Subgroups — C₂₃×#61880 #19209506

Order 5888.

2302 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3497	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3391	C₂	1	8	yes	maximal, complemented
4	C₂₃×#3358	C₂	1	8	yes	maximal, complemented
5	C₂₃×#3205	C₂	1	6	yes	maximal, complemented
6	C₂₃×#3100	C₂	1	6	yes	maximal, complemented
7	C₂₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×2₋¹⁺⁴	—	2	1	no	—
9	C₂²×#41×C₂₃	—	2	1	no	—
10	C₂₃×#550	—	2	4	no	—
11	C₂₃×#550	C₂²	1	8	yes	—
12	C₂₃×#550	C₂²	1	4	yes	complemented
13	D₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
14	C₂₃×#548	—	2	4	no	—
15	C₂₃×#537	—	2	3	no	—
16	C₂₃×#532	C₂²	1	6	yes	—
17	C₂×C₂₃×#127	C₂²	1	3	yes	—
18	C₂×C₂₃×#121	—	2	3	no	—
19	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₂₃	C₂²	1	3	yes	—
21	C₂₃×#493	C₂²	1	12	yes	complemented
22	C₂₃×#486	C₂²	1	12	yes	complemented
23	C₂₃×#480	C₂²	1	4	yes	—
24	C₂₃×#474	C₂²	1	12	yes	complemented
25	C₂₃×#459	C₂²	1	24	yes	complemented
26	C₂₃×#459	C₂²	1	12	yes	complemented
27	C₂₃×#447	C₂²	1	12	yes	—
28	C₈×Q₈×C₂₃	C₂²	1	8	yes	complemented
29	C₄×D₈×C₂₃	C₂²	1	24	yes	complemented
30	C₂₃×#431	C₂²	1	6	yes	—
31	C₂₃×2₋¹⁺⁴	—	4	8	no	—
32	C₂×#41×C₂₃	—	2	4	no	—
33	C₂×#41×C₂₃	—	2	8	no	—
34	C₂×#41×C₂₃	—	2	6	no	—
35	C₂×#41×C₂₃	—	4	8	no	—
36	C₂²×Q₈×C₂₃	—	2	4	no	—
37	C₂²×Q₈×C₂₃	D₄	1	1	yes	characteristic
38	C₂²×D₄×C₂₃	—	2	3	no	—
39	C₂²×D₄×C₂₃	—	2	3	no	—
40	C₂²×D₄×C₂₃	C₂³	1	3	yes	—
41	C₂³×C₄×C₂₃	—	2	3	no	—
42	C₂×D₈×C₂₃	C₂³	1	24	yes	—
43	C₂₃×C₄⋊Q₈	C₂³	1	6	yes	—
44	C₂₃×#127	—	2	6	no	—
45	C₂₃×#127	C₂³	1	12	yes	—
46	C₂₃×#127	C₂³	1	6	yes	—
47	C₂₃×#124	—	2	6	no	—
48	C₂₃×C₂²⋊Q₈	—	2	12	no	—
49	C₂₃×#121	—	2	6	no	—
50	C₂₃×#121	—	2	12	no	—
51	C₂₃×#121	—	2	24	no	—
52	C₂₃×#121	C₂³	1	4	yes	—
53	C₄×Q₈×C₂₃	D₄	1	4	yes	—
54	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
55	C₄×D₄×C₂₃	—	2	6	no	—
56	C₄×D₄×C₂₃	—	2	6	no	—
57	C₄×D₄×C₂₃	C₂³	1	12	yes	—
58	C₄×D₄×C₂₃	C₂³	1	12	yes	—
59	C₂×C₄⋊C₄×C₂₃	D₄	1	3	yes	—
60	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊C₄×C₂₃	—	2	3	no	—
62	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
63	C₂₃×#107	C₂³	1	8	yes	—
64	C₂₃×C₄⋊C₈	C₂³	1	12	yes	—
65	C₂₃×#102	C₂³	1	8	yes	—
66	C₂₃×#102	C₂³	1	24	yes	—
67	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
68	C₄×C₈×C₂₃	C₂³	1	12	yes	—
69	C₂⁴×C₂₃	—	2	3	no	—
70	#41×C₂₃	—	4	32	no	—
71	#41×C₂₃	—	4	16	no	—
72	#41×C₂₃	—	4	24	no	—
73	C₂×Q₈×C₂₃	—	2	6	no	—
74	C₂×Q₈×C₂₃	—	4	16	no	—
75	C₂×Q₈×C₂₃	D₈	1	8	yes	complemented
76	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
77	C₂×D₄×C₂₃	—	2	4	no	—
78	C₂×D₄×C₂₃	—	2	3	no	—
79	C₂×D₄×C₂₃	—	2	6	no	—
80	C₂×D₄×C₂₃	—	2	12	no	—
81	C₂×D₄×C₂₃	—	2	6	no	—
82	C₂×D₄×C₂₃	—	2	12	no	—
83	C₂×D₄×C₂₃	—	4	6	no	—
84	C₂×D₄×C₂₃	—	4	12	no	—
85	C₂×D₄×C₂₃	—	8	6	no	—
86	C₂×D₄×C₂₃	C₂⁴	1	6	yes	—
87	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
88	C₂²×C₄×C₂₃	—	2	1	no	—
89	C₂²×C₄×C₂₃	—	2	12	no	—
90	C₂²×C₄×C₂₃	—	2	12	no	—
91	C₂²×C₄×C₂₃	—	4	12	no	—
92	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
93	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
94	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
95	D₈×C₂₃	—	4	24	no	—
96	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
97	C₄⋊C₄×C₂₃	—	2	1	no	—
98	C₄⋊C₄×C₂₃	—	2	6	no	—
99	C₄⋊C₄×C₂₃	C₂×D₄	1	12	yes	—
100	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
101	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
102	C₂²⋊C₄×C₂₃	—	2	12	no	—
103	C₂²⋊C₄×C₂₃	—	2	4	no	—
104	C₂²⋊C₄×C₂₃	—	2	12	no	—
105	C₄²×C₂₃	C₂×D₄	1	6	yes	—
106	C₄²×C₂₃	C₂⁴	1	6	yes	—
107	#61880	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₂₃	—	2	4	no	—
109	C₂³×C₂₃	—	2	6	no	—
110	C₂³×C₂₃	—	4	6	no	—
111	C₂³×C₂₃	—	4	6	no	—
112	C₂³×C₂₃	—	4	6	no	—
113	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₂₃	—	4	12	no	—
115	Q₈×C₂₃	C₂×D₈	1	16	yes	—
116	D₄×C₂₃	—	2	8	no	—
117	D₄×C₂₃	—	2	12	no	—
118	D₄×C₂₃	—	4	6	no	—
119	D₄×C₂₃	—	4	8	no	—
120	D₄×C₂₃	—	4	24	no	—
121	D₄×C₂₃	—	4	12	no	—
122	D₄×C₂₃	—	8	6	no	—
123	D₄×C₂₃	—	8	12	no	—
124	C₂×C₄×C₂₃	—	2	2	no	—
125	C₂×C₄×C₂₃	—	2	2	no	—
126	C₂×C₄×C₂₃	—	2	4	no	—
127	C₂×C₄×C₂₃	—	4	3	no	—
128	C₂×C₄×C₂₃	—	4	2	no	—
129	C₂×C₄×C₂₃	—	4	24	no	—
130	C₂×C₄×C₂₃	—	4	24	no	—
131	C₂×C₄×C₂₃	C₂×D₈	1	12	yes	—
132	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
134	C₂×C₄×C₂₃	C₂²×D₄	1	6	yes	—
135	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
136	C₈×C₂₃	—	2	8	no	—
137	#3497	C₂×C₂₃	1	1	yes	characteristic, complemented
138	C₂×#550	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#3391	C₂×C₂₃	1	8	yes	complemented
140	#3358	C₂×C₂₃	1	8	yes	complemented
141	#3205	C₂×C₂₃	1	6	yes	complemented
142	#3100	C₂×C₂₃	1	6	yes	complemented
143	#2982	C₂×C₂₃	1	1	yes	characteristic, complemented
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	2	1	no	—
146	C₂²×C₂₃	—	4	6	no	—
147	C₂²×C₂₃	—	4	4	no	—
148	C₂²×C₂₃	—	4	6	no	—
149	C₂²×C₂₃	—	4	4	no	—
150	C₂²×C₂₃	—	4	4	no	—
151	C₂²×C₂₃	—	4	6	no	—
152	C₂²×C₂₃	—	8	12	no	—
153	C₂²×C₂₃	C₂²×D₈	1	2	yes	—
154	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
155	C₄×C₂₃	—	2	6	no	—
156	C₄×C₂₃	—	2	1	no	—
157	C₄×C₂₃	—	4	2	no	—
158	C₄×C₂₃	—	4	4	no	—
159	C₄×C₂₃	C₂²×D₈	1	12	yes	—
160	C₄×C₂₃	#585	1	2	yes	—
161	C₂×2₋¹⁺⁴	—	2	1	no	—
162	C₂²×#41	—	2	1	no	—
163	#550	—	2	4	no	—
164	#550	C₂²×C₂₃	1	8	yes	—
165	#550	C₂²×C₂₃	1	4	yes	complemented
166	D₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
167	#548	—	2	4	no	—
168	#537	—	2	3	no	—
169	#532	C₂²×C₂₃	1	6	yes	—
170	C₂×#127	C₂²×C₂₃	1	3	yes	—
171	C₂×#121	—	2	3	no	—
172	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
173	C₂×C₄×D₄	C₂²×C₂₃	1	3	yes	—
174	#493	C₂²×C₂₃	1	12	yes	complemented
175	#486	C₂²×C₂₃	1	12	yes	complemented
176	#480	C₂²×C₂₃	1	4	yes	—
177	#474	C₂²×C₂₃	1	12	yes	complemented
178	#459	C₂²×C₂₃	1	12	yes	complemented
179	#459	C₂²×C₂₃	1	24	yes	complemented
180	#447	C₂²×C₂₃	1	12	yes	—
181	C₈×Q₈	C₂²×C₂₃	1	8	yes	complemented
182	C₄×D₈	C₂²×C₂₃	1	24	yes	complemented
183	#431	C₂²×C₂₃	1	6	yes	—
184	C₂×C₂₃	—	2	2	no	—
185	C₂×C₂₃	—	8	4	no	—
186	C₂×C₂₃	—	8	6	no	—
187	C₂×C₂₃	#3487	1	1	yes	characteristic
188	C₂×C₂₃	C₂³×D₈	1	1	yes	characteristic
189	C₂×C₂₃	#3585	1	1	yes	characteristic
190	2₋¹⁺⁴	—	4	8	no	—
191	C₂×#41	—	2	4	no	—
192	C₂×#41	—	2	8	no	—
193	C₂×#41	—	2	6	no	—
194	C₂×#41	—	4	8	no	—
195	C₂²×Q₈	—	2	4	no	—
196	C₂²×Q₈	D₄×C₂₃	1	1	yes	characteristic
197	C₂²×D₄	—	2	3	no	—
198	C₂²×D₄	—	2	3	no	—
199	C₂²×D₄	C₂³×C₂₃	1	3	yes	—
200	C₂³×C₄	—	2	3	no	—
201	C₂×D₈	C₂³×C₂₃	1	24	yes	—
202	C₄⋊Q₈	C₂³×C₂₃	1	6	yes	—
203	#127	—	2	6	no	—
204	#127	C₂³×C₂₃	1	12	yes	—
205	#127	C₂³×C₂₃	1	6	yes	—
206	#124	—	2	6	no	—
207	C₂²⋊Q₈	—	2	12	no	—
208	#121	—	2	6	no	—
209	#121	—	2	24	no	—
210	#121	—	2	12	no	—
211	#121	C₂³×C₂₃	1	4	yes	—
212	C₄×Q₈	D₄×C₂₃	1	4	yes	—
213	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
214	C₄×D₄	—	2	6	no	—
215	C₄×D₄	—	2	6	no	—
216	C₄×D₄	C₂³×C₂₃	1	12	yes	—
217	C₄×D₄	C₂³×C₂₃	1	12	yes	—
218	C₂×C₄⋊C₄	D₄×C₂₃	1	3	yes	—
219	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
220	C₂×C₂²⋊C₄	—	2	3	no	—
221	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
222	#107	C₂³×C₂₃	1	8	yes	—
223	C₄⋊C₈	C₂³×C₂₃	1	12	yes	—
224	#102	C₂³×C₂₃	1	8	yes	—
225	#102	C₂³×C₂₃	1	24	yes	—
226	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
227	C₄×C₈	C₂³×C₂₃	1	12	yes	—
228	C₂₃	#61880	1	1	yes	characteristic, direct product factor, Sylow
229	C₂⁴	—	2	3	no	—
230	#41	—	4	32	no	—
231	#41	—	4	16	no	—
232	#41	—	4	24	no	—
233	C₂×Q₈	—	2	6	no	—
234	C₂×Q₈	—	4	16	no	—
235	C₂×Q₈	D₈×C₂₃	1	8	yes	complemented
236	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	3	no	—
239	C₂×D₄	—	2	6	no	—
240	C₂×D₄	—	2	12	no	—
241	C₂×D₄	—	2	6	no	—
242	C₂×D₄	—	2	12	no	—
243	C₂×D₄	—	4	6	no	—
244	C₂×D₄	—	4	12	no	—
245	C₂×D₄	—	8	6	no	—
246	C₂×D₄	C₂⁴×C₂₃	1	6	yes	—
247	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	12	no	—
250	C₂²×C₄	—	2	12	no	—
251	C₂²×C₄	—	4	12	no	—
252	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
253	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
254	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
255	D₈	—	4	24	no	—
256	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	6	no	—
259	C₄⋊C₄	C₂×D₄×C₂₃	1	12	yes	—
260	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
261	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
262	C₂²⋊C₄	—	2	12	no	—
263	C₂²⋊C₄	—	2	4	no	—
264	C₂²⋊C₄	—	2	12	no	—
265	C₄²	C₂×D₄×C₂₃	1	6	yes	—
266	C₄²	C₂⁴×C₂₃	1	6	yes	—
267	C₂³	—	2	4	no	—
268	C₂³	—	2	6	no	—
269	C₂³	—	4	6	no	—
270	C₂³	—	4	6	no	—
271	C₂³	—	4	6	no	—
272	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
273	Q₈	—	4	12	no	—
274	Q₈	C₂×D₈×C₂₃	1	16	yes	—
275	D₄	—	2	8	no	—
276	D₄	—	2	12	no	—
277	D₄	—	4	6	no	—
278	D₄	—	4	8	no	—
279	D₄	—	4	24	no	—
280	D₄	—	4	12	no	—
281	D₄	—	8	12	no	—
282	D₄	—	8	6	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	3	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	24	no	—
289	C₂×C₄	—	4	24	no	—
290	C₂×C₄	C₂×D₈×C₂₃	1	12	yes	—
291	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
293	C₂×C₄	C₂²×D₄×C₂₃	1	6	yes	—
294	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
295	C₈	—	2	8	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	4	6	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	4	4	no	—
301	C₂²	—	4	6	no	—
302	C₂²	—	4	6	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	8	12	no	—
305	C₂²	C₂²×D₈×C₂₃	1	2	yes	—
306	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	6	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	4	no	—
311	C₄	C₂²×D₈×C₂₃	1	12	yes	—
312	C₄	C₂₃×#585	1	2	yes	—
313	C₂	—	2	2	no	—
314	C₂	—	8	6	no	—
315	C₂	—	8	4	no	—
316	C₂	C₂₃×#3487	1	1	yes	characteristic
317	C₂	C₂³×D₈×C₂₃	1	1	yes	characteristic
318	C₂	C₂₃×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3497	C₂	semidirect
2. C₂×C₂₃×#550	C₂	semidirect
3. C₂₃×#3391	C₂	semidirect
4. C₂₃×#3358	C₂	semidirect
5. C₂₃×#3205	C₂	semidirect
6. C₂₃×#3100	C₂	semidirect
7. C₂₃×#2982	C₂	semidirect
12. C₂₃×#550	C₂²	semidirect
13. D₄×Q₈×C₂₃	C₂²	semidirect
21. C₂₃×#493	C₂²	semidirect
22. C₂₃×#486	C₂²	semidirect
24. C₂₃×#474	C₂²	semidirect
25. C₂₃×#459	C₂²	semidirect
26. C₂₃×#459	C₂²	semidirect
28. C₈×Q₈×C₂₃	C₂²	semidirect
29. C₄×D₈×C₂₃	C₂²	semidirect
75. C₂×Q₈×C₂₃	D₈	semidirect
107. #61880	C₂₃	direct (N × H)
137. #3497	C₂×C₂₃	semidirect
138. C₂×#550	C₂×C₂₃	semidirect
139. #3391	C₂×C₂₃	semidirect
140. #3358	C₂×C₂₃	semidirect
141. #3205	C₂×C₂₃	semidirect
142. #3100	C₂×C₂₃	semidirect
143. #2982	C₂×C₂₃	semidirect
165. #550	C₂²×C₂₃	semidirect
166. D₄×Q₈	C₂²×C₂₃	semidirect
174. #493	C₂²×C₂₃	semidirect
175. #486	C₂²×C₂₃	semidirect
177. #474	C₂²×C₂₃	semidirect
178. #459	C₂²×C₂₃	semidirect
179. #459	C₂²×C₂₃	semidirect
181. C₈×Q₈	C₂²×C₂₃	semidirect
182. C₄×D₈	C₂²×C₂₃	semidirect
228. C₂₃	#61880	direct (N × H)
235. C₂×Q₈	D₈×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#3497	2
C₂×C₂₃×#550	2
C₂₃×#3391	2
C₂₃×#3358	2
C₂₃×#3205	2
C₂₃×#3100	2
C₂₃×#2982	2
#61880	23

Subgroups — C23×#61880 #19209506

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#61880 #19209506