Subgroups — C₂₃×#29127 #19176753

Order 5888.

842 conjugacy classes of subgroups.

The subgroup lattice has 320 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, complemented
2	C₄×C₂₃×#127	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2099	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2041	C₂	1	2	yes	maximal
5	C₂₃×#1936	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1804	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1804	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₂₃	—	2	2	no	—
11	C₂×C₄×D₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₂₃×#102	C₂²	1	4	yes	—
14	C₂×C₂₃×#102	C₂²	1	4	yes	—
15	C₂×C₂₃×#97	C₂²	1	2	yes	—
16	C₂₃×#390	—	2	2	no	—
17	C₂₃×#390	C₂²	1	2	yes	—
18	C₂₃×#389	C₂²	1	2	yes	—
19	C₂₃×#389	C₂²	1	2	yes	—
20	C₂₃×#387	—	2	2	no	—
21	C₂₃×#387	—	2	2	no	—
22	C₂₃×#384	C₂²	1	1	yes	characteristic
23	C₂₃×#382	C₂²	1	1	yes	characteristic
24	C₂₃×#382	C₂²	1	1	yes	characteristic
25	C₄³×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#340	C₂²	1	4	yes	—
27	C₂₃×#340	C₂²	1	4	yes	—
28	C₂₃×#336	C₂²	1	2	yes	—
29	C₂²×D₄×C₂₃	—	2	2	no	—
30	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₂₃	—	2	2	no	—
32	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
33	C₂₃×#127	—	2	4	no	—
34	C₄×D₄×C₂₃	—	2	4	no	—
35	C₄×D₄×C₂₃	—	2	4	no	—
36	C₄×D₄×C₂₃	—	4	8	no	—
37	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
43	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
45	C₂×C₄²×C₂₃	D₄	1	2	yes	—
46	C₂×C₄²×C₂₃	D₄	1	2	yes	—
47	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
50	C₂₃×C₄⋊C₈	—	2	4	no	—
51	C₂₃×#102	—	2	8	no	—
52	C₂₃×#102	—	4	4	no	—
53	C₂₃×#97	—	4	2	no	—
54	C₂₃×#95	—	2	1	no	—
55	C₂₃×#95	—	2	1	no	—
56	C₂₃×#95	—	2	1	no	—
57	C₂₃×#95	—	2	1	no	—
58	C₂₃×#95	—	2	2	no	—
59	C₂₃×#95	—	2	2	no	—
60	C₂⁴×C₂₃	—	2	2	no	—
61	C₂×D₄×C₂₃	—	2	8	no	—
62	C₂×D₄×C₂₃	—	2	2	no	—
63	C₂×D₄×C₂₃	—	2	2	no	—
64	C₂×D₄×C₂₃	—	2	2	no	—
65	C₂×D₄×C₂₃	—	2	2	no	—
66	C₂×D₄×C₂₃	—	2	2	no	—
67	C₂×D₄×C₂₃	—	4	4	no	—
68	C₂×D₄×C₂₃	—	4	4	no	—
69	C₂×D₄×C₂₃	#41	1	4	yes	—
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	—	4	8	no	—
74	C₂²×C₄×C₂₃	—	4	4	no	—
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	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₂₃	—	2	4	no	—
81	C₂×C₈×C₂₃	—	2	4	no	—
82	C₂×C₈×C₂₃	—	2	4	no	—
83	C₄⋊C₄×C₂₃	—	2	4	no	—
84	C₄⋊C₄×C₂₃	—	2	2	no	—
85	C₄⋊C₄×C₂₃	—	2	2	no	—
86	C₄⋊C₄×C₂₃	—	2	2	no	—
87	C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₄⋊C₄×C₂₃	—	4	4	no	—
89	C₄⋊C₄×C₂₃	—	4	4	no	—
90	C₄⋊C₄×C₂₃	—	4	4	no	—
91	C₂²⋊C₄×C₂₃	—	4	8	no	—
92	C₂²⋊C₄×C₂₃	—	4	4	no	—
93	C₄²×C₂₃	—	2	4	no	—
94	C₄²×C₂₃	—	2	2	no	—
95	C₄²×C₂₃	—	2	1	no	—
96	C₄²×C₂₃	—	2	1	no	—
97	C₄²×C₂₃	—	4	2	no	—
98	C₄²×C₂₃	#41	1	4	yes	—
99	#29127	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₂₃	—	2	4	no	—
101	C₂³×C₂₃	—	4	2	no	—
102	C₂³×C₂₃	—	4	2	no	—
103	C₂³×C₂₃	—	4	4	no	—
104	C₂³×C₂₃	—	4	4	no	—
105	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
106	D₄×C₂₃	—	2	4	no	—
107	D₄×C₂₃	—	2	4	no	—
108	D₄×C₂₃	—	4	16	no	—
109	D₄×C₂₃	—	4	2	no	—
110	D₄×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	2	no	—
115	C₂×C₄×C₂₃	—	2	2	no	—
116	C₂×C₄×C₂₃	—	2	2	no	—
117	C₂×C₄×C₂₃	—	2	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	4	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	—	8	8	no	—
127	C₂×C₄×C₂₃	#124	1	2	yes	—
128	C₂×C₄×C₂₃	#124	1	2	yes	—
129	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
130	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
131	C₂×C₄×C₂₃	C₂×#41	1	2	yes	—
132	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
134	C₈×C₂₃	—	4	4	no	—
135	#2864	C₂×C₂₃	1	1	yes	characteristic, complemented
136	C₄×#127	C₂×C₂₃	1	1	yes	characteristic
137	#2099	C₂×C₂₃	1	2	yes	complemented
138	#2041	C₂×C₂₃	1	2	yes	—
139	#1936	C₂×C₂₃	1	4	yes	complemented
140	#1804	C₂×C₂₃	1	2	yes	complemented
141	#1804	C₂×C₂₃	1	2	yes	complemented
142	#1767	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂²×C₂₃	—	4	4	no	—
144	C₂²×C₂₃	—	4	8	no	—
145	C₂²×C₂₃	—	8	2	no	—
146	C₂²×C₂₃	—	8	2	no	—
147	C₂²×C₂₃	—	8	4	no	—
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	C₂×AGL(1,ℤ/8)	1	2	yes	—
152	C₂²×C₂₃	#575	1	2	yes	—
153	C₄×C₂₃	—	2	4	no	—
154	C₄×C₂₃	—	2	2	no	—
155	C₄×C₂₃	—	2	2	no	—
156	C₄×C₂₃	—	4	2	no	—
157	C₄×C₂₃	—	8	2	no	—
158	C₄×C₂₃	—	8	2	no	—
159	C₄×C₂₃	—	8	2	no	—
160	C₄×C₂₃	#552	1	2	yes	—
161	C₄×C₂₃	#553	1	2	yes	—
162	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
163	C₂×C₄×D₄	—	2	2	no	—
164	C₂×C₄×D₄	C₂²×C₂₃	1	2	yes	—
165	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
166	C₂×#102	C₂²×C₂₃	1	4	yes	—
167	C₂×#102	C₂²×C₂₃	1	4	yes	—
168	C₂×#97	C₂²×C₂₃	1	2	yes	—
169	#390	—	2	2	no	—
170	#390	C₂²×C₂₃	1	2	yes	—
171	#389	C₂²×C₂₃	1	2	yes	—
172	#389	C₂²×C₂₃	1	2	yes	—
173	#387	—	2	2	no	—
174	#387	—	2	2	no	—
175	#384	C₂²×C₂₃	1	1	yes	characteristic
176	#382	C₂²×C₂₃	1	1	yes	characteristic
177	#382	C₂²×C₂₃	1	1	yes	characteristic
178	C₄³	C₂²×C₂₃	1	1	yes	characteristic
179	#340	C₂²×C₂₃	1	4	yes	—
180	#340	C₂²×C₂₃	1	4	yes	—
181	#336	C₂²×C₂₃	1	2	yes	—
182	C₂×C₂₃	—	8	4	no	—
183	C₂×C₂₃	#2562	1	1	yes	characteristic
184	C₂×C₂₃	#3114	1	2	yes	—
185	C₂×C₂₃	#3135	1	1	yes	characteristic
186	C₂×C₂₃	#3138	1	1	yes	characteristic
187	C₂×C₂₃	#3251	1	2	yes	—
188	C₂²×D₄	—	2	2	no	—
189	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
190	C₂³×C₄	—	2	2	no	—
191	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
192	#127	—	2	4	no	—
193	C₄×D₄	—	2	4	no	—
194	C₄×D₄	—	2	4	no	—
195	C₄×D₄	—	4	8	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	4	no	—
203	C₂×C₂²⋊C₄	—	2	2	no	—
204	C₂×C₄²	D₄×C₂₃	1	2	yes	—
205	C₂×C₄²	D₄×C₂₃	1	2	yes	—
206	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	4	no	—
210	#102	—	2	8	no	—
211	#102	—	4	4	no	—
212	#97	—	4	2	no	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	#95	—	2	1	no	—
216	#95	—	2	1	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	C₂₃	#29127	1	1	yes	characteristic, direct product factor, Sylow
220	C₂⁴	—	2	2	no	—
221	C₂×D₄	—	2	8	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	#41×C₂₃	1	4	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	4	8	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₄⋊C₄	—	2	4	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	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₂²⋊C₄	—	4	8	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	4	2	no	—
258	C₄²	#41×C₂₃	1	4	yes	—
259	C₂³	—	2	4	no	—
260	C₂³	—	4	4	no	—
261	C₂³	—	4	4	no	—
262	C₂³	—	4	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
265	D₄	—	2	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	16	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	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	2	no	—
277	C₂×C₄	—	4	4	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	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	8	no	—
286	C₂×C₄	C₂₃×#124	1	2	yes	—
287	C₂×C₄	C₂₃×#124	1	2	yes	—
288	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
289	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
290	C₂×C₄	C₂×#41×C₂₃	1	2	yes	—
291	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
292	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
293	C₈	—	4	4	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	8	4	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	C₂₃×#525	1	1	yes	characteristic
300	C₂²	C₂×C₂₃×#124	1	1	yes	characteristic
301	C₂²	C₂₃×#537	1	1	yes	characteristic
302	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
303	C₂²	C₂₃×#575	1	2	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₂₃×#552	1	2	yes	—
312	C₄	C₂₃×#553	1	2	yes	—
313	C₂	—	8	4	no	—
314	C₂	C₂₃×#2562	1	1	yes	characteristic
315	C₂	C₂₃×#3114	1	2	yes	—
316	C₂	C₂₃×#3135	1	1	yes	characteristic
317	C₂	C₂₃×#3138	1	1	yes	characteristic
318	C₂	C₂₃×#3251	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#2864	C₂	semidirect
3. C₂₃×#2099	C₂	semidirect
5. C₂₃×#1936	C₂	semidirect
6. C₂₃×#1804	C₂	semidirect
7. C₂₃×#1804	C₂	semidirect
8. C₂₃×#1767	C₂	semidirect
99. #29127	C₂₃	direct (N × H)
135. #2864	C₂×C₂₃	semidirect
137. #2099	C₂×C₂₃	semidirect
139. #1936	C₂×C₂₃	semidirect
140. #1804	C₂×C₂₃	semidirect
141. #1804	C₂×C₂₃	semidirect
142. #1767	C₂×C₂₃	semidirect
219. C₂₃	#29127	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
C₄×C₂₃×#127	2
C₂₃×#2099	2
C₂₃×#2041	2
C₂₃×#1936	2
C₂₃×#1804	2
C₂₃×#1767	2
#29127	23

Subgroups — C23×#29127 #19176753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#29127 #19176753