Subgroups — C₂₃×#9892 #19157518

Order 5888.

432 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2864	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1295	C₂	1	2	yes	maximal
5	C₂₃×#1295	C₂	1	2	yes	maximal
6	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
7	C₂×C₂₃×#97	—	2	1	no	—
8	C₂×C₂₃×#97	C₂²	1	2	yes	—
9	C₂×C₂₃×#97	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₂₃	—	2	1	no	—
11	C₂₃×#389	C₄	1	1	yes	characteristic
12	C₂₃×#389	C₄	1	1	yes	characteristic
13	C₂₃×#389	C₄	1	1	yes	characteristic
14	C₂₃×#389	C₄	1	1	yes	characteristic
15	C₂₃×#387	—	2	1	no	—
16	C₂₃×#387	—	2	1	no	—
17	C₂₃×#387	—	2	1	no	—
18	C₂₃×#387	—	2	1	no	—
19	C₂₃×#382	C₂²	1	1	yes	characteristic
20	C₂₃×#382	C₂²	1	1	yes	characteristic
21	C₄³×C₂₃	C₂²	1	1	yes	characteristic
22	C₂₃×#336	—	2	1	no	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
40	C₂₃×C₄⋊C₈	—	2	4	no	—
41	C₂₃×#97	—	2	4	no	—
42	C₂₃×#97	—	2	2	no	—
43	C₂₃×#97	—	2	4	no	—
44	C₂₃×#97	Q₈	1	4	yes	—
45	C₄×C₈×C₂₃	—	2	4	no	—
46	C₂₃×#95	—	2	1	no	—
47	C₂₃×#95	—	2	1	no	—
48	C₂₃×#95	—	2	1	no	—
49	C₂₃×#95	—	2	1	no	—
50	C₂₃×#95	—	2	1	no	—
51	C₂₃×#95	—	2	1	no	—
52	C₂₃×#95	—	2	1	no	—
53	C₂₃×#95	—	2	1	no	—
54	C₂²×C₄×C₂₃	—	2	1	no	—
55	C₂²×C₄×C₂₃	—	2	1	no	—
56	C₂²×C₄×C₂₃	—	2	1	no	—
57	C₂²×C₄×C₂₃	—	2	1	no	—
58	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₂₃	—	2	4	no	—
66	C₂×C₈×C₂₃	—	2	4	no	—
67	C₂×C₈×C₂₃	—	2	4	no	—
68	C₂×C₈×C₂₃	—	2	4	no	—
69	C₂×C₈×C₂₃	—	4	2	no	—
70	C₂×C₈×C₂₃	—	4	2	no	—
71	C₄⋊C₄×C₂₃	—	4	1	no	—
72	C₄⋊C₄×C₂₃	—	4	1	no	—
73	C₄⋊C₄×C₂₃	—	4	1	no	—
74	C₄⋊C₄×C₂₃	—	4	1	no	—
75	C₄⋊C₄×C₂₃	—	4	2	no	—
76	C₄⋊C₄×C₂₃	—	4	2	no	—
77	C₄⋊C₄×C₂₃	—	4	2	no	—
78	C₄⋊C₄×C₂₃	—	4	2	no	—
79	C₄²×C₂₃	—	2	2	no	—
80	C₄²×C₂₃	—	2	2	no	—
81	C₄²×C₂₃	—	2	2	no	—
82	C₄²×C₂₃	—	4	1	no	—
83	C₄²×C₂₃	—	4	1	no	—
84	C₄²×C₂₃	—	4	1	no	—
85	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
86	C₄²×C₂₃	#41	1	2	yes	—
87	#9892	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	2	1	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	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	1	no	—
103	C₂×C₄×C₂₃	—	2	1	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	—	4	1	no	—
109	C₂×C₄×C₂₃	—	4	1	no	—
110	C₂×C₄×C₂₃	—	4	1	no	—
111	C₂×C₄×C₂₃	—	4	1	no	—
112	C₂×C₄×C₂₃	—	4	1	no	—
113	C₂×C₄×C₂₃	—	4	1	no	—
114	C₂×C₄×C₂₃	—	4	1	no	—
115	C₂×C₄×C₂₃	—	4	1	no	—
116	C₂×C₄×C₂₃	#104	1	2	yes	—
117	C₂×C₄×C₂₃	#104	1	2	yes	—
118	C₂×C₄×C₂₃	#104	1	2	yes	—
119	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
120	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
122	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
123	C₈×C₂₃	—	2	8	no	—
124	C₈×C₂₃	—	4	4	no	—
125	#2864	C₂×C₂₃	1	1	yes	characteristic
126	#1767	C₂×C₂₃	1	1	yes	characteristic
127	C₄×#97	C₂×C₂₃	1	1	yes	characteristic
128	#1295	C₂×C₂₃	1	2	yes	—
129	#1295	C₂×C₂₃	1	2	yes	—
130	C₂²×C₂₃	#391	1	1	yes	characteristic
131	C₂²×C₂₃	#413	1	1	yes	characteristic
132	C₂²×C₂₃	#413	1	1	yes	characteristic
133	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
134	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
135	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
136	C₂²×C₂₃	#421	1	1	yes	characteristic
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	2	1	no	—
139	C₄×C₂₃	—	2	2	no	—
140	C₄×C₂₃	—	2	2	no	—
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	2	2	no	—
143	C₄×C₂₃	—	2	2	no	—
144	C₄×C₂₃	—	4	1	no	—
145	C₄×C₂₃	—	8	1	no	—
146	C₄×C₂₃	—	8	1	no	—
147	C₄×C₂₃	—	8	1	no	—
148	C₄×C₂₃	—	8	1	no	—
149	C₂×C₄⋊C₈	—	2	1	no	—
150	C₂×#97	—	2	1	no	—
151	C₂×#97	C₂²×C₂₃	1	2	yes	—
152	C₂×#97	C₂²×C₂₃	1	2	yes	—
153	C₂×C₄×C₈	—	2	1	no	—
154	#389	C₄×C₂₃	1	1	yes	characteristic
155	#389	C₄×C₂₃	1	1	yes	characteristic
156	#389	C₄×C₂₃	1	1	yes	characteristic
157	#389	C₄×C₂₃	1	1	yes	characteristic
158	#387	—	2	1	no	—
159	#387	—	2	1	no	—
160	#387	—	2	1	no	—
161	#387	—	2	1	no	—
162	#382	C₂²×C₂₃	1	1	yes	characteristic
163	#382	C₂²×C₂₃	1	1	yes	characteristic
164	C₄³	C₂²×C₂₃	1	1	yes	characteristic
165	#336	—	2	1	no	—
166	C₂×C₂₃	#1524	1	1	yes	characteristic
167	C₂×C₂₃	#1524	1	1	yes	characteristic
168	C₂×C₂₃	#1529	1	1	yes	characteristic
169	C₂×C₂₃	#1532	1	1	yes	characteristic
170	C₂×C₂₃	#1986	1	1	yes	characteristic
171	C₂×C₂₃	#1994	1	1	yes	characteristic
172	C₂×C₂₃	#1998	1	1	yes	characteristic
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	1	no	—
176	C₂×C₄⋊C₄	—	2	1	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
180	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
184	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
185	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
186	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
187	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
188	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
190	C₄⋊C₈	—	2	4	no	—
191	#97	—	2	4	no	—
192	#97	—	2	2	no	—
193	#97	—	2	4	no	—
194	#97	Q₈×C₂₃	1	4	yes	—
195	C₄×C₈	—	2	4	no	—
196	#95	—	2	1	no	—
197	#95	—	2	1	no	—
198	#95	—	2	1	no	—
199	#95	—	2	1	no	—
200	#95	—	2	1	no	—
201	#95	—	2	1	no	—
202	#95	—	2	1	no	—
203	#95	—	2	1	no	—
204	C₂₃	#9892	1	1	yes	characteristic, direct product factor, Sylow
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
211	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	4	2	no	—
221	C₂×C₈	—	4	2	no	—
222	C₄⋊C₄	—	4	1	no	—
223	C₄⋊C₄	—	4	1	no	—
224	C₄⋊C₄	—	4	1	no	—
225	C₄⋊C₄	—	4	1	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	2	no	—
232	C₄²	—	2	2	no	—
233	C₄²	—	4	1	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	1	no	—
236	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
237	C₄²	#41×C₂₃	1	2	yes	—
238	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	4	1	no	—
266	C₂×C₄	C₂₃×#104	1	2	yes	—
267	C₂×C₄	C₂₃×#104	1	2	yes	—
268	C₂×C₄	C₂₃×#104	1	2	yes	—
269	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
270	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
271	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
272	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
273	C₈	—	2	8	no	—
274	C₈	—	4	4	no	—
275	C₂²	C₂₃×#391	1	1	yes	characteristic
276	C₂²	C₂₃×#413	1	1	yes	characteristic
277	C₂²	C₂₃×#413	1	1	yes	characteristic
278	C₂²	C₂×C₂₃×#104	1	1	yes	characteristic
279	C₂²	C₂×C₂₃×#104	1	1	yes	characteristic
280	C₂²	C₂×C₂₃×#104	1	1	yes	characteristic
281	C₂²	C₂₃×#421	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	1	no	—
290	C₄	—	8	1	no	—
291	C₄	—	8	1	no	—
292	C₄	—	8	1	no	—
293	C₄	—	8	1	no	—
294	C₂	C₂₃×#1524	1	1	yes	characteristic
295	C₂	C₂₃×#1524	1	1	yes	characteristic
296	C₂	C₂₃×#1529	1	1	yes	characteristic
297	C₂	C₂₃×#1532	1	1	yes	characteristic
298	C₂	C₂₃×#1986	1	1	yes	characteristic
299	C₂	C₂₃×#1994	1	1	yes	characteristic
300	C₂	C₂₃×#1998	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
87. #9892	C₂₃	direct (N × H)
204. C₂₃	#9892	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#2864	2
C₂₃×#1767	2
C₄×C₂₃×#97	2
C₂₃×#1295	2
#9892	23

Subgroups — C23×#9892 #19157518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9892 #19157518