Subgroups — C₂₃×#2682 #15670301

Order 2944.

668 conjugacy classes of subgroups.

The subgroup lattice has 300 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	2	yes	maximal, complemented
2	C₂×C₂₃×#121	C₂	1	2	yes	maximal, complemented
3	C₂₃×#396	C₂	1	2	yes	maximal, complemented
4	C₂₃×#394	C₂	1	2	yes	maximal, complemented
5	C₂₃×#392	C₂	1	2	yes	maximal, complemented
6	C₂₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#390	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#389	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#385	C₂	1	2	yes	maximal, complemented
10	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
11	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
15	C₂₃×#123	—	2	8	no	—
16	C₂₃×#121	—	2	8	no	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
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	1	yes	characteristic
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	2	yes	—
27	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
30	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
31	C₂₃×#95	C₂²	1	2	yes	—
32	C₂₃×#95	C₂²	1	2	yes	—
33	C₂⁴×C₂₃	C₂³	1	2	yes	—
34	C₂⁴×C₂₃	C₂³	1	2	yes	—
35	C₂×D₄×C₂₃	—	2	4	no	—
36	C₂×D₄×C₂₃	—	2	4	no	—
37	C₂×D₄×C₂₃	—	2	4	no	—
38	C₂×D₄×C₂₃	—	2	2	no	—
39	C₂×D₄×C₂₃	—	4	1	no	—
40	C₂×D₄×C₂₃	—	4	1	no	—
41	C₂×D₄×C₂₃	—	4	1	no	—
42	C₂×D₄×C₂₃	—	4	1	no	—
43	C₂×D₄×C₂₃	—	4	2	no	—
44	C₂×D₄×C₂₃	—	4	2	no	—
45	C₂×D₄×C₂₃	—	4	2	no	—
46	C₂×D₄×C₂₃	D₄	1	4	yes	complemented
47	C₂²×C₄×C₂₃	—	2	4	no	—
48	C₂²×C₄×C₂₃	—	2	4	no	—
49	C₂²×C₄×C₂₃	—	4	2	no	—
50	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
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	C₄⋊C₄×C₂₃	—	2	4	no	—
58	C₄⋊C₄×C₂₃	—	2	4	no	—
59	C₄⋊C₄×C₂₃	—	4	1	no	—
60	C₄⋊C₄×C₂₃	—	4	1	no	—
61	C₄⋊C₄×C₂₃	—	4	1	no	—
62	C₄⋊C₄×C₂₃	D₄	1	4	yes	complemented
63	C₂²⋊C₄×C₂₃	—	2	4	no	—
64	C₂²⋊C₄×C₂₃	—	2	4	no	—
65	C₂²⋊C₄×C₂₃	—	2	4	no	—
66	C₂²⋊C₄×C₂₃	—	2	4	no	—
67	C₂²⋊C₄×C₂₃	—	2	4	no	—
68	C₂²⋊C₄×C₂₃	—	2	4	no	—
69	C₂²⋊C₄×C₂₃	—	4	2	no	—
70	C₄²×C₂₃	—	4	1	no	—
71	C₂³×C₂₃	—	2	2	no	—
72	C₂³×C₂₃	—	2	2	no	—
73	C₂³×C₂₃	—	2	2	no	—
74	C₂³×C₂₃	—	2	2	no	—
75	C₂³×C₂₃	—	2	2	no	—
76	C₂³×C₂₃	—	2	2	no	—
77	C₂³×C₂₃	—	2	2	no	—
78	C₂³×C₂₃	—	2	2	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	2	no	—
84	C₂³×C₂₃	C₂×D₄	1	4	yes	—
85	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
86	D₄×C₂₃	—	4	4	no	—
87	D₄×C₂₃	—	4	8	no	—
88	D₄×C₂₃	—	4	4	no	—
89	D₄×C₂₃	—	4	8	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	2	2	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	—	4	8	no	—
108	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
109	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
111	C₂×C₄×C₂₃	#41	1	2	yes	—
112	#2682	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₂₃	—	2	4	no	—
114	C₂²×C₂₃	—	2	4	no	—
115	C₂²×C₂₃	—	2	4	no	—
116	C₂²×C₂₃	—	4	2	no	—
117	C₂²×C₂₃	—	4	2	no	—
118	C₂²×C₂₃	—	4	2	no	—
119	C₂²×C₂₃	—	4	2	no	—
120	C₂²×C₂₃	—	4	2	no	—
121	C₂²×C₂₃	—	4	2	no	—
122	C₂²×C₂₃	—	4	2	no	—
123	C₂²×C₂₃	—	4	2	no	—
124	C₂²×C₂₃	—	4	2	no	—
125	C₂²×C₂₃	—	4	2	no	—
126	C₂²×C₂₃	—	4	2	no	—
127	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
129	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
130	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	—	2	4	no	—
138	C₄×C₂₃	—	4	2	no	—
139	C₄×C₂₃	—	4	2	no	—
140	C₄×C₂₃	—	4	2	no	—
141	C₂×#123	C₂×C₂₃	1	2	yes	complemented
142	C₂×#121	C₂×C₂₃	1	2	yes	complemented
143	#396	C₂×C₂₃	1	2	yes	complemented
144	#394	C₂×C₂₃	1	2	yes	complemented
145	#392	C₂×C₂₃	1	2	yes	complemented
146	#390	C₂×C₂₃	1	1	yes	characteristic, complemented
147	#390	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#389	C₂×C₂₃	1	1	yes	characteristic, complemented
149	#385	C₂×C₂₃	1	2	yes	complemented
150	C₂×C₂₃	—	4	4	no	—
151	C₂×C₂₃	—	8	2	no	—
152	C₂×C₂₃	#534	1	1	yes	characteristic
153	C₂×C₂₃	#537	1	1	yes	characteristic
154	C₂×C₂₃	#538	1	1	yes	characteristic
155	C₂×C₂₃	D₄²	1	1	yes	characteristic
156	C₂×C₂₃	#547	1	1	yes	characteristic
157	C₂×C₂₃	#550	1	1	yes	characteristic
158	C₂×C₂₃	#562	1	1	yes	characteristic
159	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
160	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
161	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
162	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
163	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
164	#123	—	2	8	no	—
165	#121	—	2	8	no	—
166	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
167	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
168	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
169	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
173	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
174	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
177	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
178	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
179	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
180	#95	C₂²×C₂₃	1	2	yes	—
181	#95	C₂²×C₂₃	1	2	yes	—
182	C₂₃	#2682	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	C₂³×C₂₃	1	2	yes	—
184	C₂⁴	C₂³×C₂₃	1	2	yes	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	2	4	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	4	1	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂×D₄	—	4	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂×D₄	D₄×C₂₃	1	4	yes	complemented
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
201	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
202	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
203	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₄⋊C₄	—	4	1	no	—
212	C₄⋊C₄	D₄×C₂₃	1	4	yes	complemented
213	C₂²⋊C₄	—	2	4	no	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₂²⋊C₄	—	2	4	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₄²	—	4	1	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₂×D₄×C₂₃	1	4	yes	—
235	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
236	D₄	—	4	4	no	—
237	D₄	—	4	8	no	—
238	D₄	—	4	4	no	—
239	D₄	—	4	8	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	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	2	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
259	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
260	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
261	C₂×C₄	#41×C₂₃	1	2	yes	—
262	C₂²	—	2	4	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
277	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
278	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
279	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
280	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
281	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
282	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	2	2	no	—
286	C₄	—	2	4	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	2	no	—
290	C₂	—	4	4	no	—
291	C₂	—	8	2	no	—
292	C₂	C₂₃×#534	1	1	yes	characteristic
293	C₂	C₂₃×#537	1	1	yes	characteristic
294	C₂	C₂₃×#538	1	1	yes	characteristic
295	C₂	D₄²×C₂₃	1	1	yes	characteristic
296	C₂	C₂₃×#547	1	1	yes	characteristic
297	C₂	C₂₃×#550	1	1	yes	characteristic
298	C₂	C₂₃×#562	1	1	yes	characteristic

	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×#396	C₂	semidirect
4. C₂₃×#394	C₂	semidirect
5. C₂₃×#392	C₂	semidirect
6. C₂₃×#390	C₂	semidirect
7. C₂₃×#390	C₂	semidirect
8. C₂₃×#389	C₂	semidirect
9. C₂₃×#385	C₂	semidirect
46. C₂×D₄×C₂₃	D₄	semidirect
62. C₄⋊C₄×C₂₃	D₄	semidirect
112. #2682	C₂₃	direct (N × H)
141. C₂×#123	C₂×C₂₃	semidirect
142. C₂×#121	C₂×C₂₃	semidirect
143. #396	C₂×C₂₃	semidirect
144. #394	C₂×C₂₃	semidirect
145. #392	C₂×C₂₃	semidirect
146. #390	C₂×C₂₃	semidirect
147. #390	C₂×C₂₃	semidirect
148. #389	C₂×C₂₃	semidirect
149. #385	C₂×C₂₃	semidirect
182. C₂₃	#2682	direct (N × H)
196. C₂×D₄	D₄×C₂₃	semidirect
212. C₄⋊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×#396	2
C₂₃×#394	2
C₂₃×#392	2
C₂₃×#390	2
C₂₃×#389	2
C₂₃×#385	2
#2682	23

Subgroups — C23×#2682 #15670301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2682 #15670301