Subgroups — C₂₃×#10813 #19158439

Order 5888.

674 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#344	C₂	1	2	yes	maximal
3	C₂×C₂₃×#339	C₂	1	2	yes	maximal
4	C₂×C₂₃×#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₂₃	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#117	C₂²	1	2	yes	—
10	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
11	C₄×M₁₆×C₂₃	—	2	4	no	—
12	C₂×C₂₃×#97	C₄	1	2	yes	complemented
13	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	complemented
14	C₂₃×#344	—	2	4	no	—
15	C₂₃×#339	—	2	4	no	—
16	C₂₃×#338	C₄	1	4	yes	complemented
17	C₂₃×H(3,ℤ/4)	C₄	1	4	yes	complemented
18	C₂³×C₄×C₂₃	—	2	1	no	—
19	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₂₃	—	2	2	no	—
21	C₂×M₁₆×C₂₃	—	2	2	no	—
22	C₂×M₁₆×C₂₃	—	2	2	no	—
23	C₂×M₁₆×C₂₃	—	2	2	no	—
24	C₂×M₁₆×C₂₃	—	2	2	no	—
25	C₂×M₁₆×C₂₃	—	2	2	no	—
26	C₂×M₁₆×C₂₃	—	2	2	no	—
27	C₂×M₁₆×C₂₃	—	2	2	no	—
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
30	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
31	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
32	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
33	C₂×M₁₆×C₂₃	Q₈	1	2	yes	—
34	C₂²×C₈×C₂₃	—	2	2	no	—
35	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
37	C₂₃×#117	—	2	2	no	—
38	C₂₃×#117	—	2	2	no	—
39	C₂₃×#117	—	2	2	no	—
40	C₂₃×#117	C₂×C₄	1	4	yes	—
41	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
42	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
43	C₂×C₄²×C₂₃	—	2	2	no	—
44	C₂×C₄²×C₂₃	—	2	1	no	—
45	C₂×C₄²×C₂₃	—	2	2	no	—
46	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₂₃	D₄	1	4	yes	—
49	C₂₃×#97	—	2	4	no	—
50	C₄×C₈×C₂₃	—	2	4	no	—
51	C₂⁴×C₂₃	#41	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	—	2	1	no	—
53	C₂²×C₄×C₂₃	—	2	1	no	—
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	4	4	no	—
56	C₂²×C₄×C₂₃	—	4	2	no	—
57	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	#41	1	2	yes	—
64	C₂²×C₄×C₂₃	#41	1	2	yes	—
65	C₂²×C₄×C₂₃	#41	1	2	yes	—
66	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
67	M₁₆×C₂₃	—	2	4	no	—
68	M₁₆×C₂₃	—	2	4	no	—
69	M₁₆×C₂₃	—	2	4	no	—
70	M₁₆×C₂₃	—	2	4	no	—
71	M₁₆×C₂₃	—	2	4	no	—
72	M₁₆×C₂₃	—	2	4	no	—
73	M₁₆×C₂₃	—	4	2	no	—
74	M₁₆×C₂₃	—	4	2	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	2	2	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	2	no	—
79	C₂×C₈×C₂₃	—	2	4	no	—
80	C₂×C₈×C₂₃	—	4	2	no	—
81	C₂×C₈×C₂₃	—	4	2	no	—
82	C₂×C₈×C₂₃	C₄²	1	4	yes	—
83	C₂×C₈×C₂₃	C₄⋊C₄	1	4	yes	—
84	C₄⋊C₄×C₂₃	—	2	4	no	—
85	C₄⋊C₄×C₂₃	—	4	2	no	—
86	C₂²⋊C₄×C₂₃	—	2	4	no	—
87	C₂²⋊C₄×C₂₃	—	4	2	no	—
88	C₄²×C₂₃	—	2	1	no	—
89	C₄²×C₂₃	—	2	1	no	—
90	C₄²×C₂₃	—	2	4	no	—
91	C₄²×C₂₃	—	2	4	no	—
92	C₄²×C₂₃	—	4	2	no	—
93	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
95	#10813	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₂₃	—	4	2	no	—
97	C₂³×C₂₃	#117	1	1	yes	characteristic
98	C₂³×C₂₃	C₄×D₄	1	1	yes	characteristic
99	C₂³×C₂₃	C₄×Q₈	1	1	yes	characteristic
100	C₂³×C₂₃	#123	1	2	yes	—
101	C₂³×C₂₃	#126	1	2	yes	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	4	8	no	—
106	C₂×C₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	—	8	4	no	—
109	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#117	1	2	yes	—
114	C₂×C₄×C₂₃	#117	1	2	yes	—
115	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₂₃	#121	1	2	yes	—
122	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₂₃	#123	1	2	yes	—
124	C₂×C₄×C₂₃	#124	1	2	yes	—
125	C₂×C₄×C₂₃	#125	1	2	yes	—
126	C₂×C₄×C₂₃	#126	1	2	yes	—
127	C₈×C₂₃	—	2	4	no	—
128	C₈×C₂₃	—	2	4	no	—
129	C₈×C₂₃	—	4	4	no	—
130	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
131	C₂×#344	C₂×C₂₃	1	2	yes	—
132	C₂×#339	C₂×C₂₃	1	2	yes	—
133	C₂×#338	C₂×C₂₃	1	1	yes	characteristic
134	C₂×H(3,ℤ/4)	C₂×C₂₃	1	1	yes	characteristic
135	C₂²×C₂₃	—	2	2	no	—
136	C₂²×C₂₃	—	2	4	no	—
137	C₂²×C₂₃	—	4	2	no	—
138	C₂²×C₂₃	—	4	1	no	—
139	C₂²×C₂₃	—	4	1	no	—
140	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₂₃	#381	1	1	yes	characteristic
142	C₂²×C₂₃	#383	1	1	yes	characteristic
143	C₂²×C₂₃	#387	1	2	yes	—
144	C₂²×C₂₃	#388	1	2	yes	—
145	C₄×C₂₃	—	2	4	no	—
146	C₄×C₂₃	—	4	2	no	—
147	C₄×C₂₃	—	8	4	no	—
148	C₄×C₂₃	#376	1	1	yes	characteristic
149	C₄×C₂₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
150	C₄×C₂₃	#380	1	1	yes	characteristic
151	C₄×C₂₃	#382	1	1	yes	characteristic
152	C₄×C₂₃	#387	1	2	yes	—
153	C₄×C₂₃	#388	1	2	yes	—
154	C₂²×M₁₆	C₂²×C₂₃	1	2	yes	—
155	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
156	C₂²×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
157	C₂×#117	C₂²×C₂₃	1	2	yes	—
158	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
159	C₄×M₁₆	—	2	4	no	—
160	C₂×#97	C₄×C₂₃	1	2	yes	complemented
161	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	complemented
162	#344	—	2	4	no	—
163	#339	—	2	4	no	—
164	#338	C₄×C₂₃	1	4	yes	complemented
165	H(3,ℤ/4)	C₄×C₂₃	1	4	yes	complemented
166	C₂×C₂₃	—	2	1	no	—
167	C₂×C₂₃	—	2	2	no	—
168	C₂×C₂₃	—	2	1	no	—
169	C₂×C₂₃	—	2	2	no	—
170	C₂×C₂₃	#1434	1	1	yes	characteristic
171	C₂×C₂₃	#1841	1	2	yes	—
172	C₂³×C₄	—	2	1	no	—
173	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
184	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
185	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
186	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
187	C₂×M₁₆	Q₈×C₂₃	1	2	yes	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
190	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	C₂×C₄×C₂₃	1	4	yes	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	2	2	no	—
200	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
201	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
202	C₂×C₄²	D₄×C₂₃	1	4	yes	—
203	#97	—	2	4	no	—
204	C₄×C₈	—	2	4	no	—
205	C₂₃	#10813	1	1	yes	characteristic, direct product factor, Sylow
206	C₂⁴	#41×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	4	2	no	—
212	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
216	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	#41×C₂₃	1	2	yes	—
219	C₂²×C₄	#41×C₂₃	1	2	yes	—
220	C₂²×C₄	#41×C₂₃	1	2	yes	—
221	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	C₄²×C₂₃	1	4	yes	—
238	C₂×C₈	C₄⋊C₄×C₂₃	1	4	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	2	4	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
249	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
250	C₂³	—	4	2	no	—
251	C₂³	C₂₃×#117	1	1	yes	characteristic
252	C₂³	C₄×D₄×C₂₃	1	1	yes	characteristic
253	C₂³	C₄×Q₈×C₂₃	1	1	yes	characteristic
254	C₂³	C₂₃×#123	1	2	yes	—
255	C₂³	C₂₃×#126	1	2	yes	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
264	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
266	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
267	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
268	C₂×C₄	C₂₃×#117	1	2	yes	—
269	C₂×C₄	C₂₃×#117	1	2	yes	—
270	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
271	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
272	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
273	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
274	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
275	C₂×C₄	C₂₃×#121	1	2	yes	—
276	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
277	C₂×C₄	C₂₃×#123	1	2	yes	—
278	C₂×C₄	C₂₃×#124	1	2	yes	—
279	C₂×C₄	C₂₃×#125	1	2	yes	—
280	C₂×C₄	C₂₃×#126	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	4	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
290	C₂²	C₂₃×#381	1	1	yes	characteristic
291	C₂²	C₂₃×#383	1	1	yes	characteristic
292	C₂²	C₂₃×#387	1	2	yes	—
293	C₂²	C₂₃×#388	1	2	yes	—
294	C₄	—	2	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	8	4	no	—
297	C₄	C₂₃×#376	1	1	yes	characteristic
298	C₄	C₄×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
299	C₄	C₂₃×#380	1	1	yes	characteristic
300	C₄	C₂₃×#382	1	1	yes	characteristic
301	C₄	C₂₃×#387	1	2	yes	—
302	C₄	C₂₃×#388	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	2	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	C₂₃×#1434	1	1	yes	characteristic
308	C₂	C₂₃×#1841	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₂₃	#117
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
socle	C₂²×C₂₃	#383
prime radical	C₂⁴×C₂₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂²×M₁₆×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₂×C₂₃×#97	C₄	semidirect
13. C₂×C₄×C₈×C₂₃	C₄	semidirect
16. C₂₃×#338	C₄	semidirect
17. C₂₃×H(3,ℤ/4)	C₄	semidirect
95. #10813	C₂₃	direct (N × H)
160. C₂×#97	C₄×C₂₃	semidirect
161. C₂×C₄×C₈	C₄×C₂₃	semidirect
164. #338	C₄×C₂₃	semidirect
165. H(3,ℤ/4)	C₄×C₂₃	semidirect
205. C₂₃	#10813	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×M₁₆×C₂₃	2
C₂×C₂₃×#344	2
C₂×C₂₃×#339	2
C₂×C₂₃×#338	2
C₂×C₂₃×H(3,ℤ/4)	2
#10813	23

Subgroups — C23×#10813 #19158439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10813 #19158439