Subgroups — C₂₃×#60645 #19208271

Order 5888.

3158 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#575	C₂	1	2	yes	maximal, complemented
2	C₂²×C₂₃×#137	C₂	1	2	yes	maximal, complemented
3	C₂₃×#3449	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#3156	C₂	1	8	yes	maximal, complemented
6	C₂₃×#3154	C₂	1	8	yes	maximal, complemented
7	C₂₃×#3153	C₂	1	4	yes	maximal, complemented
8	C₂₃×#3152	C₂	1	4	yes	maximal, complemented
9	C₂₃×#2873	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂²×#41×C₂₃	C₂²	1	2	yes	—
12	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#575	—	2	16	no	—
14	C₂×C₂₃×#137	—	2	16	no	—
15	C₂×C₂₃×#137	C₂²	1	8	yes	complemented
16	C₂×C₂₃×#137	C₂²	1	8	yes	complemented
17	C₂×C₂₃×#137	C₂²	1	4	yes	complemented
18	C₂×C₂₃×#137	C₂²	1	4	yes	complemented
19	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
20	C₂×C₂₃×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
21	C₂×C₂₃×#135	C₂²	1	8	yes	—
22	C₂²×Q₁₆×C₂₃	C₂²	1	4	yes	complemented
23	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	complemented
24	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
25	C₂₃×#537	—	2	2	no	—
26	C₂₃×#536	—	2	4	no	—
27	C₂₃×#536	C₂²	1	8	yes	complemented
28	C₂₃×#536	C₂²	1	4	yes	complemented
29	C₂₃×#535	C₂²	1	4	yes	complemented
30	C₂₃×#534	—	2	2	no	—
31	C₂₃×#532	C₂²	1	4	yes	—
32	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
33	C₂×C₂₃×#124	C₂²	1	2	yes	complemented
34	C₂₃×#525	—	2	4	no	—
35	C₂×C₂₃×#123	—	2	2	no	—
36	C₂×C₂₃×C₂²⋊Q₈	—	2	2	no	—
37	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
38	C₂₃×#497	C₂²	1	16	yes	complemented
39	C₂₃×#497	C₂²	1	8	yes	complemented
40	C₂₃×#496	C₂²	1	8	yes	complemented
41	C₂₃×#495	C₂²	1	16	yes	complemented
42	C₂₃×#494	C₂²	1	8	yes	complemented
43	C₂₃×#492	C₂²	1	8	yes	complemented
44	C₂₃×#405	C₂²	1	8	yes	complemented
45	C₄×M₁₆×C₂₃	C₂²	1	4	yes	complemented
46	C₂×#41×C₂₃	—	2	2	no	—
47	C₂×#41×C₂₃	—	2	2	no	—
48	C₂×#41×C₂₃	—	2	2	no	—
49	C₂×#41×C₂₃	—	2	8	no	—
50	C₂×#41×C₂₃	—	2	8	no	—
51	C₂×#41×C₂₃	—	2	8	no	—
52	C₂×#41×C₂₃	C₂³	1	4	yes	—
53	C₂×#41×C₂₃	C₂³	1	4	yes	—
54	C₂×#41×C₂₃	C₂³	1	4	yes	—
55	C₂²×Q₈×C₂₃	—	2	1	no	—
56	C₂²×Q₈×C₂₃	—	2	2	no	—
57	C₂²×Q₈×C₂₃	—	2	8	no	—
58	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
59	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
60	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
61	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₂₃	—	2	1	no	—
63	C₂²×D₄×C₂₃	—	2	2	no	—
64	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
65	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₂₃	—	2	1	no	—
67	C₂³×C₄×C₂₃	—	2	1	no	—
68	C₂³×C₄×C₂₃	—	2	2	no	—
69	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₂₃×#137	—	2	32	no	—
71	C₂₃×#137	—	2	32	no	—
72	C₂₃×#137	—	2	16	no	—
73	C₂₃×#137	—	2	16	no	—
74	C₂₃×AGL(1,ℤ/8)	—	2	16	no	—
75	C₂₃×AGL(1,ℤ/8)	—	2	16	no	—
76	C₂₃×#135	—	2	32	no	—
77	C₂×Q₁₆×C₂₃	—	2	16	no	—
78	C₂×Q₁₆×C₂₃	C₂³	1	16	yes	—
79	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
80	C₂×SD₁₆×C₂₃	—	2	16	no	—
81	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
82	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
83	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
84	C₂×D₈×C₂₃	C₂³	1	8	yes	—
85	C₂×M₁₆×C₂₃	D₄	1	16	yes	complemented
86	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
87	C₂×M₁₆×C₂₃	C₂³	1	4	yes	—
88	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
89	C₂₃×C₄⋊Q₈	—	2	4	no	—
90	C₂₃×C₄⋊Q₈	C₂³	1	8	yes	—
91	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
92	C₂₃×#127	C₂³	1	4	yes	—
93	C₂₃×#124	—	2	4	no	—
94	C₂₃×#124	C₂³	1	8	yes	—
95	C₂₃×#124	C₂³	1	8	yes	—
96	C₂₃×#123	—	2	16	no	—
97	C₂₃×#123	—	2	8	no	—
98	C₂₃×C₂²⋊Q₈	—	2	16	no	—
99	C₂₃×C₂²⋊Q₈	—	2	8	no	—
100	C₂₃×#121	—	2	8	no	—
101	C₂₃×#121	—	2	8	no	—
102	C₂₃×#120	—	2	8	no	—
103	C₄×D₄×C₂₃	—	2	8	no	—
104	C₂₃×#117	D₄	1	4	yes	—
105	C₂₃×#117	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
107	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
108	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
109	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
110	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
111	C₂₃×#97	C₂³	1	8	yes	—
112	C₄×C₈×C₂₃	C₂³	1	8	yes	—
113	C₂⁴×C₂₃	—	2	1	no	—
114	C₂⁴×C₂₃	—	2	2	no	—
115	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
116	#41×C₂₃	—	2	8	no	—
117	#41×C₂₃	—	2	8	no	—
118	#41×C₂₃	—	2	8	no	—
119	#41×C₂₃	—	2	8	no	—
120	#41×C₂₃	—	2	8	no	—
121	#41×C₂₃	—	2	8	no	—
122	#41×C₂₃	—	4	8	no	—
123	#41×C₂₃	—	4	8	no	—
124	#41×C₂₃	—	4	8	no	—
125	C₂×Q₈×C₂₃	—	2	4	no	—
126	C₂×Q₈×C₂₃	—	2	2	no	—
127	C₂×Q₈×C₂₃	—	2	2	no	—
128	C₂×Q₈×C₂₃	—	2	1	no	—
129	C₂×Q₈×C₂₃	—	2	8	no	—
130	C₂×Q₈×C₂₃	—	2	4	no	—
131	C₂×Q₈×C₂₃	—	2	8	no	—
132	C₂×Q₈×C₂₃	—	2	8	no	—
133	C₂×Q₈×C₂₃	—	2	4	no	—
134	C₂×Q₈×C₂₃	—	2	4	no	—
135	C₂×Q₈×C₂₃	—	4	4	no	—
136	C₂×Q₈×C₂₃	—	4	8	no	—
137	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
138	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
139	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
140	C₂×D₄×C₂₃	—	2	4	no	—
141	C₂×D₄×C₂₃	—	2	4	no	—
142	C₂×D₄×C₂₃	—	2	4	no	—
143	C₂×D₄×C₂₃	—	2	4	no	—
144	C₂×D₄×C₂₃	—	2	4	no	—
145	C₂×D₄×C₂₃	—	2	4	no	—
146	C₂×D₄×C₂₃	—	2	2	no	—
147	C₂×D₄×C₂₃	—	2	1	no	—
148	C₂×D₄×C₂₃	—	2	8	no	—
149	C₂×D₄×C₂₃	—	4	4	no	—
150	C₂×D₄×C₂₃	—	4	8	no	—
151	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
152	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₂₃	—	2	2	no	—
154	C₂²×C₄×C₂₃	—	2	1	no	—
155	C₂²×C₄×C₂₃	—	2	2	no	—
156	C₂²×C₄×C₂₃	—	2	4	no	—
157	C₂²×C₄×C₂₃	—	2	4	no	—
158	C₂²×C₄×C₂₃	—	2	4	no	—
159	C₂²×C₄×C₂₃	—	2	4	no	—
160	C₂²×C₄×C₂₃	—	4	4	no	—
161	C₂²×C₄×C₂₃	—	4	4	no	—
162	C₂²×C₄×C₂₃	—	4	8	no	—
163	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
164	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₂₃	C₂×D₄	1	8	yes	—
167	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
169	Q₁₆×C₂₃	—	2	32	no	—
170	Q₁₆×C₂₃	—	2	16	no	—
171	SD₁₆×C₂₃	—	2	32	no	—
172	SD₁₆×C₂₃	—	2	16	no	—
173	SD₁₆×C₂₃	—	2	16	no	—
174	D₈×C₂₃	—	2	16	no	—
175	M₁₆×C₂₃	C₂×D₄	1	16	yes	—
176	M₁₆×C₂₃	C₂×D₄	1	16	yes	—
177	C₂×C₈×C₂₃	C₂×D₄	1	16	yes	—
178	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
179	C₄⋊C₄×C₂₃	—	2	16	no	—
180	C₄⋊C₄×C₂₃	—	2	8	no	—
181	C₄⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
182	C₂²⋊C₄×C₂₃	—	2	16	no	—
183	C₂²⋊C₄×C₂₃	—	2	16	no	—
184	C₂²⋊C₄×C₂₃	C₂×D₄	1	8	yes	—
185	C₄²×C₂₃	C₂×D₄	1	4	yes	—
186	C₄²×C₂₃	C₂⁴	1	4	yes	—
187	#60645	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂³×C₂₃	—	2	2	no	—
189	C₂³×C₂₃	—	2	4	no	—
190	C₂³×C₂₃	—	4	2	no	—
191	C₂³×C₂₃	—	4	4	no	—
192	C₂³×C₂₃	—	4	2	no	—
193	C₂³×C₂₃	—	4	4	no	—
194	C₂³×C₂₃	—	8	4	no	—
195	C₂³×C₂₃	#127	1	4	yes	—
196	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
197	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
198	Q₈×C₂₃	—	2	8	no	—
199	Q₈×C₂₃	—	2	4	no	—
200	Q₈×C₂₃	—	2	4	no	—
201	Q₈×C₂₃	—	2	4	no	—
202	Q₈×C₂₃	—	4	8	no	—
203	Q₈×C₂₃	—	4	4	no	—
204	Q₈×C₂₃	—	4	4	no	—
205	Q₈×C₂₃	—	4	2	no	—
206	D₄×C₂₃	—	2	4	no	—
207	D₄×C₂₃	—	2	4	no	—
208	D₄×C₂₃	—	2	4	no	—
209	D₄×C₂₃	—	4	8	no	—
210	D₄×C₂₃	—	4	8	no	—
211	D₄×C₂₃	—	4	8	no	—
212	D₄×C₂₃	—	4	4	no	—
213	D₄×C₂₃	—	4	2	no	—
214	D₄×C₂₃	—	8	8	no	—
215	C₂×C₄×C₂₃	—	2	4	no	—
216	C₂×C₄×C₂₃	—	2	4	no	—
217	C₂×C₄×C₂₃	—	2	2	no	—
218	C₂×C₄×C₂₃	—	4	4	no	—
219	C₂×C₄×C₂₃	—	4	2	no	—
220	C₂×C₄×C₂₃	—	4	4	no	—
221	C₂×C₄×C₂₃	—	4	4	no	—
222	C₂×C₄×C₂₃	—	4	8	no	—
223	C₂×C₄×C₂₃	—	4	8	no	—
224	C₂×C₄×C₂₃	—	4	8	no	—
225	C₂×C₄×C₂₃	—	4	8	no	—
226	C₂×C₄×C₂₃	#127	1	4	yes	—
227	C₂×C₄×C₂₃	#127	1	8	yes	—
228	C₂×C₄×C₂₃	C₂²×D₄	1	8	yes	—
229	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
230	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
232	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
233	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
234	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
235	C₈×C₂₃	C₂²×D₄	1	16	yes	—
236	C₂×#575	C₂×C₂₃	1	2	yes	complemented
237	C₂²×#137	C₂×C₂₃	1	2	yes	complemented
238	#3449	C₂×C₂₃	1	1	yes	characteristic, complemented
239	C₂×#536	C₂×C₂₃	1	1	yes	characteristic, complemented
240	#3156	C₂×C₂₃	1	8	yes	complemented
241	#3154	C₂×C₂₃	1	8	yes	complemented
242	#3153	C₂×C₂₃	1	4	yes	complemented
243	#3152	C₂×C₂₃	1	4	yes	complemented
244	#2873	C₂×C₂₃	1	1	yes	characteristic, complemented
245	C₂²×C₂₃	—	2	2	no	—
246	C₂²×C₂₃	—	2	4	no	—
247	C₂²×C₂₃	—	4	4	no	—
248	C₂²×C₂₃	—	4	8	no	—
249	C₂²×C₂₃	—	4	4	no	—
250	C₂²×C₂₃	—	4	2	no	—
251	C₂²×C₂₃	—	4	4	no	—
252	C₂²×C₂₃	—	8	4	no	—
253	C₂²×C₂₃	—	8	8	no	—
254	C₂²×C₂₃	C₂×#127	1	4	yes	—
255	C₂²×C₂₃	C₂×#127	1	2	yes	—
256	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
257	C₄×C₂₃	—	2	8	no	—
258	C₄×C₂₃	—	4	4	no	—
259	C₄×C₂₃	—	4	2	no	—
260	C₄×C₂₃	—	4	4	no	—
261	C₄×C₂₃	C₂×#127	1	2	yes	—
262	C₄×C₂₃	C₂×#127	1	4	yes	—
263	C₄×C₂₃	C₂³×D₄	1	2	yes	—
264	C₂²×#41	C₂²×C₂₃	1	1	yes	characteristic
265	C₂²×#41	C₂²×C₂₃	1	2	yes	—
266	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
267	#575	—	2	16	no	—
268	C₂×#137	—	2	16	no	—
269	C₂×#137	C₂²×C₂₃	1	4	yes	complemented
270	C₂×#137	C₂²×C₂₃	1	8	yes	complemented
271	C₂×#137	C₂²×C₂₃	1	8	yes	complemented
272	C₂×#137	C₂²×C₂₃	1	4	yes	complemented
273	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	4	yes	complemented
274	C₂×AGL(1,ℤ/8)	C₂²×C₂₃	1	4	yes	complemented
275	C₂×#135	C₂²×C₂₃	1	8	yes	—
276	C₂²×Q₁₆	C₂²×C₂₃	1	4	yes	complemented
277	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	complemented
278	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
279	#537	—	2	2	no	—
280	#536	—	2	4	no	—
281	#536	C₂²×C₂₃	1	8	yes	complemented
282	#536	C₂²×C₂₃	1	4	yes	complemented
283	#535	C₂²×C₂₃	1	4	yes	complemented
284	#534	—	2	2	no	—
285	#532	C₂²×C₂₃	1	4	yes	—
286	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
287	C₂×#124	C₂²×C₂₃	1	2	yes	complemented
288	#525	—	2	4	no	—
289	C₂×#123	—	2	2	no	—
290	C₂×C₂²⋊Q₈	—	2	2	no	—
291	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
292	#497	C₂²×C₂₃	1	16	yes	complemented
293	#497	C₂²×C₂₃	1	8	yes	complemented
294	#496	C₂²×C₂₃	1	8	yes	complemented
295	#495	C₂²×C₂₃	1	16	yes	complemented
296	#494	C₂²×C₂₃	1	8	yes	complemented
297	#492	C₂²×C₂₃	1	8	yes	complemented
298	#405	C₂²×C₂₃	1	8	yes	complemented
299	C₄×M₁₆	C₂²×C₂₃	1	4	yes	complemented
300	C₂×C₂₃	—	2	2	no	—
301	C₂×C₂₃	—	2	4	no	—
302	C₂×C₂₃	—	8	4	no	—
303	C₂×C₂₃	—	8	2	no	—
304	C₂×C₂₃	C₂²×#127	1	1	yes	characteristic
305	C₂×C₂₃	#3586	1	2	yes	—
306	C₂×#41	—	2	2	no	—
307	C₂×#41	—	2	2	no	—
308	C₂×#41	—	2	2	no	—
309	C₂×#41	—	2	8	no	—
310	C₂×#41	—	2	8	no	—
311	C₂×#41	—	2	8	no	—
312	C₂×#41	C₂³×C₂₃	1	4	yes	—
313	C₂×#41	C₂³×C₂₃	1	4	yes	—
314	C₂×#41	C₂³×C₂₃	1	4	yes	—
315	C₂²×Q₈	—	2	1	no	—
316	C₂²×Q₈	—	2	2	no	—
317	C₂²×Q₈	—	2	8	no	—
318	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
319	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
320	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
321	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
322	C₂²×D₄	—	2	1	no	—
323	C₂²×D₄	—	2	2	no	—
324	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
325	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
326	C₂³×C₄	—	2	1	no	—
327	C₂³×C₄	—	2	1	no	—
328	C₂³×C₄	—	2	2	no	—
329	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
330	#137	—	2	32	no	—
331	#137	—	2	32	no	—
332	#137	—	2	16	no	—
333	#137	—	2	16	no	—
334	AGL(1,ℤ/8)	—	2	16	no	—
335	AGL(1,ℤ/8)	—	2	16	no	—
336	#135	—	2	32	no	—
337	C₂×Q₁₆	—	2	16	no	—
338	C₂×Q₁₆	C₂³×C₂₃	1	16	yes	—
339	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
340	C₂×SD₁₆	—	2	16	no	—
341	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
342	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
343	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
344	C₂×D₈	C₂³×C₂₃	1	8	yes	—
345	C₂×M₁₆	D₄×C₂₃	1	16	yes	complemented
346	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
347	C₂×M₁₆	C₂³×C₂₃	1	4	yes	—
348	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
349	C₄⋊Q₈	—	2	4	no	—
350	C₄⋊Q₈	C₂³×C₂₃	1	8	yes	—
351	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
352	#127	C₂³×C₂₃	1	4	yes	—
353	#124	—	2	4	no	—
354	#124	C₂³×C₂₃	1	8	yes	—
355	#124	C₂³×C₂₃	1	8	yes	—
356	#123	—	2	16	no	—
357	#123	—	2	8	no	—
358	C₂²⋊Q₈	—	2	16	no	—
359	C₂²⋊Q₈	—	2	8	no	—
360	#121	—	2	8	no	—
361	#121	—	2	8	no	—
362	#120	—	2	8	no	—
363	C₄×D₄	—	2	8	no	—
364	#117	D₄×C₂₃	1	4	yes	—
365	#117	C₂³×C₂₃	1	4	yes	—
366	C₂×C₄⋊C₄	—	2	4	no	—
367	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
368	C₂×C₂²⋊C₄	—	2	4	no	—
369	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
370	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
371	#97	C₂³×C₂₃	1	8	yes	—
372	C₄×C₈	C₂³×C₂₃	1	8	yes	—
373	C₂₃	#60645	1	1	yes	characteristic, direct product factor, Sylow
374	C₂⁴	—	2	1	no	—
375	C₂⁴	—	2	2	no	—
376	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
377	#41	—	2	8	no	—
378	#41	—	2	8	no	—
379	#41	—	2	8	no	—
380	#41	—	2	8	no	—
381	#41	—	2	8	no	—
382	#41	—	2	8	no	—
383	#41	—	4	8	no	—
384	#41	—	4	8	no	—
385	#41	—	4	8	no	—
386	C₂×Q₈	—	2	4	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	—	2	8	no	—
391	C₂×Q₈	—	2	4	no	—
392	C₂×Q₈	—	2	8	no	—
393	C₂×Q₈	—	2	8	no	—
394	C₂×Q₈	—	2	4	no	—
395	C₂×Q₈	—	2	4	no	—
396	C₂×Q₈	—	4	4	no	—
397	C₂×Q₈	—	4	8	no	—
398	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
399	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
400	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
401	C₂×D₄	—	2	4	no	—
402	C₂×D₄	—	2	4	no	—
403	C₂×D₄	—	2	4	no	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	—	2	8	no	—
410	C₂×D₄	—	4	4	no	—
411	C₂×D₄	—	4	8	no	—
412	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
413	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	2	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	4	4	no	—
422	C₂²×C₄	—	4	4	no	—
423	C₂²×C₄	—	4	8	no	—
424	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
425	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
426	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
427	C₂²×C₄	C₂×D₄×C₂₃	1	8	yes	—
428	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
429	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
430	Q₁₆	—	2	32	no	—
431	Q₁₆	—	2	16	no	—
432	SD₁₆	—	2	32	no	—
433	SD₁₆	—	2	16	no	—
434	SD₁₆	—	2	16	no	—
435	D₈	—	2	16	no	—
436	M₁₆	C₂×D₄×C₂₃	1	16	yes	—
437	M₁₆	C₂×D₄×C₂₃	1	16	yes	—
438	C₂×C₈	C₂×D₄×C₂₃	1	16	yes	—
439	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
440	C₄⋊C₄	—	2	16	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
443	C₂²⋊C₄	—	2	16	no	—
444	C₂²⋊C₄	—	2	16	no	—
445	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
446	C₄²	C₂×D₄×C₂₃	1	4	yes	—
447	C₄²	C₂⁴×C₂₃	1	4	yes	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	4	no	—
450	C₂³	—	4	2	no	—
451	C₂³	—	4	4	no	—
452	C₂³	—	4	2	no	—
453	C₂³	—	4	4	no	—
454	C₂³	—	8	4	no	—
455	C₂³	C₂₃×#127	1	4	yes	—
456	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
457	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
458	Q₈	—	2	8	no	—
459	Q₈	—	2	4	no	—
460	Q₈	—	2	4	no	—
461	Q₈	—	2	4	no	—
462	Q₈	—	4	8	no	—
463	Q₈	—	4	4	no	—
464	Q₈	—	4	4	no	—
465	Q₈	—	4	2	no	—
466	D₄	—	2	4	no	—
467	D₄	—	2	4	no	—
468	D₄	—	2	4	no	—
469	D₄	—	4	8	no	—
470	D₄	—	4	8	no	—
471	D₄	—	4	8	no	—
472	D₄	—	4	4	no	—
473	D₄	—	4	2	no	—
474	D₄	—	8	8	no	—
475	C₂×C₄	—	2	4	no	—
476	C₂×C₄	—	2	4	no	—
477	C₂×C₄	—	2	2	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	4	no	—
481	C₂×C₄	—	4	4	no	—
482	C₂×C₄	—	4	8	no	—
483	C₂×C₄	—	4	8	no	—
484	C₂×C₄	—	4	8	no	—
485	C₂×C₄	—	4	8	no	—
486	C₂×C₄	C₂₃×#127	1	4	yes	—
487	C₂×C₄	C₂₃×#127	1	8	yes	—
488	C₂×C₄	C₂²×D₄×C₂₃	1	8	yes	—
489	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
490	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
491	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
492	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
493	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
494	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
495	C₈	C₂²×D₄×C₂₃	1	16	yes	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	4	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	4	8	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	8	4	no	—
504	C₂²	—	8	8	no	—
505	C₂²	C₂×C₂₃×#127	1	4	yes	—
506	C₂²	C₂×C₂₃×#127	1	2	yes	—
507	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
508	C₄	—	2	8	no	—
509	C₄	—	4	4	no	—
510	C₄	—	4	2	no	—
511	C₄	—	4	4	no	—
512	C₄	C₂×C₂₃×#127	1	4	yes	—
513	C₄	C₂×C₂₃×#127	1	2	yes	—
514	C₄	C₂³×D₄×C₂₃	1	2	yes	—
515	C₂	—	2	2	no	—
516	C₂	—	2	4	no	—
517	C₂	—	8	4	no	—
518	C₂	—	8	2	no	—
519	C₂	C₂²×C₂₃×#127	1	1	yes	characteristic
520	C₂	C₂₃×#3586	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 ◁ C₂×C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#575	C₂	semidirect
2. C₂²×C₂₃×#137	C₂	semidirect
3. C₂₃×#3449	C₂	semidirect
4. C₂×C₂₃×#536	C₂	semidirect
5. C₂₃×#3156	C₂	semidirect
6. C₂₃×#3154	C₂	semidirect
7. C₂₃×#3153	C₂	semidirect
8. C₂₃×#3152	C₂	semidirect
9. C₂₃×#2873	C₂	semidirect
15. C₂×C₂₃×#137	C₂²	semidirect
16. C₂×C₂₃×#137	C₂²	semidirect
17. C₂×C₂₃×#137	C₂²	semidirect
18. C₂×C₂₃×#137	C₂²	semidirect
19. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
20. C₂×C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
22. C₂²×Q₁₆×C₂₃	C₂²	semidirect
23. C₂²×SD₁₆×C₂₃	C₂²	semidirect
27. C₂₃×#536	C₂²	semidirect
28. C₂₃×#536	C₂²	semidirect
29. C₂₃×#535	C₂²	semidirect
32. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
33. C₂×C₂₃×#124	C₂²	semidirect
38. C₂₃×#497	C₂²	semidirect
39. C₂₃×#497	C₂²	semidirect
40. C₂₃×#496	C₂²	semidirect
41. C₂₃×#495	C₂²	semidirect
42. C₂₃×#494	C₂²	semidirect
43. C₂₃×#492	C₂²	semidirect
44. C₂₃×#405	C₂²	semidirect
45. C₄×M₁₆×C₂₃	C₂²	semidirect
85. C₂×M₁₆×C₂₃	D₄	semidirect
187. #60645	C₂₃	direct (N × H)
236. C₂×#575	C₂×C₂₃	semidirect
237. C₂²×#137	C₂×C₂₃	semidirect
238. #3449	C₂×C₂₃	semidirect
239. C₂×#536	C₂×C₂₃	semidirect
240. #3156	C₂×C₂₃	semidirect
241. #3154	C₂×C₂₃	semidirect
242. #3153	C₂×C₂₃	semidirect
243. #3152	C₂×C₂₃	semidirect
244. #2873	C₂×C₂₃	semidirect
269. C₂×#137	C₂²×C₂₃	semidirect
270. C₂×#137	C₂²×C₂₃	semidirect
271. C₂×#137	C₂²×C₂₃	semidirect
272. C₂×#137	C₂²×C₂₃	semidirect
273. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
274. C₂×AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
276. C₂²×Q₁₆	C₂²×C₂₃	semidirect
277. C₂²×SD₁₆	C₂²×C₂₃	semidirect
281. #536	C₂²×C₂₃	semidirect
282. #536	C₂²×C₂₃	semidirect
283. #535	C₂²×C₂₃	semidirect
286. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
287. C₂×#124	C₂²×C₂₃	semidirect
292. #497	C₂²×C₂₃	semidirect
293. #497	C₂²×C₂₃	semidirect
294. #496	C₂²×C₂₃	semidirect
295. #495	C₂²×C₂₃	semidirect
296. #494	C₂²×C₂₃	semidirect
297. #492	C₂²×C₂₃	semidirect
298. #405	C₂²×C₂₃	semidirect
299. C₄×M₁₆	C₂²×C₂₃	semidirect
345. C₂×M₁₆	D₄×C₂₃	semidirect
373. C₂₃	#60645	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#575	2
C₂²×C₂₃×#137	2
C₂₃×#3449	2
C₂×C₂₃×#536	2
C₂₃×#3156	2
C₂₃×#3154	2
C₂₃×#3153	2
C₂₃×#3152	2
C₂₃×#2873	2
#60645	23

Subgroups — C23×#60645 #19208271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60645 #19208271