Subgroups — C₂₃×#60693 #19208319

Order 5888.

1998 conjugacy classes of subgroups.

The subgroup lattice has 432 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×#3185	C₂	1	2	yes	maximal, complemented
3	C₂₃×#3184	C₂	1	2	yes	maximal, complemented
4	C₂₃×#3183	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3182	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×#3054	C₂	1	4	yes	maximal, complemented
10	C₂₃×#2943	C₂	1	4	yes	maximal, complemented
11	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₂₃×#135	C₂²	1	4	yes	—
13	C₂²×M₁₆×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×#470	C₂²	1	8	yes	—
37	C₂₃×#469	C₂²	1	8	yes	—
38	C₂₃×#468	C₂²	1	16	yes	—
39	C₂₃×#442	C₂²	1	8	yes	complemented
40	C₂₃×#441	C₂²	1	8	yes	complemented
41	C₂₃×#440	C₂²	1	16	yes	complemented
42	C₂₃×#427	C₂²	1	4	yes	—
43	C₂₃×#416	C₂²	1	8	yes	—
44	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
45	C₄×M₁₆×C₂₃	C₂²	1	4	yes	—
46	C₄×M₁₆×C₂₃	C₂²	1	2	yes	—
47	C₂×C₂₃×#97	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	C₂×M₁₆×C₂₃	—	2	8	no	—
59	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
60	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
61	C₂²×C₈×C₂₃	C₂³	1	2	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	4	yes	—
95	C₂×C₄²×C₂₃	D₄	1	2	yes	—
96	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
97	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
98	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
99	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
100	C₂₃×#107	C₂³	1	8	yes	—
101	C₂₃×#106	C₂³	1	8	yes	—
102	C₂₃×#103	C₂³	1	16	yes	—
103	C₂₃×#102	C₂³	1	16	yes	—
104	C₂₃×#97	C₂³	1	4	yes	—
105	C₂₃×#97	C₂³	1	4	yes	—
106	C₄×C₈×C₂₃	C₂³	1	2	yes	—
107	C₄×C₈×C₂₃	C₂³	1	2	yes	—
108	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
109	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
110	C₄×C₈×C₂₃	C₂³	1	2	yes	—
111	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
112	#41×C₂₃	—	2	16	no	—
113	#41×C₂₃	—	4	16	no	—
114	C₂×Q₈×C₂₃	—	2	4	no	—
115	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
116	C₂×D₄×C₂₃	—	2	4	no	—
117	C₂×D₄×C₂₃	—	2	8	no	—
118	C₂×D₄×C₂₃	—	2	4	no	—
119	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₂₃	—	2	4	no	—
121	C₂²×C₄×C₂₃	—	2	8	no	—
122	C₂²×C₄×C₂₃	—	2	4	no	—
123	C₂²×C₄×C₂₃	—	4	4	no	—
124	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
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	4	yes	—
130	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
131	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
132	Q₁₆×C₂₃	—	2	8	no	—
133	SD₁₆×C₂₃	—	2	16	no	—
134	D₈×C₂₃	—	2	8	no	—
135	M₁₆×C₂₃	—	2	8	no	—
136	M₁₆×C₂₃	—	2	8	no	—
137	C₂×C₈×C₂₃	—	2	4	no	—
138	C₂×C₈×C₂₃	#41	1	8	yes	complemented
139	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
140	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
141	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
142	C₄⋊C₄×C₂₃	—	2	4	no	—
143	C₄⋊C₄×C₂₃	—	2	4	no	—
144	C₄⋊C₄×C₂₃	—	2	8	no	—
145	C₄⋊C₄×C₂₃	—	2	8	no	—
146	C₄⋊C₄×C₂₃	—	2	8	no	—
147	C₄⋊C₄×C₂₃	C₂⁴	1	8	yes	—
148	C₂²⋊C₄×C₂₃	—	2	8	no	—
149	C₂²⋊C₄×C₂₃	—	2	4	no	—
150	C₂²⋊C₄×C₂₃	—	2	8	no	—
151	C₂²⋊C₄×C₂₃	—	2	8	no	—
152	C₂²⋊C₄×C₂₃	—	2	8	no	—
153	C₄²×C₂₃	—	2	8	no	—
154	C₄²×C₂₃	—	2	4	no	—
155	C₄²×C₂₃	C₂×D₄	1	4	yes	—
156	C₄²×C₂₃	C₂×D₄	1	4	yes	—
157	C₄²×C₂₃	C₂×D₄	1	2	yes	—
158	C₄²×C₂₃	C₂×D₄	1	2	yes	—
159	C₄²×C₂₃	C₂⁴	1	2	yes	—
160	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
161	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
162	#60693	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₂₃	—	2	2	no	—
164	C₂³×C₂₃	—	2	4	no	—
165	C₂³×C₂₃	—	4	2	no	—
166	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
167	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
168	Q₈×C₂₃	—	2	8	no	—
169	Q₈×C₂₃	—	4	8	no	—
170	D₄×C₂₃	—	2	8	no	—
171	D₄×C₂₃	—	4	16	no	—
172	D₄×C₂₃	—	4	8	no	—
173	D₄×C₂₃	—	4	8	no	—
174	C₂×C₄×C₂₃	—	2	2	no	—
175	C₂×C₄×C₂₃	—	2	4	no	—
176	C₂×C₄×C₂₃	—	2	4	no	—
177	C₂×C₄×C₂₃	—	2	8	no	—
178	C₂×C₄×C₂₃	—	2	4	no	—
179	C₂×C₄×C₂₃	—	2	4	no	—
180	C₂×C₄×C₂₃	—	2	8	no	—
181	C₂×C₄×C₂₃	—	4	2	no	—
182	C₂×C₄×C₂₃	—	4	4	no	—
183	C₂×C₄×C₂₃	—	4	16	no	—
184	C₂×C₄×C₂₃	—	4	8	no	—
185	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
186	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
187	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
188	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
189	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₂₃	C₂×#41	1	4	yes	—
191	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
192	C₈×C₂₃	—	2	4	no	—
193	C₈×C₂₃	C₂×#41	1	4	yes	—
194	C₈×C₂₃	C₂×#41	1	4	yes	—
195	#3444	C₂×C₂₃	1	2	yes	complemented
196	#3185	C₂×C₂₃	1	2	yes	complemented
197	#3184	C₂×C₂₃	1	2	yes	complemented
198	#3183	C₂×C₂₃	1	8	yes	complemented
199	#3182	C₂×C₂₃	1	4	yes	complemented
200	#3160	C₂×C₂₃	1	1	yes	characteristic, complemented
201	#3147	C₂×C₂₃	1	1	yes	characteristic, complemented
202	#3131	C₂×C₂₃	1	2	yes	complemented
203	#3054	C₂×C₂₃	1	4	yes	complemented
204	#2943	C₂×C₂₃	1	4	yes	complemented
205	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
206	C₂²×C₂₃	—	2	2	no	—
207	C₂²×C₂₃	—	2	2	no	—
208	C₂²×C₂₃	—	2	4	no	—
209	C₂²×C₂₃	—	4	4	no	—
210	C₂²×C₂₃	—	4	4	no	—
211	C₂²×C₂₃	—	4	8	no	—
212	C₂²×C₂₃	#532	1	2	yes	—
213	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
214	C₄×C₂₃	—	2	2	no	—
215	C₄×C₂₃	—	2	4	no	—
216	C₄×C₂₃	—	2	2	no	—
217	C₄×C₂₃	—	4	4	no	—
218	C₄×C₂₃	—	4	8	no	—
219	C₄×C₂₃	#532	1	2	yes	—
220	C₄×C₂₃	#575	1	4	yes	—
221	C₄×C₂₃	C₂²×#41	1	2	yes	—
222	C₂×#135	C₂²×C₂₃	1	4	yes	—
223	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
224	#533	C₂²×C₂₃	1	4	yes	—
225	#533	C₂²×C₂₃	1	2	yes	—
226	#532	C₂²×C₂₃	1	2	yes	—
227	#532	C₂²×C₂₃	1	4	yes	—
228	#529	—	2	4	no	—
229	#529	—	2	4	no	—
230	#529	C₂²×C₂₃	1	8	yes	—
231	#525	—	2	4	no	—
232	#525	C₂²×C₂₃	1	4	yes	—
233	C₄×#41	—	2	4	no	—
234	C₄×#41	C₂²×C₂₃	1	4	yes	—
235	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
236	#500	C₂²×C₂₃	1	2	yes	complemented
237	#499	C₂²×C₂₃	1	4	yes	complemented
238	#498	C₂²×C₂₃	1	2	yes	complemented
239	#495	C₂²×C₂₃	1	4	yes	complemented
240	#494	C₂²×C₂₃	1	1	yes	characteristic, complemented
241	#493	C₂²×C₂₃	1	1	yes	characteristic, complemented
242	#492	C₂²×C₂₃	1	2	yes	complemented
243	#488	C₂²×C₂₃	1	8	yes	complemented
244	#487	C₂²×C₂₃	1	4	yes	complemented
245	#486	C₂²×C₂₃	1	4	yes	complemented
246	#470	C₂²×C₂₃	1	8	yes	—
247	#469	C₂²×C₂₃	1	8	yes	—
248	#468	C₂²×C₂₃	1	16	yes	—
249	#442	C₂²×C₂₃	1	8	yes	complemented
250	#441	C₂²×C₂₃	1	8	yes	complemented
251	#440	C₂²×C₂₃	1	16	yes	complemented
252	#427	C₂²×C₂₃	1	4	yes	—
253	#416	C₂²×C₂₃	1	8	yes	—
254	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
255	C₄×M₁₆	C₂²×C₂₃	1	4	yes	—
256	C₄×M₁₆	C₂²×C₂₃	1	2	yes	—
257	C₂×#97	C₂²×C₂₃	1	2	yes	—
258	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
259	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
260	C₂×C₂₃	—	2	2	no	—
261	C₂×C₂₃	—	4	2	no	—
262	C₂×C₂₃	—	8	4	no	—
263	C₂×C₂₃	C₂×#532	1	1	yes	characteristic
264	C₂×C₂₃	C₂×#575	1	2	yes	—
265	C₂×#41	—	2	4	no	—
266	C₂×#41	C₂³×C₂₃	1	4	yes	—
267	C₂³×C₄	D₄×C₂₃	1	2	yes	—
268	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
269	#135	—	2	16	no	—
270	C₂×Q₁₆	C₂³×C₂₃	1	4	yes	—
271	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
272	C₂×D₈	C₂³×C₂₃	1	4	yes	—
273	C₂×M₁₆	—	2	8	no	—
274	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
275	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
276	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
277	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
278	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
279	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
280	#127	C₂³×C₂₃	1	2	yes	—
281	#126	—	2	8	no	—
282	#125	—	2	4	no	—
283	#125	C₂³×C₂₃	1	4	yes	—
284	#124	—	2	4	no	—
285	#124	C₂³×C₂₃	1	4	yes	—
286	#123	—	2	8	no	—
287	#123	—	2	4	no	—
288	C₂²⋊Q₈	—	2	4	no	—
289	C₂²⋊Q₈	—	2	4	no	—
290	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
291	#121	—	2	4	no	—
292	#121	—	2	4	no	—
293	#121	C₂³×C₂₃	1	8	yes	—
294	#120	—	2	4	no	—
295	C₄×Q₈	—	2	4	no	—
296	C₄×Q₈	—	2	4	no	—
297	C₄×Q₈	C₂³×C₂₃	1	8	yes	—
298	C₄×D₄	—	2	8	no	—
299	C₄×D₄	—	2	8	no	—
300	C₄×D₄	—	2	8	no	—
301	C₄×D₄	—	2	4	no	—
302	C₄×D₄	—	2	4	no	—
303	C₄×D₄	C₂³×C₂₃	1	8	yes	—
304	#117	—	2	4	no	—
305	#117	—	2	8	no	—
306	#117	—	2	4	no	—
307	#117	C₂³×C₂₃	1	4	yes	—
308	C₂×C₄²	—	2	4	no	—
309	C₂×C₄²	D₄×C₂₃	1	4	yes	—
310	C₂×C₄²	D₄×C₂₃	1	2	yes	—
311	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
312	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
313	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
315	#107	C₂³×C₂₃	1	8	yes	—
316	#106	C₂³×C₂₃	1	8	yes	—
317	#103	C₂³×C₂₃	1	16	yes	—
318	#102	C₂³×C₂₃	1	16	yes	—
319	#97	C₂³×C₂₃	1	4	yes	—
320	#97	C₂³×C₂₃	1	4	yes	—
321	C₄×C₈	C₂³×C₂₃	1	2	yes	—
322	C₄×C₈	C₂³×C₂₃	1	2	yes	—
323	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
324	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
325	C₄×C₈	C₂³×C₂₃	1	2	yes	—
326	C₂₃	#60693	1	1	yes	characteristic, direct product factor, Sylow
327	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
328	#41	—	2	16	no	—
329	#41	—	4	16	no	—
330	C₂×Q₈	—	2	4	no	—
331	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
332	C₂×D₄	—	2	4	no	—
333	C₂×D₄	—	2	8	no	—
334	C₂×D₄	—	2	4	no	—
335	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
336	C₂²×C₄	—	2	4	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	8	no	—
339	C₂²×C₄	—	4	4	no	—
340	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
341	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
343	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
344	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
345	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
346	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
347	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
348	Q₁₆	—	2	8	no	—
349	SD₁₆	—	2	16	no	—
350	D₈	—	2	8	no	—
351	M₁₆	—	2	8	no	—
352	M₁₆	—	2	8	no	—
353	C₂×C₈	—	2	4	no	—
354	C₂×C₈	#41×C₂₃	1	8	yes	complemented
355	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
356	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
357	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
358	C₄⋊C₄	—	2	4	no	—
359	C₄⋊C₄	—	2	4	no	—
360	C₄⋊C₄	—	2	8	no	—
361	C₄⋊C₄	—	2	8	no	—
362	C₄⋊C₄	—	2	8	no	—
363	C₄⋊C₄	C₂⁴×C₂₃	1	8	yes	—
364	C₂²⋊C₄	—	2	8	no	—
365	C₂²⋊C₄	—	2	4	no	—
366	C₂²⋊C₄	—	2	8	no	—
367	C₂²⋊C₄	—	2	8	no	—
368	C₂²⋊C₄	—	2	8	no	—
369	C₄²	—	2	8	no	—
370	C₄²	—	2	4	no	—
371	C₄²	C₂×D₄×C₂₃	1	2	yes	—
372	C₄²	C₂×D₄×C₂₃	1	4	yes	—
373	C₄²	C₂×D₄×C₂₃	1	2	yes	—
374	C₄²	C₂×D₄×C₂₃	1	4	yes	—
375	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
376	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
377	C₄²	C₂⁴×C₂₃	1	2	yes	—
378	C₂³	—	2	2	no	—
379	C₂³	—	2	4	no	—
380	C₂³	—	4	2	no	—
381	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
382	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
383	Q₈	—	2	8	no	—
384	Q₈	—	4	8	no	—
385	D₄	—	2	8	no	—
386	D₄	—	4	16	no	—
387	D₄	—	4	8	no	—
388	D₄	—	4	8	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	8	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	2	4	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	8	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	4	4	no	—
398	C₂×C₄	—	4	16	no	—
399	C₂×C₄	—	4	8	no	—
400	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
401	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
402	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
403	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
404	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
405	C₂×C₄	C₂×#41×C₂₃	1	4	yes	—
406	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
407	C₈	—	2	4	no	—
408	C₈	C₂×#41×C₂₃	1	4	yes	—
409	C₈	C₂×#41×C₂₃	1	4	yes	—
410	C₂²	—	2	2	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	4	no	—
413	C₂²	—	4	4	no	—
414	C₂²	—	4	4	no	—
415	C₂²	—	4	8	no	—
416	C₂²	C₂₃×#532	1	2	yes	—
417	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
418	C₄	—	2	2	no	—
419	C₄	—	2	4	no	—
420	C₄	—	2	2	no	—
421	C₄	—	4	4	no	—
422	C₄	—	4	8	no	—
423	C₄	C₂₃×#532	1	2	yes	—
424	C₄	C₂₃×#575	1	4	yes	—
425	C₄	C₂²×#41×C₂₃	1	2	yes	—
426	C₂	—	2	2	no	—
427	C₂	—	4	2	no	—
428	C₂	—	8	4	no	—
429	C₂	C₂×C₂₃×#532	1	1	yes	characteristic
430	C₂	C₂×C₂₃×#575	1	2	yes	—

	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 ◁ 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×#3185	C₂	semidirect
3. C₂₃×#3184	C₂	semidirect
4. C₂₃×#3183	C₂	semidirect
5. C₂₃×#3182	C₂	semidirect
6. C₂₃×#3160	C₂	semidirect
7. C₂₃×#3147	C₂	semidirect
8. C₂₃×#3131	C₂	semidirect
9. C₂₃×#3054	C₂	semidirect
10. C₂₃×#2943	C₂	semidirect
11. C₂×C₄×M₁₆×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. C₂₃×#442	C₂²	semidirect
40. C₂₃×#441	C₂²	semidirect
41. C₂₃×#440	C₂²	semidirect
138. C₂×C₈×C₂₃	#41	semidirect
162. #60693	C₂₃	direct (N × H)
195. #3444	C₂×C₂₃	semidirect
196. #3185	C₂×C₂₃	semidirect
197. #3184	C₂×C₂₃	semidirect
198. #3183	C₂×C₂₃	semidirect
199. #3182	C₂×C₂₃	semidirect
200. #3160	C₂×C₂₃	semidirect
201. #3147	C₂×C₂₃	semidirect
202. #3131	C₂×C₂₃	semidirect
203. #3054	C₂×C₂₃	semidirect
204. #2943	C₂×C₂₃	semidirect
205. C₂×C₄×M₁₆	C₂×C₂₃	semidirect
236. #500	C₂²×C₂₃	semidirect
237. #499	C₂²×C₂₃	semidirect
238. #498	C₂²×C₂₃	semidirect
239. #495	C₂²×C₂₃	semidirect
240. #494	C₂²×C₂₃	semidirect
241. #493	C₂²×C₂₃	semidirect
242. #492	C₂²×C₂₃	semidirect
243. #488	C₂²×C₂₃	semidirect
244. #487	C₂²×C₂₃	semidirect
245. #486	C₂²×C₂₃	semidirect
249. #442	C₂²×C₂₃	semidirect
250. #441	C₂²×C₂₃	semidirect
251. #440	C₂²×C₂₃	semidirect
326. C₂₃	#60693	direct (N × H)
354. C₂×C₈	#41×C₂₃	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×#3185	2
C₂₃×#3184	2
C₂₃×#3183	2
C₂₃×#3182	2
C₂₃×#3160	2
C₂₃×#3147	2
C₂₃×#3131	2
C₂₃×#3054	2
C₂₃×#2943	2
C₂×C₄×M₁₆×C₂₃	2
#60693	23

Subgroups — C23×#60693 #19208319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60693 #19208319