Subgroups — C₂₃×#11236 #19158862

Order 5888.

654 conjugacy classes of subgroups.

The subgroup lattice has 308 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	C₂×C₂₃×#424	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1740	C₂	1	2	yes	maximal
4	C₂₃×#1735	C₂	1	2	yes	maximal
5	C₂×C₂₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#117	C₂²	1	2	yes	—
9	C₂×C₂₃×#117	C₂²	1	2	yes	—
10	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
11	C₂₃×#428	—	2	4	no	—
12	C₂₃×#427	—	2	2	no	—
13	C₂₃×#424	—	2	2	no	—
14	C₂₃×#424	C₄	1	4	yes	complemented
15	C₂×C₂₃×C₄⋊C₈	—	2	2	no	—
16	C₂₃×#342	—	2	4	no	—
17	C₂₃×#340	—	2	4	no	—
18	C₂₃×H(3,ℤ/4)	—	2	2	no	—
19	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₂₃	—	2	2	no	—
21	C₂×M₁₆×C₂₃	—	2	2	no	—
22	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
23	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	2	1	no	—
26	C₂²×C₈×C₂₃	—	2	1	no	—
27	C₂²×C₈×C₂₃	—	2	2	no	—
28	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₂₃	D₄	1	2	yes	—
30	C₂₃×#117	—	2	1	no	—
31	C₂₃×#117	—	2	1	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	—	2	2	no	—
38	C₂₃×#117	—	2	2	no	—
39	C₂₃×#117	C₂×C₄	1	2	yes	—
40	C₂₃×#117	Q₈	1	4	yes	—
41	C₂₃×#117	Q₈	1	2	yes	—
42	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	D₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
48	C₂×C₄²×C₂₃	—	2	1	no	—
49	C₂×C₄²×C₂₃	—	2	2	no	—
50	C₂×C₄²×C₂₃	—	2	2	no	—
51	C₂₃×#107	—	2	4	no	—
52	C₂₃×#106	—	2	4	no	—
53	C₂₃×C₄⋊C₈	—	2	8	no	—
54	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	4	no	—
57	C₂²×C₄×C₂₃	—	4	2	no	—
58	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂×D₄	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×Q₈	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	#41	1	2	yes	—
67	C₂²×C₄×C₂₃	#41	1	2	yes	—
68	M₁₆×C₂₃	—	2	4	no	—
69	M₁₆×C₂₃	—	2	4	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	—	4	1	no	—
75	C₂×C₈×C₂₃	—	4	2	no	—
76	C₂×C₈×C₂₃	—	4	1	no	—
77	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
78	C₄⋊C₄×C₂₃	—	2	2	no	—
79	C₄⋊C₄×C₂₃	—	2	2	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	2	no	—
84	C₂²⋊C₄×C₂₃	—	2	2	no	—
85	C₂²⋊C₄×C₂₃	—	2	4	no	—
86	C₂²⋊C₄×C₂₃	—	4	2	no	—
87	C₂²⋊C₄×C₂₃	—	4	2	no	—
88	C₂²⋊C₄×C₂₃	—	4	2	no	—
89	C₂²⋊C₄×C₂₃	C₄⋊C₄	1	4	yes	—
90	C₄²×C₂₃	—	2	2	no	—
91	C₄²×C₂₃	—	2	2	no	—
92	C₄²×C₂₃	—	2	4	no	—
93	C₄²×C₂₃	—	4	2	no	—
94	C₄²×C₂₃	—	4	2	no	—
95	C₄²×C₂₃	—	4	2	no	—
96	#11236	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₂₃	—	4	2	no	—
98	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₂₃	#121	1	1	yes	characteristic
100	C₂³×C₂₃	#121	1	2	yes	—
101	C₂³×C₂₃	#121	1	1	yes	characteristic
102	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	8	4	no	—
108	C₂×C₄×C₂₃	—	8	4	no	—
109	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₂₃	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#121	1	2	yes	—
115	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
120	C₂×C₄×C₂₃	#123	1	2	yes	—
121	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
122	C₂×C₄×C₂₃	#125	1	2	yes	—
123	C₂×C₄×C₂₃	#125	1	2	yes	—
124	C₂×C₄×C₂₃	#126	1	2	yes	—
125	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
127	C₈×C₂₃	—	2	4	no	—
128	C₈×C₂₃	—	4	2	no	—
129	#2911	C₂×C₂₃	1	1	yes	characteristic
130	C₂×#424	C₂×C₂₃	1	1	yes	characteristic
131	#1740	C₂×C₂₃	1	2	yes	—
132	#1735	C₂×C₂₃	1	2	yes	—
133	C₂×H(3,ℤ/4)	C₂×C₂₃	1	1	yes	characteristic
134	C₂²×C₂₃	—	2	2	no	—
135	C₂²×C₂₃	—	2	4	no	—
136	C₂²×C₂₃	—	4	1	no	—
137	C₂²×C₂₃	—	4	1	no	—
138	C₂²×C₂₃	—	4	2	no	—
139	C₂²×C₂₃	#380	1	1	yes	characteristic
140	C₂²×C₂₃	#389	1	2	yes	—
141	C₂²×C₂₃	#394	1	1	yes	characteristic
142	C₂²×C₂₃	#396	1	2	yes	—
143	C₂²×C₂₃	#398	1	1	yes	characteristic
144	C₄×C₂₃	—	4	4	no	—
145	C₄×C₂₃	—	8	4	no	—
146	C₄×C₂₃	—	8	4	no	—
147	C₄×C₂₃	#386	1	1	yes	characteristic
148	C₄×C₂₃	#387	1	2	yes	—
149	C₄×C₂₃	#391	1	1	yes	characteristic
150	C₄×C₂₃	#395	1	1	yes	characteristic
151	C₄×C₂₃	#398	1	2	yes	—
152	C₄×C₂₃	#400	1	1	yes	characteristic
153	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
154	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
155	C₂×#117	C₂²×C₂₃	1	2	yes	—
156	C₂×#117	C₂²×C₂₃	1	2	yes	—
157	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
158	#428	—	2	4	no	—
159	#427	—	2	2	no	—
160	#424	—	2	2	no	—
161	#424	C₄×C₂₃	1	4	yes	complemented
162	C₂×C₄⋊C₈	—	2	2	no	—
163	#342	—	2	4	no	—
164	#340	—	2	4	no	—
165	H(3,ℤ/4)	—	2	2	no	—
166	C₂×C₂₃	—	2	1	no	—
167	C₂×C₂₃	—	2	2	no	—
168	C₂×C₂₃	—	2	2	no	—
169	C₂×C₂₃	—	2	1	no	—
170	C₂×C₂₃	#1441	1	1	yes	characteristic
171	C₂×C₂₃	#2053	1	2	yes	—
172	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
176	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
177	C₂²×C₈	—	2	2	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
182	C₂²×C₈	D₄×C₂₃	1	2	yes	—
183	#117	—	2	1	no	—
184	#117	—	2	1	no	—
185	#117	—	2	2	no	—
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	2	yes	—
193	#117	Q₈×C₂₃	1	4	yes	—
194	#117	Q₈×C₂₃	1	2	yes	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	D₄×C₂₃	1	4	yes	—
198	C₂×C₂²⋊C₄	—	2	2	no	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
201	C₂×C₄²	—	2	1	no	—
202	C₂×C₄²	—	2	2	no	—
203	C₂×C₄²	—	2	2	no	—
204	#107	—	2	4	no	—
205	#106	—	2	4	no	—
206	C₄⋊C₈	—	2	8	no	—
207	C₂₃	#11236	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
216	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
217	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
220	C₂²×C₄	#41×C₂₃	1	2	yes	—
221	C₂²×C₄	#41×C₂₃	1	2	yes	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	2	2	no	—
239	C₂²⋊C₄	—	2	4	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₂²⋊C₄	C₄⋊C₄×C₂₃	1	4	yes	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	4	2	no	—
248	C₄²	—	4	2	no	—
249	C₄²	—	4	2	no	—
250	C₂³	—	4	2	no	—
251	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
252	C₂³	C₂₃×#121	1	1	yes	characteristic
253	C₂³	C₂₃×#121	1	2	yes	—
254	C₂³	C₂₃×#121	1	1	yes	characteristic
255	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
263	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
264	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
265	C₂×C₄	C₄×Q₈×C₂₃	1	2	yes	—
266	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
267	C₂×C₄	C₂₃×#121	1	2	yes	—
268	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
269	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
270	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
271	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
272	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
273	C₂×C₄	C₂₃×#123	1	2	yes	—
274	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
275	C₂×C₄	C₂₃×#125	1	2	yes	—
276	C₂×C₄	C₂₃×#125	1	2	yes	—
277	C₂×C₄	C₂₃×#126	1	2	yes	—
278	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
279	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
280	C₈	—	2	4	no	—
281	C₈	—	4	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	C₂₃×#380	1	1	yes	characteristic
288	C₂²	C₂₃×#389	1	2	yes	—
289	C₂²	C₂₃×#394	1	1	yes	characteristic
290	C₂²	C₂₃×#396	1	2	yes	—
291	C₂²	C₂₃×#398	1	1	yes	characteristic
292	C₄	—	4	4	no	—
293	C₄	—	8	4	no	—
294	C₄	—	8	4	no	—
295	C₄	C₂₃×#386	1	1	yes	characteristic
296	C₄	C₂₃×#387	1	2	yes	—
297	C₄	C₂₃×#391	1	1	yes	characteristic
298	C₄	C₂₃×#395	1	1	yes	characteristic
299	C₄	C₂₃×#398	1	2	yes	—
300	C₄	C₂₃×#400	1	1	yes	characteristic
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	—	2	2	no	—
304	C₂	—	2	1	no	—
305	C₂	C₂₃×#1441	1	1	yes	characteristic
306	C₂	C₂₃×#2053	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#124
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	#394
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
14. C₂₃×#424	C₄	semidirect
96. #11236	C₂₃	direct (N × H)
161. #424	C₄×C₂₃	semidirect
207. C₂₃	#11236	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×#2911	2
C₂×C₂₃×#424	2
C₂₃×#1740	2
C₂₃×#1735	2
C₂×C₂₃×H(3,ℤ/4)	2
#11236	23

Subgroups — C23×#11236 #19158862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11236 #19158862