Subgroups — C₂₃×#60045 #19207671

Order 5888.

2490 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2804	C₂	1	3	yes	maximal, complemented
2	C₂₃×#2679	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#2598	C₂	1	6	yes	maximal, complemented
4	C₂₃×#2572	C₂	1	6	yes	maximal, complemented
5	C₂₃×#2404	C₂	1	6	yes	maximal, complemented
6	C₂₃×#2358	C₂	1	6	yes	maximal, complemented
7	C₂₃×#2297	C₂	1	3	yes	maximal, complemented
8	C₂²×#41×C₂₃	C₂²	1	3	yes	—
9	C₂×C₂₃×#120	C₂²	1	6	yes	—
10	C₂×C₂₃×#120	C₂²	1	2	yes	complemented
11	C₂×C₄×D₄×C₂₃	C₂²	1	6	yes	—
12	C₂×C₂₃×#117	C₂²	1	3	yes	complemented
13	C₂²×C₂²⋊C₄×C₂₃	C₂²	1	6	yes	complemented
14	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic, complemented
15	C₂₃×#400	C₂²	1	6	yes	complemented
16	C₂₃×#399	C₂²	1	2	yes	complemented
17	C₂₃×#394	C₂²	1	6	yes	—
18	C₂₃×#393	C₂²	1	12	yes	complemented
19	C₂₃×#392	C₂²	1	6	yes	—
20	C₂₃×#391	C₂²	1	3	yes	complemented
21	C₂₃×#391	C₂²	1	3	yes	complemented
22	C₂₃×#390	C₂²	1	3	yes	—
23	C₂₃×#390	C₂²	1	3	yes	—
24	C₂₃×#388	C₂²	1	12	yes	complemented
25	C₂₃×#386	C₂²	1	6	yes	—
26	C₂₃×#386	C₂²	1	12	yes	—
27	C₂₃×#386	C₂²	1	12	yes	complemented
28	C₂₃×#386	C₂²	1	6	yes	complemented
29	C₂₃×#385	C₂²	1	12	yes	complemented
30	C₂₃×#383	C₂²	1	6	yes	complemented
31	C₂₃×#381	C₂²	1	6	yes	complemented
32	C₄×C₂²⋊C₄×C₂₃	C₂²	1	12	yes	complemented
33	C₂⁵×C₂₃	C₂³	1	2	yes	—
34	C₂×#41×C₂₃	—	2	12	no	—
35	C₂×#41×C₂₃	—	4	6	no	—
36	C₂×#41×C₂₃	—	4	6	no	—
37	C₂²×Q₈×C₂₃	C₂³	1	3	yes	—
38	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
39	C₂²×D₄×C₂₃	C₂³	1	6	yes	—
40	C₂²×D₄×C₂₃	C₂³	1	3	yes	—
41	C₂³×C₄×C₂₃	C₂³	1	3	yes	—
42	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
43	C₂³×C₄×C₂₃	C₂³	1	6	yes	—
44	C₂₃×#120	—	2	24	no	—
45	C₂₃×#120	—	4	4	no	—
46	C₄×D₄×C₂₃	—	4	12	no	—
47	C₂₃×#117	—	4	6	no	—
48	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
49	C₂×C₄⋊C₄×C₂₃	C₂³	1	6	yes	—
50	C₂×C₄⋊C₄×C₂₃	C₂³	1	3	yes	—
51	C₂×C₂²⋊C₄×C₂₃	—	4	12	no	—
52	C₂×C₂²⋊C₄×C₂₃	C₂³	1	12	yes	—
53	C₂×C₂²⋊C₄×C₂₃	C₂³	1	12	yes	—
54	C₂×C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
55	C₂×C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
56	C₂×C₂²⋊C₄×C₂₃	C₂³	1	12	yes	—
57	C₂×C₂²⋊C₄×C₂₃	C₂³	1	6	yes	—
58	C₂×C₄²×C₂₃	—	4	2	no	—
59	C₂×C₄²×C₂₃	C₂³	1	6	yes	—
60	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
61	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
62	C₂₃×#95	C₂³	1	12	yes	—
63	C₂₃×#95	C₂³	1	12	yes	—
64	C₂₃×#95	C₂³	1	6	yes	—
65	C₂₃×#95	C₂³	1	2	yes	—
66	C₂₃×#95	C₂³	1	12	yes	—
67	C₂⁴×C₂₃	—	2	12	no	—
68	C₂⁴×C₂₃	—	4	6	no	—
69	C₂⁴×C₂₃	—	4	2	no	—
70	C₂⁴×C₂₃	C₂⁴	1	6	yes	—
71	C₂⁴×C₂₃	C₂⁴	1	1	yes	characteristic
72	#41×C₂₃	—	4	48	no	—
73	C₂×Q₈×C₂₃	—	2	6	no	—
74	C₂×Q₈×C₂₃	—	4	3	no	—
75	C₂×Q₈×C₂₃	—	4	3	no	—
76	C₂×D₄×C₂₃	—	2	24	no	—
77	C₂×D₄×C₂₃	—	2	12	no	—
78	C₂×D₄×C₂₃	—	2	12	no	—
79	C₂×D₄×C₂₃	—	4	6	no	—
80	C₂×D₄×C₂₃	—	4	3	no	—
81	C₂×D₄×C₂₃	—	4	3	no	—
82	C₂×D₄×C₂₃	—	4	6	no	—
83	C₂×D₄×C₂₃	#41	1	12	yes	complemented
84	C₂²×C₄×C₂₃	—	2	6	no	—
85	C₂²×C₄×C₂₃	—	2	6	no	—
86	C₂²×C₄×C₂₃	—	2	6	no	—
87	C₂²×C₄×C₂₃	—	2	12	no	—
88	C₂²×C₄×C₂₃	—	2	12	no	—
89	C₂²×C₄×C₂₃	—	4	6	no	—
90	C₂²×C₄×C₂₃	—	4	6	no	—
91	C₂²×C₄×C₂₃	—	4	12	no	—
92	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
93	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
94	C₂²×C₄×C₂₃	C₂⁴	1	6	yes	—
95	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
96	C₂²×C₄×C₂₃	C₂⁴	1	3	yes	—
97	C₄⋊C₄×C₂₃	—	4	3	no	—
98	C₄⋊C₄×C₂₃	—	4	3	no	—
99	C₄⋊C₄×C₂₃	—	4	6	no	—
100	C₂²⋊C₄×C₂₃	—	2	12	no	—
101	C₂²⋊C₄×C₂₃	—	2	24	no	—
102	C₂²⋊C₄×C₂₃	—	2	24	no	—
103	C₂²⋊C₄×C₂₃	—	4	12	no	—
104	C₂²⋊C₄×C₂₃	—	4	6	no	—
105	C₂²⋊C₄×C₂₃	—	4	6	no	—
106	C₄²×C₂₃	—	2	12	no	—
107	C₄²×C₂₃	—	4	3	no	—
108	C₄²×C₂₃	—	4	1	no	—
109	#60045	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₂₃	—	2	12	no	—
111	C₂³×C₂₃	—	2	3	no	—
112	C₂³×C₂₃	—	2	12	no	—
113	C₂³×C₂₃	—	2	3	no	—
114	C₂³×C₂₃	—	2	6	no	—
115	C₂³×C₂₃	—	2	6	no	—
116	C₂³×C₂₃	—	4	24	no	—
117	C₂³×C₂₃	—	8	12	no	—
118	C₂³×C₂₃	—	8	4	no	—
119	C₂³×C₂₃	C₂×#41	1	12	yes	—
120	C₂³×C₂₃	H(5,𝔽₂)	1	2	yes	—
121	C₂³×C₂₃	C₂⁵	1	1	yes	characteristic
122	Q₈×C₂₃	—	4	12	no	—
123	D₄×C₂₃	—	4	12	no	—
124	D₄×C₂₃	—	4	24	no	—
125	D₄×C₂₃	—	4	24	no	—
126	C₂×C₄×C₂₃	—	2	24	no	—
127	C₂×C₄×C₂₃	—	2	3	no	—
128	C₂×C₄×C₂₃	—	2	3	no	—
129	C₂×C₄×C₂₃	—	2	3	no	—
130	C₂×C₄×C₂₃	—	2	3	no	—
131	C₂×C₄×C₂₃	—	2	12	no	—
132	C₂×C₄×C₂₃	—	2	6	no	—
133	C₂×C₄×C₂₃	—	2	6	no	—
134	C₂×C₄×C₂₃	—	2	6	no	—
135	C₂×C₄×C₂₃	—	2	12	no	—
136	C₂×C₄×C₂₃	—	2	6	no	—
137	C₂×C₄×C₂₃	—	2	6	no	—
138	C₂×C₄×C₂₃	—	4	24	no	—
139	C₂×C₄×C₂₃	—	8	12	no	—
140	C₂×C₄×C₂₃	C₂×#41	1	6	yes	—
141	C₂×C₄×C₂₃	H(5,𝔽₂)	1	6	yes	—
142	#2804	C₂×C₂₃	1	3	yes	complemented
143	#2679	C₂×C₂₃	1	1	yes	characteristic, complemented
144	#2598	C₂×C₂₃	1	6	yes	complemented
145	#2572	C₂×C₂₃	1	6	yes	complemented
146	#2404	C₂×C₂₃	1	6	yes	complemented
147	#2358	C₂×C₂₃	1	6	yes	complemented
148	#2297	C₂×C₂₃	1	3	yes	complemented
149	C₂²×C₂₃	—	2	24	no	—
150	C₂²×C₂₃	—	2	6	no	—
151	C₂²×C₂₃	—	2	12	no	—
152	C₂²×C₂₃	—	4	3	no	—
153	C₂²×C₂₃	—	4	6	no	—
154	C₂²×C₂₃	—	4	12	no	—
155	C₂²×C₂₃	—	4	6	no	—
156	C₂²×C₂₃	—	4	1	no	—
157	C₂²×C₂₃	—	8	16	no	—
158	C₂²×C₂₃	C₂²×#41	1	3	yes	—
159	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	3	yes	—
160	C₂²×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
161	C₄×C₂₃	—	2	6	no	—
162	C₄×C₂₃	—	2	6	no	—
163	C₄×C₂₃	—	4	6	no	—
164	C₄×C₂₃	—	4	6	no	—
165	C₄×C₂₃	—	4	6	no	—
166	C₂²×#41	C₂²×C₂₃	1	3	yes	—
167	C₂×#120	C₂²×C₂₃	1	6	yes	—
168	C₂×#120	C₂²×C₂₃	1	2	yes	complemented
169	C₂×C₄×D₄	C₂²×C₂₃	1	6	yes	—
170	C₂×#117	C₂²×C₂₃	1	3	yes	complemented
171	C₂²×C₂²⋊C₄	C₂²×C₂₃	1	6	yes	complemented
172	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic, complemented
173	#400	C₂²×C₂₃	1	6	yes	complemented
174	#399	C₂²×C₂₃	1	2	yes	complemented
175	#394	C₂²×C₂₃	1	6	yes	—
176	#393	C₂²×C₂₃	1	12	yes	complemented
177	#392	C₂²×C₂₃	1	6	yes	—
178	#391	C₂²×C₂₃	1	3	yes	complemented
179	#391	C₂²×C₂₃	1	3	yes	complemented
180	#390	C₂²×C₂₃	1	3	yes	—
181	#390	C₂²×C₂₃	1	3	yes	—
182	#388	C₂²×C₂₃	1	12	yes	complemented
183	#386	C₂²×C₂₃	1	12	yes	—
184	#386	C₂²×C₂₃	1	6	yes	—
185	#386	C₂²×C₂₃	1	6	yes	complemented
186	#386	C₂²×C₂₃	1	12	yes	complemented
187	#385	C₂²×C₂₃	1	12	yes	complemented
188	#383	C₂²×C₂₃	1	6	yes	complemented
189	#381	C₂²×C₂₃	1	6	yes	complemented
190	C₄×C₂²⋊C₄	C₂²×C₂₃	1	12	yes	complemented
191	C₂×C₂₃	—	4	2	no	—
192	C₂×C₂₃	—	4	12	no	—
193	C₂×C₂₃	#3481	1	3	yes	—
194	C₂×C₂₃	#3490	1	3	yes	—
195	C₂×C₂₃	#3549	1	1	yes	characteristic
196	C₂⁵	C₂³×C₂₃	1	2	yes	—
197	C₂×#41	—	2	12	no	—
198	C₂×#41	—	4	6	no	—
199	C₂×#41	—	4	6	no	—
200	C₂²×Q₈	C₂³×C₂₃	1	3	yes	—
201	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
202	C₂²×D₄	C₂³×C₂₃	1	6	yes	—
203	C₂²×D₄	C₂³×C₂₃	1	3	yes	—
204	C₂³×C₄	C₂³×C₂₃	1	3	yes	—
205	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
206	C₂³×C₄	C₂³×C₂₃	1	6	yes	—
207	#120	—	2	24	no	—
208	#120	—	4	4	no	—
209	C₄×D₄	—	4	12	no	—
210	#117	—	4	6	no	—
211	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
212	C₂×C₄⋊C₄	C₂³×C₂₃	1	3	yes	—
213	C₂×C₄⋊C₄	C₂³×C₂₃	1	6	yes	—
214	C₂×C₂²⋊C₄	—	4	12	no	—
215	C₂×C₂²⋊C₄	C₂³×C₂₃	1	12	yes	—
216	C₂×C₂²⋊C₄	C₂³×C₂₃	1	12	yes	—
217	C₂×C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
218	C₂×C₂²⋊C₄	C₂³×C₂₃	1	12	yes	—
219	C₂×C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
220	C₂×C₂²⋊C₄	C₂³×C₂₃	1	6	yes	—
221	C₂×C₄²	—	4	2	no	—
222	C₂×C₄²	C₂³×C₂₃	1	6	yes	—
223	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
224	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
225	#95	C₂³×C₂₃	1	12	yes	—
226	#95	C₂³×C₂₃	1	12	yes	—
227	#95	C₂³×C₂₃	1	6	yes	—
228	#95	C₂³×C₂₃	1	2	yes	—
229	#95	C₂³×C₂₃	1	12	yes	—
230	C₂₃	#60045	1	1	yes	characteristic, direct product factor, Sylow
231	C₂⁴	—	2	12	no	—
232	C₂⁴	—	4	6	no	—
233	C₂⁴	—	4	2	no	—
234	C₂⁴	C₂⁴×C₂₃	1	6	yes	—
235	C₂⁴	C₂⁴×C₂₃	1	1	yes	characteristic
236	#41	—	4	48	no	—
237	C₂×Q₈	—	2	6	no	—
238	C₂×Q₈	—	4	3	no	—
239	C₂×Q₈	—	4	3	no	—
240	C₂×D₄	—	2	24	no	—
241	C₂×D₄	—	2	12	no	—
242	C₂×D₄	—	2	12	no	—
243	C₂×D₄	—	4	6	no	—
244	C₂×D₄	—	4	3	no	—
245	C₂×D₄	—	4	3	no	—
246	C₂×D₄	—	4	6	no	—
247	C₂×D₄	#41×C₂₃	1	12	yes	complemented
248	C₂²×C₄	—	2	6	no	—
249	C₂²×C₄	—	2	6	no	—
250	C₂²×C₄	—	2	6	no	—
251	C₂²×C₄	—	2	12	no	—
252	C₂²×C₄	—	2	12	no	—
253	C₂²×C₄	—	4	6	no	—
254	C₂²×C₄	—	4	6	no	—
255	C₂²×C₄	—	4	12	no	—
256	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
257	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
258	C₂²×C₄	C₂⁴×C₂₃	1	6	yes	—
259	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
260	C₂²×C₄	C₂⁴×C₂₃	1	3	yes	—
261	C₄⋊C₄	—	4	3	no	—
262	C₄⋊C₄	—	4	3	no	—
263	C₄⋊C₄	—	4	6	no	—
264	C₂²⋊C₄	—	2	12	no	—
265	C₂²⋊C₄	—	2	24	no	—
266	C₂²⋊C₄	—	2	24	no	—
267	C₂²⋊C₄	—	4	12	no	—
268	C₂²⋊C₄	—	4	6	no	—
269	C₂²⋊C₄	—	4	6	no	—
270	C₄²	—	2	12	no	—
271	C₄²	—	4	3	no	—
272	C₄²	—	4	1	no	—
273	C₂³	—	2	12	no	—
274	C₂³	—	2	3	no	—
275	C₂³	—	2	12	no	—
276	C₂³	—	2	3	no	—
277	C₂³	—	2	6	no	—
278	C₂³	—	2	6	no	—
279	C₂³	—	4	24	no	—
280	C₂³	—	8	12	no	—
281	C₂³	—	8	4	no	—
282	C₂³	C₂×#41×C₂₃	1	12	yes	—
283	C₂³	C₂₃×H(5,𝔽₂)	1	2	yes	—
284	C₂³	C₂⁵×C₂₃	1	1	yes	characteristic
285	Q₈	—	4	12	no	—
286	D₄	—	4	12	no	—
287	D₄	—	4	24	no	—
288	D₄	—	4	24	no	—
289	C₂×C₄	—	2	24	no	—
290	C₂×C₄	—	2	3	no	—
291	C₂×C₄	—	2	3	no	—
292	C₂×C₄	—	2	3	no	—
293	C₂×C₄	—	2	3	no	—
294	C₂×C₄	—	2	12	no	—
295	C₂×C₄	—	2	6	no	—
296	C₂×C₄	—	2	6	no	—
297	C₂×C₄	—	2	6	no	—
298	C₂×C₄	—	2	12	no	—
299	C₂×C₄	—	2	6	no	—
300	C₂×C₄	—	2	6	no	—
301	C₂×C₄	—	4	24	no	—
302	C₂×C₄	—	8	12	no	—
303	C₂×C₄	C₂×#41×C₂₃	1	6	yes	—
304	C₂×C₄	C₂₃×H(5,𝔽₂)	1	6	yes	—
305	C₂²	—	2	24	no	—
306	C₂²	—	2	6	no	—
307	C₂²	—	2	12	no	—
308	C₂²	—	4	6	no	—
309	C₂²	—	4	3	no	—
310	C₂²	—	4	12	no	—
311	C₂²	—	4	6	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	8	16	no	—
314	C₂²	C₂²×#41×C₂₃	1	3	yes	—
315	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	3	yes	—
316	C₂²	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
317	C₄	—	2	6	no	—
318	C₄	—	2	6	no	—
319	C₄	—	4	6	no	—
320	C₄	—	4	6	no	—
321	C₄	—	4	6	no	—
322	C₂	—	4	12	no	—
323	C₂	—	4	2	no	—
324	C₂	C₂₃×#3481	1	3	yes	—
325	C₂	C₂₃×#3490	1	3	yes	—
326	C₂	C₂₃×#3549	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2804	C₂	semidirect
2. C₂₃×#2679	C₂	semidirect
3. C₂₃×#2598	C₂	semidirect
4. C₂₃×#2572	C₂	semidirect
5. C₂₃×#2404	C₂	semidirect
6. C₂₃×#2358	C₂	semidirect
7. C₂₃×#2297	C₂	semidirect
10. C₂×C₂₃×#120	C₂²	semidirect
12. C₂×C₂₃×#117	C₂²	semidirect
13. C₂²×C₂²⋊C₄×C₂₃	C₂²	semidirect
14. C₂²×C₄²×C₂₃	C₂²	semidirect
15. C₂₃×#400	C₂²	semidirect
16. C₂₃×#399	C₂²	semidirect
18. C₂₃×#393	C₂²	semidirect
20. C₂₃×#391	C₂²	semidirect
21. C₂₃×#391	C₂²	semidirect
24. C₂₃×#388	C₂²	semidirect
27. C₂₃×#386	C₂²	semidirect
28. C₂₃×#386	C₂²	semidirect
29. C₂₃×#385	C₂²	semidirect
30. C₂₃×#383	C₂²	semidirect
31. C₂₃×#381	C₂²	semidirect
32. C₄×C₂²⋊C₄×C₂₃	C₂²	semidirect
83. C₂×D₄×C₂₃	#41	semidirect
109. #60045	C₂₃	direct (N × H)
142. #2804	C₂×C₂₃	semidirect
143. #2679	C₂×C₂₃	semidirect
144. #2598	C₂×C₂₃	semidirect
145. #2572	C₂×C₂₃	semidirect
146. #2404	C₂×C₂₃	semidirect
147. #2358	C₂×C₂₃	semidirect
148. #2297	C₂×C₂₃	semidirect
168. C₂×#120	C₂²×C₂₃	semidirect
170. C₂×#117	C₂²×C₂₃	semidirect
171. C₂²×C₂²⋊C₄	C₂²×C₂₃	semidirect
172. C₂²×C₄²	C₂²×C₂₃	semidirect
173. #400	C₂²×C₂₃	semidirect
174. #399	C₂²×C₂₃	semidirect
176. #393	C₂²×C₂₃	semidirect
178. #391	C₂²×C₂₃	semidirect
179. #391	C₂²×C₂₃	semidirect
182. #388	C₂²×C₂₃	semidirect
185. #386	C₂²×C₂₃	semidirect
186. #386	C₂²×C₂₃	semidirect
187. #385	C₂²×C₂₃	semidirect
188. #383	C₂²×C₂₃	semidirect
189. #381	C₂²×C₂₃	semidirect
190. C₄×C₂²⋊C₄	C₂²×C₂₃	semidirect
230. C₂₃	#60045	direct (N × H)
247. C₂×D₄	#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×#2804	2
C₂₃×#2679	2
C₂₃×#2598	2
C₂₃×#2572	2
C₂₃×#2404	2
C₂₃×#2358	2
C₂₃×#2297	2
#60045	23

Subgroups — C23×#60045 #19207671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#60045 #19207671