Subgroups — C₂₃×#11343 #19158969

Order 5888.

666 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#430	C₂	1	2	yes	maximal
2	C₂₃×#2905	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1742	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#341	C₂	1	2	yes	maximal
5	C₂₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
11	C₂₃×#430	—	2	4	no	—
12	C₂₃×#430	—	2	4	no	—
13	C₂×C₂₃×#108	—	2	4	no	—
14	C₂₃×#424	—	2	4	no	—
15	C₂₃×#423	—	2	2	no	—
16	C₂₃×#343	C₄	1	4	yes	—
17	C₂₃×#341	—	2	4	no	—
18	C₂₃×#336	—	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	—	2	2	no	—
23	C₂×M₁₆×C₂₃	—	2	2	no	—
24	C₂×M₁₆×C₂₃	—	2	1	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	2	2	no	—
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	—	2	2	no	—
30	C₂×M₁₆×C₂₃	—	2	2	no	—
31	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
32	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
33	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
34	C₂×M₁₆×C₂₃	D₄	1	4	yes	—
35	C₂×M₁₆×C₂₃	Q₈	1	4	yes	—
36	C₂²×C₈×C₂₃	—	2	2	no	—
37	C₂²×C₈×C₂₃	—	2	1	no	—
38	C₂²×C₈×C₂₃	—	2	2	no	—
39	C₂²×C₈×C₂₃	—	2	1	no	—
40	C₂²×C₈×C₂₃	—	2	2	no	—
41	C₂²×C₈×C₂₃	—	2	1	no	—
42	C₂²×C₈×C₂₃	—	2	2	no	—
43	C₂²×C₈×C₂₃	Q₈	1	2	yes	—
44	C₂₃×#117	—	2	2	no	—
45	C₂₃×#117	—	2	2	no	—
46	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₂₃	Q₈	1	2	yes	—
48	C₂×C₄²×C₂₃	D₄	1	2	yes	—
49	C₂₃×#108	—	2	16	no	—
50	C₂₃×C₄⋊C₈	—	2	8	no	—
51	C₂⁴×C₂₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×Q₈	1	2	yes	—
59	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	#41	1	2	yes	—
62	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
63	M₁₆×C₂₃	—	2	4	no	—
64	M₁₆×C₂₃	—	2	4	no	—
65	M₁₆×C₂₃	—	2	4	no	—
66	M₁₆×C₂₃	—	2	4	no	—
67	M₁₆×C₂₃	—	2	4	no	—
68	M₁₆×C₂₃	—	2	4	no	—
69	M₁₆×C₂₃	—	4	2	no	—
70	M₁₆×C₂₃	—	4	2	no	—
71	M₁₆×C₂₃	—	4	2	no	—
72	M₁₆×C₂₃	C₄⋊C₄	1	8	yes	—
73	C₂×C₈×C₂₃	—	2	4	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	—	2	4	no	—
76	C₂×C₈×C₂₃	—	2	4	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	4	no	—
79	C₂×C₈×C₂₃	—	2	2	no	—
80	C₂×C₈×C₂₃	—	4	2	no	—
81	C₂×C₈×C₂₃	—	4	1	no	—
82	C₂×C₈×C₂₃	—	4	2	no	—
83	C₂×C₈×C₂₃	—	4	2	no	—
84	C₂×C₈×C₂₃	—	4	1	no	—
85	C₂×C₈×C₂₃	—	4	1	no	—
86	C₂×C₈×C₂₃	—	4	1	no	—
87	C₄⋊C₄×C₂₃	—	2	4	no	—
88	C₂²⋊C₄×C₂₃	—	2	4	no	—
89	C₄²×C₂₃	—	2	2	no	—
90	C₄²×C₂₃	—	2	2	no	—
91	#11343	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₂₃	—	4	2	no	—
93	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₂₃	C₂²⋊Q₈	1	2	yes	—
95	C₂³×C₂₃	#125	1	1	yes	characteristic
96	C₂³×C₂₃	C₄⋊Q₈	1	2	yes	—
97	C₂³×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	4	4	no	—
100	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₂₃	#120	1	2	yes	—
105	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
106	C₂×C₄×C₂₃	#121	1	2	yes	—
107	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#123	1	2	yes	—
113	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#125	1	2	yes	—
115	C₂×C₄×C₂₃	#127	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
118	C₈×C₂₃	—	2	8	no	—
119	C₈×C₂₃	—	4	4	no	—
120	C₈×C₂₃	—	4	2	no	—
121	C₈×C₂₃	—	4	2	no	—
122	C₂×#430	C₂×C₂₃	1	2	yes	—
123	#2905	C₂×C₂₃	1	1	yes	characteristic
124	#1742	C₂×C₂₃	1	1	yes	characteristic
125	C₂×#341	C₂×C₂₃	1	2	yes	—
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	#384	1	1	yes	characteristic
133	C₂²×C₂₃	#385	1	1	yes	characteristic
134	C₂²×C₂₃	#389	1	1	yes	characteristic
135	C₂²×C₂₃	#395	1	2	yes	—
136	C₂²×C₂₃	#398	1	2	yes	—
137	C₄×C₂₃	—	4	4	no	—
138	C₄×C₂₃	#385	1	2	yes	—
139	C₄×C₂₃	#389	1	2	yes	—
140	C₄×C₂₃	#392	1	1	yes	characteristic
141	C₄×C₂₃	#395	1	1	yes	characteristic
142	C₄×C₂₃	#398	1	1	yes	characteristic
143	C₄×C₂₃	#399	1	1	yes	characteristic
144	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
145	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
146	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
147	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
149	#430	—	2	4	no	—
150	#430	—	2	4	no	—
151	C₂×#108	—	2	4	no	—
152	#424	—	2	4	no	—
153	#423	—	2	2	no	—
154	#343	C₄×C₂₃	1	4	yes	—
155	#341	—	2	4	no	—
156	#336	—	2	2	no	—
157	C₂×C₂₃	—	2	1	no	—
158	C₂×C₂₃	—	2	1	no	—
159	C₂×C₂₃	—	2	2	no	—
160	C₂×C₂₃	—	2	2	no	—
161	C₂×C₂₃	#1444	1	1	yes	characteristic
162	C₂×C₂₃	#1858	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	—	2	2	no	—
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	1	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
176	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
177	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
178	C₂×M₁₆	D₄×C₂₃	1	4	yes	—
179	C₂×M₁₆	Q₈×C₂₃	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	Q₈×C₂₃	1	2	yes	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
191	C₂×C₂²⋊C₄	Q₈×C₂₃	1	2	yes	—
192	C₂×C₄²	D₄×C₂₃	1	2	yes	—
193	#108	—	2	16	no	—
194	C₄⋊C₈	—	2	8	no	—
195	C₂₃	#11343	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	C₂×Q₈×C₂₃	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
198	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	#41×C₂₃	1	2	yes	—
207	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
208	M₁₆	—	2	4	no	—
209	M₁₆	—	2	4	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	C₄⋊C₄×C₂₃	1	8	yes	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	4	1	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
238	C₂³	C₂₃×C₂²⋊Q₈	1	2	yes	—
239	C₂³	C₂₃×#125	1	1	yes	characteristic
240	C₂³	C₂₃×C₄⋊Q₈	1	2	yes	—
241	C₂³	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
245	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
247	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
248	C₂×C₄	C₂₃×#120	1	2	yes	—
249	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
250	C₂×C₄	C₂₃×#121	1	2	yes	—
251	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
252	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
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	2	yes	—
257	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
258	C₂×C₄	C₂₃×#125	1	2	yes	—
259	C₂×C₄	C₂₃×#127	1	1	yes	characteristic
260	C₂×C₄	C₂₃×C₄⋊Q₈	1	2	yes	—
261	C₂×C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
262	C₈	—	2	8	no	—
263	C₈	—	4	4	no	—
264	C₈	—	4	2	no	—
265	C₈	—	4	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₂₃×#384	1	1	yes	characteristic
272	C₂²	C₂₃×#385	1	1	yes	characteristic
273	C₂²	C₂₃×#389	1	1	yes	characteristic
274	C₂²	C₂₃×#395	1	2	yes	—
275	C₂²	C₂₃×#398	1	2	yes	—
276	C₄	—	4	4	no	—
277	C₄	C₂₃×#385	1	2	yes	—
278	C₄	C₂₃×#389	1	2	yes	—
279	C₄	C₂₃×#392	1	1	yes	characteristic
280	C₄	C₂₃×#395	1	1	yes	characteristic
281	C₄	C₂₃×#398	1	1	yes	characteristic
282	C₄	C₂₃×#399	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	2	no	—
286	C₂	—	2	2	no	—
287	C₂	C₂₃×#1444	1	1	yes	characteristic
288	C₂	C₂₃×#1858	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#120
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	#385
prime radical	C₂⁴×C₂₃	C₂×Q₈
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
91. #11343	C₂₃	direct (N × H)
195. C₂₃	#11343	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#430	2
C₂₃×#2905	2
C₂₃×#1742	2
C₂×C₂₃×#341	2
C₂₃×#1729	2
#11343	23

Subgroups — C23×#11343 #19158969

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#11343 #19158969