#41	1	4	yes	complemented
170	C₂×Q₈×C₂₃	—	2	4	no	—
171	C₂×Q₈×C₂₃	—	2	3	no	—
172	C₂×Q₈×C₂₃	—	2	3	no	—
173	C₂×Q₈×C₂₃	—	4	4	no	—
174	C₂×Q₈×C₂₃	C₂×D₄	1	3	yes	—
175	C₂×Q₈×C₂₃	C₂×D₄	1	1	yes	characteristic
176	C₂×Q₈×C₂₃	C₂⁴	1	3	yes	—
177	C₂×Q₈×C₂₃	C₂⁴	1	1	yes	characteristic
178	C₂×D₄×C₂₃	—	2	1	no	—
179	C₂×D₄×C₂₃	—	2	3	no	—
180	C₂×D₄×C₂₃	—	2	3	no	—
181	C₂×D₄×C₂₃	—	2	3	no	—
182	C₂×D₄×C₂₃	C₂⁴	1	3	yes	—
183	C₂×D₄×C₂₃	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₂₃	—	2	1	no	—
185	C₂²×C₄×C₂₃	—	2	3	no	—
186	C₂²×C₄×C₂₃	—	2	3	no	—
187	C₂²×C₄×C₂₃	—	2	3	no	—
188	C₂²×C₄×C₂₃	—	2	3	no	—
189	C₂²×C₄×C₂₃	—	2	3	no	—
190	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
191	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
192	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
193	Q₁₆×C₂₃	—	2	6	no	—
194	SD₁₆×C₂₃	—	2	6	no	—
195	SD₁₆×C₂₃	—	2	6	no	—
196	D₈×C₂₃	—	2	6	no	—
197	C₂×C₈×C₂₃	—	2	6	no	—
198	C₂×C₈×C₂₃	—	2	2	no	—
199	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
200	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
201	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
202	C₂×C₈×C₂₃	C₂⁴	1	3	yes	—
203	C₄⋊C₄×C₂₃	—	2	6	no	—
204	C₄⋊C₄×C₂₃	—	2	3	no	—
205	C₄⋊C₄×C₂₃	—	2	3	no	—
206	C₄⋊C₄×C₂₃	—	2	3	no	—
207	C₄⋊C₄×C₂₃	—	2	3	no	—
208	C₄⋊C₄×C₂₃	—	2	3	no	—
209	C₄⋊C₄×C₂₃	—	2	3	no	—
210	C₄⋊C₄×C₂₃	—	2	3	no	—
211	C₄⋊C₄×C₂₃	—	2	3	no	—
212	C₄⋊C₄×C₂₃	—	2	6	no	—
213	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
214	C₄⋊C₄×C₂₃	C₂×D₄	1	3	yes	—
215	C₄⋊C₄×C₂₃	C₂×D₄	1	6	yes	—
216	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
217	C₄⋊C₄×C₂₃	C₂⁴	1	3	yes	—
218	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
219	C₄⋊C₄×C₂₃	C₂⁴	1	1	yes	characteristic
220	C₂²⋊C₄×C₂₃	—	2	1	no	—
221	C₂²⋊C₄×C₂₃	—	2	3	no	—
222	C₂²⋊C₄×C₂₃	—	2	6	no	—
223	C₂²⋊C₄×C₂₃	—	2	3	no	—
224	C₂²⋊C₄×C₂₃	—	2	3	no	—
225	C₂²⋊C₄×C₂₃	—	2	3	no	—
226	C₂²⋊C₄×C₂₃	—	2	3	no	—
227	C₂²⋊C₄×C₂₃	—	2	6	no	—
228	C₄²×C₂₃	—	2	3	no	—
229	C₄²×C₂₃	—	2	3	no	—
230	C₄²×C₂₃	—	2	3	no	—
231	C₄²×C₂₃	—	2	3	no	—
232	C₄²×C₂₃	—	2	1	no	—
233	C₄²×C₂₃	—	2	3	no	—
234	C₄²×C₂₃	C₂×D₄	1	3	yes	—
235	C₄²×C₂₃	C₂×D₄	1	3	yes	—
236	C₄²×C₂₃	C₂⁴	1	3	yes	—
237	C₄²×C₂₃	C₂⁴	1	3	yes	—
238	#61303	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
239	C₂³×C₂₃	—	2	1	no	—
240	C₂³×C₂₃	—	2	3	no	—
241	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
242	Q₈×C₂₃	—	2	6	no	—
243	Q₈×C₂₃	—	2	1	no	—
244	Q₈×C₂₃	—	2	3	no	—
245	Q₈×C₂₃	—	2	3	no	—
246	Q₈×C₂₃	—	4	6	no	—
247	Q₈×C₂₃	—	8	3	no	—
248	Q₈×C₂₃	#135	1	4	yes	complemented
249	Q₈×C₂₃	C₂×#41	1	2	yes	—
250	D₄×C₂₃	—	2	4	no	—
251	D₄×C₂₃	—	2	3	no	—
252	D₄×C₂₃	—	2	1	no	—
253	D₄×C₂₃	—	2	3	no	—
254	D₄×C₂₃	—	4	6	no	—
255	D₄×C₂₃	—	4	6	no	—
256	D₄×C₂₃	—	8	3	no	—
257	D₄×C₂₃	C₂×#41	1	2	yes	—
258	C₂×C₄×C₂₃	—	2	1	no	—
259	C₂×C₄×C₂₃	—	2	4	no	—
260	C₂×C₄×C₂₃	—	2	6	no	—
261	C₂×C₄×C₂₃	—	2	1	no	—
262	C₂×C₄×C₂₃	—	2	1	no	—
263	C₂×C₄×C₂₃	—	2	3	no	—
264	C₂×C₄×C₂₃	—	2	3	no	—
265	C₂×C₄×C₂₃	—	2	3	no	—
266	C₂×C₄×C₂₃	—	4	3	no	—
267	C₂×C₄×C₂₃	—	4	6	no	—
268	C₂×C₄×C₂₃	—	4	6	no	—
269	C₂×C₄×C₂₃	—	4	6	no	—
270	C₂×C₄×C₂₃	—	8	3	no	—
271	C₂×C₄×C₂₃	—	8	3	no	—
272	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
273	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
274	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
275	C₂×C₄×C₂₃	C₂²×D₄	1	3	yes	—
276	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
277	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
278	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
279	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
280	C₈×C₂₃	—	2	3	no	—
281	C₈×C₂₃	—	2	3	no	—
282	C₈×C₂₃	—	2	1	no	—
283	C₈×C₂₃	—	2	1	no	—
284	#3480	C₂×C₂₃	1	1	yes	characteristic, complemented
285	Q₈×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#3365	C₂×C₂₃	1	3	yes	complemented
287	#3364	C₂×C₂₃	1	3	yes	complemented
288	#3363	C₂×C₂₃	1	1	yes	characteristic, complemented
289	#3362	C₂×C₂₃	1	1	yes	characteristic, complemented
290	#3361	C₂×C₂₃	1	3	yes	complemented
291	#3360	C₂×C₂₃	1	3	yes	complemented
292	#3359	C₂×C₂₃	1	1	yes	characteristic, complemented
293	#3358	C₂×C₂₃	1	1	yes	characteristic, complemented
294	#3131	C₂×C₂₃	1	3	yes	complemented
295	#3076	C₂×C₂₃	1	3	yes	complemented
296	#3035	C₂×C₂₃	1	3	yes	complemented
297	C₂×C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
298	C₄×#135	C₂×C₂₃	1	3	yes	complemented
299	C₂²×C₂₃	—	2	1	no	—
300	C₂²×C₂₃	—	2	1	no	—
301	C₂²×C₂₃	—	2	2	no	—
302	C₂²×C₂₃	—	4	1	no	—
303	C₂²×C₂₃	—	4	1	no	—
304	C₂²×C₂₃	—	4	3	no	—
305	C₂²×C₂₃	—	4	3	no	—
306	C₂²×C₂₃	—	4	3	no	—
307	C₂²×C₂₃	#548	1	2	yes	—
308	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
309	C₄×C₂₃	—	2	3	no	—
310	C₄×C₂₃	—	2	3	no	—
311	C₄×C₂₃	—	2	2	no	—
312	C₄×C₂₃	—	2	3	no	—
313	C₄×C₂₃	—	4	1	no	—
314	C₄×C₂₃	—	4	3	no	—
315	C₄×C₂₃	—	4	3	no	—
316	C₄×C₂₃	—	4	1	no	—
317	C₄×C₂₃	—	4	3	no	—
318	C₄×C₂₃	#548	1	2	yes	—
319	C₄×C₂₃	C₂×#135	1	6	yes	—
320	C₄×C₂₃	C₂²×#41	1	1	yes	characteristic
321	C₄×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
322	C₂×#135	C₂²×C₂₃	1	3	yes	—
323	#559	C₂²×C₂₃	1	3	yes	—
324	Q₈²	C₂²×C₂₃	1	1	yes	characteristic
325	#557	C₂²×C₂₃	1	1	yes	characteristic
326	#557	C₂²×C₂₃	1	3	yes	—
327	#556	C₂²×C₂₃	1	3	yes	—
328	#554	—	2	3	no	—
329	#554	C₂²×C₂₃	1	3	yes	—
330	#553	—	2	3	no	—
331	#550	C₂²×C₂₃	1	1	yes	characteristic
332	D₄×Q₈	—	2	1	no	—
333	D₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
334	#548	—	2	1	no	—
335	#533	—	2	3	no	—
336	#533	C₂²×C₂₃	1	3	yes	—
337	#532	C₂²×C₂₃	1	3	yes	—
338	#529	—	2	3	no	—
339	C₄×#41	—	2	1	no	—
340	C₄×#41	C₂²×C₂₃	1	3	yes	—
341	C₄×#41	C₂²×C₂₃	1	3	yes	—
342	C₂×C₄×Q₈	—	2	1	no	—
343	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
344	#488	C₂²×C₂₃	1	6	yes	—
345	#488	C₂²×C₂₃	1	6	yes	—
346	#487	C₂²×C₂₃	1	3	yes	—
347	#487	C₂²×C₂₃	1	3	yes	—
348	#486	C₂²×C₂₃	1	3	yes	—
349	#486	C₂²×C₂₃	1	3	yes	—
350	#479	C₂²×C₂₃	1	6	yes	—
351	#478	C₂²×C₂₃	1	6	yes	—
352	#477	C₂²×C₂₃	1	3	yes	—
353	#476	C₂²×C₂₃	1	3	yes	—
354	#475	C₂²×C₂₃	1	3	yes	—
355	#474	C₂²×C₂₃	1	3	yes	—
356	#464	C₂²×C₂₃	1	6	yes	—
357	#463	C₂²×C₂₃	1	6	yes	—
358	#462	C₂²×C₂₃	1	3	yes	—
359	#461	C₂²×C₂₃	1	3	yes	—
360	#460	C₂²×C₂₃	1	3	yes	—
361	#459	C₂²×C₂₃	1	3	yes	—
362	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
363	C₈×Q₈	C₂²×C₂₃	1	1	yes	characteristic
364	C₈×Q₈	C₂²×C₂₃	1	3	yes	—
365	C₈×Q₈	C₂²×C₂₃	1	3	yes	—
366	C₄×Q₁₆	C₂²×C₂₃	1	3	yes	—
367	C₄×Q₁₆	C₂²×C₂₃	1	3	yes	—
368	C₄×SD₁₆	C₂²×C₂₃	1	3	yes	—
369	C₄×SD₁₆	C₂²×C₂₃	1	3	yes	—
370	C₄×SD₁₆	C₂²×C₂₃	1	3	yes	—
371	C₄×SD₁₆	C₂²×C₂₃	1	3	yes	—
372	C₄×D₈	C₂²×C₂₃	1	3	yes	—
373	C₄×D₈	C₂²×C₂₃	1	3	yes	—
374	#427	C₂²×C₂₃	1	3	yes	—
375	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	—
376	#416	C₂²×C₂₃	1	6	yes	—
377	#416	C₂²×C₂₃	1	3	yes	—
378	#416	C₂²×C₂₃	1	1	yes	characteristic
379	C₂×C₄×C₈	C₂²×C₂₃	1	3	yes	—
380	C₂×C₂₃	—	2	2	no	—
381	C₂×C₂₃	—	4	2	no	—
382	C₂×C₂₃	—	8	3	no	—
383	C₂×C₂₃	C₂×#548	1	1	yes	characteristic
384	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
385	C₂×C₂₃	#3584	1	1	yes	characteristic
386	C₂×#41	—	2	1	no	—
387	C₂×#41	C₂³×C₂₃	1	3	yes	—
388	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
389	C₂²×Q₈	—	2	1	no	—
390	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
391	#135	—	2	12	no	—
392	C₂×Q₁₆	C₂³×C₂₃	1	3	yes	—
393	C₂×SD₁₆	C₂³×C₂₃	1	3	yes	—
394	C₂×SD₁₆	C₂³×C₂₃	1	3	yes	—
395	C₂×D₈	C₂³×C₂₃	1	3	yes	—
396	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
397	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
398	C₄⋊Q₈	—	2	3	no	—
399	C₄⋊Q₈	—	2	3	no	—
400	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
401	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
402	C₄⋊Q₈	C₂³×C₂₃	1	3	yes	—
403	#127	C₂³×C₂₃	1	3	yes	—
404	#126	—	2	6	no	—
405	#125	—	2	3	no	—
406	#125	—	2	6	no	—
407	#125	C₂³×C₂₃	1	6	yes	—
408	#124	—	2	3	no	—
409	#124	C₂³×C₂₃	1	6	yes	—
410	#123	—	2	6	no	—
411	C₂²⋊Q₈	—	2	3	no	—
412	C₂²⋊Q₈	—	2	3	no	—
413	C₂²⋊Q₈	—	2	3	no	—
414	C₂²⋊Q₈	—	2	3	no	—
415	C₂²⋊Q₈	—	2	3	no	—
416	C₂²⋊Q₈	—	2	6	no	—
417	#121	—	2	3	no	—
418	#121	—	2	3	no	—
419	#121	—	2	3	no	—
420	C₄×Q₈	—	2	1	no	—
421	C₄×Q₈	—	2	1	no	—
422	C₄×Q₈	—	2	3	no	—
423	C₄×Q₈	—	2	3	no	—
424	C₄×Q₈	—	2	3	no	—
425	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
426	C₄×Q₈	D₄×C₂₃	1	3	yes	—
427	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
428	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
429	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
430	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
431	C₄×Q₈	C₂³×C₂₃	1	1	yes	characteristic
432	C₄×Q₈	C₂³×C₂₃	1	3	yes	—
433	C₄×D₄	—	2	3	no	—
434	C₄×D₄	—	2	3	no	—
435	C₄×D₄	—	2	3	no	—
436	C₄×D₄	—	2	3	no	—
437	C₄×D₄	—	2	3	no	—
438	C₄×D₄	—	2	3	no	—
439	C₄×D₄	—	2	3	no	—
440	C₄×D₄	C₂³×C₂₃	1	3	yes	—
441	C₄×D₄	C₂³×C₂₃	1	3	yes	—
442	C₄×D₄	C₂³×C₂₃	1	3	yes	—
443	C₄×D₄	C₂³×C₂₃	1	3	yes	—
444	#117	—	2	3	no	—
445	#117	—	2	3	no	—
446	#117	—	2	3	no	—
447	#117	C₂³×C₂₃	1	1	yes	characteristic
448	#117	C₂³×C₂₃	1	3	yes	—
449	C₂×C₄⋊C₄	—	2	3	no	—
450	C₂×C₄⋊C₄	D₄×C₂₃	1	3	yes	—
451	C₂×C₄²	—	2	3	no	—
452	C₂×C₄²	—	2	3	no	—
453	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
454	#107	C₂³×C₂₃	1	3	yes	—
455	#107	C₂³×C₂₃	1	3	yes	—
456	#106	C₂³×C₂₃	1	3	yes	—
457	#106	C₂³×C₂₃	1	3	yes	—
458	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
459	C₄⋊C₈	C₂³×C₂₃	1	3	yes	—
460	C₄⋊C₈	C₂³×C₂₃	1	6	yes	—
461	#103	C₂³×C₂₃	1	1	yes	characteristic
462	#103	C₂³×C₂₃	1	1	yes	characteristic
463	#103	C₂³×C₂₃	1	3	yes	—
464	#103	C₂³×C₂₃	1	3	yes	—
465	#103	C₂³×C₂₃	1	6	yes	—
466	#103	C₂³×C₂₃	1	6	yes	—
467	#102	C₂³×C₂₃	1	1	yes	characteristic
468	#102	C₂³×C₂₃	1	1	yes	characteristic
469	#102	C₂³×C₂₃	1	3	yes	—
470	#102	C₂³×C₂₃	1	3	yes	—
471	#102	C₂³×C₂₃	1	6	yes	—
472	#102	C₂³×C₂₃	1	6	yes	—
473	C₄×C₈	C₂³×C₂₃	1	3	yes	—
474	C₄×C₈	C₂³×C₂₃	1	3	yes	—
475	C₄×C₈	C₂³×C₂₃	1	3	yes	—
476	C₄×C₈	C₂³×C₂₃	1	3	yes	—
477	C₂₃	#61303	1	1	yes	characteristic, direct product factor, Sylow
478	#41	—	2	2	no	—
479	#41	—	2	6	no	—
480	#41	—	2	6	no	—
481	#41	—	4	4	no	—
482	#41	#41×C₂₃	1	4	yes	complemented
483	C₂×Q₈	—	2	4	no	—
484	C₂×Q₈	—	2	3	no	—
485	C₂×Q₈	—	2	3	no	—
486	C₂×Q₈	—	4	4	no	—
487	C₂×Q₈	C₂×D₄×C₂₃	1	3	yes	—
488	C₂×Q₈	C₂×D₄×C₂₃	1	1	yes	characteristic
489	C₂×Q₈	C₂⁴×C₂₃	1	3	yes	—
490	C₂×Q₈	C₂⁴×C₂₃	1	1	yes	characteristic
491	C₂×D₄	—	2	1	no	—
492	C₂×D₄	—	2	3	no	—
493	C₂×D₄	—	2	3	no	—
494	C₂×D₄	—	2	3	no	—
495	C₂×D₄	C₂⁴×C₂₃	1	3	yes	—
496	C₂×D₄	C₂⁴×C₂₃	1	1	yes	characteristic
497	C₂²×C₄	—	2	1	no	—
498	C₂²×C₄	—	2	3	no	—
499	C₂²×C₄	—	2	3	no	—
500	C₂²×C₄	—	2	3	no	—
501	C₂²×C₄	—	2	3	no	—
502	C₂²×C₄	—	2	3	no	—
503	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
504	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
505	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
506	Q₁₆	—	2	6	no	—
507	SD₁₆	—	2	6	no	—
508	SD₁₆	—	2	6	no	—
509	D₈	—	2	6	no	—
510	C₂×C₈	—	2	6	no	—
511	C₂×C₈	—	2	2	no	—
512	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
513	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
514	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
515	C₂×C₈	C₂⁴×C₂₃	1	3	yes	—
516	C₄⋊C₄	—	2	6	no	—
517	C₄⋊C₄	—	2	3	no	—
518	C₄⋊C₄	—	2	3	no	—
519	C₄⋊C₄	—	2	3	no	—
520	C₄⋊C₄	—	2	3	no	—
521	C₄⋊C₄	—	2	3	no	—
522	C₄⋊C₄	—	2	3	no	—
523	C₄⋊C₄	—	2	3	no	—
524	C₄⋊C₄	—	2	3	no	—
525	C₄⋊C₄	—	2	6	no	—
526	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
527	C₄⋊C₄	C₂×D₄×C₂₃	1	3	yes	—
528	C₄⋊C₄	C₂×D₄×C₂₃	1	6	yes	—
529	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
530	C₄⋊C₄	C₂⁴×C₂₃	1	3	yes	—
531	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
532	C₄⋊C₄	C₂⁴×C₂₃	1	1	yes	characteristic
533	C₂²⋊C₄	—	2	1	no	—
534	C₂²⋊C₄	—	2	3	no	—
535	C₂²⋊C₄	—	2	6	no	—
536	C₂²⋊C₄	—	2	3	no	—
537	C₂²⋊C₄	—	2	3	no	—
538	C₂²⋊C₄	—	2	3	no	—
539	C₂²⋊C₄	—	2	3	no	—
540	C₂²⋊C₄	—	2	6	no	—
541	C₄²	—	2	1	no	—
542	C₄²	—	2	3	no	—
543	C₄²	—	2	3	no	—
544	C₄²	—	2	3	no	—
545	C₄²	—	2	3	no	—
546	C₄²	—	2	3	no	—
547	C₄²	C₂×D₄×C₂₃	1	3	yes	—
548	C₄²	C₂×D₄×C₂₃	1	3	yes	—
549	C₄²	C₂⁴×C₂₃	1	3	yes	—
550	C₄²	C₂⁴×C₂₃	1	3	yes	—
551	C₂³	—	2	1	no	—
552	C₂³	—	2	3	no	—
553	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
554	Q₈	—	2	6	no	—
555	Q₈	—	2	1	no	—
556	Q₈	—	2	3	no	—
557	Q₈	—	2	3	no	—
558	Q₈	—	4	6	no	—
559	Q₈	—	8	3	no	—
560	Q₈	C₂₃×#135	1	4	yes	complemented
561	Q₈	C₂×#41×C₂₃	1	2	yes	—
562	D₄	—	2	4	no	—
563	D₄	—	2	3	no	—
564	D₄	—	2	1	no	—
565	D₄	—	2	3	no	—
566	D₄	—	4	6	no	—
567	D₄	—	4	6	no	—
568	D₄	—	8	3	no	—
569	D₄	C₂×#41×C₂₃	1	2	yes	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	6	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	3	no	—
576	C₂×C₄	—	2	3	no	—
577	C₂×C₄	—	2	3	no	—
578	C₂×C₄	—	4	3	no	—
579	C₂×C₄	—	4	6	no	—
580	C₂×C₄	—	4	6	no	—
581	C₂×C₄	—	4	6	no	—
582	C₂×C₄	—	8	3	no	—
583	C₂×C₄	—	8	3	no	—
584	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
585	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
586	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
587	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
588	C₂×C₄	C₂²×D₄×C₂₃	1	3	yes	—
589	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
590	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
591	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
592	C₈	—	2	3	no	—
593	C₈	—	2	3	no	—
594	C₈	—	2	1	no	—
595	C₈	—	2	1	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	2	1	no	—
598	C₂²	—	2	2	no	—
599	C₂²	—	4	1	no	—
600	C₂²	—	4	1	no	—
601	C₂²	—	4	3	no	—
602	C₂²	—	4	3	no	—
603	C₂²	—	4	3	no	—
604	C₂²	C₂₃×#548	1	2	yes	—
605	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
606	C₄	—	2	2	no	—
607	C₄	—	2	3	no	—
608	C₄	—	2	3	no	—
609	C₄	—	2	3	no	—
610	C₄	—	4	3	no	—
611	C₄	—	4	1	no	—
612	C₄	—	4	3	no	—
613	C₄	—	4	1	no	—
614	C₄	—	4	3	no	—
615	C₄	C₂₃×#548	1	2	yes	—
616	C₄	C₂×C₂₃×#135	1	6	yes	—
617	C₄	C₂²×#41×C₂₃	1	1	yes	characteristic
618	C₄	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
619	C₂	—	2	2	no	—
620	C₂	—	4	2	no	—
621	C₂	—	8	3	no	—
622	C₂	C₂×C₂₃×#548	1	1	yes	characteristic
623	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
624	C₂	C₂₃×#3584	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3480	C₂	semidirect
2. Q₈×#41×C₂₃	C₂	semidirect
3. C₂₃×#3365	C₂	semidirect
4. C₂₃×#3364	C₂	semidirect
5. C₂₃×#3363	C₂	semidirect
6. C₂₃×#3362	C₂	semidirect
7. C₂₃×#3361	C₂	semidirect
8. C₂₃×#3360	C₂	semidirect
9. C₂₃×#3359	C₂	semidirect
10. C₂₃×#3358	C₂	semidirect
11. C₂₃×#3131	C₂	semidirect
12. C₂₃×#3076	C₂	semidirect
13. C₂₃×#3035	C₂	semidirect
14. C₂×C₈×Q₈×C₂₃	C₂	semidirect
15. C₄×C₂₃×#135	C₂	semidirect
169. #41×C₂₃	#41	semidirect
238. #61303	C₂₃	direct (N × H)
248. Q₈×C₂₃	#135	semidirect
284. #3480	C₂×C₂₃	semidirect
285. Q₈×#41	C₂×C₂₃	semidirect
286. #3365	C₂×C₂₃	semidirect
287. #3364	C₂×C₂₃	semidirect
288. #3363	C₂×C₂₃	semidirect
289. #3362	C₂×C₂₃	semidirect
290. #3361	C₂×C₂₃	semidirect
291. #3360	C₂×C₂₃	semidirect
292. #3359	C₂×C₂₃	semidirect
293. #3358	C₂×C₂₃	semidirect
294. #3131	C₂×C₂₃	semidirect
295. #3076	C₂×C₂₃	semidirect
296. #3035	C₂×C₂₃	semidirect
297. C₂×C₈×Q₈	C₂×C₂₃	semidirect
298. C₄×#135	C₂×C₂₃	semidirect
477. C₂₃	#61303	direct (N × H)
482. #41	#41×C₂₃	semidirect
560. Q₈	C₂₃×#135	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂₃×#3480	2
Q₈×#41×C₂₃	2
C₂₃×#3365	2
C₂₃×#3364	2
C₂₃×#3363	2
C₂₃×#3362	2
C₂₃×#3361	2
C₂₃×#3360	2
C₂₃×#3359	2
C₂₃×#3358	2
C₂₃×#3131	2
C₂₃×#3076	2
C₂₃×#3035	2
C₂×C₈×Q₈×C₂₃	2
C₄×C₂₃×#135	2
#61303	23

Subgroups — C23×#61303 #19208929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61303 #19208929