Subgroups — C₂₃×#60673 #19208299

Order 5888.

2050 conjugacy classes of subgroups.

The subgroup lattice has 454 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	2	yes	maximal, complemented
2	C₂₃×#3170	C₂	1	2	yes	maximal, complemented
3	C₂₃×#3169	C₂	1	2	yes	maximal, complemented
4	C₂₃×#3168	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3167	C₂	1	4	yes	maximal, complemented
6	C₂₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#3131	C₂	1	2	yes	maximal, complemented
9	C₂₃×#3050	C₂	1	4	yes	maximal, complemented
10	C₄×C₂₃×#135	C₂	1	4	yes	maximal, complemented
11	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#135	C₂²	1	4	yes	—
13	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#533	C₂²	1	4	yes	—
15	C₂₃×#533	C₂²	1	2	yes	—
16	C₂₃×#532	C₂²	1	2	yes	—
17	C₂₃×#532	C₂²	1	4	yes	—
18	C₂₃×#529	—	2	4	no	—
19	C₂₃×#529	—	2	4	no	—
20	C₂₃×#529	C₂²	1	8	yes	—
21	C₂₃×#525	—	2	4	no	—
22	C₂₃×#525	C₂²	1	4	yes	—
23	C₄×#41×C₂₃	—	2	4	no	—
24	C₄×#41×C₂₃	C₂²	1	4	yes	—
25	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#500	C₂²	1	2	yes	complemented
27	C₂₃×#499	C₂²	1	4	yes	complemented
28	C₂₃×#498	C₂²	1	2	yes	complemented
29	C₂₃×#495	C₂²	1	4	yes	complemented
30	C₂₃×#494	C₂²	1	1	yes	characteristic, complemented
31	C₂₃×#493	C₂²	1	1	yes	characteristic, complemented
32	C₂₃×#492	C₂²	1	2	yes	complemented
33	C₂₃×#488	C₂²	1	8	yes	complemented
34	C₂₃×#487	C₂²	1	4	yes	complemented
35	C₂₃×#486	C₂²	1	4	yes	complemented
36	C₂₃×#467	C₂²	1	8	yes	—
37	C₂₃×#466	C₂²	1	8	yes	—
38	C₂₃×#465	C₂²	1	16	yes	—
39	C₄×Q₁₆×C₂₃	C₂²	1	8	yes	complemented
40	C₄×SD₁₆×C₂₃	C₂²	1	16	yes	complemented
41	C₄×D₈×C₂₃	C₂²	1	8	yes	complemented
42	C₂₃×#427	C₂²	1	4	yes	—
43	C₂₃×#416	C₂²	1	8	yes	—
44	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
45	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
46	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
47	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
48	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
49	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
50	C₂×#41×C₂₃	—	2	4	no	—
51	C₂×#41×C₂₃	C₂³	1	4	yes	—
52	C₂³×C₄×C₂₃	D₄	1	2	yes	—
53	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂₃×#135	—	2	16	no	—
55	C₂×Q₁₆×C₂₃	C₂³	1	4	yes	—
56	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
57	C₂×D₈×C₂₃	C₂³	1	4	yes	—
58	C₂²×C₈×C₂₃	—	2	8	no	—
59	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
60	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
61	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
62	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
63	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
64	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
65	C₂₃×#127	C₂³	1	2	yes	—
66	C₂₃×#126	—	2	8	no	—
67	C₂₃×#125	—	2	4	no	—
68	C₂₃×#125	C₂³	1	4	yes	—
69	C₂₃×#124	—	2	4	no	—
70	C₂₃×#124	C₂³	1	4	yes	—
71	C₂₃×#123	—	2	8	no	—
72	C₂₃×#123	—	2	4	no	—
73	C₂₃×C₂²⋊Q₈	—	2	4	no	—
74	C₂₃×C₂²⋊Q₈	—	2	4	no	—
75	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
76	C₂₃×#121	—	2	4	no	—
77	C₂₃×#121	—	2	4	no	—
78	C₂₃×#121	C₂³	1	8	yes	—
79	C₂₃×#120	—	2	4	no	—
80	C₄×Q₈×C₂₃	—	2	4	no	—
81	C₄×Q₈×C₂₃	—	2	4	no	—
82	C₄×Q₈×C₂₃	C₂³	1	8	yes	—
83	C₄×D₄×C₂₃	—	2	8	no	—
84	C₄×D₄×C₂₃	—	2	8	no	—
85	C₄×D₄×C₂₃	—	2	8	no	—
86	C₄×D₄×C₂₃	—	2	4	no	—
87	C₄×D₄×C₂₃	—	2	4	no	—
88	C₄×D₄×C₂₃	C₂³	1	8	yes	—
89	C₂₃×#117	—	2	4	no	—
90	C₂₃×#117	—	2	8	no	—
91	C₂₃×#117	—	2	4	no	—
92	C₂₃×#117	C₂³	1	4	yes	—
93	C₂×C₄²×C₂₃	—	2	4	no	—
94	C₂×C₄²×C₂₃	D₄	1	2	yes	—
95	C₂×C₄²×C₂₃	D₄	1	2	yes	—
96	C₂×C₄²×C₂₃	D₄	1	2	yes	—
97	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
98	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
100	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
101	C₂₃×#107	C₂³	1	8	yes	—
102	C₂₃×#106	C₂³	1	8	yes	—
103	C₂₃×#103	C₂³	1	16	yes	—
104	C₂₃×#102	C₂³	1	16	yes	—
105	C₄×C₈×C₂₃	C₂³	1	4	yes	—
106	C₄×C₈×C₂₃	C₂³	1	4	yes	—
107	C₄×C₈×C₂₃	C₂³	1	2	yes	—
108	C₄×C₈×C₂₃	C₂³	1	2	yes	—
109	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
110	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
111	C₄×C₈×C₂₃	C₂³	1	2	yes	—
112	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
113	#41×C₂₃	—	2	16	no	—
114	#41×C₂₃	—	4	16	no	—
115	C₂×Q₈×C₂₃	—	2	4	no	—
116	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
117	C₂×D₄×C₂₃	—	2	4	no	—
118	C₂×D₄×C₂₃	—	2	8	no	—
119	C₂×D₄×C₂₃	—	2	4	no	—
120	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
121	C₂²×C₄×C₂₃	—	2	8	no	—
122	C₂²×C₄×C₂₃	—	2	4	no	—
123	C₂²×C₄×C₂₃	—	2	4	no	—
124	C₂²×C₄×C₂₃	—	2	8	no	—
125	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
135	Q₁₆×C₂₃	—	2	8	no	—
136	SD₁₆×C₂₃	—	2	16	no	—
137	D₈×C₂₃	—	2	8	no	—
138	C₂×C₈×C₂₃	—	2	4	no	—
139	C₂×C₈×C₂₃	—	2	8	no	—
140	C₂×C₈×C₂₃	—	2	8	no	—
141	C₂×C₈×C₂₃	#41	1	8	yes	complemented
142	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
143	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
144	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
145	C₄⋊C₄×C₂₃	—	2	4	no	—
146	C₄⋊C₄×C₂₃	—	2	4	no	—
147	C₄⋊C₄×C₂₃	—	2	8	no	—
148	C₄⋊C₄×C₂₃	—	2	8	no	—
149	C₄⋊C₄×C₂₃	—	2	8	no	—
150	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
151	C₂²⋊C₄×C₂₃	—	2	8	no	—
152	C₂²⋊C₄×C₂₃	—	2	4	no	—
153	C₂²⋊C₄×C₂₃	—	2	8	no	—
154	C₂²⋊C₄×C₂₃	—	2	8	no	—
155	C₂²⋊C₄×C₂₃	—	2	8	no	—
156	C₄²×C₂₃	—	2	8	no	—
157	C₄²×C₂₃	—	2	4	no	—
158	C₄²×C₂₃	C₂×D₄	1	2	yes	—
159	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
160	C₄²×C₂₃	C₂×D₄	1	2	yes	—
161	C₄²×C₂₃	C₂×D₄	1	2	yes	—
162	C₄²×C₂₃	C₂×D₄	1	2	yes	—
163	C₄²×C₂₃	C₂×D₄	1	2	yes	—
164	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₄²×C₂₃	C₂⁴	1	2	yes	—
166	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
167	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
168	#60673	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂³×C₂₃	—	2	2	no	—
170	C₂³×C₂₃	—	2	4	no	—
171	C₂³×C₂₃	—	2	4	no	—
172	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
173	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
174	Q₈×C₂₃	—	2	8	no	—
175	Q₈×C₂₃	—	4	8	no	—
176	D₄×C₂₃	—	2	8	no	—
177	D₄×C₂₃	—	4	8	no	—
178	D₄×C₂₃	—	4	16	no	—
179	D₄×C₂₃	—	4	8	no	—
180	C₂×C₄×C₂₃	—	2	2	no	—
181	C₂×C₄×C₂₃	—	2	4	no	—
182	C₂×C₄×C₂₃	—	2	4	no	—
183	C₂×C₄×C₂₃	—	2	4	no	—
184	C₂×C₄×C₂₃	—	2	4	no	—
185	C₂×C₄×C₂₃	—	2	4	no	—
186	C₂×C₄×C₂₃	—	2	4	no	—
187	C₂×C₄×C₂₃	—	2	4	no	—
188	C₂×C₄×C₂₃	—	2	4	no	—
189	C₂×C₄×C₂₃	—	2	8	no	—
190	C₂×C₄×C₂₃	—	4	8	no	—
191	C₂×C₄×C₂₃	—	4	16	no	—
192	C₂×C₄×C₂₃	#135	1	8	yes	complemented
193	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
194	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
195	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
196	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
198	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
199	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
200	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
201	C₈×C₂₃	—	2	4	no	—
202	C₈×C₂₃	C₂×#41	1	4	yes	—
203	C₈×C₂₃	C₂×#41	1	4	yes	—
204	#3444	C₂×C₂₃	1	2	yes	complemented
205	#3170	C₂×C₂₃	1	2	yes	complemented
206	#3169	C₂×C₂₃	1	2	yes	complemented
207	#3168	C₂×C₂₃	1	8	yes	complemented
208	#3167	C₂×C₂₃	1	4	yes	complemented
209	#3160	C₂×C₂₃	1	1	yes	characteristic, complemented
210	#3147	C₂×C₂₃	1	1	yes	characteristic, complemented
211	#3131	C₂×C₂₃	1	2	yes	complemented
212	#3050	C₂×C₂₃	1	4	yes	complemented
213	C₄×#135	C₂×C₂₃	1	4	yes	complemented
214	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
215	C₂²×C₂₃	—	2	8	no	—
216	C₂²×C₂₃	—	2	2	no	—
217	C₂²×C₂₃	—	2	2	no	—
218	C₂²×C₂₃	—	2	2	no	—
219	C₂²×C₂₃	—	4	4	no	—
220	C₂²×C₂₃	—	4	8	no	—
221	C₂²×C₂₃	#532	1	2	yes	—
222	C₂²×C₂₃	C₂×#135	1	4	yes	—
223	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
224	C₄×C₂₃	—	2	2	no	—
225	C₄×C₂₃	—	2	2	no	—
226	C₄×C₂₃	—	2	2	no	—
227	C₄×C₂₃	—	4	4	no	—
228	C₄×C₂₃	—	4	8	no	—
229	C₄×C₂₃	#532	1	2	yes	—
230	C₄×C₂₃	C₂×#135	1	4	yes	—
231	C₄×C₂₃	C₂×#135	1	4	yes	—
232	C₄×C₂₃	C₂²×#41	1	2	yes	—
233	C₂×#135	C₂²×C₂₃	1	4	yes	—
234	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
235	#533	C₂²×C₂₃	1	4	yes	—
236	#533	C₂²×C₂₃	1	2	yes	—
237	#532	C₂²×C₂₃	1	2	yes	—
238	#532	C₂²×C₂₃	1	4	yes	—
239	#529	—	2	4	no	—
240	#529	—	2	4	no	—
241	#529	C₂²×C₂₃	1	8	yes	—
242	#525	—	2	4	no	—
243	#525	C₂²×C₂₃	1	4	yes	—
244	C₄×#41	—	2	4	no	—
245	C₄×#41	C₂²×C₂₃	1	4	yes	—
246	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
247	#500	C₂²×C₂₃	1	2	yes	complemented
248	#499	C₂²×C₂₃	1	4	yes	complemented
249	#498	C₂²×C₂₃	1	2	yes	complemented
250	#495	C₂²×C₂₃	1	4	yes	complemented
251	#494	C₂²×C₂₃	1	1	yes	characteristic, complemented
252	#493	C₂²×C₂₃	1	1	yes	characteristic, complemented
253	#492	C₂²×C₂₃	1	2	yes	complemented
254	#488	C₂²×C₂₃	1	8	yes	complemented
255	#487	C₂²×C₂₃	1	4	yes	complemented
256	#486	C₂²×C₂₃	1	4	yes	complemented
257	#467	C₂²×C₂₃	1	8	yes	—
258	#466	C₂²×C₂₃	1	8	yes	—
259	#465	C₂²×C₂₃	1	16	yes	—
260	C₄×Q₁₆	C₂²×C₂₃	1	8	yes	complemented
261	C₄×SD₁₆	C₂²×C₂₃	1	16	yes	complemented
262	C₄×D₈	C₂²×C₂₃	1	8	yes	complemented
263	#427	C₂²×C₂₃	1	4	yes	—
264	#416	C₂²×C₂₃	1	8	yes	—
265	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
266	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
267	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
268	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
269	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
270	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
271	C₂×C₂₃	—	2	2	no	—
272	C₂×C₂₃	—	2	4	no	—
273	C₂×C₂₃	—	8	4	no	—
274	C₂×C₂₃	C₂×#532	1	1	yes	characteristic
275	C₂×C₂₃	C₂²×#135	1	2	yes	—
276	C₂×#41	—	2	4	no	—
277	C₂×#41	C₂³×C₂₃	1	4	yes	—
278	C₂³×C₄	D₄×C₂₃	1	2	yes	—
279	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
280	#135	—	2	16	no	—
281	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
282	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
283	C₂×D₈	C₂³×C₂₃	1	4	yes	—
284	C₂²×C₈	—	2	8	no	—
285	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
286	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
287	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
288	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
289	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
290	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
291	#127	C₂³×C₂₃	1	2	yes	—
292	#126	—	2	8	no	—
293	#125	—	2	4	no	—
294	#125	C₂³×C₂₃	1	4	yes	—
295	#124	—	2	4	no	—
296	#124	C₂³×C₂₃	1	4	yes	—
297	#123	—	2	8	no	—
298	#123	—	2	4	no	—
299	C₂²⋊Q₈	—	2	4	no	—
300	C₂²⋊Q₈	—	2	4	no	—
301	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
302	#121	—	2	4	no	—
303	#121	—	2	4	no	—
304	#121	C₂³×C₂₃	1	8	yes	—
305	#120	—	2	4	no	—
306	C₄×Q₈	—	2	4	no	—
307	C₄×Q₈	—	2	4	no	—
308	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
309	C₄×D₄	—	2	8	no	—
310	C₄×D₄	—	2	8	no	—
311	C₄×D₄	—	2	8	no	—
312	C₄×D₄	—	2	4	no	—
313	C₄×D₄	—	2	4	no	—
314	C₄×D₄	C₂³×C₂₃	1	8	yes	—
315	#117	—	2	4	no	—
316	#117	—	2	8	no	—
317	#117	—	2	4	no	—
318	#117	C₂³×C₂₃	1	4	yes	—
319	C₂×C₄²	—	2	4	no	—
320	C₂×C₄²	D₄×C₂₃	1	2	yes	—
321	C₂×C₄²	D₄×C₂₃	1	2	yes	—
322	C₂×C₄²	D₄×C₂₃	1	2	yes	—
323	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
324	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
325	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
326	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
327	#107	C₂³×C₂₃	1	8	yes	—
328	#106	C₂³×C₂₃	1	8	yes	—
329	#103	C₂³×C₂₃	1	16	yes	—
330	#102	C₂³×C₂₃	1	16	yes	—
331	C₄×C₈	C₂³×C₂₃	1	4	yes	—
332	C₄×C₈	C₂³×C₂₃	1	4	yes	—
333	C₄×C₈	C₂³×C₂₃	1	2	yes	—
334	C₄×C₈	C₂³×C₂₃	1	2	yes	—
335	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
336	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
337	C₄×C₈	C₂³×C₂₃	1	2	yes	—
338	C₂₃	#60673	1	1	yes	characteristic, direct product factor, Sylow
339	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
340	#41	—	2	16	no	—
341	#41	—	4	16	no	—
342	C₂×Q₈	—	2	4	no	—
343	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
344	C₂×D₄	—	2	4	no	—
345	C₂×D₄	—	2	8	no	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
348	C₂²×C₄	—	2	8	no	—
349	C₂²×C₄	—	2	4	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	8	no	—
352	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
355	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
356	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
357	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
358	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
359	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
360	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
361	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
362	Q₁₆	—	2	8	no	—
363	SD₁₆	—	2	16	no	—
364	D₈	—	2	8	no	—
365	C₂×C₈	—	2	4	no	—
366	C₂×C₈	—	2	8	no	—
367	C₂×C₈	—	2	8	no	—
368	C₂×C₈	#41×C₂₃	1	8	yes	complemented
369	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
370	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
371	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
372	C₄⋊C₄	—	2	4	no	—
373	C₄⋊C₄	—	2	4	no	—
374	C₄⋊C₄	—	2	8	no	—
375	C₄⋊C₄	—	2	8	no	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
378	C₂²⋊C₄	—	2	8	no	—
379	C₂²⋊C₄	—	2	4	no	—
380	C₂²⋊C₄	—	2	8	no	—
381	C₂²⋊C₄	—	2	8	no	—
382	C₂²⋊C₄	—	2	8	no	—
383	C₄²	—	2	8	no	—
384	C₄²	—	2	4	no	—
385	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
386	C₄²	C₂×D₄×C₂₃	1	2	yes	—
387	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
388	C₄²	C₂×D₄×C₂₃	1	2	yes	—
389	C₄²	C₂×D₄×C₂₃	1	2	yes	—
390	C₄²	C₂×D₄×C₂₃	1	2	yes	—
391	C₄²	C₂×D₄×C₂₃	1	2	yes	—
392	C₄²	C₂⁴×C₂₃	1	2	yes	—
393	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
394	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
395	C₂³	—	2	4	no	—
396	C₂³	—	2	2	no	—
397	C₂³	—	2	4	no	—
398	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
399	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
400	Q₈	—	2	8	no	—
401	Q₈	—	4	8	no	—
402	D₄	—	2	8	no	—
403	D₄	—	4	16	no	—
404	D₄	—	4	8	no	—
405	D₄	—	4	8	no	—
406	C₂×C₄	—	2	4	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	4	no	—
409	C₂×C₄	—	2	4	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	4	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	8	no	—
416	C₂×C₄	—	4	16	no	—
417	C₂×C₄	—	4	8	no	—
418	C₂×C₄	C₂₃×#135	1	8	yes	complemented
419	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
420	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
421	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
422	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
423	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
424	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
426	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
427	C₈	—	2	4	no	—
428	C₈	C₂×#41×C₂₃	1	4	yes	—
429	C₈	C₂×#41×C₂₃	1	4	yes	—
430	C₂²	—	2	8	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	2	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	4	4	no	—
435	C₂²	—	4	8	no	—
436	C₂²	C₂₃×#532	1	2	yes	—
437	C₂²	C₂×C₂₃×#135	1	4	yes	—
438	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
439	C₄	—	2	2	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	2	no	—
442	C₄	—	4	4	no	—
443	C₄	—	4	8	no	—
444	C₄	C₂₃×#532	1	2	yes	—
445	C₄	C₂×C₂₃×#135	1	4	yes	—
446	C₄	C₂×C₂₃×#135	1	4	yes	—
447	C₄	C₂²×#41×C₂₃	1	2	yes	—
448	C₂	—	2	4	no	—
449	C₂	—	2	2	no	—
450	C₂	—	8	4	no	—
451	C₂	C₂×C₂₃×#532	1	1	yes	characteristic
452	C₂	C₂²×C₂₃×#135	1	2	yes	—

	subgroup	G/H
center	C₄²×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
socle	C₂²×C₂₃	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×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. C₂₃×#3170	C₂	semidirect
3. C₂₃×#3169	C₂	semidirect
4. C₂₃×#3168	C₂	semidirect
5. C₂₃×#3167	C₂	semidirect
6. C₂₃×#3160	C₂	semidirect
7. C₂₃×#3147	C₂	semidirect
8. C₂₃×#3131	C₂	semidirect
9. C₂₃×#3050	C₂	semidirect
10. C₄×C₂₃×#135	C₂	semidirect
11. C₂²×C₄×C₈×C₂₃	C₂	semidirect
26. C₂₃×#500	C₂²	semidirect
27. C₂₃×#499	C₂²	semidirect
28. C₂₃×#498	C₂²	semidirect
29. C₂₃×#495	C₂²	semidirect
30. C₂₃×#494	C₂²	semidirect
31. C₂₃×#493	C₂²	semidirect
32. C₂₃×#492	C₂²	semidirect
33. C₂₃×#488	C₂²	semidirect
34. C₂₃×#487	C₂²	semidirect
35. C₂₃×#486	C₂²	semidirect
39. C₄×Q₁₆×C₂₃	C₂²	semidirect
40. C₄×SD₁₆×C₂₃	C₂²	semidirect
41. C₄×D₈×C₂₃	C₂²	semidirect
141. C₂×C₈×C₂₃	#41	semidirect
168. #60673	C₂₃	direct (N × H)
192. C₂×C₄×C₂₃	#135	semidirect
204. #3444	C₂×C₂₃	semidirect
205. #3170	C₂×C₂₃	semidirect
206. #3169	C₂×C₂₃	semidirect
207. #3168	C₂×C₂₃	semidirect
208. #3167	C₂×C₂₃	semidirect
209. #3160	C₂×C₂₃	semidirect
210. #3147	C₂×C₂₃	semidirect
211. #3131	C₂×C₂₃	semidirect
212. #3050	C₂×C₂₃	semidirect
213. C₄×#135	C₂×C₂₃	semidirect
214. C₂²×C₄×C₈	C₂×C₂₃	semidirect
247. #500	C₂²×C₂₃	semidirect
248. #499	C₂²×C₂₃	semidirect
249. #498	C₂²×C₂₃	semidirect
250. #495	C₂²×C₂₃	semidirect
251. #494	C₂²×C₂₃	semidirect
252. #493	C₂²×C₂₃	semidirect
253. #492	C₂²×C₂₃	semidirect
254. #488	C₂²×C₂₃	semidirect
255. #487	C₂²×C₂₃	semidirect
256. #486	C₂²×C₂₃	semidirect
260. C₄×Q₁₆	C₂²×C₂₃	semidirect
261. C₄×SD₁₆	C₂²×C₂₃	semidirect
262. C₄×D₈	C₂²×C₂₃	semidirect
338. C₂₃	#60673	direct (N × H)
368. C₂×C₈	#41×C₂₃	semidirect
418. C₂×C₄	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×#3444	2
C₂₃×#3170	2
C₂₃×#3169	2
C₂₃×#3168	2
C₂₃×#3167	2
C₂₃×#3160	2
C₂₃×#3147	2
C₂₃×#3131	2
C₂₃×#3050	2
C₄×C₂₃×#135	2
C₂²×C₄×C₈×C₂₃	2
#60673	23

Subgroups — C23×#60673 #19208299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60673 #19208299