Subgroups — C₂₃×#11466 #19159092

Order 5888.

438 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2112	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1322	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1314	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₂₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
8	C₂×C₂₃×M₃₂	—	2	2	no	—
9	C₈×D₄×C₂₃	—	2	2	no	—
10	C₈×D₄×C₂₃	—	2	2	no	—
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×C₂²⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic
15	C₂₃×C₂²⋊C₁₆	—	2	2	no	—
16	C₂₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
17	C₂²×D₄×C₂₃	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₂₃	—	2	1	no	—
21	C₂²×C₈×C₂₃	—	2	1	no	—
22	C₂²×C₈×C₂₃	—	4	2	no	—
23	C₂²×C₈×C₂₃	—	4	2	no	—
24	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₂₃	D₄	1	2	yes	—
28	C₂²×C₈×C₂₃	D₄	1	2	yes	—
29	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
30	C₄×D₄×C₂₃	—	2	2	no	—
31	C₄×D₄×C₂₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₃	C₈	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₂₃	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₂₃	C₈	1	1	yes	characteristic
35	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂₃×M₃₂	—	4	4	no	—
37	C₂×C₁₆×C₂₃	—	2	4	no	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×C₂²⋊C₈	—	2	2	no	—
40	C₂₃×C₂²⋊C₈	—	2	2	no	—
41	C₄×C₈×C₂₃	—	2	1	no	—
42	C₄×C₈×C₂₃	—	2	1	no	—
43	C₂⁴×C₂₃	C₂×C₈	1	1	yes	characteristic
44	C₂⁴×C₂₃	C₂×C₈	1	1	yes	characteristic
45	C₂×D₄×C₂₃	—	2	2	no	—
46	C₂×D₄×C₂₃	—	2	2	no	—
47	C₂×D₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	—	2	1	no	—
49	C₂²×C₄×C₂₃	—	2	1	no	—
50	C₂²×C₄×C₂₃	—	4	1	no	—
51	C₂²×C₄×C₂₃	—	4	1	no	—
52	C₂²×C₄×C₂₃	—	4	1	no	—
53	C₂²×C₄×C₂₃	—	4	1	no	—
54	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	M₁₆	1	2	yes	—
60	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₂₃	—	2	4	no	—
62	C₂×C₈×C₂₃	—	2	2	no	—
63	C₂×C₈×C₂₃	—	2	1	no	—
64	C₂×C₈×C₂₃	—	2	1	no	—
65	C₂×C₈×C₂₃	—	4	2	no	—
66	C₂×C₈×C₂₃	—	4	1	no	—
67	C₂×C₈×C₂₃	—	4	2	no	—
68	C₂×C₈×C₂₃	—	4	2	no	—
69	C₂×C₈×C₂₃	—	4	1	no	—
70	C₂×C₈×C₂₃	—	4	1	no	—
71	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
74	C₄⋊C₄×C₂₃	—	2	2	no	—
75	C₂²⋊C₄×C₂₃	—	2	2	no	—
76	C₂²⋊C₄×C₂₃	—	2	2	no	—
77	C₄²×C₂₃	—	2	1	no	—
78	C₄²×C₂₃	—	2	1	no	—
79	C₁₆×C₂₃	—	4	4	no	—
80	#11466	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₂₃	—	2	1	no	—
82	C₂³×C₂₃	—	2	1	no	—
83	C₂³×C₂₃	—	2	1	no	—
84	C₂³×C₂₃	—	2	1	no	—
85	C₂³×C₂₃	—	4	1	no	—
86	C₂³×C₂₃	—	4	1	no	—
87	C₂³×C₂₃	—	4	1	no	—
88	C₂³×C₂₃	—	4	1	no	—
89	C₂³×C₂₃	C₂²⋊C₈	1	2	yes	—
90	C₂³×C₂₃	M₃₂	1	2	yes	complemented
91	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
92	D₄×C₂₃	—	2	8	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	4	1	no	—
102	C₂×C₄×C₂₃	—	4	2	no	—
103	C₂×C₄×C₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	4	1	no	—
105	C₂×C₄×C₂₃	—	4	1	no	—
106	C₂×C₄×C₂₃	—	4	1	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
109	C₂×C₄×C₂₃	M₃₂	1	2	yes	complemented
110	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
112	C₈×C₂₃	—	2	2	no	—
113	C₈×C₂₃	—	2	1	no	—
114	C₈×C₂₃	—	4	1	no	—
115	C₈×C₂₃	—	4	1	no	—
116	C₈×C₂₃	#99	1	2	yes	—
117	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
118	#2112	C₂×C₂₃	1	2	yes	complemented
119	#1322	C₂×C₂₃	1	2	yes	complemented
120	#1314	C₂×C₂₃	1	2	yes	complemented
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	—	2	1	no	—
123	C₂²×C₂₃	—	2	2	no	—
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	—	4	2	no	—
130	C₂²×C₂₃	—	4	2	no	—
131	C₂²×C₂₃	—	4	1	no	—
132	C₂²×C₂₃	—	4	1	no	—
133	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₂²×C₂₃	C₂×M₃₂	1	1	yes	characteristic
135	C₂²×C₂₃	C₂×M₃₂	1	1	yes	characteristic
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	2	2	no	—
139	C₄×C₂₃	—	4	1	no	—
140	C₄×C₂₃	—	4	1	no	—
141	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
142	C₄×C₂₃	#413	1	1	yes	characteristic
143	C₂³×C₈	C₄×C₂₃	1	1	yes	characteristic
144	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
145	C₂×C₄×D₄	C₄×C₂₃	1	1	yes	characteristic
146	C₂×M₃₂	—	2	2	no	—
147	C₈×D₄	—	2	2	no	—
148	C₈×D₄	—	2	2	no	—
149	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×C₂²⋊C₈	C₄×C₂₃	1	1	yes	characteristic
151	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
152	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic
153	C₂²⋊C₁₆	—	2	2	no	—
154	C₂²⋊C₁₆	C₂²×C₂₃	1	4	yes	—
155	C₂×C₂₃	—	2	1	no	—
156	C₂×C₂₃	—	2	1	no	—
157	C₂×C₂₃	—	4	2	no	—
158	C₂×C₂₃	—	4	2	no	—
159	C₂×C₂₃	#1459	1	1	yes	characteristic
160	C₂×C₂₃	#2112	1	1	yes	characteristic
161	C₂×C₂₃	#2115	1	1	yes	characteristic
162	C₂²×D₄	C₈×C₂₃	1	1	yes	characteristic
163	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
164	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	2	1	no	—
167	C₂²×C₈	—	4	2	no	—
168	C₂²×C₈	—	4	2	no	—
169	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
170	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
172	C₂²×C₈	D₄×C₂₃	1	2	yes	—
173	C₂²×C₈	D₄×C₂₃	1	2	yes	—
174	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
175	C₄×D₄	—	2	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₂×C₄⋊C₄	C₈×C₂₃	1	1	yes	characteristic
178	C₂×C₂²⋊C₄	C₈×C₂₃	1	1	yes	characteristic
179	C₂×C₂²⋊C₄	C₈×C₂₃	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
181	M₃₂	—	4	4	no	—
182	C₂×C₁₆	—	2	4	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₂²⋊C₈	—	2	2	no	—
185	C₂²⋊C₈	—	2	2	no	—
186	C₄×C₈	—	2	1	no	—
187	C₄×C₈	—	2	1	no	—
188	C₂₃	#11466	1	1	yes	characteristic, direct product factor, Sylow
189	C₂⁴	C₂×C₈×C₂₃	1	1	yes	characteristic
190	C₂⁴	C₂×C₈×C₂₃	1	1	yes	characteristic
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	1	no	—
200	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
201	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	M₁₆×C₂₃	1	2	yes	—
206	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
219	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
220	C₄⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	2	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₄²	—	2	1	no	—
224	C₄²	—	2	1	no	—
225	C₁₆	—	4	4	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	1	no	—
229	C₂³	—	2	1	no	—
230	C₂³	—	4	1	no	—
231	C₂³	—	4	1	no	—
232	C₂³	—	4	1	no	—
233	C₂³	—	4	1	no	—
234	C₂³	C₂₃×C₂²⋊C₈	1	2	yes	—
235	C₂³	C₂₃×M₃₂	1	2	yes	complemented
236	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
237	D₄	—	2	8	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
254	C₂×C₄	C₂₃×M₃₂	1	2	yes	complemented
255	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	2	1	no	—
259	C₈	—	4	1	no	—
260	C₈	—	4	1	no	—
261	C₈	C₂₃×#99	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	1	no	—
274	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
275	C₂²	C₂×C₂₃×M₃₂	1	1	yes	characteristic
276	C₂²	C₂×C₂₃×M₃₂	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	1	no	—
281	C₄	—	4	1	no	—
282	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
283	C₄	C₂₃×#413	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	4	2	no	—
287	C₂	—	4	2	no	—
288	C₂	C₂₃×#1459	1	1	yes	characteristic
289	C₂	C₂₃×#2112	1	1	yes	characteristic
290	C₂	C₂₃×#2115	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×#2112	C₂	semidirect
3. C₂₃×#1322	C₂	semidirect
4. C₂₃×#1314	C₂	semidirect
80. #11466	C₂₃	direct (N × H)
90. C₂³×C₂₃	M₃₂	semidirect
109. C₂×C₄×C₂₃	M₃₂	semidirect
118. #2112	C₂×C₂₃	semidirect
119. #1322	C₂×C₂₃	semidirect
120. #1314	C₂×C₂₃	semidirect
188. C₂₃	#11466	direct (N × H)
235. C₂³	C₂₃×M₃₂	semidirect
254. C₂×C₄	C₂₃×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₂₃	2
C₂₃×#2112	2
C₂₃×#1322	2
C₂₃×#1314	2
#11466	23

Subgroups — C23×#11466 #19159092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11466 #19159092