Subgroups — C₂₃×#11272 #19158898

Order 5888.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 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	C₂×C₂₃×#344	C₂	1	2	yes	maximal
3	C₂₃×#1738	C₂	1	2	yes	maximal
4	C₂×C₂₃×#341	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×#108	—	2	4	no	—
12	C₂×C₂₃×#108	—	2	2	no	—
13	C₂₃×#344	—	2	4	no	—
14	C₂₃×#341	—	2	4	no	—
15	C₂₃×#341	C₄	1	4	yes	complemented
16	C₂₃×#336	—	2	2	no	—
17	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₂₃	—	2	2	no	—
19	C₂×M₁₆×C₂₃	—	2	1	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	—	2	2	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	2	2	no	—
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	—	2	2	no	—
30	C₂×M₁₆×C₂₃	—	2	2	no	—
31	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
33	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
34	C₂²×C₈×C₂₃	—	2	2	no	—
35	C₂²×C₈×C₂₃	—	2	2	no	—
36	C₂²×C₈×C₂₃	—	2	1	no	—
37	C₂²×C₈×C₂₃	—	2	2	no	—
38	C₂²×C₈×C₂₃	—	2	2	no	—
39	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₂₃	D₄	1	2	yes	—
41	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
42	C₂₃×#117	—	2	2	no	—
43	C₂₃×#117	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
45	C₂×C₂²⋊C₄×C₂₃	Q₈	1	2	yes	—
46	C₂×C₄²×C₂₃	—	2	1	no	—
47	C₂₃×#108	—	2	8	no	—
48	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	#41	1	2	yes	—
56	C₂²×C₄×C₂₃	#41	1	2	yes	—
57	C₂²×C₄×C₂₃	#41	1	2	yes	—
58	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
60	M₁₆×C₂₃	—	2	4	no	—
61	M₁₆×C₂₃	—	2	4	no	—
62	M₁₆×C₂₃	—	2	4	no	—
63	M₁₆×C₂₃	—	2	4	no	—
64	M₁₆×C₂₃	—	2	4	no	—
65	M₁₆×C₂₃	—	4	2	no	—
66	M₁₆×C₂₃	—	4	2	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	—	2	4	no	—
77	C₂×C₈×C₂₃	—	4	2	no	—
78	C₂×C₈×C₂₃	—	4	2	no	—
79	C₂×C₈×C₂₃	—	4	2	no	—
80	C₂×C₈×C₂₃	—	4	2	no	—
81	C₂×C₈×C₂₃	—	4	1	no	—
82	C₂×C₈×C₂₃	—	4	1	no	—
83	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₂₃	—	4	2	no	—
85	C₂²⋊C₄×C₂₃	—	2	4	no	—
86	C₄²×C₂₃	—	4	1	no	—
87	C₄²×C₂₃	—	4	1	no	—
88	#11272	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₂₃	—	4	2	no	—
90	C₂³×C₂₃	C₄×Q₈	1	1	yes	characteristic
91	C₂³×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
93	C₂³×C₂₃	#125	1	2	yes	—
94	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₂₃	—	4	4	no	—
96	C₂×C₄×C₂₃	—	4	2	no	—
97	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	#117	1	2	yes	—
99	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
102	C₂×C₄×C₂₃	#121	1	2	yes	—
103	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#123	1	2	yes	—
107	C₂×C₄×C₂₃	#123	1	2	yes	—
108	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₂₃	#124	1	2	yes	—
111	C₂×C₄×C₂₃	#125	1	2	yes	—
112	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#126	1	2	yes	—
114	C₂×C₄×C₂₃	#126	1	2	yes	—
115	C₈×C₂₃	—	2	4	no	—
116	C₈×C₂₃	—	4	4	no	—
117	C₈×C₂₃	—	4	4	no	—
118	C₈×C₂₃	—	4	2	no	—
119	C₂²×#108	C₂×C₂₃	1	1	yes	characteristic
120	C₂×#344	C₂×C₂₃	1	2	yes	—
121	#1738	C₂×C₂₃	1	2	yes	—
122	C₂×#341	C₂×C₂₃	1	1	yes	characteristic
123	#1729	C₂×C₂₃	1	1	yes	characteristic
124	C₂²×C₂₃	—	2	2	no	—
125	C₂²×C₂₃	—	2	2	no	—
126	C₂²×C₂₃	—	2	2	no	—
127	C₂²×C₂₃	—	4	2	no	—
128	C₂²×C₂₃	—	4	2	no	—
129	C₂²×C₂₃	#387	1	2	yes	—
130	C₂²×C₂₃	#389	1	1	yes	characteristic
131	C₂²×C₂₃	#393	1	1	yes	characteristic
132	C₂²×C₂₃	#398	1	2	yes	—
133	C₂²×C₂₃	#400	1	1	yes	characteristic
134	C₄×C₂₃	—	8	2	no	—
135	C₄×C₂₃	#382	1	1	yes	characteristic
136	C₄×C₂₃	#385	1	1	yes	characteristic
137	C₄×C₂₃	#388	1	2	yes	—
138	C₄×C₂₃	#394	1	1	yes	characteristic
139	C₄×C₂₃	#397	1	1	yes	characteristic
140	C₄×C₂₃	#400	1	2	yes	—
141	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
142	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
143	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
144	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
146	C₂×#108	—	2	4	no	—
147	C₂×#108	—	2	2	no	—
148	#344	—	2	4	no	—
149	#341	—	2	4	no	—
150	#341	C₄×C₂₃	1	4	yes	complemented
151	#336	—	2	2	no	—
152	C₂×C₂₃	—	2	1	no	—
153	C₂×C₂₃	—	2	2	no	—
154	C₂×C₂₃	—	2	2	no	—
155	C₂×C₂₃	—	2	1	no	—
156	C₂×C₂₃	#1442	1	1	yes	characteristic
157	C₂×C₂₃	#1953	1	2	yes	—
158	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	1	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	—	2	2	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	C₂×C₄×C₂₃	1	4	yes	—
173	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
174	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	2	2	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	2	2	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	1	no	—
186	C₂×C₂²⋊C₄	Q₈×C₂₃	1	2	yes	—
187	C₂×C₄²	—	2	1	no	—
188	#108	—	2	8	no	—
189	C₂₃	#11272	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	#41×C₂₃	1	2	yes	—
198	C₂²×C₄	#41×C₂₃	1	2	yes	—
199	C₂²×C₄	#41×C₂₃	1	2	yes	—
200	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	2	4	no	—
206	M₁₆	—	2	4	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	1	no	—
224	C₂×C₈	—	4	1	no	—
225	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
226	C₄⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	2	4	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₂³	—	4	2	no	—
231	C₂³	C₄×Q₈×C₂₃	1	1	yes	characteristic
232	C₂³	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
233	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
234	C₂³	C₂₃×#125	1	2	yes	—
235	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂₃×#117	1	2	yes	—
240	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
241	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
242	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
243	C₂×C₄	C₂₃×#121	1	2	yes	—
244	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
245	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
246	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
247	C₂×C₄	C₂₃×#123	1	2	yes	—
248	C₂×C₄	C₂₃×#123	1	2	yes	—
249	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
250	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
251	C₂×C₄	C₂₃×#124	1	2	yes	—
252	C₂×C₄	C₂₃×#125	1	2	yes	—
253	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
254	C₂×C₄	C₂₃×#126	1	2	yes	—
255	C₂×C₄	C₂₃×#126	1	2	yes	—
256	C₈	—	2	4	no	—
257	C₈	—	4	4	no	—
258	C₈	—	4	4	no	—
259	C₈	—	4	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	C₂₃×#387	1	2	yes	—
266	C₂²	C₂₃×#389	1	1	yes	characteristic
267	C₂²	C₂₃×#393	1	1	yes	characteristic
268	C₂²	C₂₃×#398	1	2	yes	—
269	C₂²	C₂₃×#400	1	1	yes	characteristic
270	C₄	—	8	2	no	—
271	C₄	C₂₃×#382	1	1	yes	characteristic
272	C₄	C₂₃×#385	1	1	yes	characteristic
273	C₄	C₂₃×#388	1	2	yes	—
274	C₄	C₂₃×#394	1	1	yes	characteristic
275	C₄	C₂₃×#397	1	1	yes	characteristic
276	C₄	C₂₃×#400	1	2	yes	—
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	—	2	1	no	—
281	C₂	C₂₃×#1442	1	1	yes	characteristic
282	C₂	C₂₃×#1953	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	#389
prime radical	C₂⁴×C₂₃	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₂₃×#341	C₄	semidirect
88. #11272	C₂₃	direct (N × H)
150. #341	C₄×C₂₃	semidirect
189. C₂₃	#11272	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
C₂×C₂₃×#344	2
C₂₃×#1738	2
C₂×C₂₃×#341	2
C₂₃×#1729	2
#11272	23

Subgroups — C23×#11272 #19158898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11272 #19158898