Subgroups — C₂₃×#32864 #19180490

Order 5888.

630 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2644	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2088	C₂	1	2	yes	maximal, complemented
3	C₂₃×#2086	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2067	C₂	1	2	yes	maximal, complemented
5	C₂₃×#2065	C₂	1	2	yes	maximal, complemented
6	C₂₃×#2065	C₂	1	2	yes	maximal, complemented
7	C₂₃×#2054	C₂	1	2	yes	maximal
8	C₂₃×#2054	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1843	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#106	C₂²	1	2	yes	—
13	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₂₃×#102	C₂²	1	2	yes	—
18	C₂×C₂₃×#102	C₂²	1	2	yes	—
19	C₂×C₂₃×#102	C₂²	1	2	yes	—
20	C₂₃×#397	—	2	2	no	—
21	C₂₃×#397	—	2	2	no	—
22	C₂₃×#397	—	2	2	no	—
23	C₂₃×#396	—	2	2	no	—
24	C₂₃×#390	C₂²	1	2	yes	—
25	C₂₃×#390	C₂²	1	2	yes	—
26	C₂₃×#390	C₂²	1	2	yes	—
27	C₂₃×#389	C₂²	1	2	yes	—
28	C₂₃×#389	C₂²	1	2	yes	—
29	C₂₃×#389	C₂²	1	2	yes	—
30	C₂₃×#376	C₂²	1	1	yes	characteristic
31	C₂₃×#340	C₂²	1	2	yes	—
32	C₂₃×#340	C₂²	1	1	yes	characteristic
33	C₂₃×#340	C₂²	1	1	yes	characteristic
34	C₂₃×#340	C₂²	1	2	yes	—
35	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
36	C₂×D₈×C₂₃	—	4	2	no	—
37	C₂×D₈×C₂₃	—	4	1	no	—
38	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
47	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
53	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
55	C₂₃×#107	—	4	1	no	—
56	C₂₃×#106	—	4	2	no	—
57	C₂₃×C₄⋊C₈	—	4	1	no	—
58	C₂₃×C₄⋊C₈	—	4	2	no	—
59	C₂₃×C₄⋊C₈	—	4	2	no	—
60	C₂₃×C₄⋊C₈	—	4	1	no	—
61	C₂₃×#102	—	4	2	no	—
62	C₂₃×#102	—	4	2	no	—
63	C₂₃×#102	—	4	2	no	—
64	C₂₃×#95	—	2	2	no	—
65	C₂₃×#95	—	2	2	no	—
66	C₂₃×#95	—	2	2	no	—
67	C₂₃×#95	D₄	1	2	yes	—
68	C₂₃×#95	D₄	1	1	yes	characteristic
69	C₂₃×#95	D₄	1	1	yes	characteristic
70	C₂⁴×C₂₃	—	2	2	no	—
71	C₂×D₄×C₂₃	—	2	2	no	—
72	C₂×D₄×C₂₃	—	2	2	no	—
73	C₂×D₄×C₂₃	—	2	2	no	—
74	C₂×D₄×C₂₃	—	2	2	no	—
75	C₂×D₄×C₂₃	—	2	2	no	—
76	C₂×D₄×C₂₃	—	2	2	no	—
77	C₂²×C₄×C₂₃	—	2	2	no	—
78	C₂²×C₄×C₂₃	—	2	2	no	—
79	C₂²×C₄×C₂₃	—	2	2	no	—
80	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
85	D₈×C₂₃	—	4	4	no	—
86	C₂×C₈×C₂₃	—	2	2	no	—
87	C₂×C₈×C₂₃	—	2	2	no	—
88	C₂×C₈×C₂₃	—	2	2	no	—
89	C₂×C₈×C₂₃	—	2	2	no	—
90	C₂×C₈×C₂₃	—	2	1	no	—
91	C₂×C₈×C₂₃	—	2	2	no	—
92	C₂×C₈×C₂₃	—	2	1	no	—
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	—	8	2	no	—
100	C₄⋊C₄×C₂₃	—	8	2	no	—
101	C₄⋊C₄×C₂₃	—	8	2	no	—
102	C₂²⋊C₄×C₂₃	—	8	2	no	—
103	C₂²⋊C₄×C₂₃	—	8	2	no	—
104	C₂²⋊C₄×C₂₃	—	8	2	no	—
105	C₂²⋊C₄×C₂₃	—	8	2	no	—
106	C₂²⋊C₄×C₂₃	—	8	2	no	—
107	C₂²⋊C₄×C₂₃	—	8	2	no	—
108	C₄²×C₂₃	—	2	2	no	—
109	C₄²×C₂₃	—	2	1	no	—
110	C₄²×C₂₃	—	2	2	no	—
111	C₄²×C₂₃	—	2	1	no	—
112	#32864	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₂₃	—	4	2	no	—
114	C₂³×C₂₃	—	4	2	no	—
115	C₂³×C₂₃	—	4	2	no	—
116	C₂³×C₂₃	—	4	2	no	—
117	C₂³×C₂₃	—	4	2	no	—
118	C₂³×C₂₃	—	4	2	no	—
119	C₂³×C₂₃	—	4	2	no	—
120	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
121	D₄×C₂₃	—	4	2	no	—
122	D₄×C₂₃	—	4	2	no	—
123	D₄×C₂₃	—	4	2	no	—
124	D₄×C₂₃	—	4	2	no	—
125	C₂×C₄×C₂₃	—	2	2	no	—
126	C₂×C₄×C₂₃	—	2	2	no	—
127	C₂×C₄×C₂₃	—	2	1	no	—
128	C₂×C₄×C₂₃	—	2	1	no	—
129	C₂×C₄×C₂₃	—	2	2	no	—
130	C₂×C₄×C₂₃	—	2	1	no	—
131	C₂×C₄×C₂₃	—	2	2	no	—
132	C₂×C₄×C₂₃	—	2	1	no	—
133	C₂×C₄×C₂₃	—	2	2	no	—
134	C₂×C₄×C₂₃	—	2	2	no	—
135	C₂×C₄×C₂₃	—	2	2	no	—
136	C₂×C₄×C₂₃	—	4	2	no	—
137	C₂×C₄×C₂₃	—	4	2	no	—
138	C₂×C₄×C₂₃	—	4	2	no	—
139	C₂×C₄×C₂₃	—	4	2	no	—
140	C₂×C₄×C₂₃	—	4	2	no	—
141	C₂×C₄×C₂₃	—	4	2	no	—
142	C₂×C₄×C₂₃	—	4	2	no	—
143	C₂×C₄×C₂₃	—	4	2	no	—
144	C₂×C₄×C₂₃	—	4	2	no	—
145	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
146	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
148	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
149	C₈×C₂₃	—	4	2	no	—
150	C₈×C₂₃	—	4	1	no	—
151	C₈×C₂₃	—	4	1	no	—
152	#2644	C₂×C₂₃	1	2	yes	complemented
153	#2088	C₂×C₂₃	1	2	yes	complemented
154	#2086	C₂×C₂₃	1	1	yes	characteristic, complemented
155	#2067	C₂×C₂₃	1	2	yes	complemented
156	#2065	C₂×C₂₃	1	2	yes	complemented
157	#2065	C₂×C₂₃	1	2	yes	complemented
158	#2054	C₂×C₂₃	1	2	yes	—
159	#2054	C₂×C₂₃	1	1	yes	characteristic
160	#1843	C₂×C₂₃	1	1	yes	characteristic, complemented
161	C₂²×C₂₃	—	8	2	no	—
162	C₂²×C₂₃	—	8	2	no	—
163	C₂²×C₂₃	—	8	2	no	—
164	C₂²×C₂₃	—	8	2	no	—
165	C₂²×C₂₃	—	8	2	no	—
166	C₂²×C₂₃	—	8	2	no	—
167	C₂²×C₂₃	—	8	2	no	—
168	C₂²×C₂₃	#537	1	2	yes	—
169	C₂²×C₂₃	#537	1	1	yes	characteristic
170	C₂²×C₂₃	#576	1	1	yes	characteristic
171	C₂²×C₂₃	#576	1	1	yes	characteristic
172	C₂²×C₂₃	#577	1	2	yes	—
173	C₄×C₂₃	—	4	2	no	—
174	C₄×C₂₃	—	4	2	no	—
175	C₄×C₂₃	—	4	1	no	—
176	C₄×C₂₃	—	4	1	no	—
177	C₄×C₂₃	—	8	2	no	—
178	C₄×C₂₃	—	8	2	no	—
179	C₄×C₂₃	—	8	2	no	—
180	C₄×C₂₃	#560	1	1	yes	characteristic
181	C₄×C₂₃	#563	1	2	yes	—
182	C₄×C₂₃	#563	1	1	yes	characteristic
183	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
184	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
185	C₂×#106	C₂²×C₂₃	1	2	yes	—
186	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
187	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
188	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
190	C₂×#102	C₂²×C₂₃	1	2	yes	—
191	C₂×#102	C₂²×C₂₃	1	2	yes	—
192	C₂×#102	C₂²×C₂₃	1	2	yes	—
193	#397	—	2	2	no	—
194	#397	—	2	2	no	—
195	#397	—	2	2	no	—
196	#396	—	2	2	no	—
197	#390	C₂²×C₂₃	1	2	yes	—
198	#390	C₂²×C₂₃	1	2	yes	—
199	#390	C₂²×C₂₃	1	2	yes	—
200	#389	C₂²×C₂₃	1	2	yes	—
201	#389	C₂²×C₂₃	1	2	yes	—
202	#389	C₂²×C₂₃	1	2	yes	—
203	#376	C₂²×C₂₃	1	1	yes	characteristic
204	#340	C₂²×C₂₃	1	2	yes	—
205	#340	C₂²×C₂₃	1	1	yes	characteristic
206	#340	C₂²×C₂₃	1	1	yes	characteristic
207	#340	C₂²×C₂₃	1	2	yes	—
208	C₂×C₂₃	—	16	2	no	—
209	C₂×C₂₃	#2823	1	1	yes	characteristic
210	C₂×C₂₃	#3257	1	2	yes	—
211	C₂×C₂₃	#3257	1	2	yes	—
212	C₂×C₂₃	#3259	1	1	yes	characteristic
213	C₂×C₂₃	#3263	1	1	yes	characteristic
214	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
215	C₂×D₈	—	4	2	no	—
216	C₂×D₈	—	4	1	no	—
217	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
219	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
225	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
234	#107	—	4	1	no	—
235	#106	—	4	2	no	—
236	C₄⋊C₈	—	4	1	no	—
237	C₄⋊C₈	—	4	2	no	—
238	C₄⋊C₈	—	4	2	no	—
239	C₄⋊C₈	—	4	1	no	—
240	#102	—	4	2	no	—
241	#102	—	4	2	no	—
242	#102	—	4	2	no	—
243	#95	—	2	2	no	—
244	#95	—	2	2	no	—
245	#95	—	2	2	no	—
246	#95	D₄×C₂₃	1	2	yes	—
247	#95	D₄×C₂₃	1	1	yes	characteristic
248	#95	D₄×C₂₃	1	1	yes	characteristic
249	C₂₃	#32864	1	1	yes	characteristic, direct product factor, Sylow
250	C₂⁴	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
263	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
265	D₈	—	4	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	8	2	no	—
280	C₄⋊C₄	—	8	2	no	—
281	C₄⋊C₄	—	8	2	no	—
282	C₂²⋊C₄	—	8	2	no	—
283	C₂²⋊C₄	—	8	2	no	—
284	C₂²⋊C₄	—	8	2	no	—
285	C₂²⋊C₄	—	8	2	no	—
286	C₂²⋊C₄	—	8	2	no	—
287	C₂²⋊C₄	—	8	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
325	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
326	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
327	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
328	C₈	—	4	1	no	—
329	C₈	—	4	2	no	—
330	C₈	—	4	1	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	C₂₃×#537	1	2	yes	—
339	C₂²	C₂₃×#537	1	1	yes	characteristic
340	C₂²	C₂₃×#576	1	1	yes	characteristic
341	C₂²	C₂₃×#576	1	1	yes	characteristic
342	C₂²	C₂₃×#577	1	2	yes	—
343	C₄	—	4	2	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	2	no	—
350	C₄	C₂₃×#560	1	1	yes	characteristic
351	C₄	C₂₃×#563	1	2	yes	—
352	C₄	C₂₃×#563	1	1	yes	characteristic
353	C₂	—	16	2	no	—
354	C₂	C₂₃×#2823	1	1	yes	characteristic
355	C₂	C₂₃×#3257	1	2	yes	—
356	C₂	C₂₃×#3257	1	2	yes	—
357	C₂	C₂₃×#3259	1	1	yes	characteristic
358	C₂	C₂₃×#3263	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2644	C₂	semidirect
2. C₂₃×#2088	C₂	semidirect
3. C₂₃×#2086	C₂	semidirect
4. C₂₃×#2067	C₂	semidirect
5. C₂₃×#2065	C₂	semidirect
6. C₂₃×#2065	C₂	semidirect
9. C₂₃×#1843	C₂	semidirect
112. #32864	C₂₃	direct (N × H)
152. #2644	C₂×C₂₃	semidirect
153. #2088	C₂×C₂₃	semidirect
154. #2086	C₂×C₂₃	semidirect
155. #2067	C₂×C₂₃	semidirect
156. #2065	C₂×C₂₃	semidirect
157. #2065	C₂×C₂₃	semidirect
160. #1843	C₂×C₂₃	semidirect
249. C₂₃	#32864	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2644	2
C₂₃×#2088	2
C₂₃×#2086	2
C₂₃×#2067	2
C₂₃×#2065	2
C₂₃×#2054	2
C₂₃×#1843	2
#32864	23

Subgroups — C23×#32864 #19180490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#32864 #19180490