Subgroups — C₂₃×#32691 #19180317

Order 5888.

610 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2584	C₂	1	2	yes	maximal
2	C₂₃×#2087	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2086	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#2059	C₂	1	2	yes	maximal
5	C₂₃×#2059	C₂	1	2	yes	maximal
6	C₂₃×#1985	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1943	C₂	1	2	yes	maximal
8	C₂₃×#1921	C₂	1	2	yes	maximal
9	C₂₃×#1918	C₂	1	2	yes	maximal
10	C₂×C₂₃×#125	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×#107	C₂²	1	2	yes	—
15	C₂×C₂₃×#106	C₂²	1	2	yes	—
16	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₂₃×#97	C₂²	1	1	yes	characteristic
18	C₂₃×#398	—	2	2	no	—
19	C₂₃×#395	—	2	2	no	—
20	C₂₃×#389	—	2	2	no	—
21	C₂₃×#389	—	2	2	no	—
22	C₂₃×#389	C₂²	1	2	yes	—
23	C₂₃×#389	C₂²	1	2	yes	—
24	C₂₃×#389	C₂²	1	2	yes	—
25	C₂₃×#389	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
27	C₂₃×#340	C₂²	1	1	yes	characteristic
28	C₂₃×#340	C₂²	1	2	yes	—
29	C₂₃×#340	C₂²	1	2	yes	—
30	C₂₃×#340	C₂²	1	1	yes	characteristic
31	C₂₃×#340	C₂²	1	2	yes	—
32	C₂₃×#340	C₂²	1	2	yes	—
33	C₂₃×#336	C₂²	1	2	yes	—
34	C₂₃×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
39	C₂₃×#125	—	4	2	no	—
40	C₂₃×#125	—	4	2	no	—
41	C₄×Q₈×C₂₃	—	4	1	no	—
42	C₄×Q₈×C₂₃	—	4	1	no	—
43	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
48	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₂₃	—	2	2	no	—
56	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
57	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
58	C₂₃×#107	—	4	1	no	—
59	C₂₃×#107	—	4	1	no	—
60	C₂₃×#107	Q₈	1	8	yes	complemented
61	C₂₃×#106	—	4	2	no	—
62	C₂₃×C₄⋊C₈	—	4	1	no	—
63	C₂₃×#97	—	4	1	no	—
64	C₂₃×#95	—	2	2	no	—
65	C₂₃×#95	—	2	2	no	—
66	C₂₃×#95	—	2	2	no	—
67	C₂₃×#95	—	2	2	no	—
68	C₂₃×#95	—	2	2	no	—
69	C₂×Q₈×C₂₃	—	4	1	no	—
70	C₂×Q₈×C₂₃	—	4	2	no	—
71	C₂²×C₄×C₂₃	—	2	2	no	—
72	C₂²×C₄×C₂₃	—	2	2	no	—
73	C₂²×C₄×C₂₃	—	2	2	no	—
74	C₂²×C₄×C₂₃	—	2	2	no	—
75	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₂₃	—	2	2	no	—
81	C₂×C₈×C₂₃	—	2	2	no	—
82	C₂×C₈×C₂₃	—	2	1	no	—
83	C₂×C₈×C₂₃	—	2	1	no	—
84	C₂×C₈×C₂₃	—	2	2	no	—
85	C₂×C₈×C₂₃	—	2	2	no	—
86	C₂×C₈×C₂₃	C₂×Q₈	1	4	yes	—
87	C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₄⋊C₄×C₂₃	—	2	2	no	—
90	C₄⋊C₄×C₂₃	—	2	2	no	—
91	C₄⋊C₄×C₂₃	—	2	2	no	—
92	C₄⋊C₄×C₂₃	—	2	2	no	—
93	C₄⋊C₄×C₂₃	—	4	1	no	—
94	C₄⋊C₄×C₂₃	—	4	2	no	—
95	C₄⋊C₄×C₂₃	—	4	4	no	—
96	C₄⋊C₄×C₂₃	—	4	4	no	—
97	C₄⋊C₄×C₂₃	—	4	4	no	—
98	C₄⋊C₄×C₂₃	—	4	4	no	—
99	C₄⋊C₄×C₂₃	—	8	2	no	—
100	C₄⋊C₄×C₂₃	—	8	2	no	—
101	C₄⋊C₄×C₂₃	C₂×Q₈	1	4	yes	—
102	C₄⋊C₄×C₂₃	C₂×Q₈	1	4	yes	—
103	C₄²×C₂₃	—	2	1	no	—
104	C₄²×C₂₃	—	2	2	no	—
105	C₄²×C₂₃	—	2	2	no	—
106	C₄²×C₂₃	—	2	1	no	—
107	C₄²×C₂₃	—	8	2	no	—
108	#32691	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₂₃	—	4	4	no	—
111	C₂×C₄×C₂₃	—	2	2	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	2	1	no	—
114	C₂×C₄×C₂₃	—	2	1	no	—
115	C₂×C₄×C₂₃	—	2	4	no	—
116	C₂×C₄×C₂₃	—	2	4	no	—
117	C₂×C₄×C₂₃	—	2	2	no	—
118	C₂×C₄×C₂₃	—	2	2	no	—
119	C₂×C₄×C₂₃	—	2	2	no	—
120	C₂×C₄×C₂₃	—	2	2	no	—
121	C₂×C₄×C₂₃	—	2	1	no	—
122	C₂×C₄×C₂₃	—	2	2	no	—
123	C₂×C₄×C₂₃	—	2	1	no	—
124	C₂×C₄×C₂₃	—	4	2	no	—
125	C₂×C₄×C₂₃	—	4	2	no	—
126	C₂×C₄×C₂₃	—	4	2	no	—
127	C₂×C₄×C₂₃	—	4	2	no	—
128	C₂×C₄×C₂₃	—	4	2	no	—
129	C₂×C₄×C₂₃	—	4	2	no	—
130	C₂×C₄×C₂₃	—	4	2	no	—
131	C₂×C₄×C₂₃	—	4	2	no	—
132	C₂×C₄×C₂₃	—	4	2	no	—
133	C₂×C₄×C₂₃	—	4	2	no	—
134	C₂×C₄×C₂₃	C₂²×Q₈	1	2	yes	—
135	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
136	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
138	C₈×C₂₃	—	2	4	no	—
139	C₈×C₂₃	—	4	1	no	—
140	C₈×C₂₃	—	4	1	no	—
141	#2584	C₂×C₂₃	1	2	yes	—
142	#2087	C₂×C₂₃	1	1	yes	characteristic
143	#2086	C₂×C₂₃	1	1	yes	characteristic
144	#2059	C₂×C₂₃	1	2	yes	—
145	#2059	C₂×C₂₃	1	2	yes	—
146	#1985	C₂×C₂₃	1	1	yes	characteristic
147	#1943	C₂×C₂₃	1	2	yes	—
148	#1921	C₂×C₂₃	1	2	yes	—
149	#1918	C₂×C₂₃	1	2	yes	—
150	C₂²×C₂₃	#534	1	1	yes	characteristic
151	C₂²×C₂₃	D₄×Q₈	1	2	yes	—
152	C₂²×C₂₃	#575	1	2	yes	—
153	C₂²×C₂₃	#576	1	1	yes	characteristic
154	C₂²×C₂₃	#578	1	1	yes	characteristic
155	C₄×C₂₃	—	4	4	no	—
156	C₄×C₂₃	—	4	4	no	—
157	C₄×C₂₃	—	4	2	no	—
158	C₄×C₂₃	—	4	1	no	—
159	C₄×C₂₃	—	4	2	no	—
160	C₄×C₂₃	—	4	1	no	—
161	C₄×C₂₃	—	8	2	no	—
162	C₄×C₂₃	—	8	2	no	—
163	C₄×C₂₃	#544	1	2	yes	—
164	C₄×C₂₃	Q₈²	1	1	yes	characteristic
165	C₄×C₂₃	#563	1	1	yes	characteristic
166	C₂×#125	C₂²×C₂₃	1	2	yes	—
167	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
168	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
169	C₂×#107	C₂²×C₂₃	1	1	yes	characteristic
170	C₂×#107	C₂²×C₂₃	1	2	yes	—
171	C₂×#106	C₂²×C₂₃	1	2	yes	—
172	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×#97	C₂²×C₂₃	1	1	yes	characteristic
174	#398	—	2	2	no	—
175	#395	—	2	2	no	—
176	#389	—	2	2	no	—
177	#389	—	2	2	no	—
178	#389	C₂²×C₂₃	1	2	yes	—
179	#389	C₂²×C₂₃	1	2	yes	—
180	#389	C₂²×C₂₃	1	2	yes	—
181	#389	C₂²×C₂₃	1	2	yes	—
182	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
183	#340	C₂²×C₂₃	1	2	yes	—
184	#340	C₂²×C₂₃	1	2	yes	—
185	#340	C₂²×C₂₃	1	2	yes	—
186	#340	C₂²×C₂₃	1	2	yes	—
187	#340	C₂²×C₂₃	1	1	yes	characteristic
188	#340	C₂²×C₂₃	1	1	yes	characteristic
189	#336	C₂²×C₂₃	1	2	yes	—
190	#336	C₂²×C₂₃	1	2	yes	—
191	C₂×C₂₃	#2734	1	1	yes	characteristic
192	C₂×C₂₃	#3197	1	1	yes	characteristic
193	C₂×C₂₃	#3203	1	1	yes	characteristic
194	C₂×C₂₃	#3389	1	2	yes	—
195	C₂×C₂₃	#3390	1	2	yes	—
196	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
200	#125	—	4	2	no	—
201	#125	—	4	2	no	—
202	C₄×Q₈	—	4	1	no	—
203	C₄×Q₈	—	4	1	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
211	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
218	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
219	#107	—	4	1	no	—
220	#107	—	4	1	no	—
221	#107	Q₈×C₂₃	1	8	yes	complemented
222	#106	—	4	2	no	—
223	C₄⋊C₈	—	4	1	no	—
224	#97	—	4	1	no	—
225	#95	—	2	2	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	C₂₃	#32691	1	1	yes	characteristic, direct product factor, Sylow
231	C₂×Q₈	—	4	1	no	—
232	C₂×Q₈	—	4	2	no	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×Q₈×C₂₃	1	4	yes	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	4	no	—
259	C₄⋊C₄	—	4	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	8	2	no	—
262	C₄⋊C₄	—	8	2	no	—
263	C₄⋊C₄	C₂×Q₈×C₂₃	1	4	yes	—
264	C₄⋊C₄	C₂×Q₈×C₂₃	1	4	yes	—
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	8	2	no	—
270	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
271	Q₈	—	4	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	C₂²×Q₈×C₂₃	1	2	yes	—
296	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
297	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
298	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
299	C₈	—	2	4	no	—
300	C₈	—	4	1	no	—
301	C₈	—	4	1	no	—
302	C₂²	C₂₃×#534	1	1	yes	characteristic
303	C₂²	D₄×Q₈×C₂₃	1	2	yes	—
304	C₂²	C₂₃×#575	1	2	yes	—
305	C₂²	C₂₃×#576	1	1	yes	characteristic
306	C₂²	C₂₃×#578	1	1	yes	characteristic
307	C₄	—	4	1	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	4	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	C₂₃×#544	1	2	yes	—
316	C₄	Q₈²×C₂₃	1	1	yes	characteristic
317	C₄	C₂₃×#563	1	1	yes	characteristic
318	C₂	C₂₃×#2734	1	1	yes	characteristic
319	C₂	C₂₃×#3197	1	1	yes	characteristic
320	C₂	C₂₃×#3203	1	1	yes	characteristic
321	C₂	C₂₃×#3389	1	2	yes	—
322	C₂	C₂₃×#3390	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
60. C₂₃×#107	Q₈	semidirect
108. #32691	C₂₃	direct (N × H)
221. #107	Q₈×C₂₃	semidirect
230. C₂₃	#32691	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×#2584	2
C₂₃×#2087	2
C₂₃×#2086	2
C₂₃×#2059	2
C₂₃×#1985	2
C₂₃×#1943	2
C₂₃×#1921	2
C₂₃×#1918	2
#32691	23

Subgroups — C23×#32691 #19180317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#32691 #19180317