Subgroups — C₂₃×#22730 #19170356

Order 5888.

730 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#446	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2355	C₂	1	2	yes	maximal
3	C₂₃×#1949	C₂	1	2	yes	maximal
4	C₂₃×#1949	C₂	1	2	yes	maximal
5	C₂₃×#1948	C₂	1	2	yes	maximal
6	C₂₃×#1847	C₂	1	2	yes	maximal
7	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#1776	C₂	1	2	yes	maximal
9	C₄×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#446	C₄	1	8	yes	complemented
12	C₂×C₂₃×#107	C₂²	1	2	yes	—
13	C₂×C₂₃×#107	C₂²	1	2	yes	—
14	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
15	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#106	C₂²	1	2	yes	—
17	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
19	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
20	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₂₃×#97	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#391	—	2	2	no	—
24	C₂₃×#387	—	2	2	no	—
25	C₂₃×#387	—	2	2	no	—
26	C₂₃×#384	C₂²	1	2	yes	—
27	C₂₃×#382	C₂²	1	2	yes	—
28	C₂₃×#382	C₂²	1	2	yes	—
29	C₄×C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
33	C₂₃×#340	C₂²	1	2	yes	—
34	C₂₃×#340	C₂²	1	2	yes	—
35	C₂₃×#340	C₂²	1	2	yes	—
36	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
40	C₄×Q₈×C₂₃	C₂×C₄	1	4	yes	—
41	C₄×Q₈×C₂₃	Q₈	1	4	yes	—
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₂₃	—	2	2	no	—
51	C₂×C₄²×C₂₃	—	2	2	no	—
52	C₂×C₄²×C₂₃	—	2	2	no	—
53	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
55	C₂₃×#107	—	2	4	no	—
56	C₂₃×#107	—	4	2	no	—
57	C₂₃×#106	—	2	2	no	—
58	C₂₃×#106	—	4	1	no	—
59	C₂₃×#106	—	4	2	no	—
60	C₂₃×#106	—	4	1	no	—
61	C₂₃×#106	Q₈	1	4	yes	complemented
62	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
63	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
64	C₂₃×#97	C₂×C₄	1	8	yes	—
65	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
66	C₂₃×#95	—	2	2	no	—
67	C₂₃×#95	—	2	2	no	—
68	C₂₃×#95	—	2	2	no	—
69	C₂₃×#95	—	2	2	no	—
70	C₂₃×#95	—	2	2	no	—
71	C₂₃×#95	—	2	2	no	—
72	C₂₃×#95	—	2	2	no	—
73	C₂×Q₈×C₂₃	—	4	2	no	—
74	C₂×Q₈×C₂₃	C₄⋊C₄	1	4	yes	—
75	C₂²×C₄×C₂₃	—	2	2	no	—
76	C₂²×C₄×C₂₃	—	2	2	no	—
77	C₂²×C₄×C₂₃	—	2	2	no	—
78	C₂²×C₄×C₂₃	—	2	2	no	—
79	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
84	C₂×C₈×C₂₃	—	2	1	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	—	2	1	no	—
87	C₂×C₈×C₂₃	—	2	2	no	—
88	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₂₃	#41	1	2	yes	—
93	C₄⋊C₄×C₂₃	—	2	2	no	—
94	C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₄⋊C₄×C₂₃	—	2	2	no	—
96	C₄⋊C₄×C₂₃	—	2	2	no	—
97	C₄⋊C₄×C₂₃	—	2	2	no	—
98	C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₄⋊C₄×C₂₃	—	4	4	no	—
101	C₄⋊C₄×C₂₃	—	8	2	no	—
102	C₄⋊C₄×C₂₃	C₄⋊C₄	1	8	yes	—
103	C₄⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
104	C₄⋊C₄×C₂₃	C₂×Q₈	1	4	yes	—
105	C₄²×C₂₃	—	4	4	no	—
106	C₄²×C₂₃	—	8	2	no	—
107	C₄²×C₂₃	—	8	2	no	—
108	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
110	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
111	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
112	#22730	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₂₃	—	4	4	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	2	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	4	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	—	4	2	no	—
124	C₂×C₄×C₂₃	—	4	2	no	—
125	C₂×C₄×C₂₃	—	4	2	no	—
126	C₂×C₄×C₂₃	—	4	2	no	—
127	C₂×C₄×C₂₃	—	4	2	no	—
128	C₂×C₄×C₂₃	—	4	2	no	—
129	C₂×C₄×C₂₃	—	4	2	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
135	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
137	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₈×C₂₃	—	2	2	no	—
144	C₈×C₂₃	—	2	4	no	—
145	C₈×C₂₃	C₄×Q₈	1	4	yes	—
146	C₂×#446	C₂×C₂₃	1	1	yes	characteristic
147	#2355	C₂×C₂₃	1	2	yes	—
148	#1949	C₂×C₂₃	1	2	yes	—
149	#1949	C₂×C₂₃	1	2	yes	—
150	#1948	C₂×C₂₃	1	2	yes	—
151	#1847	C₂×C₂₃	1	2	yes	—
152	#1846	C₂×C₂₃	1	1	yes	characteristic
153	#1776	C₂×C₂₃	1	2	yes	—
154	C₄×#106	C₂×C₂₃	1	1	yes	characteristic
155	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
156	C₂²×C₂₃	D₄×Q₈	1	1	yes	characteristic
157	C₂²×C₂₃	#550	1	1	yes	characteristic
158	C₂²×C₂₃	#575	1	2	yes	—
159	C₂²×C₂₃	#577	1	2	yes	—
160	C₄×C₂₃	—	2	4	no	—
161	C₄×C₂₃	—	2	2	no	—
162	C₄×C₂₃	—	2	2	no	—
163	C₄×C₂₃	—	2	4	no	—
164	C₄×C₂₃	—	4	4	no	—
165	C₄×C₂₃	—	8	2	no	—
166	C₄×C₂₃	—	8	2	no	—
167	C₄×C₂₃	—	8	2	no	—
168	C₄×C₂₃	C₂×C₄×Q₈	1	1	yes	characteristic
169	C₄×C₂₃	#520	1	1	yes	characteristic
170	C₄×C₂₃	#557	1	1	yes	characteristic
171	C₄×C₂₃	Q₈²	1	1	yes	characteristic
172	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
173	#446	C₄×C₂₃	1	8	yes	complemented
174	C₂×#107	C₂²×C₂₃	1	2	yes	—
175	C₂×#107	C₂²×C₂₃	1	2	yes	—
176	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
177	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
178	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
179	C₂×#106	C₂²×C₂₃	1	2	yes	—
180	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
181	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
182	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
183	C₂×#97	C₂²×C₂₃	1	2	yes	—
184	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
185	#391	—	2	2	no	—
186	#387	—	2	2	no	—
187	#387	—	2	2	no	—
188	#384	C₂²×C₂₃	1	2	yes	—
189	#382	C₂²×C₂₃	1	2	yes	—
190	#382	C₂²×C₂₃	1	2	yes	—
191	C₄×C₄⋊C₄	—	2	2	no	—
192	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
193	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
194	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
195	#340	C₂²×C₂₃	1	2	yes	—
196	#340	C₂²×C₂₃	1	2	yes	—
197	#340	C₂²×C₂₃	1	2	yes	—
198	C₂×C₂₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
199	C₂×C₂₃	#2911	1	1	yes	characteristic
200	C₂×C₂₃	#2912	1	1	yes	characteristic
201	C₂×C₂₃	#3388	1	2	yes	—
202	C₂×C₂₃	#3401	1	2	yes	—
203	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
206	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
207	C₄×Q₈	C₂×C₄×C₂₃	1	4	yes	—
208	C₄×Q₈	Q₈×C₂₃	1	4	yes	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
212	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
216	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
222	#107	—	2	4	no	—
223	#107	—	4	2	no	—
224	#106	—	2	2	no	—
225	#106	—	4	1	no	—
226	#106	—	4	2	no	—
227	#106	—	4	1	no	—
228	#106	Q₈×C₂₃	1	4	yes	complemented
229	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
230	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
231	#97	C₂×C₄×C₂₃	1	8	yes	—
232	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	#95	—	2	2	no	—
236	#95	—	2	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	C₂₃	#22730	1	1	yes	characteristic, direct product factor, Sylow
241	C₂×Q₈	—	4	2	no	—
242	C₂×Q₈	C₄⋊C₄×C₂₃	1	4	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
249	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
251	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
257	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
258	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
259	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
260	C₂×C₈	#41×C₂₃	1	2	yes	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	4	no	—
269	C₄⋊C₄	—	8	2	no	—
270	C₄⋊C₄	C₄⋊C₄×C₂₃	1	8	yes	—
271	C₄⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
272	C₄⋊C₄	C₂×Q₈×C₂₃	1	4	yes	—
273	C₄²	—	4	4	no	—
274	C₄²	—	8	2	no	—
275	C₄²	—	8	2	no	—
276	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
277	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
278	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
279	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
280	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
281	Q₈	—	4	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
301	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
302	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
304	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
305	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
308	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
309	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	2	4	no	—
312	C₈	C₄×Q₈×C₂₃	1	4	yes	—
313	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
314	C₂²	D₄×Q₈×C₂₃	1	1	yes	characteristic
315	C₂²	C₂₃×#550	1	1	yes	characteristic
316	C₂²	C₂₃×#575	1	2	yes	—
317	C₂²	C₂₃×#577	1	2	yes	—
318	C₄	—	2	4	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	4	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₂×C₄×Q₈×C₂₃	1	1	yes	characteristic
327	C₄	C₂₃×#520	1	1	yes	characteristic
328	C₄	C₂₃×#557	1	1	yes	characteristic
329	C₄	Q₈²×C₂₃	1	1	yes	characteristic
330	C₂	Q₈×C₄⋊C₄×C₂₃	1	1	yes	characteristic
331	C₂	C₂₃×#2911	1	1	yes	characteristic
332	C₂	C₂₃×#2912	1	1	yes	characteristic
333	C₂	C₂₃×#3388	1	2	yes	—
334	C₂	C₂₃×#3401	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
11. C₂₃×#446	C₄	semidirect
61. C₂₃×#106	Q₈	semidirect
112. #22730	C₂₃	direct (N × H)
173. #446	C₄×C₂₃	semidirect
228. #106	Q₈×C₂₃	semidirect
240. C₂₃	#22730	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×#446	2
C₂₃×#2355	2
C₂₃×#1949	2
C₂₃×#1948	2
C₂₃×#1847	2
C₂₃×#1846	2
C₂₃×#1776	2
C₄×C₂₃×#106	2
#22730	23

Subgroups — C23×#22730 #19170356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22730 #19170356