Subgroups — C₂₃×#2661 #15670280

Order 2944.

598 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#123	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#398	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#397	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#394	C₂	1	2	yes	maximal, complemented
6	C₂₃×#394	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#392	C₂	1	2	yes	maximal, complemented
8	C₂₃×#392	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#390	C₂	1	2	yes	maximal, complemented
10	C₂₃×#388	C₂	1	2	yes	maximal, complemented
11	C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
15	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#123	—	2	4	no	—
18	C₂₃×#121	—	2	4	no	—
19	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
30	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
32	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
33	C₂₃×#95	C₂²	1	1	yes	characteristic
34	C₂₃×#95	C₂²	1	1	yes	characteristic
35	C₂₃×#95	C₂²	1	2	yes	—
36	C₂₃×#95	C₂²	1	2	yes	—
37	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
38	C₂⁴×C₂₃	C₂³	1	2	yes	—
39	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
40	C₂×D₄×C₂₃	—	2	4	no	—
41	C₂×D₄×C₂₃	—	2	2	no	—
42	C₂×D₄×C₂₃	—	2	2	no	—
43	C₂×D₄×C₂₃	—	2	4	no	—
44	C₂×D₄×C₂₃	—	4	2	no	—
45	C₂×D₄×C₂₃	—	4	2	no	—
46	C₂×D₄×C₂₃	—	4	2	no	—
47	C₂×D₄×C₂₃	—	4	2	no	—
48	C₂×D₄×C₂₃	—	4	2	no	—
49	C₂×D₄×C₂₃	—	4	2	no	—
50	C₂²×C₄×C₂₃	—	4	2	no	—
51	C₂²×C₄×C₂₃	D₄	1	4	yes	complemented
52	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	—	2	2	no	—
62	C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₄⋊C₄×C₂₃	—	4	2	no	—
64	C₂²⋊C₄×C₂₃	—	2	2	no	—
65	C₂²⋊C₄×C₂₃	—	2	4	no	—
66	C₂²⋊C₄×C₂₃	—	2	2	no	—
67	C₂²⋊C₄×C₂₃	—	2	2	no	—
68	C₂²⋊C₄×C₂₃	—	2	4	no	—
69	C₂²⋊C₄×C₂₃	—	2	4	no	—
70	C₂²⋊C₄×C₂₃	—	4	2	no	—
71	C₂²⋊C₄×C₂₃	—	4	2	no	—
72	C₂²⋊C₄×C₂₃	—	4	2	no	—
73	C₄²×C₂₃	—	4	2	no	—
74	C₂³×C₂₃	—	2	2	no	—
75	C₂³×C₂₃	—	2	2	no	—
76	C₂³×C₂₃	—	2	1	no	—
77	C₂³×C₂₃	—	2	1	no	—
78	C₂³×C₂₃	—	2	1	no	—
79	C₂³×C₂₃	—	2	2	no	—
80	C₂³×C₂₃	—	2	2	no	—
81	C₂³×C₂₃	—	2	2	no	—
82	C₂³×C₂₃	—	2	2	no	—
83	C₂³×C₂₃	—	2	1	no	—
84	C₂³×C₂₃	—	2	2	no	—
85	C₂³×C₂₃	—	2	2	no	—
86	C₂³×C₂₃	—	2	2	no	—
87	C₂³×C₂₃	—	2	2	no	—
88	C₂³×C₂₃	—	2	2	no	—
89	C₂³×C₂₃	—	2	1	no	—
90	C₂³×C₂₃	C₂×D₄	1	2	yes	—
91	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
92	D₄×C₂₃	—	4	4	no	—
93	D₄×C₂₃	—	4	8	no	—
94	D₄×C₂₃	—	4	4	no	—
95	D₄×C₂₃	—	4	8	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	1	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	2	no	—
110	C₂×C₄×C₂₃	—	2	2	no	—
111	C₂×C₄×C₂₃	—	2	1	no	—
112	C₂×C₄×C₂₃	—	2	2	no	—
113	C₂×C₄×C₂₃	—	4	4	no	—
114	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
116	C₂×C₄×C₂₃	#41	1	4	yes	—
117	#2661	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₂₃	—	2	2	no	—
119	C₂²×C₂₃	—	2	2	no	—
120	C₂²×C₂₃	—	2	2	no	—
121	C₂²×C₂₃	—	4	2	no	—
122	C₂²×C₂₃	—	4	2	no	—
123	C₂²×C₂₃	—	4	1	no	—
124	C₂²×C₂₃	—	4	2	no	—
125	C₂²×C₂₃	—	4	2	no	—
126	C₂²×C₂₃	—	4	1	no	—
127	C₂²×C₂₃	—	4	2	no	—
128	C₂²×C₂₃	—	4	1	no	—
129	C₂²×C₂₃	—	4	2	no	—
130	C₂²×C₂₃	—	4	2	no	—
131	C₂²×C₂₃	—	4	2	no	—
132	C₂²×C₂₃	—	4	2	no	—
133	C₂²×C₂₃	—	4	2	no	—
134	C₂²×C₂₃	—	4	2	no	—
135	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
136	C₂²×C₂₃	C₂×#41	1	2	yes	—
137	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
138	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
139	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
140	C₄×C₂₃	—	2	2	no	—
141	C₄×C₂₃	—	2	2	no	—
142	C₄×C₂₃	—	2	4	no	—
143	C₄×C₂₃	—	4	1	no	—
144	C₄×C₂₃	—	4	1	no	—
145	C₄×C₂₃	—	4	1	no	—
146	C₄×C₂₃	—	4	2	no	—
147	C₄×C₂₃	—	4	2	no	—
148	C₂×#123	C₂×C₂₃	1	1	yes	characteristic, complemented
149	C₂×#121	C₂×C₂₃	1	1	yes	characteristic, complemented
150	#398	C₂×C₂₃	1	1	yes	characteristic, complemented
151	#397	C₂×C₂₃	1	1	yes	characteristic, complemented
152	#394	C₂×C₂₃	1	1	yes	characteristic, complemented
153	#394	C₂×C₂₃	1	2	yes	complemented
154	#392	C₂×C₂₃	1	1	yes	characteristic, complemented
155	#392	C₂×C₂₃	1	2	yes	complemented
156	#390	C₂×C₂₃	1	2	yes	complemented
157	#388	C₂×C₂₃	1	2	yes	complemented
158	C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
159	C₂×C₂₃	—	4	2	no	—
160	C₂×C₂₃	—	8	2	no	—
161	C₂×C₂₃	—	8	1	no	—
162	C₂×C₂₃	#532	1	1	yes	characteristic
163	C₂×C₂₃	#534	1	1	yes	characteristic
164	C₂×C₂₃	#535	1	1	yes	characteristic
165	C₂×C₂₃	#538	1	2	yes	—
166	C₂×C₂₃	#540	1	2	yes	—
167	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
168	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
169	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
170	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
171	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
172	#123	—	2	4	no	—
173	#121	—	2	4	no	—
174	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
177	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
178	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
179	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
182	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
183	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
184	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
185	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
186	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
187	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
188	#95	C₂²×C₂₃	1	1	yes	characteristic
189	#95	C₂²×C₂₃	1	1	yes	characteristic
190	#95	C₂²×C₂₃	1	2	yes	—
191	#95	C₂²×C₂₃	1	2	yes	—
192	C₂₃	#2661	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
194	C₂⁴	C₂³×C₂₃	1	2	yes	—
195	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	2	no	—
202	C₂×D₄	—	4	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	D₄×C₂₃	1	4	yes	complemented
208	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
209	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
211	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
214	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
215	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	4	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	4	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₄²	—	4	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	1	no	—
246	C₂³	C₂×D₄×C₂₃	1	2	yes	—
247	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
248	D₄	—	4	4	no	—
249	D₄	—	4	8	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	8	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
271	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
272	C₂×C₄	#41×C₂₃	1	4	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
291	C₂²	C₂×#41×C₂₃	1	2	yes	—
292	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
293	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
294	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	4	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	2	no	—
303	C₂	—	4	2	no	—
304	C₂	—	8	2	no	—
305	C₂	—	8	1	no	—
306	C₂	C₂₃×#532	1	1	yes	characteristic
307	C₂	C₂₃×#534	1	1	yes	characteristic
308	C₂	C₂₃×#535	1	1	yes	characteristic
309	C₂	C₂₃×#538	1	2	yes	—
310	C₂	C₂₃×#540	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#123	C₂	semidirect
2. C₂×C₂₃×#121	C₂	semidirect
3. C₂₃×#398	C₂	semidirect
4. C₂₃×#397	C₂	semidirect
5. C₂₃×#394	C₂	semidirect
6. C₂₃×#394	C₂	semidirect
7. C₂₃×#392	C₂	semidirect
8. C₂₃×#392	C₂	semidirect
9. C₂₃×#390	C₂	semidirect
10. C₂₃×#388	C₂	semidirect
11. C₄×C₂²⋊C₄×C₂₃	C₂	semidirect
51. C₂²×C₄×C₂₃	D₄	semidirect
117. #2661	C₂₃	direct (N × H)
148. C₂×#123	C₂×C₂₃	semidirect
149. C₂×#121	C₂×C₂₃	semidirect
150. #398	C₂×C₂₃	semidirect
151. #397	C₂×C₂₃	semidirect
152. #394	C₂×C₂₃	semidirect
153. #394	C₂×C₂₃	semidirect
154. #392	C₂×C₂₃	semidirect
155. #392	C₂×C₂₃	semidirect
156. #390	C₂×C₂₃	semidirect
157. #388	C₂×C₂₃	semidirect
158. C₄×C₂²⋊C₄	C₂×C₂₃	semidirect
192. C₂₃	#2661	direct (N × H)
207. C₂²×C₄	D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#123	2
C₂×C₂₃×#121	2
C₂₃×#398	2
C₂₃×#397	2
C₂₃×#394	2
C₂₃×#392	2
C₂₃×#390	2
C₂₃×#388	2
C₄×C₂²⋊C₄×C₂₃	2
#2661	23

Subgroups — C23×#2661 #15670280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2661 #15670280