Subgroups — C₂₃×#11335 #19158961

Order 5888.

790 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2911	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2296	C₂	1	2	yes	maximal
3	C₂₃×#1733	C₂	1	2	yes	maximal
4	C₂×C₂₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#117	C₂²	1	2	yes	—
9	C₂×C₂₃×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×#428	—	2	4	no	—
12	C₂₃×#427	—	2	2	no	—
13	C₂₃×#389	—	2	4	no	—
14	C₂₃×#385	—	2	4	no	—
15	C₄×C₄⋊C₄×C₂₃	—	2	4	no	—
16	C₂₃×#339	—	2	4	no	—
17	C₂₃×H(3,ℤ/4)	C₄	1	4	yes	complemented
18	C₂₃×#336	—	2	2	no	—
19	C₂³×C₄×C₂₃	D₄	1	2	yes	—
20	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₂₃	—	2	2	no	—
22	C₂×M₁₆×C₂₃	—	2	2	no	—
23	C₂×M₁₆×C₂₃	—	2	1	no	—
24	C₂×M₁₆×C₂₃	—	2	2	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
27	C₂×M₁₆×C₂₃	Q₈	1	2	yes	—
28	C₂×M₁₆×C₂₃	Q₈	1	2	yes	—
29	C₂²×C₈×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	D₄	1	2	yes	—
31	C₂₃×#117	—	2	2	no	—
32	C₂₃×#117	—	2	2	no	—
33	C₂₃×#117	—	2	2	no	—
34	C₂₃×#117	—	2	2	no	—
35	C₂₃×#117	—	2	2	no	—
36	C₂₃×#117	—	2	2	no	—
37	C₂₃×#117	C₂×C₄	1	4	yes	—
38	C₂₃×#117	Q₈	1	4	yes	—
39	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
41	C₂×C₄⋊C₄×C₂₃	D₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄²×C₂₃	—	2	1	no	—
45	C₂×C₄²×C₂₃	—	2	2	no	—
46	C₂×C₄²×C₂₃	—	2	2	no	—
47	C₂×C₄²×C₂₃	—	2	2	no	—
48	C₂×C₄²×C₂₃	—	2	1	no	—
49	C₂×C₄²×C₂₃	D₄	1	4	yes	—
50	C₂₃×#107	—	2	4	no	—
51	C₂₃×#106	—	2	4	no	—
52	C₂₃×#95	—	2	8	no	—
53	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	4	no	—
58	C₂²×C₄×C₂₃	—	4	4	no	—
59	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	#41	1	2	yes	—
67	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
69	M₁₆×C₂₃	—	2	4	no	—
70	M₁₆×C₂₃	—	2	4	no	—
71	M₁₆×C₂₃	—	2	4	no	—
72	M₁₆×C₂₃	—	4	2	no	—
73	C₂×C₈×C₂₃	—	2	2	no	—
74	C₂×C₈×C₂₃	—	2	2	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	4	1	no	—
77	C₂×C₈×C₂₃	—	4	1	no	—
78	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	complemented
79	C₄⋊C₄×C₂₃	—	2	4	no	—
80	C₄⋊C₄×C₂₃	—	2	4	no	—
81	C₄⋊C₄×C₂₃	—	2	4	no	—
82	C₄⋊C₄×C₂₃	—	2	4	no	—
83	C₄⋊C₄×C₂₃	—	4	16	no	—
84	C₂²⋊C₄×C₂₃	—	2	4	no	—
85	C₂²⋊C₄×C₂₃	—	2	4	no	—
86	C₂²⋊C₄×C₂₃	—	4	2	no	—
87	C₂²⋊C₄×C₂₃	—	4	2	no	—
88	C₄²×C₂₃	—	2	4	no	—
89	C₄²×C₂₃	—	4	1	no	—
90	C₄²×C₂₃	—	4	2	no	—
91	C₄²×C₂₃	—	4	2	no	—
92	C₄²×C₂₃	—	4	1	no	—
93	C₄²×C₂₃	—	4	2	no	—
94	C₄²×C₂₃	C₄⋊C₄	1	8	yes	complemented
95	#11335	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₂₃	—	4	2	no	—
97	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
98	C₂³×C₂₃	#120	1	2	yes	—
99	C₂³×C₂₃	#121	1	1	yes	characteristic
100	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₂₃	—	2	8	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	2	8	no	—
105	C₂×C₄×C₂₃	—	4	8	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	8	no	—
110	C₂×C₄×C₂₃	—	4	4	no	—
111	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
116	C₂×C₄×C₂₃	#121	1	2	yes	—
117	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
122	C₂×C₄×C₂₃	#123	1	2	yes	—
123	C₂×C₄×C₂₃	#125	1	2	yes	—
124	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
125	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
128	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
129	C₈×C₂₃	—	2	4	no	—
130	C₈×C₂₃	—	4	2	no	—
131	#2911	C₂×C₂₃	1	1	yes	characteristic
132	#2296	C₂×C₂₃	1	2	yes	—
133	#1733	C₂×C₂₃	1	2	yes	—
134	C₂×H(3,ℤ/4)	C₂×C₂₃	1	1	yes	characteristic
135	#1728	C₂×C₂₃	1	1	yes	characteristic
136	C₂²×C₂₃	—	2	4	no	—
137	C₂²×C₂₃	—	2	2	no	—
138	C₂²×C₂₃	—	4	1	no	—
139	C₂²×C₂₃	—	4	2	no	—
140	C₂²×C₂₃	—	4	1	no	—
141	C₂²×C₂₃	#385	1	2	yes	—
142	C₂²×C₂₃	#389	1	1	yes	characteristic
143	C₂²×C₂₃	#392	1	1	yes	characteristic
144	C₂²×C₂₃	#395	1	2	yes	—
145	C₂²×C₂₃	#398	1	1	yes	characteristic
146	C₄×C₂₃	—	4	8	no	—
147	C₄×C₂₃	—	4	4	no	—
148	C₄×C₂₃	—	8	4	no	—
149	C₄×C₂₃	#384	1	1	yes	characteristic
150	C₄×C₂₃	#385	1	1	yes	characteristic
151	C₄×C₂₃	#389	1	2	yes	—
152	C₄×C₂₃	#395	1	1	yes	characteristic
153	C₄×C₂₃	#398	1	2	yes	—
154	C₄×C₂₃	#399	1	1	yes	characteristic
155	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
156	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×#117	C₂²×C₂₃	1	2	yes	—
158	C₂×#117	C₂²×C₂₃	1	2	yes	—
159	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
160	#428	—	2	4	no	—
161	#427	—	2	2	no	—
162	#389	—	2	4	no	—
163	#385	—	2	4	no	—
164	C₄×C₄⋊C₄	—	2	4	no	—
165	#339	—	2	4	no	—
166	H(3,ℤ/4)	C₄×C₂₃	1	4	yes	complemented
167	#336	—	2	2	no	—
168	C₂×C₂₃	—	2	2	no	—
169	C₂×C₂₃	—	2	1	no	—
170	C₂×C₂₃	—	2	2	no	—
171	C₂×C₂₃	—	2	1	no	—
172	C₂×C₂₃	#1444	1	1	yes	characteristic
173	C₂×C₂₃	#2010	1	2	yes	—
174	C₂³×C₄	D₄×C₂₃	1	2	yes	—
175	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
182	C₂×M₁₆	Q₈×C₂₃	1	2	yes	—
183	C₂×M₁₆	Q₈×C₂₃	1	2	yes	—
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	D₄×C₂₃	1	2	yes	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	C₂×C₄×C₂₃	1	4	yes	—
193	#117	Q₈×C₂₃	1	4	yes	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	8	no	—
196	C₂×C₄⋊C₄	D₄×C₂₃	1	4	yes	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	1	no	—
204	C₂×C₄²	D₄×C₂₃	1	4	yes	—
205	#107	—	2	4	no	—
206	#106	—	2	4	no	—
207	#95	—	2	8	no	—
208	C₂₃	#11335	1	1	yes	characteristic, direct product factor, Sylow
209	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	4	4	no	—
215	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
218	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
219	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
221	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
222	C₂²×C₄	#41×C₂₃	1	2	yes	—
223	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
224	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	complemented
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	4	16	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	—	4	2	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	4	1	no	—
246	C₄²	—	4	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	4	1	no	—
249	C₄²	—	4	2	no	—
250	C₄²	C₄⋊C₄×C₂₃	1	8	yes	complemented
251	C₂³	—	4	2	no	—
252	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
253	C₂³	C₂₃×#120	1	2	yes	—
254	C₂³	C₂₃×#121	1	1	yes	characteristic
255	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
256	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
268	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
269	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
270	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
271	C₂×C₄	C₂₃×#121	1	2	yes	—
272	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
273	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
274	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
275	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
276	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
277	C₂×C₄	C₂₃×#123	1	2	yes	—
278	C₂×C₄	C₂₃×#125	1	2	yes	—
279	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
280	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
281	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
282	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
283	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
284	C₈	—	2	4	no	—
285	C₈	—	4	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₂₃×#385	1	2	yes	—
292	C₂²	C₂₃×#389	1	1	yes	characteristic
293	C₂²	C₂₃×#392	1	1	yes	characteristic
294	C₂²	C₂₃×#395	1	2	yes	—
295	C₂²	C₂₃×#398	1	1	yes	characteristic
296	C₄	—	4	8	no	—
297	C₄	—	4	4	no	—
298	C₄	—	8	4	no	—
299	C₄	C₂₃×#384	1	1	yes	characteristic
300	C₄	C₂₃×#385	1	1	yes	characteristic
301	C₄	C₂₃×#389	1	2	yes	—
302	C₄	C₂₃×#395	1	1	yes	characteristic
303	C₄	C₂₃×#398	1	2	yes	—
304	C₄	C₂₃×#399	1	1	yes	characteristic
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	1	no	—
309	C₂	C₂₃×#1444	1	1	yes	characteristic
310	C₂	C₂₃×#2010	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#127
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	#392
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
17. C₂₃×H(3,ℤ/4)	C₄	semidirect
78. C₂×C₈×C₂₃	C₄⋊C₄	semidirect
94. C₄²×C₂₃	C₄⋊C₄	semidirect
95. #11335	C₂₃	direct (N × H)
166. H(3,ℤ/4)	C₄×C₂₃	semidirect
208. C₂₃	#11335	direct (N × H)
234. C₂×C₈	C₄⋊C₄×C₂₃	semidirect
250. C₄²	C₄⋊C₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#2911	2
C₂₃×#2296	2
C₂₃×#1733	2
C₂×C₂₃×H(3,ℤ/4)	2
C₂₃×#1728	2
#11335	23

Subgroups — C23×#11335 #19158961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11335 #19158961