Subgroups — C₂₃×#11271 #19158897

Order 5888.

634 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#343	C₂	1	2	yes	maximal
3	C₂×C₂₃×#341	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1736	C₂	1	2	yes	maximal
5	C₂₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₂₃×#106	—	2	2	no	—
12	C₂×C₂₃×#106	—	2	4	no	—
13	C₂₃×#343	—	2	4	no	—
14	C₂₃×#341	C₄	1	4	yes	complemented
15	C₂₃×#340	—	2	4	no	—
16	C₂₃×#336	—	2	2	no	—
17	C₂³×C₄×C₂₃	—	2	2	no	—
18	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₂₃	—	2	2	no	—
20	C₂×M₁₆×C₂₃	—	2	1	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	—	2	2	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₂₃	Q₈	1	2	yes	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂²×C₈×C₂₃	—	2	2	no	—
31	C₂²×C₈×C₂₃	—	2	1	no	—
32	C₂²×C₈×C₂₃	—	2	2	no	—
33	C₂²×C₈×C₂₃	C₂×C₄	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	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₂₃	D₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
46	C₂×C₄²×C₂₃	—	2	1	no	—
47	C₂₃×#106	—	2	8	no	—
48	C₂⁴×C₂₃	#41	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	—	2	4	no	—
51	C₂²×C₄×C₂₃	—	4	2	no	—
52	C₂²×C₄×C₂₃	—	4	2	no	—
53	C₂²×C₄×C₂₃	—	8	4	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	—	2	4	no	—
67	M₁₆×C₂₃	—	4	2	no	—
68	M₁₆×C₂₃	—	4	2	no	—
69	M₁₆×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	2	4	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	4	1	no	—
77	C₂×C₈×C₂₃	—	4	2	no	—
78	C₂×C₈×C₂₃	—	4	2	no	—
79	C₂×C₈×C₂₃	—	4	1	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	C₄⋊C₄×C₂₃	—	2	4	no	—
84	C₄⋊C₄×C₂₃	—	4	2	no	—
85	C₄⋊C₄×C₂₃	—	4	2	no	—
86	C₂²⋊C₄×C₂₃	—	4	2	no	—
87	C₄²×C₂₃	—	4	1	no	—
88	C₄²×C₂₃	—	4	1	no	—
89	#11271	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	2	4	no	—
91	C₂³×C₂₃	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₂₃	#121	1	1	yes	characteristic
93	C₂³×C₂₃	#123	1	2	yes	—
94	C₂³×C₂₃	#125	1	1	yes	characteristic
95	C₂³×C₂₃	#126	1	2	yes	—
96	C₂×C₄×C₂₃	—	4	4	no	—
97	C₂×C₄×C₂₃	—	4	2	no	—
98	C₂×C₄×C₂₃	—	4	4	no	—
99	C₂×C₄×C₂₃	—	8	4	no	—
100	C₂×C₄×C₂₃	—	8	4	no	—
101	C₂×C₄×C₂₃	—	8	4	no	—
102	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	#117	1	2	yes	—
104	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#121	1	2	yes	—
108	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#123	1	2	yes	—
113	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#124	1	2	yes	—
116	C₂×C₄×C₂₃	#125	1	2	yes	—
117	C₂×C₄×C₂₃	#125	1	2	yes	—
118	C₂×C₄×C₂₃	#126	1	2	yes	—
119	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
120	C₈×C₂₃	—	2	4	no	—
121	C₈×C₂₃	—	4	4	no	—
122	C₈×C₂₃	—	4	2	no	—
123	C₂²×#106	C₂×C₂₃	1	1	yes	characteristic
124	C₂×#343	C₂×C₂₃	1	2	yes	—
125	C₂×#341	C₂×C₂₃	1	1	yes	characteristic
126	#1736	C₂×C₂₃	1	2	yes	—
127	#1729	C₂×C₂₃	1	1	yes	characteristic
128	C₂²×C₂₃	—	2	4	no	—
129	C₂²×C₂₃	—	2	2	no	—
130	C₂²×C₂₃	—	2	2	no	—
131	C₂²×C₂₃	—	2	2	no	—
132	C₂²×C₂₃	—	2	2	no	—
133	C₂²×C₂₃	#387	1	2	yes	—
134	C₂²×C₂₃	#389	1	1	yes	characteristic
135	C₂²×C₂₃	#394	1	1	yes	characteristic
136	C₂²×C₂₃	#397	1	1	yes	characteristic
137	C₂²×C₂₃	#400	1	2	yes	—
138	C₂²×C₂₃	#471	1	4	yes	—
139	C₄×C₂₃	—	8	2	no	—
140	C₄×C₂₃	—	8	4	no	—
141	C₄×C₂₃	#382	1	1	yes	characteristic
142	C₄×C₂₃	#385	1	1	yes	characteristic
143	C₄×C₂₃	#388	1	2	yes	—
144	C₄×C₂₃	#393	1	1	yes	characteristic
145	C₄×C₂₃	#398	1	2	yes	—
146	C₄×C₂₃	#400	1	1	yes	characteristic
147	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
148	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
149	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
151	C₂²×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
152	C₂×#106	—	2	2	no	—
153	C₂×#106	—	2	4	no	—
154	#343	—	2	4	no	—
155	#341	C₄×C₂₃	1	4	yes	complemented
156	#340	—	2	4	no	—
157	#336	—	2	2	no	—
158	C₂×C₂₃	—	2	2	no	—
159	C₂×C₂₃	—	2	2	no	—
160	C₂×C₂₃	#1442	1	1	yes	characteristic
161	C₂×C₂₃	#1951	1	2	yes	—
162	C₂×C₂₃	#2046	1	2	yes	—
163	C₂×C₂₃	#2079	1	2	yes	—
164	C₂³×C₄	—	2	2	no	—
165	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
175	C₂×M₁₆	Q₈×C₂₃	1	2	yes	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
181	C₂²×C₈	D₄×C₂₃	1	2	yes	—
182	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	D₄×C₂₃	1	4	yes	—
191	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	1	no	—
193	C₂×C₄²	—	2	1	no	—
194	#106	—	2	8	no	—
195	C₂₃	#11271	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	#41×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	8	4	no	—
202	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
205	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	#41×C₂₃	1	2	yes	—
209	C₂²×C₄	#41×C₂₃	1	2	yes	—
210	C₂²×C₄	#41×C₂₃	1	2	yes	—
211	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₄²	—	4	1	no	—
236	C₄²	—	4	1	no	—
237	C₂³	—	2	4	no	—
238	C₂³	C₄×Q₈×C₂₃	1	1	yes	characteristic
239	C₂³	C₂₃×#121	1	1	yes	characteristic
240	C₂³	C₂₃×#123	1	2	yes	—
241	C₂³	C₂₃×#125	1	1	yes	characteristic
242	C₂³	C₂₃×#126	1	2	yes	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂₃×#117	1	2	yes	—
251	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
252	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
254	C₂×C₄	C₂₃×#121	1	2	yes	—
255	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
257	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
258	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
259	C₂×C₄	C₂₃×#123	1	2	yes	—
260	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
261	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
262	C₂×C₄	C₂₃×#124	1	2	yes	—
263	C₂×C₄	C₂₃×#125	1	2	yes	—
264	C₂×C₄	C₂₃×#125	1	2	yes	—
265	C₂×C₄	C₂₃×#126	1	2	yes	—
266	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
267	C₈	—	2	4	no	—
268	C₈	—	4	4	no	—
269	C₈	—	4	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	C₂₃×#387	1	2	yes	—
276	C₂²	C₂₃×#389	1	1	yes	characteristic
277	C₂²	C₂₃×#394	1	1	yes	characteristic
278	C₂²	C₂₃×#397	1	1	yes	characteristic
279	C₂²	C₂₃×#400	1	2	yes	—
280	C₂²	C₂₃×#471	1	4	yes	—
281	C₄	—	8	4	no	—
282	C₄	—	8	2	no	—
283	C₄	C₂₃×#382	1	1	yes	characteristic
284	C₄	C₂₃×#385	1	1	yes	characteristic
285	C₄	C₂₃×#388	1	2	yes	—
286	C₄	C₂₃×#393	1	1	yes	characteristic
287	C₄	C₂₃×#398	1	2	yes	—
288	C₄	C₂₃×#400	1	1	yes	characteristic
289	C₂	—	2	2	no	—
290	C₂	—	2	2	no	—
291	C₂	C₂₃×#1442	1	1	yes	characteristic
292	C₂	C₂₃×#1951	1	2	yes	—
293	C₂	C₂₃×#2046	1	2	yes	—
294	C₂	C₂₃×#2079	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₂₃×#341	C₄	semidirect
89. #11271	C₂₃	direct (N × H)
155. #341	C₄×C₂₃	semidirect
195. C₂₃	#11271	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×#106	2
C₂×C₂₃×#343	2
C₂×C₂₃×#341	2
C₂₃×#1736	2
C₂₃×#1729	2
#11271	23

Subgroups — C23×#11271 #19158897

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11271 #19158897