Subgroups — C₂₃×#11285 #19158911

Order 5888.

606 conjugacy classes of subgroups.

The subgroup lattice has 294 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×#1740	C₂	1	2	yes	maximal
3	C₂₃×#1735	C₂	1	2	yes	maximal
4	C₂×C₂₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#1729	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×#342	—	2	4	no	—
14	C₂₃×#340	—	2	4	no	—
15	C₂₃×H(3,ℤ/4)	C₄	1	4	yes	complemented
16	C₂₃×#336	—	2	2	no	—
17	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
18	C₂×M₁₆×C₂₃	—	2	2	no	—
19	C₂×M₁₆×C₂₃	—	2	2	no	—
20	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
21	C₂×M₁₆×C₂₃	Q₈	1	2	yes	—
22	C₂²×C₈×C₂₃	—	2	1	no	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	2	1	no	—
26	C₂²×C₈×C₂₃	D₄	1	2	yes	—
27	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
28	C₂₃×#117	—	2	2	no	—
29	C₂₃×#117	—	2	2	no	—
30	C₂₃×#117	—	2	2	no	—
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	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₂₃	D₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
44	C₂×C₄²×C₂₃	—	2	2	no	—
45	C₂×C₄²×C₂₃	—	2	1	no	—
46	C₂×C₄²×C₂₃	—	2	2	no	—
47	C₂₃×#107	—	2	4	no	—
48	C₂₃×#106	—	2	4	no	—
49	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	—	2	2	no	—
51	C₂²×C₄×C₂₃	—	2	4	no	—
52	C₂²×C₄×C₂₃	—	4	2	no	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	#41	1	2	yes	—
59	C₂²×C₄×C₂₃	#41	1	2	yes	—
60	C₂²×C₄×C₂₃	#41	1	2	yes	—
61	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
63	M₁₆×C₂₃	—	2	4	no	—
64	M₁₆×C₂₃	—	2	4	no	—
65	C₂×C₈×C₂₃	—	2	2	no	—
66	C₂×C₈×C₂₃	—	2	2	no	—
67	C₂×C₈×C₂₃	—	2	4	no	—
68	C₂×C₈×C₂₃	—	2	2	no	—
69	C₂×C₈×C₂₃	—	4	1	no	—
70	C₂×C₈×C₂₃	—	4	1	no	—
71	C₂×C₈×C₂₃	—	4	2	no	—
72	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
73	C₄⋊C₄×C₂₃	—	2	4	no	—
74	C₄⋊C₄×C₂₃	—	2	4	no	—
75	C₄⋊C₄×C₂₃	—	2	4	no	—
76	C₄⋊C₄×C₂₃	—	4	2	no	—
77	C₄⋊C₄×C₂₃	—	4	2	no	—
78	C₂²⋊C₄×C₂₃	—	2	4	no	—
79	C₂²⋊C₄×C₂₃	—	2	4	no	—
80	C₂²⋊C₄×C₂₃	—	4	2	no	—
81	C₂²⋊C₄×C₂₃	—	4	2	no	—
82	C₂²⋊C₄×C₂₃	—	4	2	no	—
83	C₄²×C₂₃	—	2	4	no	—
84	C₄²×C₂₃	—	4	2	no	—
85	C₄²×C₂₃	—	4	1	no	—
86	C₄²×C₂₃	—	4	1	no	—
87	C₄²×C₂₃	—	4	2	no	—
88	C₄²×C₂₃	—	4	2	no	—
89	#11285	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	4	2	no	—
91	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
92	C₂³×C₂₃	#121	1	2	yes	—
93	C₂³×C₂₃	#121	1	1	yes	characteristic
94	C₂³×C₂₃	#123	1	2	yes	—
95	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₂₃	—	4	4	no	—
97	C₂×C₄×C₂₃	—	4	4	no	—
98	C₂×C₄×C₂₃	—	4	4	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	8	4	no	—
101	C₂×C₄×C₂₃	—	8	4	no	—
102	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₃	#117	1	2	yes	—
104	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
111	C₂×C₄×C₂₃	#123	1	2	yes	—
112	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#124	1	2	yes	—
114	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#125	1	2	yes	—
116	C₂×C₄×C₂₃	#125	1	2	yes	—
117	C₂×C₄×C₂₃	#125	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#126	1	2	yes	—
119	C₂×C₄×C₂₃	#126	1	2	yes	—
120	C₈×C₂₃	—	2	4	no	—
121	C₈×C₂₃	—	4	2	no	—
122	#2911	C₂×C₂₃	1	1	yes	characteristic
123	#1740	C₂×C₂₃	1	2	yes	—
124	#1735	C₂×C₂₃	1	2	yes	—
125	C₂×H(3,ℤ/4)	C₂×C₂₃	1	1	yes	characteristic
126	#1729	C₂×C₂₃	1	1	yes	characteristic
127	C₂²×C₂₃	—	2	2	no	—
128	C₂²×C₂₃	—	2	4	no	—
129	C₂²×C₂₃	—	4	1	no	—
130	C₂²×C₂₃	—	4	2	no	—
131	C₂²×C₂₃	—	4	1	no	—
132	C₂²×C₂₃	#388	1	2	yes	—
133	C₂²×C₂₃	#389	1	1	yes	characteristic
134	C₂²×C₂₃	#394	1	1	yes	characteristic
135	C₂²×C₂₃	#397	1	1	yes	characteristic
136	C₂²×C₂₃	#398	1	2	yes	—
137	C₄×C₂₃	—	8	2	no	—
138	C₄×C₂₃	—	8	4	no	—
139	C₄×C₂₃	—	8	4	no	—
140	C₄×C₂₃	#382	1	1	yes	characteristic
141	C₄×C₂₃	#385	1	1	yes	characteristic
142	C₄×C₂₃	#387	1	2	yes	—
143	C₄×C₂₃	#393	1	1	yes	characteristic
144	C₄×C₂₃	#400	1	2	yes	—
145	C₄×C₂₃	#400	1	1	yes	characteristic
146	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
147	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×#117	C₂²×C₂₃	1	2	yes	—
149	C₂×#117	C₂²×C₂₃	1	2	yes	—
150	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
151	#428	—	2	4	no	—
152	#427	—	2	2	no	—
153	#342	—	2	4	no	—
154	#340	—	2	4	no	—
155	H(3,ℤ/4)	C₄×C₂₃	1	4	yes	complemented
156	#336	—	2	2	no	—
157	C₂×C₂₃	—	2	2	no	—
158	C₂×C₂₃	—	2	1	no	—
159	C₂×C₂₃	—	2	1	no	—
160	C₂×C₂₃	—	2	2	no	—
161	C₂×C₂₃	#1442	1	1	yes	characteristic
162	C₂×C₂₃	#2053	1	2	yes	—
163	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
167	C₂×M₁₆	Q₈×C₂₃	1	2	yes	—
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	D₄×C₂₃	1	2	yes	—
173	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
174	#117	—	2	2	no	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	—	2	2	no	—
182	#117	—	2	2	no	—
183	#117	C₂×C₄×C₂₃	1	4	yes	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	D₄×C₂₃	1	4	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	1	no	—
192	C₂×C₄²	—	2	2	no	—
193	#107	—	2	4	no	—
194	#106	—	2	4	no	—
195	C₂₃	#11285	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
203	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	#41×C₂₃	1	2	yes	—
206	C₂²×C₄	#41×C₂₃	1	2	yes	—
207	C₂²×C₄	#41×C₂₃	1	2	yes	—
208	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₄²	—	2	4	no	—
231	C₄²	—	4	2	no	—
232	C₄²	—	4	1	no	—
233	C₄²	—	4	2	no	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
238	C₂³	C₂₃×#121	1	2	yes	—
239	C₂³	C₂₃×#121	1	1	yes	characteristic
240	C₂³	C₂₃×#123	1	2	yes	—
241	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	8	4	no	—
247	C₂×C₄	—	8	4	no	—
248	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂₃×#117	1	2	yes	—
250	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
251	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
253	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
254	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
255	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
256	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
257	C₂×C₄	C₂₃×#123	1	2	yes	—
258	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
259	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
260	C₂×C₄	C₂₃×#124	1	2	yes	—
261	C₂×C₄	C₂₃×#125	1	2	yes	—
262	C₂×C₄	C₂₃×#125	1	2	yes	—
263	C₂×C₄	C₂₃×#125	1	1	yes	characteristic
264	C₂×C₄	C₂₃×#126	1	2	yes	—
265	C₂×C₄	C₂₃×#126	1	2	yes	—
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	C₂₃×#388	1	2	yes	—
274	C₂²	C₂₃×#389	1	1	yes	characteristic
275	C₂²	C₂₃×#394	1	1	yes	characteristic
276	C₂²	C₂₃×#397	1	1	yes	characteristic
277	C₂²	C₂₃×#398	1	2	yes	—
278	C₄	—	8	2	no	—
279	C₄	—	8	4	no	—
280	C₄	—	8	4	no	—
281	C₄	C₂₃×#382	1	1	yes	characteristic
282	C₄	C₂₃×#385	1	1	yes	characteristic
283	C₄	C₂₃×#387	1	2	yes	—
284	C₄	C₂₃×#393	1	1	yes	characteristic
285	C₄	C₂₃×#400	1	2	yes	—
286	C₄	C₂₃×#400	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	C₂₃×#1442	1	1	yes	characteristic
292	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
15. C₂₃×H(3,ℤ/4)	C₄	semidirect
89. #11285	C₂₃	direct (N × H)
155. H(3,ℤ/4)	C₄×C₂₃	semidirect
195. C₂₃	#11285	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
C₂₃×#1740	2
C₂₃×#1735	2
C₂×C₂₃×H(3,ℤ/4)	2
C₂₃×#1729	2
#11285	23

Subgroups — C23×#11285 #19158911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11285 #19158911