Subgroups — C₂₃×#19927 #19167553

Order 5888.

606 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3384	C₂	1	2	yes	maximal
2	C₂₃×#3299	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1692	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1683	C₂	1	2	yes	maximal, complemented
6	C₂₃×#1649	C₂	1	2	yes	maximal, complemented
7	C₂₃×#1632	C₂	1	2	yes	maximal, complemented
8	C₂₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
10	C₂₃×#554	C₂²	1	2	yes	—
11	D₄×Q₈×C₂₃	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	C₂²	1	2	yes	—
17	C₂₃×#484	—	2	2	no	—
18	C₂₃×#478	—	2	2	no	—
19	C₂₃×#477	C₂²	1	2	yes	—
20	C₂₃×#476	—	2	2	no	—
21	C₂₃×#476	C₂²	1	2	yes	—
22	C₂₃×#474	C₂²	1	2	yes	—
23	C₂₃×#467	—	2	2	no	—
24	C₂₃×#462	C₂²	1	2	yes	—
25	C₂₃×#451	—	2	2	no	—
26	C₂₃×#446	C₂²	1	2	yes	—
27	C₂₃×#442	—	2	2	no	—
28	C₄×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₂₃	C₂²	1	1	yes	characteristic
30	C₈×D₄×C₂₃	C₂²	1	2	yes	—
31	C₂×C₂₃×#103	—	2	2	no	—
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×D₄⋊C₈	C₂²	1	2	yes	—
38	C₈²×C₂₃	C₂²	1	1	yes	characteristic
39	C₂²×Q₈×C₂₃	—	2	2	no	—
40	C₂×Q₁₆×C₂₃	—	2	2	no	—
41	C₂×Q₁₆×C₂₃	—	4	1	no	—
42	C₂×Q₁₆×C₂₃	D₄	1	1	yes	characteristic
43	C₂×D₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₂₃	—	2	2	no	—
45	C₂₃×C₄⋊Q₈	—	2	2	no	—
46	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
48	C₂₃×#125	—	2	2	no	—
49	C₂₃×C₂²⋊Q₈	—	2	2	no	—
50	C₂₃×C₂²⋊Q₈	—	2	2	no	—
51	C₂₃×C₂²⋊Q₈	—	4	2	no	—
52	C₂₃×C₂²⋊Q₈	—	4	2	no	—
53	C₄×Q₈×C₂₃	C₂³	1	2	yes	—
54	C₄×D₄×C₂₃	—	2	2	no	—
55	C₄×D₄×C₂₃	—	2	2	no	—
56	C₄×D₄×C₂₃	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
58	C₂₃×#107	—	2	2	no	—
59	C₂₃×#107	—	2	2	no	—
60	C₂₃×#107	—	4	1	no	—
61	C₂₃×#107	D₄	1	1	yes	characteristic
62	C₂₃×#107	D₄	1	1	yes	characteristic
63	C₂₃×#106	—	2	2	no	—
64	C₂₃×#106	—	2	2	no	—
65	C₂₃×#106	—	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	D₄	1	2	yes	—
74	C₂₃×#102	—	2	2	no	—
75	C₂₃×#102	—	2	2	no	—
76	C₂₃×#102	D₄	1	2	yes	—
77	C₂₃×C₂²⋊C₈	—	2	2	no	—
78	C₂₃×#97	—	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	—	4	4	no	—
86	C₂×Q₈×C₂₃	—	8	2	no	—
87	C₂×Q₈×C₂₃	C₂×D₄	1	2	yes	—
88	C₂×D₄×C₂₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₂₃	—	2	2	no	—
90	C₂²×C₄×C₂₃	—	4	2	no	—
91	C₂²×C₄×C₂₃	—	4	2	no	—
92	Q₁₆×C₂₃	—	4	1	no	—
93	Q₁₆×C₂₃	—	8	2	no	—
94	Q₁₆×C₂₃	—	8	2	no	—
95	D₈×C₂₃	Q₁₆	1	4	yes	complemented
96	C₂×C₈×C₂₃	—	2	1	no	—
97	C₂×C₈×C₂₃	—	2	2	no	—
98	C₂×C₈×C₂₃	—	2	2	no	—
99	C₂×C₈×C₂₃	—	8	2	no	—
100	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₂₃	—	2	2	no	—
105	C₄⋊C₄×C₂₃	—	2	2	no	—
106	C₄⋊C₄×C₂₃	—	2	2	no	—
107	C₄⋊C₄×C₂₃	—	2	2	no	—
108	C₄⋊C₄×C₂₃	—	2	2	no	—
109	C₄⋊C₄×C₂₃	—	4	2	no	—
110	C₄⋊C₄×C₂₃	—	4	2	no	—
111	C₄⋊C₄×C₂₃	—	4	2	no	—
112	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
113	C₄⋊C₄×C₂₃	C₂×D₄	1	2	yes	—
114	C₂²⋊C₄×C₂₃	—	2	2	no	—
115	C₂²⋊C₄×C₂₃	—	4	2	no	—
116	C₂²⋊C₄×C₂₃	—	4	2	no	—
117	C₄²×C₂₃	—	2	2	no	—
118	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
119	#19927	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂³×C₂₃	—	2	2	no	—
121	Q₈×C₂₃	—	2	2	no	—
122	Q₈×C₂₃	—	2	2	no	—
123	Q₈×C₂₃	—	4	2	no	—
124	Q₈×C₂₃	—	4	2	no	—
125	Q₈×C₂₃	—	8	2	no	—
126	Q₈×C₂₃	—	8	4	no	—
127	D₄×C₂₃	—	2	2	no	—
128	D₄×C₂₃	C₂×Q₁₆	1	4	yes	—
129	C₂×C₄×C₂₃	—	2	2	no	—
130	C₂×C₄×C₂₃	—	2	2	no	—
131	C₂×C₄×C₂₃	—	2	2	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	4	2	no	—
134	C₂×C₄×C₂₃	—	4	2	no	—
135	C₂×C₄×C₂₃	—	4	2	no	—
136	C₂×C₄×C₂₃	—	8	4	no	—
137	C₂×C₄×C₂₃	—	16	2	no	—
138	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₈×C₂₃	—	2	2	no	—
142	C₈×C₂₃	—	2	1	no	—
143	C₈×C₂₃	—	2	1	no	—
144	C₈×C₂₃	—	2	1	no	—
145	C₈×C₂₃	—	4	2	no	—
146	C₈×C₂₃	—	4	2	no	—
147	C₈×C₂₃	C₂×Q₁₆	1	2	yes	—
148	#3384	C₂×C₂₃	1	2	yes	—
149	#3299	C₂×C₂₃	1	2	yes	complemented
150	#1707	C₂×C₂₃	1	1	yes	characteristic, complemented
151	#1692	C₂×C₂₃	1	2	yes	complemented
152	#1683	C₂×C₂₃	1	2	yes	complemented
153	#1649	C₂×C₂₃	1	2	yes	complemented
154	#1632	C₂×C₂₃	1	2	yes	complemented
155	#1591	C₂×C₂₃	1	1	yes	characteristic, complemented
156	C₈×D₈	C₂×C₂₃	1	1	yes	characteristic
157	C₂²×C₂₃	—	2	4	no	—
158	C₂²×C₂₃	—	4	2	no	—
159	C₂²×C₂₃	—	4	2	no	—
160	C₂²×C₂₃	D₄²	1	1	yes	characteristic
161	C₄×C₂₃	—	2	1	no	—
162	C₄×C₂₃	—	4	2	no	—
163	C₄×C₂₃	—	4	2	no	—
164	C₄×C₂₃	—	4	2	no	—
165	C₄×C₂₃	—	8	2	no	—
166	C₄×C₂₃	—	8	2	no	—
167	C₄×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
168	C₄×C₂₃	#575	1	1	yes	characteristic
169	C₄×C₂₃	#576	1	1	yes	characteristic
170	C₄×C₂₃	#578	1	1	yes	characteristic
171	#554	C₂²×C₂₃	1	2	yes	—
172	D₄×Q₈	C₂²×C₂₃	1	2	yes	—
173	#501	—	2	2	no	—
174	#500	C₂²×C₂₃	1	1	yes	characteristic
175	#498	C₂²×C₂₃	1	2	yes	—
176	#494	C₂²×C₂₃	1	1	yes	characteristic
177	#487	C₂²×C₂₃	1	2	yes	—
178	#484	—	2	2	no	—
179	#478	—	2	2	no	—
180	#477	C₂²×C₂₃	1	2	yes	—
181	#476	—	2	2	no	—
182	#476	C₂²×C₂₃	1	2	yes	—
183	#474	C₂²×C₂₃	1	2	yes	—
184	#467	—	2	2	no	—
185	#462	C₂²×C₂₃	1	2	yes	—
186	#451	—	2	2	no	—
187	#446	C₂²×C₂₃	1	2	yes	—
188	#442	—	2	2	no	—
189	C₄×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
190	C₄×D₈	C₂²×C₂₃	1	1	yes	characteristic
191	C₈×D₄	C₂²×C₂₃	1	2	yes	—
192	C₂×#103	—	2	2	no	—
193	#335	C₂²×C₂₃	1	1	yes	characteristic
194	#335	C₂²×C₂₃	1	1	yes	characteristic
195	#333	C₂²×C₂₃	1	2	yes	—
196	#332	C₂²×C₂₃	1	2	yes	—
197	Q₈⋊C₈	C₂²×C₂₃	1	2	yes	—
198	D₄⋊C₈	C₂²×C₂₃	1	2	yes	—
199	C₈²	C₂²×C₂₃	1	1	yes	characteristic
200	C₂×C₂₃	—	4	4	no	—
201	C₂×C₂₃	#3273	1	1	yes	characteristic
202	C₂×C₂₃	D₄×Q₁₆	1	1	yes	characteristic
203	C₂×C₂₃	#3291	1	1	yes	characteristic
204	C₂²×Q₈	—	2	2	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	—	4	1	no	—
207	C₂×Q₁₆	D₄×C₂₃	1	1	yes	characteristic
208	C₂×D₈	D₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₈	—	2	2	no	—
210	C₄⋊Q₈	—	2	2	no	—
211	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
212	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
213	#125	—	2	2	no	—
214	C₂²⋊Q₈	—	2	2	no	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	4	2	no	—
217	C₂²⋊Q₈	—	4	2	no	—
218	C₄×Q₈	C₂³×C₂₃	1	2	yes	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	C₂³×C₂₃	1	2	yes	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	#107	—	2	2	no	—
224	#107	—	2	2	no	—
225	#107	—	4	1	no	—
226	#107	D₄×C₂₃	1	1	yes	characteristic
227	#107	D₄×C₂₃	1	1	yes	characteristic
228	#106	—	2	2	no	—
229	#106	—	2	2	no	—
230	#106	—	2	2	no	—
231	C₄⋊C₈	—	2	2	no	—
232	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
233	C₄⋊C₈	C₂³×C₂₃	1	2	yes	—
234	#103	—	2	2	no	—
235	#103	—	2	2	no	—
236	#103	—	2	2	no	—
237	#103	—	2	2	no	—
238	#103	D₄×C₂₃	1	2	yes	—
239	#102	—	2	2	no	—
240	#102	—	2	2	no	—
241	#102	D₄×C₂₃	1	2	yes	—
242	C₂²⋊C₈	—	2	2	no	—
243	#97	—	2	2	no	—
244	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
245	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
247	C₂₃	#19927	1	1	yes	characteristic, direct product factor, Sylow
248	C₂×Q₈	—	2	2	no	—
249	C₂×Q₈	—	2	2	no	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×Q₈	—	4	4	no	—
252	C₂×Q₈	—	8	2	no	—
253	C₂×Q₈	C₂×D₄×C₂₃	1	2	yes	—
254	C₂×D₄	C₂×D₄×C₂₃	1	2	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	2	no	—
258	Q₁₆	—	4	1	no	—
259	Q₁₆	—	8	2	no	—
260	Q₁₆	—	8	2	no	—
261	D₈	Q₁₆×C₂₃	1	4	yes	complemented
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	8	2	no	—
266	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
267	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
268	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
279	C₄⋊C₄	C₂×D₄×C₂₃	1	2	yes	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
285	C₂³	—	2	2	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	8	4	no	—
292	D₄	—	2	2	no	—
293	D₄	C₂×Q₁₆×C₂₃	1	4	yes	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	—	16	2	no	—
303	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
305	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	C₂×Q₁₆×C₂₃	1	2	yes	—
313	C₂²	—	2	4	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	D₄²×C₂₃	1	1	yes	characteristic
317	C₄	—	2	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
324	C₄	C₂₃×#575	1	1	yes	characteristic
325	C₄	C₂₃×#576	1	1	yes	characteristic
326	C₄	C₂₃×#578	1	1	yes	characteristic
327	C₂	—	4	4	no	—
328	C₂	C₂₃×#3273	1	1	yes	characteristic
329	C₂	D₄×Q₁₆×C₂₃	1	1	yes	characteristic
330	C₂	C₂₃×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂×D₈×C₂₃	D₄
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
2. C₂₃×#3299	C₂	semidirect
3. C₂₃×#1707	C₂	semidirect
4. C₂₃×#1692	C₂	semidirect
5. C₂₃×#1683	C₂	semidirect
6. C₂₃×#1649	C₂	semidirect
7. C₂₃×#1632	C₂	semidirect
8. C₂₃×#1591	C₂	semidirect
95. D₈×C₂₃	Q₁₆	semidirect
119. #19927	C₂₃	direct (N × H)
149. #3299	C₂×C₂₃	semidirect
150. #1707	C₂×C₂₃	semidirect
151. #1692	C₂×C₂₃	semidirect
152. #1683	C₂×C₂₃	semidirect
153. #1649	C₂×C₂₃	semidirect
154. #1632	C₂×C₂₃	semidirect
155. #1591	C₂×C₂₃	semidirect
247. C₂₃	#19927	direct (N × H)
261. D₈	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×#3384	2
C₂₃×#3299	2
C₂₃×#1707	2
C₂₃×#1692	2
C₂₃×#1683	2
C₂₃×#1649	2
C₂₃×#1632	2
C₂₃×#1591	2
C₈×D₈×C₂₃	2
#19927	23

Subgroups — C23×#19927 #19167553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#19927 #19167553