Subgroups — C₂₃×#19931 #19167557

Order 5888.

554 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×#3387	C₂	1	2	yes	maximal
2	C₂₃×#3363	C₂	1	2	yes	maximal
3	C₂₃×#1707	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1692	C₂	1	2	yes	maximal
5	C₂₃×#1684	C₂	1	2	yes	maximal
6	C₂₃×#1648	C₂	1	2	yes	maximal
7	C₂₃×#1633	C₂	1	2	yes	maximal
8	C₂₃×#1591	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
10	Q₈²×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#557	C₂²	1	2	yes	—
12	C₂₃×#501	—	2	2	no	—
13	C₂₃×#500	C₂²	1	1	yes	characteristic
14	C₂₃×#498	C₂²	1	2	yes	—
15	C₂₃×#494	C₂²	1	1	yes	characteristic
16	C₂₃×#487	—	2	2	no	—
17	C₂₃×#487	C₂²	1	2	yes	—
18	C₂₃×#479	—	2	2	no	—
19	C₂₃×#477	—	2	2	no	—
20	C₂₃×#477	C₂²	1	2	yes	—
21	C₂₃×#477	C₂²	1	2	yes	—
22	C₂₃×#475	—	2	2	no	—
23	C₂₃×#475	C₂²	1	2	yes	—
24	C₂₃×#462	—	2	2	no	—
25	C₂₃×#462	C₂²	1	2	yes	—
26	C₂₃×#446	C₂²	1	2	yes	—
27	C₈×Q₈×C₂₃	C₂²	1	2	yes	—
28	C₂₃×#441	—	2	2	no	—
29	C₄×Q₁₆×C₂₃	—	2	2	no	—
30	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
31	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
32	C₂₃×#335	C₂²	1	1	yes	characteristic
33	C₂₃×#335	C₂²	1	1	yes	characteristic
34	C₂₃×#333	C₂²	1	2	yes	—
35	C₂₃×#332	C₂²	1	2	yes	—
36	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₂₃×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₂₃	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆×C₂₃	—	2	2	no	—
40	C₂×Q₁₆×C₂₃	—	4	1	no	—
41	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
42	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
43	C₂₃×C₄⋊Q₈	—	2	2	no	—
44	C₂₃×C₄⋊Q₈	—	2	2	no	—
45	C₂₃×C₄⋊Q₈	—	2	2	no	—
46	C₂₃×C₄⋊Q₈	—	4	2	no	—
47	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
49	C₂₃×#125	—	2	2	no	—
50	C₂₃×#125	—	4	2	no	—
51	C₄×Q₈×C₂₃	—	2	2	no	—
52	C₄×Q₈×C₂₃	—	2	2	no	—
53	C₄×Q₈×C₂₃	—	2	2	no	—
54	C₄×Q₈×C₂₃	—	2	2	no	—
55	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
56	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
57	C₂₃×#107	—	2	2	no	—
58	C₂₃×#107	—	2	2	no	—
59	C₂₃×#107	—	4	1	no	—
60	C₂₃×#107	D₄	1	1	yes	characteristic
61	C₂₃×#107	D₄	1	1	yes	characteristic
62	C₂₃×#106	—	2	2	no	—
63	C₂₃×#106	—	2	2	no	—
64	C₂₃×#106	—	2	2	no	—
65	C₂₃×C₄⋊C₈	—	2	2	no	—
66	C₂₃×C₄⋊C₈	—	2	2	no	—
67	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
68	C₂₃×C₄⋊C₈	C₂³	1	2	yes	—
69	C₂₃×#103	—	2	2	no	—
70	C₂₃×#103	—	2	2	no	—
71	C₂₃×#103	—	2	2	no	—
72	C₂₃×#103	—	2	2	no	—
73	C₂₃×#103	—	2	2	no	—
74	C₂₃×#103	—	2	2	no	—
75	C₂₃×#103	D₄	1	2	yes	—
76	C₂₃×#103	D₄	1	2	yes	—
77	C₂₃×#97	—	2	2	no	—
78	C₄×C₈×C₂₃	—	2	2	no	—
79	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
80	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
82	C₂×Q₈×C₂₃	—	2	2	no	—
83	C₂×Q₈×C₂₃	—	2	2	no	—
84	C₂×Q₈×C₂₃	—	4	2	no	—
85	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
86	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
87	Q₁₆×C₂₃	—	4	1	no	—
88	Q₁₆×C₂₃	—	4	4	no	—
89	Q₁₆×C₂₃	—	8	2	no	—
90	Q₁₆×C₂₃	Q₁₆	1	4	yes	complemented
91	C₂×C₈×C₂₃	—	2	1	no	—
92	C₂×C₈×C₂₃	—	2	2	no	—
93	C₂×C₈×C₂₃	—	2	2	no	—
94	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₄⋊C₄×C₂₃	—	2	2	no	—
99	C₄⋊C₄×C₂₃	—	2	2	no	—
100	C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₄⋊C₄×C₂₃	—	2	2	no	—
102	C₄⋊C₄×C₂₃	—	2	2	no	—
103	C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₄⋊C₄×C₂₃	—	4	2	no	—
105	C₄⋊C₄×C₂₃	—	4	2	no	—
106	C₄⋊C₄×C₂₃	—	4	2	no	—
107	C₄⋊C₄×C₂₃	—	4	2	no	—
108	C₄⋊C₄×C₂₃	—	4	2	no	—
109	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
110	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
111	C₄²×C₂₃	—	2	2	no	—
112	C₄²×C₂₃	—	2	2	no	—
113	C₄²×C₂₃	—	4	2	no	—
114	C₄²×C₂₃	—	4	2	no	—
115	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
116	#19931	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
117	Q₈×C₂₃	—	2	2	no	—
118	Q₈×C₂₃	—	2	2	no	—
119	Q₈×C₂₃	—	2	2	no	—
120	Q₈×C₂₃	—	4	2	no	—
121	Q₈×C₂₃	—	4	2	no	—
122	Q₈×C₂₃	—	8	2	no	—
123	Q₈×C₂₃	—	16	2	no	—
124	Q₈×C₂₃	C₂×Q₁₆	1	4	yes	—
125	C₂×C₄×C₂₃	—	2	2	no	—
126	C₂×C₄×C₂₃	—	2	2	no	—
127	C₂×C₄×C₂₃	—	2	2	no	—
128	C₂×C₄×C₂₃	—	2	2	no	—
129	C₂×C₄×C₂₃	—	4	2	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
134	C₈×C₂₃	—	2	2	no	—
135	C₈×C₂₃	—	2	1	no	—
136	C₈×C₂₃	—	2	1	no	—
137	C₈×C₂₃	—	2	1	no	—
138	C₈×C₂₃	—	4	2	no	—
139	C₈×C₂₃	—	4	2	no	—
140	C₈×C₂₃	C₂×Q₁₆	1	2	yes	—
141	#3387	C₂×C₂₃	1	2	yes	—
142	#3363	C₂×C₂₃	1	2	yes	—
143	#1707	C₂×C₂₃	1	1	yes	characteristic
144	#1692	C₂×C₂₃	1	2	yes	—
145	#1684	C₂×C₂₃	1	2	yes	—
146	#1648	C₂×C₂₃	1	2	yes	—
147	#1633	C₂×C₂₃	1	2	yes	—
148	#1591	C₂×C₂₃	1	1	yes	characteristic
149	C₈×Q₁₆	C₂×C₂₃	1	1	yes	characteristic
150	C₂²×C₂₃	D₄²	1	1	yes	characteristic
151	C₄×C₂₃	—	2	4	no	—
152	C₄×C₂₃	—	2	1	no	—
153	C₄×C₂₃	—	4	2	no	—
154	C₄×C₂₃	—	4	2	no	—
155	C₄×C₂₃	—	4	2	no	—
156	C₄×C₂₃	—	8	2	no	—
157	C₄×C₂₃	—	8	2	no	—
158	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
159	C₄×C₂₃	#575	1	1	yes	characteristic
160	C₄×C₂₃	#578	1	1	yes	characteristic
161	C₄×C₂₃	#578	1	1	yes	characteristic
162	Q₈²	C₂²×C₂₃	1	2	yes	—
163	#557	C₂²×C₂₃	1	2	yes	—
164	#501	—	2	2	no	—
165	#500	C₂²×C₂₃	1	1	yes	characteristic
166	#498	C₂²×C₂₃	1	2	yes	—
167	#494	C₂²×C₂₃	1	1	yes	characteristic
168	#487	—	2	2	no	—
169	#487	C₂²×C₂₃	1	2	yes	—
170	#479	—	2	2	no	—
171	#477	—	2	2	no	—
172	#477	C₂²×C₂₃	1	2	yes	—
173	#477	C₂²×C₂₃	1	2	yes	—
174	#475	—	2	2	no	—
175	#475	C₂²×C₂₃	1	2	yes	—
176	#462	—	2	2	no	—
177	#462	C₂²×C₂₃	1	2	yes	—
178	#446	C₂²×C₂₃	1	2	yes	—
179	C₈×Q₈	C₂²×C₂₃	1	2	yes	—
180	#441	—	2	2	no	—
181	C₄×Q₁₆	—	2	2	no	—
182	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
183	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
184	#335	C₂²×C₂₃	1	1	yes	characteristic
185	#335	C₂²×C₂₃	1	1	yes	characteristic
186	#333	C₂²×C₂₃	1	2	yes	—
187	#332	C₂²×C₂₃	1	2	yes	—
188	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
189	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
190	C₈²	C₂²×C₂₃	1	1	yes	characteristic
191	C₂×C₂₃	#3277	1	1	yes	characteristic
192	C₂×C₂₃	D₄×Q₁₆	1	1	yes	characteristic
193	C₂×C₂₃	#3291	1	1	yes	characteristic
194	C₂×Q₁₆	—	2	2	no	—
195	C₂×Q₁₆	—	4	1	no	—
196	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
197	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	4	2	no	—
202	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
203	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
204	#125	—	2	2	no	—
205	#125	—	4	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
211	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
212	#107	—	2	2	no	—
213	#107	—	2	2	no	—
214	#107	—	4	1	no	—
215	#107	D₄×C₂₃	1	1	yes	characteristic
216	#107	D₄×C₂₃	1	1	yes	characteristic
217	#106	—	2	2	no	—
218	#106	—	2	2	no	—
219	#106	—	2	2	no	—
220	C₄⋊C₈	—	2	2	no	—
221	C₄⋊C₈	—	2	2	no	—
222	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
223	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
224	#103	—	2	2	no	—
225	#103	—	2	2	no	—
226	#103	—	2	2	no	—
227	#103	—	2	2	no	—
228	#103	—	2	2	no	—
229	#103	—	2	2	no	—
230	#103	D₄×C₂₃	1	2	yes	—
231	#103	D₄×C₂₃	1	2	yes	—
232	#97	—	2	2	no	—
233	C₄×C₈	—	2	2	no	—
234	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
235	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
236	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
237	C₂₃	#19931	1	1	yes	characteristic, direct product factor, Sylow
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
242	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
243	Q₁₆	—	4	1	no	—
244	Q₁₆	—	4	4	no	—
245	Q₁₆	—	8	2	no	—
246	Q₁₆	Q₁₆×C₂₃	1	4	yes	complemented
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
253	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
266	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	4	2	no	—
271	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
272	Q₈	—	2	2	no	—
273	Q₈	—	2	2	no	—
274	Q₈	—	2	2	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	8	2	no	—
278	Q₈	—	16	2	no	—
279	Q₈	C₂×Q₁₆×C₂₃	1	4	yes	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
288	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	4	2	no	—
294	C₈	—	4	2	no	—
295	C₈	C₂×Q₁₆×C₂₃	1	2	yes	—
296	C₂²	D₄²×C₂₃	1	1	yes	characteristic
297	C₄	—	2	4	no	—
298	C₄	—	2	1	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
305	C₄	C₂₃×#575	1	1	yes	characteristic
306	C₄	C₂₃×#578	1	1	yes	characteristic
307	C₄	C₂₃×#578	1	1	yes	characteristic
308	C₂	C₂₃×#3277	1	1	yes	characteristic
309	C₂	D₄×Q₁₆×C₂₃	1	1	yes	characteristic
310	C₂	C₂₃×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
90. Q₁₆×C₂₃	Q₁₆	semidirect
116. #19931	C₂₃	direct (N × H)
237. C₂₃	#19931	direct (N × H)
246. Q₁₆	Q₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3387	2
C₂₃×#3363	2
C₂₃×#1707	2
C₂₃×#1692	2
C₂₃×#1684	2
C₂₃×#1648	2
C₂₃×#1633	2
C₂₃×#1591	2
C₈×Q₁₆×C₂₃	2
#19931	23

Subgroups — C23×#19931 #19167557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19931 #19167557