Subgroups — C₂₃×#11258 #19158884

Order 5888.

654 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#108	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1735	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂₃×#339	C₂	1	2	yes	maximal
4	C₂₃×#1729	C₂	1	2	yes	maximal
5	C₂×C₂₃×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#108	—	2	4	no	—
12	C₂×C₂₃×#108	—	2	2	no	—
13	C₂₃×#340	C₄	1	4	yes	complemented
14	C₂₃×#339	—	2	4	no	—
15	C₂₃×#336	—	2	2	no	—
16	C₂₃×#336	—	2	4	no	—
17	C₂³×C₄×C₂₃	—	2	1	no	—
18	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₂₃	—	2	2	no	—
20	C₂×M₁₆×C₂₃	—	2	2	no	—
21	C₂×M₁₆×C₂₃	—	2	2	no	—
22	C₂×M₁₆×C₂₃	—	2	2	no	—
23	C₂×M₁₆×C₂₃	—	2	2	no	—
24	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂²×C₈×C₂₃	—	2	2	no	—
27	C₂²×C₈×C₂₃	—	2	1	no	—
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	—	2	2	no	—
31	C₂²×C₈×C₂₃	—	2	2	no	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₂₃	D₄	1	2	yes	—
34	C₂²×C₈×C₂₃	D₄	1	2	yes	—
35	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
36	C₂₃×#117	—	2	2	no	—
37	C₂₃×#117	—	2	2	no	—
38	C₂₃×#117	—	2	2	no	—
39	C₂₃×#117	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₄²×C₂₃	—	2	2	no	—
43	C₂×C₄²×C₂₃	—	2	2	no	—
44	C₂×C₄²×C₂₃	—	2	1	no	—
45	C₂×C₄²×C₂₃	—	2	2	no	—
46	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
47	C₂₃×#108	—	2	8	no	—
48	C₂⁴×C₂₃	#41	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	—	2	4	no	—
50	C₂²×C₄×C₂₃	—	2	2	no	—
51	C₂²×C₄×C₂₃	—	2	4	no	—
52	C₂²×C₄×C₂₃	—	4	2	no	—
53	C₂²×C₄×C₂₃	—	4	2	no	—
54	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	#41	1	2	yes	—
61	C₂²×C₄×C₂₃	#41	1	2	yes	—
62	C₂²×C₄×C₂₃	#41	1	2	yes	—
63	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
64	M₁₆×C₂₃	—	2	4	no	—
65	M₁₆×C₂₃	—	2	4	no	—
66	M₁₆×C₂₃	—	4	2	no	—
67	M₁₆×C₂₃	—	4	2	no	—
68	C₂×C₈×C₂₃	—	2	4	no	—
69	C₂×C₈×C₂₃	—	2	2	no	—
70	C₂×C₈×C₂₃	—	2	4	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	—	4	2	no	—
77	C₂×C₈×C₂₃	—	4	1	no	—
78	C₂×C₈×C₂₃	—	4	1	no	—
79	C₂×C₈×C₂₃	—	4	2	no	—
80	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
81	C₄⋊C₄×C₂₃	—	2	4	no	—
82	C₄⋊C₄×C₂₃	—	2	4	no	—
83	C₂²⋊C₄×C₂₃	—	4	2	no	—
84	C₂²⋊C₄×C₂₃	—	4	2	no	—
85	C₄²×C₂₃	—	2	4	no	—
86	C₄²×C₂₃	—	4	1	no	—
87	C₄²×C₂₃	—	4	2	no	—
88	C₄²×C₂₃	—	4	2	no	—
89	C₄²×C₂₃	—	4	1	no	—
90	#11258	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	—	2	2	no	—
92	C₂³×C₂₃	—	2	2	no	—
93	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
94	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂³×C₂₃	#124	1	2	yes	—
96	C₂³×C₂₃	#125	1	1	yes	characteristic
97	C₂³×C₂₃	#126	1	2	yes	—
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	4	8	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#117	1	2	yes	—
107	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
110	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₂₃	#121	1	2	yes	—
112	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#123	1	2	yes	—
116	C₂×C₄×C₂₃	#123	1	2	yes	—
117	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
119	C₂×C₄×C₂₃	#125	1	2	yes	—
120	C₂×C₄×C₂₃	#125	1	2	yes	—
121	C₂×C₄×C₂₃	#126	1	2	yes	—
122	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
123	C₈×C₂₃	—	2	4	no	—
124	C₈×C₂₃	—	4	4	no	—
125	C₈×C₂₃	—	4	2	no	—
126	C₂²×#108	C₂×C₂₃	1	1	yes	characteristic
127	#1735	C₂×C₂₃	1	1	yes	characteristic
128	C₂×#339	C₂×C₂₃	1	2	yes	—
129	#1729	C₂×C₂₃	1	2	yes	—
130	C₂×#336	C₂×C₂₃	1	1	yes	characteristic
131	C₂²×C₂₃	—	2	4	no	—
132	C₂²×C₂₃	—	2	4	no	—
133	C₂²×C₂₃	—	2	1	no	—
134	C₂²×C₂₃	—	2	1	no	—
135	C₂²×C₂₃	—	2	1	no	—
136	C₂²×C₂₃	—	2	1	no	—
137	C₂²×C₂₃	#388	1	2	yes	—
138	C₂²×C₂₃	#389	1	1	yes	characteristic
139	C₂²×C₂₃	#393	1	1	yes	characteristic
140	C₂²×C₂₃	#400	1	2	yes	—
141	C₂²×C₂₃	#400	1	1	yes	characteristic
142	C₂²×C₂₃	#443	1	2	yes	—
143	C₂²×C₂₃	#444	1	2	yes	—
144	C₄×C₂₃	—	4	4	no	—
145	C₄×C₂₃	—	8	4	no	—
146	C₄×C₂₃	#382	1	1	yes	characteristic
147	C₄×C₂₃	#385	1	1	yes	characteristic
148	C₄×C₂₃	#387	1	2	yes	—
149	C₄×C₂₃	#394	1	1	yes	characteristic
150	C₄×C₂₃	#397	1	1	yes	characteristic
151	C₄×C₂₃	#398	1	2	yes	—
152	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
153	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
154	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
155	C₂×#117	C₂²×C₂₃	1	2	yes	—
156	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×#108	—	2	4	no	—
158	C₂×#108	—	2	2	no	—
159	#340	C₄×C₂₃	1	4	yes	complemented
160	#339	—	2	4	no	—
161	#336	—	2	2	no	—
162	#336	—	2	4	no	—
163	C₂×C₂₃	—	2	1	no	—
164	C₂×C₂₃	—	2	2	no	—
165	C₂×C₂₃	—	2	1	no	—
166	C₂×C₂₃	#1442	1	1	yes	characteristic
167	C₂×C₂₃	#1929	1	4	yes	—
168	C₂×C₂₃	#1945	1	1	yes	characteristic
169	C₂×C₂₃	#1946	1	1	yes	characteristic
170	C₂³×C₄	—	2	1	no	—
171	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
186	C₂²×C₈	D₄×C₂₃	1	2	yes	—
187	C₂²×C₈	D₄×C₂₃	1	2	yes	—
188	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
200	#108	—	2	8	no	—
201	C₂₃	#11258	1	1	yes	characteristic, direct product factor, Sylow
202	C₂⁴	#41×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
214	C₂²×C₄	#41×C₂₃	1	2	yes	—
215	C₂²×C₄	#41×C₂₃	1	2	yes	—
216	C₂²×C₄	#41×C₂₃	1	2	yes	—
217	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	2	no	—
242	C₄²	—	4	2	no	—
243	C₄²	—	4	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
247	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
248	C₂³	C₂₃×#124	1	2	yes	—
249	C₂³	C₂₃×#125	1	1	yes	characteristic
250	C₂³	C₂₃×#126	1	2	yes	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	8	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂₃×#117	1	2	yes	—
260	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
261	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
262	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
263	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
264	C₂×C₄	C₂₃×#121	1	2	yes	—
265	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
266	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
267	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
268	C₂×C₄	C₂₃×#123	1	2	yes	—
269	C₂×C₄	C₂₃×#123	1	2	yes	—
270	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
271	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
272	C₂×C₄	C₂₃×#125	1	2	yes	—
273	C₂×C₄	C₂₃×#125	1	2	yes	—
274	C₂×C₄	C₂₃×#126	1	2	yes	—
275	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	4	4	no	—
278	C₈	—	4	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	C₂₃×#388	1	2	yes	—
286	C₂²	C₂₃×#389	1	1	yes	characteristic
287	C₂²	C₂₃×#393	1	1	yes	characteristic
288	C₂²	C₂₃×#400	1	2	yes	—
289	C₂²	C₂₃×#400	1	1	yes	characteristic
290	C₂²	C₂₃×#443	1	2	yes	—
291	C₂²	C₂₃×#444	1	2	yes	—
292	C₄	—	4	4	no	—
293	C₄	—	8	4	no	—
294	C₄	C₂₃×#382	1	1	yes	characteristic
295	C₄	C₂₃×#385	1	1	yes	characteristic
296	C₄	C₂₃×#387	1	2	yes	—
297	C₄	C₂₃×#394	1	1	yes	characteristic
298	C₄	C₂₃×#397	1	1	yes	characteristic
299	C₄	C₂₃×#398	1	2	yes	—
300	C₂	—	2	1	no	—
301	C₂	—	2	2	no	—
302	C₂	—	2	1	no	—
303	C₂	C₂₃×#1442	1	1	yes	characteristic
304	C₂	C₂₃×#1929	1	4	yes	—
305	C₂	C₂₃×#1945	1	1	yes	characteristic
306	C₂	C₂₃×#1946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂³×C₂₃	C₄×D₄
prime radical	C₂⁴×C₂₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂₃×#340	C₄	semidirect
90. #11258	C₂₃	direct (N × H)
159. #340	C₄×C₂₃	semidirect
201. C₂₃	#11258	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#108	2
C₂₃×#1735	2
C₂×C₂₃×#339	2
C₂₃×#1729	2
C₂×C₂₃×#336	2
#11258	23

Subgroups — C23×#11258 #19158884

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11258 #19158884