Subgroups — C₂₃×#18546 #19166172

Order 5888.

1004 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#3152	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3112	C₂	1	2	yes	maximal
4	C₂₃×#1633	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1623	C₂	1	4	yes	maximal, complemented
6	C₂₃×#1558	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1491	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#137	—	2	2	no	—
9	Q₈²×C₂₃	—	2	2	no	—
10	Q₈²×C₂₃	—	2	4	no	—
11	Q₈²×C₂₃	C₂²	1	4	yes	—
12	C₂₃×#532	C₂²	1	1	yes	characteristic
13	C₂×C₂₃×C₄⋊Q₈	—	2	2	no	—
14	C₂×C₂₃×C₄⋊Q₈	—	2	2	no	—
15	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₂₃	—	2	2	no	—
17	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#497	—	2	2	no	—
19	C₂₃×#494	C₂²	1	2	yes	—
20	C₂₃×#492	C₂²	1	2	yes	—
21	C₂₃×#480	—	2	2	no	—
22	C₂₃×#477	C₂²	1	4	yes	—
23	C₂₃×#460	C₂²	1	4	yes	—
24	C₂₃×#448	—	2	2	no	—
25	C₂₃×#440	—	2	4	no	—
26	C₂₃×#432	C₂²	1	2	yes	—
27	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
28	C₂₃×#333	C₂²	1	4	yes	—
29	C₂₃×Q₈⋊C₈	C₂²	1	8	yes	—
30	C₂×#41×C₂₃	—	2	1	no	—
31	C₂²×Q₈×C₂₃	—	2	2	no	—
32	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
33	C₂²×Q₈×C₂₃	D₄	1	2	yes	—
34	C₂₃×#137	—	4	8	no	—
35	C₂×Q₁₆×C₂₃	—	2	4	no	—
36	C₂×SD₁₆×C₂₃	—	2	4	no	—
37	C₂×SD₁₆×C₂₃	—	2	4	no	—
38	C₂×M₁₆×C₂₃	—	2	1	no	—
39	C₂₃×C₄⋊Q₈	—	2	8	no	—
40	C₂₃×C₄⋊Q₈	—	2	8	no	—
41	C₂₃×C₄⋊Q₈	—	2	8	no	—
42	C₂₃×C₄⋊Q₈	—	2	8	no	—
43	C₂₃×C₄⋊Q₈	—	2	2	no	—
44	C₂₃×C₄⋊Q₈	D₄	1	2	yes	—
45	C₂₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
46	C₂₃×C₄⋊Q₈	C₂³	1	2	yes	—
47	C₂₃×#127	C₂³	1	1	yes	characteristic
48	C₂₃×#124	—	2	1	no	—
49	C₂₃×#121	—	2	2	no	—
50	C₄×Q₈×C₂₃	—	2	8	no	—
51	C₄×Q₈×C₂₃	—	2	8	no	—
52	C₄×Q₈×C₂₃	—	2	2	no	—
53	C₄×Q₈×C₂₃	—	2	2	no	—
54	C₄×Q₈×C₂₃	D₄	1	4	yes	—
55	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
56	C₄×D₄×C₂₃	—	2	2	no	—
57	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
58	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
59	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₂₃	D₄	1	2	yes	—
61	C₂×C₄²×C₂₃	—	2	2	no	—
62	C₂×C₄²×C₂₃	—	2	2	no	—
63	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
64	C₂₃×#107	—	2	4	no	—
65	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
66	C₂₃×#103	—	2	4	no	—
67	C₂₃×#102	—	2	4	no	—
68	C₂₃×C₂²⋊C₈	—	2	2	no	—
69	C₂₃×#97	—	2	1	no	—
70	C₂₃×#97	—	2	2	no	—
71	C₄×C₈×C₂₃	C₂³	1	2	yes	—
72	#41×C₂₃	—	4	4	no	—
73	C₂×Q₈×C₂₃	—	2	8	no	—
74	C₂×Q₈×C₂₃	—	2	8	no	—
75	C₂×Q₈×C₂₃	—	2	2	no	—
76	C₂×Q₈×C₂₃	—	2	2	no	—
77	C₂×Q₈×C₂₃	—	2	1	no	—
78	C₂×Q₈×C₂₃	—	4	4	no	—
79	C₂×Q₈×C₂₃	—	8	4	no	—
80	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
81	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
82	C₂×D₄×C₂₃	—	2	2	no	—
83	C₂×D₄×C₂₃	—	2	1	no	—
84	C₂×D₄×C₂₃	—	4	1	no	—
85	C₂²×C₄×C₂₃	—	2	2	no	—
86	C₂²×C₄×C₂₃	—	2	2	no	—
87	C₂²×C₄×C₂₃	—	2	2	no	—
88	C₂²×C₄×C₂₃	—	4	1	no	—
89	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
91	Q₁₆×C₂₃	—	4	8	no	—
92	SD₁₆×C₂₃	—	4	8	no	—
93	SD₁₆×C₂₃	—	4	8	no	—
94	M₁₆×C₂₃	—	2	4	no	—
95	C₂×C₈×C₂₃	—	2	2	no	—
96	C₂×C₈×C₂₃	—	2	4	no	—
97	C₄⋊C₄×C₂₃	—	2	16	no	—
98	C₄⋊C₄×C₂₃	—	2	4	no	—
99	C₄⋊C₄×C₂₃	—	2	4	no	—
100	C₄⋊C₄×C₂₃	—	2	2	no	—
101	C₄⋊C₄×C₂₃	—	2	4	no	—
102	C₄⋊C₄×C₂₃	—	2	4	no	—
103	C₄⋊C₄×C₂₃	—	2	2	no	—
104	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
105	C₂²⋊C₄×C₂₃	—	4	2	no	—
106	C₄²×C₂₃	—	2	8	no	—
107	C₄²×C₂₃	—	2	4	no	—
108	C₄²×C₂₃	—	2	4	no	—
109	C₄²×C₂₃	C₂×D₄	1	2	yes	—
110	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₂₃	C₂⁴	1	1	yes	characteristic
112	#18546	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₂₃	—	4	1	no	—
114	C₂³×C₂₃	#120	1	1	yes	characteristic
115	Q₈×C₂₃	—	2	8	no	—
116	Q₈×C₂₃	—	2	4	no	—
117	Q₈×C₂₃	—	4	4	no	—
118	Q₈×C₂₃	—	4	2	no	—
119	Q₈×C₂₃	—	8	2	no	—
120	Q₈×C₂₃	—	8	8	no	—
121	Q₈×C₂₃	#137	1	8	yes	complemented
122	D₄×C₂₃	—	4	2	no	—
123	D₄×C₂₃	—	4	4	no	—
124	D₄×C₂₃	—	8	2	no	—
125	C₂×C₄×C₂₃	—	2	8	no	—
126	C₂×C₄×C₂₃	—	2	4	no	—
127	C₂×C₄×C₂₃	—	2	2	no	—
128	C₂×C₄×C₂₃	—	2	4	no	—
129	C₂×C₄×C₂₃	—	2	4	no	—
130	C₂×C₄×C₂₃	—	2	4	no	—
131	C₂×C₄×C₂₃	—	4	4	no	—
132	C₂×C₄×C₂₃	—	4	4	no	—
133	C₂×C₄×C₂₃	—	4	1	no	—
134	C₂×C₄×C₂₃	—	8	2	no	—
135	C₂×C₄×C₂₃	#120	1	2	yes	—
136	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
137	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
139	C₈×C₂₃	—	2	4	no	—
140	C₈×C₂₃	—	4	4	no	—
141	C₂×Q₈²	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#3152	C₂×C₂₃	1	1	yes	characteristic
143	#3112	C₂×C₂₃	1	2	yes	—
144	#1633	C₂×C₂₃	1	4	yes	complemented
145	#1623	C₂×C₂₃	1	4	yes	complemented
146	#1558	C₂×C₂₃	1	1	yes	characteristic, complemented
147	#1491	C₂×C₂₃	1	2	yes	complemented
148	C₂²×C₂₃	—	2	2	no	—
149	C₂²×C₂₃	—	8	2	no	—
150	C₂²×C₂₃	—	8	1	no	—
151	C₂²×C₂₃	#453	1	2	yes	—
152	C₂²×C₂₃	C₂×#120	1	1	yes	characteristic
153	C₄×C₂₃	—	2	8	no	—
154	C₄×C₂₃	—	2	2	no	—
155	C₄×C₂₃	—	2	1	no	—
156	C₄×C₂₃	—	2	1	no	—
157	C₄×C₂₃	—	4	4	no	—
158	C₄×C₂₃	—	4	4	no	—
159	C₄×C₂₃	—	8	1	no	—
160	C₄×C₂₃	C₂×#137	1	4	yes	—
161	C₂×#137	—	2	2	no	—
162	Q₈²	—	2	2	no	—
163	Q₈²	—	2	4	no	—
164	Q₈²	C₂²×C₂₃	1	4	yes	—
165	#532	C₂²×C₂₃	1	1	yes	characteristic
166	C₂×C₄⋊Q₈	—	2	2	no	—
167	C₂×C₄⋊Q₈	—	2	2	no	—
168	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
169	C₂×C₄×Q₈	—	2	2	no	—
170	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	—
171	#497	—	2	2	no	—
172	#494	C₂²×C₂₃	1	2	yes	—
173	#492	C₂²×C₂₃	1	2	yes	—
174	#480	—	2	2	no	—
175	#477	C₂²×C₂₃	1	4	yes	—
176	#460	C₂²×C₂₃	1	4	yes	—
177	#448	—	2	2	no	—
178	#440	—	2	4	no	—
179	#432	C₂²×C₂₃	1	2	yes	—
180	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
181	#333	C₂²×C₂₃	1	4	yes	—
182	Q₈⋊C₈	C₂²×C₂₃	1	8	yes	—
183	C₂×C₂₃	—	2	2	no	—
184	C₂×C₂₃	—	16	1	no	—
185	C₂×C₂₃	#3004	1	2	yes	—
186	C₂×C₂₃	C₂×#453	1	1	yes	characteristic
187	C₂×#41	—	2	1	no	—
188	C₂²×Q₈	—	2	2	no	—
189	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
190	C₂²×Q₈	D₄×C₂₃	1	2	yes	—
191	#137	—	4	8	no	—
192	C₂×Q₁₆	—	2	4	no	—
193	C₂×SD₁₆	—	2	4	no	—
194	C₂×SD₁₆	—	2	4	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₄⋊Q₈	—	2	8	no	—
197	C₄⋊Q₈	—	2	8	no	—
198	C₄⋊Q₈	—	2	8	no	—
199	C₄⋊Q₈	—	2	8	no	—
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	D₄×C₂₃	1	2	yes	—
202	C₄⋊Q₈	C₂³×C₂₃	1	1	yes	characteristic
203	C₄⋊Q₈	C₂³×C₂₃	1	2	yes	—
204	#127	C₂³×C₂₃	1	1	yes	characteristic
205	#124	—	2	1	no	—
206	#121	—	2	2	no	—
207	C₄×Q₈	—	2	8	no	—
208	C₄×Q₈	—	2	8	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	D₄×C₂₃	1	4	yes	—
212	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
213	C₄×D₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	4	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	D₄×C₂₃	1	2	yes	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
221	#107	—	2	4	no	—
222	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
223	#103	—	2	4	no	—
224	#102	—	2	4	no	—
225	C₂²⋊C₈	—	2	2	no	—
226	#97	—	2	1	no	—
227	#97	—	2	2	no	—
228	C₄×C₈	C₂³×C₂₃	1	2	yes	—
229	C₂₃	#18546	1	1	yes	characteristic, direct product factor, Sylow
230	#41	—	4	4	no	—
231	C₂×Q₈	—	2	8	no	—
232	C₂×Q₈	—	2	8	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	1	no	—
236	C₂×Q₈	—	4	4	no	—
237	C₂×Q₈	—	8	4	no	—
238	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
239	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	4	1	no	—
247	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
248	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
249	Q₁₆	—	4	8	no	—
250	SD₁₆	—	4	8	no	—
251	SD₁₆	—	4	8	no	—
252	M₁₆	—	2	4	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	4	no	—
255	C₄⋊C₄	—	2	16	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	4	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₄²	—	2	8	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	4	no	—
267	C₄²	C₂×D₄×C₂₃	1	2	yes	—
268	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
269	C₄²	C₂⁴×C₂₃	1	1	yes	characteristic
270	C₂³	—	4	1	no	—
271	C₂³	C₂₃×#120	1	1	yes	characteristic
272	Q₈	—	2	8	no	—
273	Q₈	—	2	4	no	—
274	Q₈	—	4	4	no	—
275	Q₈	—	4	2	no	—
276	Q₈	—	8	2	no	—
277	Q₈	—	8	8	no	—
278	Q₈	C₂₃×#137	1	8	yes	complemented
279	D₄	—	4	2	no	—
280	D₄	—	4	4	no	—
281	D₄	—	8	2	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	C₂₃×#120	1	2	yes	—
293	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
296	C₈	—	2	4	no	—
297	C₈	—	4	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	1	no	—
301	C₂²	C₂₃×#453	1	2	yes	—
302	C₂²	C₂×C₂₃×#120	1	1	yes	characteristic
303	C₄	—	2	8	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	4	no	—
308	C₄	—	4	4	no	—
309	C₄	—	8	1	no	—
310	C₄	C₂×C₂₃×#137	1	4	yes	—
311	C₂	—	2	2	no	—
312	C₂	—	16	1	no	—
313	C₂	C₂₃×#3004	1	2	yes	—
314	C₂	C₂×C₂₃×#453	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₂₃
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈²×C₂₃	C₂	semidirect
4. C₂₃×#1633	C₂	semidirect
5. C₂₃×#1623	C₂	semidirect
6. C₂₃×#1558	C₂	semidirect
7. C₂₃×#1491	C₂	semidirect
112. #18546	C₂₃	direct (N × H)
121. Q₈×C₂₃	#137	semidirect
141. C₂×Q₈²	C₂×C₂₃	semidirect
144. #1633	C₂×C₂₃	semidirect
145. #1623	C₂×C₂₃	semidirect
146. #1558	C₂×C₂₃	semidirect
147. #1491	C₂×C₂₃	semidirect
229. C₂₃	#18546	direct (N × H)
278. Q₈	C₂₃×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₂₃	2
C₂₃×#3152	2
C₂₃×#3112	2
C₂₃×#1633	2
C₂₃×#1623	2
C₂₃×#1558	2
C₂₃×#1491	2
#18546	23

Subgroups — C23×#18546 #19166172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#18546 #19166172