Subgroups — C₂₃×#33438 #19181064

Order 5888.

682 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#504	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2169	C₂	1	2	yes	maximal, complemented
4	C₂₃×#2168	C₂	1	4	yes	maximal, complemented
5	D₄×C₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#2113	C₂	1	2	yes	maximal, complemented
8	C₄×C₂₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#131	C₂²	1	2	yes	—
10	C₂₃×#532	C₄	1	1	yes	characteristic
11	C₂₃×#504	—	2	8	no	—
12	C₂×C₂₃×M₃₂	C₂²	1	4	yes	—
13	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₂₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₂₃	C₂²	1	4	yes	—
17	C₂₃×#436	C₄	1	2	yes	—
18	C₂₃×#436	C₄	1	2	yes	—
19	C₈×D₄×C₂₃	C₂²	1	2	yes	—
20	C₈×D₄×C₂₃	C₂²	1	2	yes	—
21	C₂₃×#423	C₄	1	1	yes	characteristic
22	C₂₃×#408	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₂₃×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₂₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₂₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
28	C₂₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
29	C₂₃×#346	C₂²	1	2	yes	—
30	C₄×C₁₆×C₂₃	C₂²	1	2	yes	—
31	C₂×#41×C₂₃	C₂×C₄	1	2	yes	—
32	C₂₃×#131	—	2	4	no	—
33	C₂₃×#131	—	2	4	no	—
34	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₂₃	C₂×C₄	1	4	yes	—
36	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
38	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
40	C₂₃×#127	C₈	1	1	yes	characteristic
41	C₂₃×#124	C₈	1	2	yes	—
42	C₂₃×#121	C₈	1	2	yes	—
43	C₂₃×#121	C₈	1	2	yes	—
44	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
45	C₄×D₄×C₂₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
47	C₂₃×M₃₂	—	2	2	no	—
48	C₂₃×M₃₂	—	2	8	no	—
49	C₂₃×M₃₂	D₄	1	4	yes	complemented
50	C₂×C₁₆×C₂₃	—	2	4	no	—
51	C₂×C₁₆×C₂₃	—	2	8	no	—
52	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
53	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
54	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
55	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
56	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
57	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
58	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
59	C₂₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
60	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	#41×C₂₃	—	2	4	no	—
65	#41×C₂₃	—	2	4	no	—
66	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	—
67	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
68	C₂×D₄×C₂₃	C₂×C₈	1	4	yes	—
69	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
70	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	M₁₆×C₂₃	—	2	4	no	—
76	M₁₆×C₂₃	—	2	8	no	—
77	C₂×C₈×C₂₃	—	2	2	no	—
78	C₂×C₈×C₂₃	—	2	4	no	—
79	C₂×C₈×C₂₃	—	2	4	no	—
80	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₂₃	#41	1	2	yes	—
86	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
88	C₄⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
89	C₂²⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
90	C₂²⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
91	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
92	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
95	C₁₆×C₂₃	—	2	2	no	—
96	C₁₆×C₂₃	—	2	2	no	—
97	C₁₆×C₂₃	—	2	2	no	—
98	C₁₆×C₂₃	C₂×D₄	1	4	yes	—
99	#33438	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₂₃	C₂²×C₈	1	4	yes	—
101	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
102	Q₈×C₂₃	—	2	4	no	—
103	D₄×C₂₃	—	2	8	no	—
104	D₄×C₂₃	—	2	4	no	—
105	D₄×C₂₃	—	2	4	no	—
106	D₄×C₂₃	—	4	2	no	—
107	C₂×C₄×C₂₃	—	2	4	no	—
108	C₂×C₄×C₂₃	—	2	4	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	4	1	no	—
111	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂²×C₈	1	4	yes	—
116	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₂₃	#131	1	2	yes	—
119	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
120	C₈×C₂₃	—	2	4	no	—
121	C₈×C₂₃	—	2	1	no	—
122	C₈×C₂₃	—	2	1	no	—
123	C₈×C₂₃	C₄×D₄	1	2	yes	—
124	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
125	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
126	C₂×#504	C₂×C₂₃	1	2	yes	complemented
127	#2931	C₂×C₂₃	1	1	yes	characteristic
128	#2169	C₂×C₂₃	1	2	yes	complemented
129	#2168	C₂×C₂₃	1	4	yes	complemented
130	D₄×C₁₆	C₂×C₂₃	1	2	yes	complemented
131	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
132	#2113	C₂×C₂₃	1	2	yes	complemented
133	C₄×M₃₂	C₂×C₂₃	1	1	yes	characteristic, complemented
134	C₂²×C₂₃	—	2	1	no	—
135	C₂²×C₂₃	—	2	1	no	—
136	C₂²×C₂₃	—	2	4	no	—
137	C₂²×C₂₃	—	2	4	no	—
138	C₂²×C₂₃	—	2	4	no	—
139	C₂²×C₂₃	C₈×D₄	1	2	yes	—
140	C₂²×C₂₃	C₂³×C₈	1	1	yes	characteristic
141	C₄×C₂₃	—	2	1	no	—
142	C₄×C₂₃	—	2	1	no	—
143	C₄×C₂₃	—	2	1	no	—
144	C₄×C₂₃	—	2	4	no	—
145	C₄×C₂₃	C₈×D₄	1	2	yes	—
146	C₄×C₂₃	#504	1	2	yes	complemented
147	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
149	C₂×#131	C₂²×C₂₃	1	2	yes	—
150	#532	C₄×C₂₃	1	1	yes	characteristic
151	#504	—	2	8	no	—
152	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
153	C₂×M₃₂	C₂²×C₂₃	1	4	yes	—
154	C₂×M₃₂	C₂²×C₂₃	1	1	yes	characteristic
155	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
156	C₂²×C₁₆	C₂²×C₂₃	1	4	yes	—
157	#436	C₄×C₂₃	1	2	yes	—
158	#436	C₄×C₂₃	1	2	yes	—
159	C₈×D₄	C₂²×C₂₃	1	2	yes	—
160	C₈×D₄	C₂²×C₂₃	1	2	yes	—
161	#423	C₄×C₂₃	1	1	yes	characteristic
162	#408	C₄×C₂₃	1	2	yes	—
163	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
164	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
165	C₄⋊C₁₆	C₂²×C₂₃	1	2	yes	—
166	C₄⋊C₁₆	C₂²×C₂₃	1	1	yes	characteristic
167	C₂²⋊C₁₆	C₂²×C₂₃	1	4	yes	—
168	C₂²⋊C₁₆	C₂²×C₂₃	1	4	yes	—
169	#346	C₂²×C₂₃	1	2	yes	—
170	C₄×C₁₆	C₂²×C₂₃	1	2	yes	—
171	C₂×C₂₃	—	2	2	no	—
172	C₂×C₂₃	—	4	4	no	—
173	C₂×C₂₃	C₂×C₈×D₄	1	1	yes	characteristic
174	C₂×C₂₃	C₂×#504	1	1	yes	characteristic
175	C₂×C₂₃	#3407	1	1	yes	characteristic
176	C₂×#41	C₂×C₄×C₂₃	1	2	yes	—
177	#131	—	2	4	no	—
178	#131	—	2	4	no	—
179	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
180	C₂×M₁₆	C₂×C₄×C₂₃	1	4	yes	—
181	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
183	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
184	C₄⋊Q₈	C₈×C₂₃	1	1	yes	characteristic
185	#127	C₈×C₂₃	1	1	yes	characteristic
186	#124	C₈×C₂₃	1	2	yes	—
187	#121	C₈×C₂₃	1	2	yes	—
188	#121	C₈×C₂₃	1	2	yes	—
189	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
190	C₄×D₄	C₂×C₄×C₂₃	1	2	yes	—
191	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
192	M₃₂	—	2	2	no	—
193	M₃₂	—	2	8	no	—
194	M₃₂	D₄×C₂₃	1	4	yes	complemented
195	C₂×C₁₆	—	2	4	no	—
196	C₂×C₁₆	—	2	8	no	—
197	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
198	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
199	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
200	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
201	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
202	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
203	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
204	C₂²⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
205	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
206	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
207	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
208	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
209	C₂₃	#33438	1	1	yes	characteristic, direct product factor, Sylow
210	#41	—	2	4	no	—
211	#41	—	2	4	no	—
212	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	—
213	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
214	C₂×D₄	C₂×C₈×C₂₃	1	4	yes	—
215	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
216	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
217	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
220	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	8	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
228	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
229	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
231	C₂×C₈	#41×C₂₃	1	2	yes	—
232	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
233	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
234	C₄⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
235	C₂²⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
236	C₂²⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
237	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
238	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
239	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
240	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
241	C₁₆	—	2	2	no	—
242	C₁₆	—	2	2	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	C₂×D₄×C₂₃	1	4	yes	—
245	C₂³	C₂²×C₈×C₂₃	1	4	yes	—
246	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	D₄	—	2	8	no	—
249	D₄	—	2	4	no	—
250	D₄	—	2	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
257	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
258	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₈×C₂₃	1	4	yes	—
261	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
262	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
263	C₂×C₄	C₂₃×#131	1	2	yes	—
264	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
265	C₈	—	2	1	no	—
266	C₈	—	2	1	no	—
267	C₈	—	2	4	no	—
268	C₈	C₄×D₄×C₂₃	1	2	yes	—
269	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
270	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	C₈×D₄×C₂₃	1	2	yes	—
277	C₂²	C₂³×C₈×C₂₃	1	1	yes	characteristic
278	C₄	—	2	4	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	2	1	no	—
282	C₄	C₈×D₄×C₂₃	1	2	yes	—
283	C₄	C₂₃×#504	1	2	yes	complemented
284	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
285	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	4	4	no	—
288	C₂	C₂×C₈×D₄×C₂₃	1	1	yes	characteristic
289	C₂	C₂×C₂₃×#504	1	1	yes	characteristic
290	C₂	C₂₃×#3407	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₃	C₂⁴
commutator	C₂²	C₂³×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂²×C₂₃	C₂³×C₈
prime radical	C₂₃×#532	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#504	C₂	semidirect
3. C₂₃×#2169	C₂	semidirect
4. C₂₃×#2168	C₂	semidirect
5. D₄×C₁₆×C₂₃	C₂	semidirect
6. C₂×C₂₃×C₄⋊C₁₆	C₂	semidirect
7. C₂₃×#2113	C₂	semidirect
8. C₄×C₂₃×M₃₂	C₂	semidirect
49. C₂₃×M₃₂	D₄	semidirect
99. #33438	C₂₃	direct (N × H)
126. C₂×#504	C₂×C₂₃	semidirect
128. #2169	C₂×C₂₃	semidirect
129. #2168	C₂×C₂₃	semidirect
130. D₄×C₁₆	C₂×C₂₃	semidirect
131. C₂×C₄⋊C₁₆	C₂×C₂₃	semidirect
132. #2113	C₂×C₂₃	semidirect
133. C₄×M₃₂	C₂×C₂₃	semidirect
146. C₄×C₂₃	#504	semidirect
194. M₃₂	D₄×C₂₃	semidirect
209. C₂₃	#33438	direct (N × H)
283. C₄	C₂₃×#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#504	2
C₂₃×#2931	2
C₂₃×#2169	2
C₂₃×#2168	2
D₄×C₁₆×C₂₃	2
C₂×C₂₃×C₄⋊C₁₆	2
C₂₃×#2113	2
C₄×C₂₃×M₃₂	2
#33438	23

Subgroups — C23×#33438 #19181064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#33438 #19181064