Subgroups — C₂₃×#23842 #19171468

Order 5888.

662 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2331	C₂	1	2	yes	maximal
2	C₂₃×#2085	C₂	1	2	yes	maximal
3	C₂₃×#2085	C₂	1	2	yes	maximal
4	C₂₃×#2058	C₂	1	2	yes	maximal
5	C₂₃×#2057	C₂	1	2	yes	maximal
6	C₂₃×#1776	C₂	1	2	yes	maximal
7	C₄×C₂₃×#107	C₂	1	1	yes	maximal, characteristic
8	C₄×C₂₃×#106	C₂	1	1	yes	maximal, characteristic
9	C₂₃×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₂₃×#107	C₂²	1	2	yes	—
12	C₂×C₂₃×#107	C₂²	1	2	yes	—
13	C₂×C₂₃×#106	C₂²	1	2	yes	—
14	C₂×C₂₃×#106	C₂²	1	2	yes	—
15	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#391	—	2	2	no	—
18	C₂₃×#391	—	2	2	no	—
19	C₂₃×#389	C₂²	1	2	yes	—
20	C₂₃×#389	C₂²	1	2	yes	—
21	C₂₃×#387	—	2	2	no	—
22	C₂₃×#387	—	2	2	no	—
23	C₂₃×#382	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂₃×#376	C₂²	1	1	yes	characteristic
27	C₂₃×#340	C₂²	1	1	yes	characteristic
28	C₂₃×#340	C₂²	1	1	yes	characteristic
29	C₂₃×#340	C₂²	1	1	yes	characteristic
30	C₂₃×#340	C₂²	1	1	yes	characteristic
31	C₂₃×#340	C₂²	1	2	yes	—
32	C₂₃×#340	C₂²	1	2	yes	—
33	C₂₃×#336	C₂²	1	2	yes	—
34	C₂₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₂₃	—	2	2	no	—
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂₃×C₄⋊Q₈	—	2	4	no	—
40	C₂₃×C₄⋊Q₈	—	2	4	no	—
41	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
45	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
48	C₂×C₄²×C₂₃	—	2	2	no	—
49	C₂×C₄²×C₂₃	—	2	2	no	—
50	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
52	C₂₃×#107	—	2	4	no	—
53	C₂₃×#107	—	2	4	no	—
54	C₂₃×#106	—	2	4	no	—
55	C₂₃×#106	—	2	4	no	—
56	C₂₃×#97	Q₈	1	4	yes	complemented
57	C₄×C₈×C₂₃	Q₈	1	4	yes	complemented
58	C₂₃×#95	—	2	2	no	—
59	C₂₃×#95	—	2	2	no	—
60	C₂₃×#95	—	2	2	no	—
61	C₂₃×#95	—	2	2	no	—
62	C₂₃×#95	—	2	2	no	—
63	C₂₃×#95	—	2	2	no	—
64	C₂₃×#95	D₄	1	1	yes	characteristic
65	C₂₃×#95	D₄	1	2	yes	—
66	C₂₃×#95	D₄	1	1	yes	characteristic
67	C₂×Q₈×C₂₃	—	2	8	no	—
68	C₂×Q₈×C₂₃	—	8	2	no	—
69	C₂×Q₈×C₂₃	—	8	2	no	—
70	C₂²×C₄×C₂₃	—	2	2	no	—
71	C₂²×C₄×C₂₃	—	2	2	no	—
72	C₂²×C₄×C₂₃	—	2	2	no	—
73	C₂²×C₄×C₂₃	—	2	2	no	—
74	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	2	no	—
83	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₂₃	C₂×Q₈	1	2	yes	—
85	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
86	C₄⋊C₄×C₂₃	—	2	8	no	—
87	C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	—	2	2	no	—
90	C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₄⋊C₄×C₂₃	—	2	2	no	—
92	C₄⋊C₄×C₂₃	—	2	2	no	—
93	C₄⋊C₄×C₂₃	—	8	2	no	—
94	C₄⋊C₄×C₂₃	—	8	2	no	—
95	C₄⋊C₄×C₂₃	#41	1	4	yes	—
96	C₄⋊C₄×C₂₃	#41	1	4	yes	—
97	C₄²×C₂₃	—	2	2	no	—
98	C₄²×C₂₃	—	2	2	no	—
99	C₄²×C₂₃	—	8	2	no	—
100	C₄²×C₂₃	—	8	2	no	—
101	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
102	C₄²×C₂₃	C₂×Q₈	1	2	yes	—
103	#23842	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₂₃	—	8	8	no	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	4	no	—
112	C₂×C₄×C₂₃	—	2	4	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	2	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	4	2	no	—
118	C₂×C₄×C₂₃	—	4	2	no	—
119	C₂×C₄×C₂₃	—	4	2	no	—
120	C₂×C₄×C₂₃	—	4	2	no	—
121	C₂×C₄×C₂₃	—	4	2	no	—
122	C₂×C₄×C₂₃	—	4	2	no	—
123	C₂×C₄×C₂₃	—	4	2	no	—
124	C₂×C₄×C₂₃	—	4	2	no	—
125	C₂×C₄×C₂₃	—	4	2	no	—
126	C₂×C₄×C₂₃	—	4	2	no	—
127	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
132	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
133	C₈×C₂₃	—	2	2	no	—
134	C₈×C₂₃	—	2	4	no	—
135	C₈×C₂₃	—	2	2	no	—
136	#2331	C₂×C₂₃	1	2	yes	—
137	#2085	C₂×C₂₃	1	2	yes	—
138	#2085	C₂×C₂₃	1	2	yes	—
139	#2058	C₂×C₂₃	1	2	yes	—
140	#2057	C₂×C₂₃	1	2	yes	—
141	#1776	C₂×C₂₃	1	2	yes	—
142	C₄×#107	C₂×C₂₃	1	1	yes	characteristic
143	C₄×#106	C₂×C₂₃	1	1	yes	characteristic
144	#1746	C₂×C₂₃	1	1	yes	characteristic
145	C₂²×C₂₃	#525	1	2	yes	—
146	C₂²×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	C₂²×C₂₃	#576	1	1	yes	characteristic
148	C₂²×C₂₃	#577	1	2	yes	—
149	C₂²×C₂₃	#578	1	1	yes	characteristic
150	C₄×C₂₃	—	2	2	no	—
151	C₄×C₂₃	—	2	2	no	—
152	C₄×C₂₃	—	4	2	no	—
153	C₄×C₂₃	—	4	4	no	—
154	C₄×C₂₃	—	4	4	no	—
155	C₄×C₂₃	—	4	2	no	—
156	C₄×C₂₃	—	8	2	no	—
157	C₄×C₂₃	—	8	2	no	—
158	C₄×C₂₃	#533	1	1	yes	characteristic
159	C₄×C₂₃	#533	1	2	yes	—
160	C₄×C₂₃	#533	1	1	yes	characteristic
161	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
162	C₂×#107	C₂²×C₂₃	1	2	yes	—
163	C₂×#107	C₂²×C₂₃	1	2	yes	—
164	C₂×#106	C₂²×C₂₃	1	2	yes	—
165	C₂×#106	C₂²×C₂₃	1	2	yes	—
166	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
167	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
168	#391	—	2	2	no	—
169	#391	—	2	2	no	—
170	#389	C₂²×C₂₃	1	2	yes	—
171	#389	C₂²×C₂₃	1	2	yes	—
172	#387	—	2	2	no	—
173	#387	—	2	2	no	—
174	#382	C₂²×C₂₃	1	2	yes	—
175	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
176	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
177	#376	C₂²×C₂₃	1	1	yes	characteristic
178	#340	C₂²×C₂₃	1	1	yes	characteristic
179	#340	C₂²×C₂₃	1	2	yes	—
180	#340	C₂²×C₂₃	1	1	yes	characteristic
181	#340	C₂²×C₂₃	1	1	yes	characteristic
182	#340	C₂²×C₂₃	1	1	yes	characteristic
183	#340	C₂²×C₂₃	1	2	yes	—
184	#336	C₂²×C₂₃	1	2	yes	—
185	#336	C₂²×C₂₃	1	2	yes	—
186	C₂×C₂₃	#2399	1	1	yes	characteristic
187	C₂×C₂₃	#3118	1	2	yes	—
188	C₂×C₂₃	#3121	1	2	yes	—
189	C₂×C₂₃	#3163	1	1	yes	characteristic
190	C₂×C₂₃	#3164	1	1	yes	characteristic
191	C₂²×Q₈	—	2	2	no	—
192	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
194	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
195	C₄⋊Q₈	—	2	4	no	—
196	C₄⋊Q₈	—	2	4	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
203	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
207	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
208	#107	—	2	4	no	—
209	#107	—	2	4	no	—
210	#106	—	2	4	no	—
211	#106	—	2	4	no	—
212	#97	Q₈×C₂₃	1	4	yes	complemented
213	C₄×C₈	Q₈×C₂₃	1	4	yes	complemented
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	D₄×C₂₃	1	1	yes	characteristic
221	#95	D₄×C₂₃	1	2	yes	—
222	#95	D₄×C₂₃	1	1	yes	characteristic
223	C₂₃	#23842	1	1	yes	characteristic, direct product factor, Sylow
224	C₂×Q₈	—	2	8	no	—
225	C₂×Q₈	—	8	2	no	—
226	C₂×Q₈	—	8	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
233	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
241	C₂×C₈	C₂×Q₈×C₂₃	1	2	yes	—
242	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
243	C₄⋊C₄	—	2	8	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	8	2	no	—
251	C₄⋊C₄	—	8	2	no	—
252	C₄⋊C₄	#41×C₂₃	1	4	yes	—
253	C₄⋊C₄	#41×C₂₃	1	4	yes	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	8	2	no	—
257	C₄²	—	8	2	no	—
258	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
259	C₄²	C₂×Q₈×C₂₃	1	2	yes	—
260	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
261	Q₈	—	8	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
284	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
285	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
286	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
287	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
288	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
289	C₈	—	2	2	no	—
290	C₈	—	2	4	no	—
291	C₈	—	2	2	no	—
292	C₂²	C₂₃×#525	1	2	yes	—
293	C₂²	C₂×C₂₃×C₄⋊Q₈	1	1	yes	characteristic
294	C₂²	C₂₃×#576	1	1	yes	characteristic
295	C₂²	C₂₃×#577	1	2	yes	—
296	C₂²	C₂₃×#578	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	2	2	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	C₂₃×#533	1	1	yes	characteristic
306	C₄	C₂₃×#533	1	2	yes	—
307	C₄	C₂₃×#533	1	1	yes	characteristic
308	C₂	C₂₃×#2399	1	1	yes	characteristic
309	C₂	C₂₃×#3118	1	2	yes	—
310	C₂	C₂₃×#3121	1	2	yes	—
311	C₂	C₂₃×#3163	1	1	yes	characteristic
312	C₂	C₂₃×#3164	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
56. C₂₃×#97	Q₈	semidirect
57. C₄×C₈×C₂₃	Q₈	semidirect
103. #23842	C₂₃	direct (N × H)
212. #97	Q₈×C₂₃	semidirect
213. C₄×C₈	Q₈×C₂₃	semidirect
223. C₂₃	#23842	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×#2331	2
C₂₃×#2085	2
C₂₃×#2058	2
C₂₃×#2057	2
C₂₃×#1776	2
C₄×C₂₃×#107	2
C₄×C₂₃×#106	2
C₂₃×#1746	2
#23842	23

Subgroups — C23×#23842 #19171468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23842 #19171468