Subgroups — C₂₃×#22672 #19170298

Order 5888.

1050 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#442	C₂	1	2	yes	maximal
2	C₂×C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2349	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1943	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1862	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1804	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₂₃	C₂²	1	4	yes	—
11	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂₃×#442	—	2	8	no	—
13	C₄×D₈×C₂₃	C₄	1	8	yes	complemented
14	C₂×C₂₃×#107	C₂²	1	2	yes	—
15	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×#106	C₂²	1	2	yes	—
17	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₂₃×#102	C₂²	1	4	yes	—
19	C₂×C₂₃×#102	C₂²	1	2	yes	—
20	C₂×C₂₃×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#389	C₂²	1	4	yes	—
23	C₂₃×#385	—	2	4	no	—
24	C₂₃×#382	C₂²	1	2	yes	—
25	C₂₃×#380	—	2	2	no	—
26	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₂₃×#95	—	2	2	no	—
28	C₂₃×#340	C₂²	1	4	yes	—
29	C₂₃×#336	C₂²	1	2	yes	—
30	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₂₃	—	2	2	no	—
32	C₂³×C₄×C₂₃	—	2	4	no	—
33	C₂×D₈×C₂₃	C₂×C₄	1	4	yes	—
34	C₂×D₈×C₂₃	D₄	1	4	yes	—
35	C₂×D₈×C₂₃	Q₈	1	4	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
39	C₄×D₄×C₂₃	—	2	8	no	—
40	C₄×D₄×C₂₃	—	2	4	no	—
41	C₄×D₄×C₂₃	—	4	4	no	—
42	C₄×D₄×C₂₃	C₂×C₄	1	8	yes	complemented
43	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
47	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
49	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
50	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
51	C₂₃×#107	—	4	2	no	—
52	C₂₃×#107	C₂×C₄	1	4	yes	complemented
53	C₂₃×#106	—	2	4	no	—
54	C₂₃×C₄⋊C₈	—	2	2	no	—
55	C₂₃×#102	—	2	8	no	—
56	C₂₃×#102	C₂×C₄	1	8	yes	complemented
57	C₂₃×#97	—	2	4	no	—
58	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	complemented
59	C₂₃×#95	—	2	2	no	—
60	C₂₃×#95	—	2	4	no	—
61	C₂₃×#95	—	2	2	no	—
62	C₂₃×#95	D₄	1	2	yes	—
63	C₂⁴×C₂₃	—	2	2	no	—
64	C₂×D₄×C₂₃	—	2	2	no	—
65	C₂×D₄×C₂₃	—	2	2	no	—
66	C₂×D₄×C₂₃	—	2	2	no	—
67	C₂×D₄×C₂₃	C₂²×C₄	1	4	yes	—
68	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
69	C₂×D₄×C₂₃	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₂₃	—	2	2	no	—
71	C₂²×C₄×C₂₃	—	2	8	no	—
72	C₂²×C₄×C₂₃	—	2	4	no	—
73	C₂²×C₄×C₂₃	—	4	8	no	—
74	C₂²×C₄×C₂₃	—	4	8	no	—
75	C₂²×C₄×C₂₃	—	8	4	no	—
76	C₂²×C₄×C₂₃	—	8	4	no	—
77	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
81	D₈×C₂₃	C₄⋊C₄	1	16	yes	complemented
82	C₂×C₈×C₂₃	—	2	1	no	—
83	C₂×C₈×C₂₃	—	2	1	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	—	2	2	no	—
87	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
91	C₄⋊C₄×C₂₃	—	2	2	no	—
92	C₄⋊C₄×C₂₃	—	2	4	no	—
93	C₄⋊C₄×C₂₃	—	2	4	no	—
94	C₄⋊C₄×C₂₃	—	4	2	no	—
95	C₄⋊C₄×C₂₃	—	8	4	no	—
96	C₄⋊C₄×C₂₃	C₂²×C₄	1	4	yes	—
97	C₂²⋊C₄×C₂₃	—	2	8	no	—
98	C₂²⋊C₄×C₂₃	—	4	8	no	—
99	C₄²×C₂₃	—	2	2	no	—
100	C₄²×C₂₃	—	2	2	no	—
101	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
102	C₄²×C₂₃	#41	1	2	yes	—
103	#22672	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	—	2	4	no	—
105	C₂³×C₂₃	—	2	4	no	—
106	C₂³×C₂₃	—	2	4	no	—
107	C₂³×C₂₃	—	4	4	no	—
108	C₂³×C₂₃	—	4	4	no	—
109	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
110	D₄×C₂₃	—	2	4	no	—
111	D₄×C₂₃	—	2	4	no	—
112	D₄×C₂₃	—	2	4	no	—
113	D₄×C₂₃	C₂×C₄⋊C₄	1	8	yes	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	2	4	no	—
118	C₂×C₄×C₂₃	—	2	4	no	—
119	C₂×C₄×C₂₃	—	2	4	no	—
120	C₂×C₄×C₂₃	—	4	4	no	—
121	C₂×C₄×C₂₃	—	4	4	no	—
122	C₂×C₄×C₂₃	—	4	4	no	—
123	C₂×C₄×C₂₃	—	4	4	no	—
124	C₂×C₄×C₂₃	—	8	8	no	—
125	C₂×C₄×C₂₃	—	8	16	no	—
126	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
134	C₈×C₂₃	—	2	2	no	—
135	C₈×C₂₃	—	4	2	no	—
136	C₈×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
137	C₂×#442	C₂×C₂₃	1	2	yes	—
138	C₂×C₄×D₈	C₂×C₂₃	1	1	yes	characteristic
139	#2349	C₂×C₂₃	1	2	yes	complemented
140	#1943	C₂×C₂₃	1	2	yes	complemented
141	#1862	C₂×C₂₃	1	4	yes	complemented
142	#1846	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#1804	C₂×C₂₃	1	2	yes	complemented
144	#1770	C₂×C₂₃	1	1	yes	characteristic, complemented
145	C₂²×C₂₃	—	2	8	no	—
146	C₂²×C₂₃	—	4	8	no	—
147	C₂²×C₂₃	—	4	4	no	—
148	C₂²×C₂₃	—	4	4	no	—
149	C₂²×C₂₃	—	4	8	no	—
150	C₂²×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂²×C₂₃	D₄²	1	1	yes	characteristic
152	C₂²×C₂₃	D₄×Q₈	1	1	yes	characteristic
153	C₂²×C₂₃	#575	1	2	yes	—
154	C₂²×C₂₃	#576	1	2	yes	—
155	C₄×C₂₃	—	2	2	no	—
156	C₄×C₂₃	—	2	2	no	—
157	C₄×C₂₃	—	4	4	no	—
158	C₄×C₂₃	—	4	4	no	—
159	C₄×C₂₃	—	8	4	no	—
160	C₄×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₄×C₂₃	#520	1	1	yes	characteristic
162	C₄×C₂₃	#547	1	1	yes	characteristic
163	C₄×C₂₃	#554	1	1	yes	characteristic
164	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
165	C₂×C₄×D₄	C₂²×C₂₃	1	4	yes	—
166	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
167	#442	—	2	8	no	—
168	C₄×D₈	C₄×C₂₃	1	8	yes	complemented
169	C₂×#107	C₂²×C₂₃	1	2	yes	—
170	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×#106	C₂²×C₂₃	1	2	yes	—
172	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×#102	C₂²×C₂₃	1	4	yes	—
174	C₂×#102	C₂²×C₂₃	1	2	yes	—
175	C₂×#97	C₂²×C₂₃	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
177	#389	C₂²×C₂₃	1	4	yes	—
178	#385	—	2	4	no	—
179	#382	C₂²×C₂₃	1	2	yes	—
180	#380	—	2	2	no	—
181	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
182	C₂×#95	—	2	2	no	—
183	#340	C₂²×C₂₃	1	4	yes	—
184	#336	C₂²×C₂₃	1	2	yes	—
185	C₂×C₂₃	—	4	8	no	—
186	C₂×C₂₃	D₄×C₄⋊C₄	1	1	yes	characteristic
187	C₂×C₂₃	#2943	1	1	yes	characteristic
188	C₂×C₂₃	#2948	1	1	yes	characteristic
189	C₂×C₂₃	#3273	1	2	yes	—
190	C₂×C₂₃	#3389	1	2	yes	—
191	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	—	2	4	no	—
194	C₂×D₈	C₂×C₄×C₂₃	1	4	yes	—
195	C₂×D₈	D₄×C₂₃	1	4	yes	—
196	C₂×D₈	Q₈×C₂₃	1	4	yes	—
197	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
198	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
200	C₄×D₄	—	2	8	no	—
201	C₄×D₄	—	2	4	no	—
202	C₄×D₄	—	4	4	no	—
203	C₄×D₄	C₂×C₄×C₂₃	1	8	yes	complemented
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
206	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
207	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
208	C₂×C₂²⋊C₄	—	2	2	no	—
209	C₂×C₂²⋊C₄	—	2	4	no	—
210	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
211	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
212	#107	—	4	2	no	—
213	#107	C₂×C₄×C₂₃	1	4	yes	complemented
214	#106	—	2	4	no	—
215	C₄⋊C₈	—	2	2	no	—
216	#102	—	2	8	no	—
217	#102	C₂×C₄×C₂₃	1	8	yes	complemented
218	#97	—	2	4	no	—
219	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	complemented
220	#95	—	2	2	no	—
221	#95	—	2	4	no	—
222	#95	—	2	2	no	—
223	#95	D₄×C₂₃	1	2	yes	—
224	C₂₃	#22672	1	1	yes	characteristic, direct product factor, Sylow
225	C₂⁴	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂²×C₄×C₂₃	1	4	yes	—
230	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
231	C₂×D₄	C₂×Q₈×C₂₃	1	4	yes	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	4	8	no	—
236	C₂²×C₄	—	4	8	no	—
237	C₂²×C₄	—	8	4	no	—
238	C₂²×C₄	—	8	4	no	—
239	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
240	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
242	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
243	D₈	C₄⋊C₄×C₂₃	1	16	yes	complemented
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
250	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
252	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	8	4	no	—
258	C₄⋊C₄	C₂²×C₄×C₂₃	1	4	yes	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₂²⋊C₄	—	4	8	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
264	C₄²	#41×C₂₃	1	2	yes	—
265	C₂³	—	2	4	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	2	4	no	—
268	C₂³	—	4	4	no	—
269	C₂³	—	4	4	no	—
270	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
271	D₄	—	2	4	no	—
272	D₄	—	2	4	no	—
273	D₄	—	2	4	no	—
274	D₄	C₂×C₄⋊C₄×C₂₃	1	8	yes	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	8	8	no	—
286	C₂×C₄	—	8	16	no	—
287	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
288	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
289	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
291	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
292	C₂×C₄	C₂×#41×C₂₃	1	1	yes	characteristic
293	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
294	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	4	2	no	—
297	C₈	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
298	C₂²	—	2	8	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	8	no	—
302	C₂²	—	4	8	no	—
303	C₂²	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
304	C₂²	D₄²×C₂₃	1	1	yes	characteristic
305	C₂²	D₄×Q₈×C₂₃	1	1	yes	characteristic
306	C₂²	C₂₃×#575	1	2	yes	—
307	C₂²	C₂₃×#576	1	2	yes	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	4	4	no	—
311	C₄	—	4	4	no	—
312	C₄	—	8	4	no	—
313	C₄	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
314	C₄	C₂₃×#520	1	1	yes	characteristic
315	C₄	C₂₃×#547	1	1	yes	characteristic
316	C₄	C₂₃×#554	1	1	yes	characteristic
317	C₂	—	4	8	no	—
318	C₂	D₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
319	C₂	C₂₃×#2943	1	1	yes	characteristic
320	C₂	C₂₃×#2948	1	1	yes	characteristic
321	C₂	C₂₃×#3273	1	2	yes	—
322	C₂	C₂₃×#3389	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄×C₄⋊C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#2349	C₂	semidirect
4. C₂₃×#1943	C₂	semidirect
5. C₂₃×#1862	C₂	semidirect
6. C₂₃×#1846	C₂	semidirect
7. C₂₃×#1804	C₂	semidirect
8. C₂₃×#1770	C₂	semidirect
13. C₄×D₈×C₂₃	C₄	semidirect
42. C₄×D₄×C₂₃	C₂×C₄	semidirect
52. C₂₃×#107	C₂×C₄	semidirect
56. C₂₃×#102	C₂×C₄	semidirect
58. C₄×C₈×C₂₃	C₂×C₄	semidirect
81. D₈×C₂₃	C₄⋊C₄	semidirect
103. #22672	C₂₃	direct (N × H)
139. #2349	C₂×C₂₃	semidirect
140. #1943	C₂×C₂₃	semidirect
141. #1862	C₂×C₂₃	semidirect
142. #1846	C₂×C₂₃	semidirect
143. #1804	C₂×C₂₃	semidirect
144. #1770	C₂×C₂₃	semidirect
168. C₄×D₈	C₄×C₂₃	semidirect
203. C₄×D₄	C₂×C₄×C₂₃	semidirect
213. #107	C₂×C₄×C₂₃	semidirect
217. #102	C₂×C₄×C₂₃	semidirect
219. C₄×C₈	C₂×C₄×C₂₃	semidirect
224. C₂₃	#22672	direct (N × H)
243. D₈	C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#442	2
C₂×C₄×D₈×C₂₃	2
C₂₃×#2349	2
C₂₃×#1943	2
C₂₃×#1862	2
C₂₃×#1846	2
C₂₃×#1804	2
C₂₃×#1770	2
#22672	23

Subgroups — C23×#22672 #19170298

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#22672 #19170298