Subgroups — C₂₃×#29138 #19176764

Order 5888.

642 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2647	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2327	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2101	C₂	1	2	yes	maximal
4	C₂₃×#2043	C₂	1	2	yes	maximal
5	C₂₃×#1935	C₂	1	2	yes	maximal
6	C₂₃×#1933	C₂	1	2	yes	maximal
7	C₂₃×#1805	C₂	1	2	yes	maximal
8	C₂₃×#1803	C₂	1	2	yes	maximal
9	C₂₃×#1768	C₂	1	1	yes	maximal, characteristic
10	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃×#103	C₂²	1	2	yes	—
14	C₂×C₂₃×#103	C₂²	1	2	yes	—
15	C₂×C₂₃×#103	C₂²	1	2	yes	—
16	C₂×C₂₃×#103	C₂²	1	2	yes	—
17	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#400	—	2	2	no	—
20	C₂₃×#398	—	2	2	no	—
21	C₂₃×#391	C₂²	1	2	yes	—
22	C₂₃×#389	C₂²	1	2	yes	—
23	C₂₃×#389	C₂²	1	2	yes	—
24	C₂₃×#387	—	2	2	no	—
25	C₂₃×#387	—	2	2	no	—
26	C₂₃×#384	C₂²	1	1	yes	characteristic
27	C₂₃×#382	C₂²	1	1	yes	characteristic
28	C₂₃×#382	C₂²	1	1	yes	characteristic
29	C₂₃×#376	C₂²	1	1	yes	characteristic
30	C₂₃×#340	C₂²	1	2	yes	—
31	C₂₃×#340	C₂²	1	2	yes	—
32	C₂₃×#340	C₂²	1	2	yes	—
33	C₂₃×#340	C₂²	1	2	yes	—
34	C₂₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
38	C₂₃×C₄⋊Q₈	—	2	4	no	—
39	C₄×Q₈×C₂₃	—	2	4	no	—
40	C₄×Q₈×C₂₃	—	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	—	2	2	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
49	C₂×C₄²×C₂₃	—	2	2	no	—
50	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
53	C₂₃×C₄⋊C₈	—	2	4	no	—
54	C₂₃×#103	—	2	4	no	—
55	C₂₃×#103	—	2	4	no	—
56	C₂₃×#103	—	4	2	no	—
57	C₂₃×#103	—	4	2	no	—
58	C₂₃×#97	—	4	1	no	—
59	C₄×C₈×C₂₃	—	4	1	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	—	2	2	no	—
65	C₂₃×#95	—	2	2	no	—
66	C₂₃×#95	D₄	1	2	yes	—
67	C₂₃×#95	D₄	1	2	yes	—
68	C₂×Q₈×C₂₃	—	2	2	no	—
69	C₂×Q₈×C₂₃	—	2	2	no	—
70	C₂×Q₈×C₂₃	—	2	2	no	—
71	C₂×Q₈×C₂₃	—	2	2	no	—
72	C₂×Q₈×C₂₃	—	2	2	no	—
73	C₂×Q₈×C₂₃	#41	1	4	yes	—
74	C₂²×C₄×C₂₃	—	2	2	no	—
75	C₂²×C₄×C₂₃	—	2	2	no	—
76	C₂²×C₄×C₂₃	—	2	2	no	—
77	C₂²×C₄×C₂₃	—	2	2	no	—
78	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₂₃	—	2	2	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	—	2	2	no	—
87	C₂×C₈×C₂₃	—	2	2	no	—
88	C₂×C₈×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	—	2	8	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	—	2	2	no	—
94	C₄⋊C₄×C₂₃	—	2	2	no	—
95	C₄⋊C₄×C₂₃	—	2	2	no	—
96	C₄⋊C₄×C₂₃	—	4	4	no	—
97	C₄⋊C₄×C₂₃	—	8	2	no	—
98	C₄⋊C₄×C₂₃	—	8	2	no	—
99	C₄⋊C₄×C₂₃	—	8	2	no	—
100	C₄²×C₂₃	—	2	2	no	—
101	C₄²×C₂₃	—	2	1	no	—
102	C₄²×C₂₃	—	2	1	no	—
103	C₄²×C₂₃	—	4	4	no	—
104	C₄²×C₂₃	#41	1	4	yes	—
105	#29138	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₂₃	—	2	4	no	—
108	Q₈×C₂₃	—	2	4	no	—
109	Q₈×C₂₃	—	4	2	no	—
110	Q₈×C₂₃	—	4	2	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	2	no	—
114	C₂×C₄×C₂₃	—	2	4	no	—
115	C₂×C₄×C₂₃	—	2	1	no	—
116	C₂×C₄×C₂₃	—	2	1	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	2	no	—
120	C₂×C₄×C₂₃	—	2	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	—	4	2	no	—
128	C₂×C₄×C₂₃	—	4	2	no	—
129	C₂×C₄×C₂₃	—	4	2	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	#124	1	4	yes	—
133	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
134	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
135	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
137	C₈×C₂₃	—	4	2	no	—
138	C₈×C₂₃	—	4	2	no	—
139	#2647	C₂×C₂₃	1	1	yes	characteristic
140	#2327	C₂×C₂₃	1	1	yes	characteristic
141	#2101	C₂×C₂₃	1	2	yes	—
142	#2043	C₂×C₂₃	1	2	yes	—
143	#1935	C₂×C₂₃	1	2	yes	—
144	#1933	C₂×C₂₃	1	2	yes	—
145	#1805	C₂×C₂₃	1	2	yes	—
146	#1803	C₂×C₂₃	1	2	yes	—
147	#1768	C₂×C₂₃	1	1	yes	characteristic
148	C₂²×C₂₃	#525	1	1	yes	characteristic
149	C₂²×C₂₃	C₂×#124	1	1	yes	characteristic
150	C₂²×C₂₃	#537	1	1	yes	characteristic
151	C₂²×C₂₃	#577	1	2	yes	—
152	C₂²×C₂₃	#578	1	2	yes	—
153	C₄×C₂₃	—	2	4	no	—
154	C₄×C₂₃	—	4	4	no	—
155	C₄×C₂₃	—	4	2	no	—
156	C₄×C₂₃	—	4	1	no	—
157	C₄×C₂₃	—	4	1	no	—
158	C₄×C₂₃	—	8	2	no	—
159	C₄×C₂₃	—	8	2	no	—
160	C₄×C₂₃	—	8	2	no	—
161	C₄×C₂₃	#552	1	2	yes	—
162	C₄×C₂₃	#553	1	2	yes	—
163	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
165	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
166	C₂×#103	C₂²×C₂₃	1	2	yes	—
167	C₂×#103	C₂²×C₂₃	1	2	yes	—
168	C₂×#103	C₂²×C₂₃	1	2	yes	—
169	C₂×#103	C₂²×C₂₃	1	2	yes	—
170	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
172	#400	—	2	2	no	—
173	#398	—	2	2	no	—
174	#391	C₂²×C₂₃	1	2	yes	—
175	#389	C₂²×C₂₃	1	2	yes	—
176	#389	C₂²×C₂₃	1	2	yes	—
177	#387	—	2	2	no	—
178	#387	—	2	2	no	—
179	#384	C₂²×C₂₃	1	1	yes	characteristic
180	#382	C₂²×C₂₃	1	1	yes	characteristic
181	#382	C₂²×C₂₃	1	1	yes	characteristic
182	#376	C₂²×C₂₃	1	1	yes	characteristic
183	#340	C₂²×C₂₃	1	2	yes	—
184	#340	C₂²×C₂₃	1	2	yes	—
185	#340	C₂²×C₂₃	1	2	yes	—
186	#340	C₂²×C₂₃	1	2	yes	—
187	#336	C₂²×C₂₃	1	2	yes	—
188	C₂×C₂₃	#2562	1	1	yes	characteristic
189	C₂×C₂₃	#3121	1	2	yes	—
190	C₂×C₂₃	#3137	1	2	yes	—
191	C₂×C₂₃	#3260	1	1	yes	characteristic
192	C₂×C₂₃	#3261	1	1	yes	characteristic
193	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
194	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
195	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
196	C₄⋊Q₈	—	2	4	no	—
197	C₄×Q₈	—	2	4	no	—
198	C₄×Q₈	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
206	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
211	C₄⋊C₈	—	2	4	no	—
212	#103	—	2	4	no	—
213	#103	—	2	4	no	—
214	#103	—	4	2	no	—
215	#103	—	4	2	no	—
216	#97	—	4	1	no	—
217	C₄×C₈	—	4	1	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	#95	—	2	2	no	—
221	#95	—	2	2	no	—
222	#95	—	2	2	no	—
223	#95	—	2	2	no	—
224	#95	D₄×C₂₃	1	2	yes	—
225	#95	D₄×C₂₃	1	2	yes	—
226	C₂₃	#29138	1	1	yes	characteristic, direct product factor, Sylow
227	C₂×Q₈	—	2	2	no	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	#41×C₂₃	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	8	2	no	—
257	C₄⋊C₄	—	8	2	no	—
258	C₄⋊C₄	—	8	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	4	4	no	—
263	C₄²	#41×C₂₃	1	4	yes	—
264	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
265	Q₈	—	2	4	no	—
266	Q₈	—	2	4	no	—
267	Q₈	—	4	2	no	—
268	Q₈	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	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	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	C₂₃×#124	1	4	yes	—
291	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
292	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
293	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
294	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
295	C₈	—	4	2	no	—
296	C₈	—	4	2	no	—
297	C₂²	C₂₃×#525	1	1	yes	characteristic
298	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
299	C₂²	C₂₃×#537	1	1	yes	characteristic
300	C₂²	C₂₃×#577	1	2	yes	—
301	C₂²	C₂₃×#578	1	2	yes	—
302	C₄	—	2	4	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	4	no	—
306	C₄	—	4	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	C₂₃×#552	1	2	yes	—
311	C₄	C₂₃×#553	1	2	yes	—
312	C₂	C₂₃×#2562	1	1	yes	characteristic
313	C₂	C₂₃×#3121	1	2	yes	—
314	C₂	C₂₃×#3137	1	2	yes	—
315	C₂	C₂₃×#3260	1	1	yes	characteristic
316	C₂	C₂₃×#3261	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
105. #29138	C₂₃	direct (N × H)
226. C₂₃	#29138	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×#2647	2
C₂₃×#2327	2
C₂₃×#2101	2
C₂₃×#2043	2
C₂₃×#1935	2
C₂₃×#1933	2
C₂₃×#1805	2
C₂₃×#1803	2
C₂₃×#1768	2
#29138	23

Subgroups — C23×#29138 #19176764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29138 #19176764