Subgroups — C₂₃×#12981 #19160607

Order 5888.

488 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#2929	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2115	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#2112	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1378	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#525	C₄	1	1	yes	characteristic
9	C₂₃×#504	—	2	4	no	—
10	C₂×C₂₃×M₃₂	—	2	2	no	—
11	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₂₃×M₃₂	C₂²	1	2	yes	—
13	C₂₃×#435	—	2	2	no	—
14	C₈×D₄×C₂₃	—	2	2	no	—
15	C₂₃×#424	C₄	1	1	yes	characteristic
16	C₂₃×#408	C₄	1	1	yes	characteristic
17	C₂₃×#407	C₄	1	1	yes	characteristic
18	C₂₃×#405	C₂²	1	1	yes	characteristic
19	C₂₃×#349	—	2	2	no	—
20	C₂₃×C₂²⋊C₁₆	—	2	4	no	—
21	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂₃×#131	—	2	2	no	—
24	C₂₃×#131	—	2	2	no	—
25	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	—	2	1	no	—
31	C₂²×C₈×C₂₃	—	2	2	no	—
32	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₂₃	D₄	1	2	yes	—
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×#123	C₈	1	2	yes	—
36	C₂₃×C₂²⋊Q₈	—	2	1	no	—
37	C₂₃×#121	—	2	1	no	—
38	C₂₃×#120	C₈	1	2	yes	—
39	C₄×D₄×C₂₃	—	2	2	no	—
40	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
41	C₂₃×M₃₂	—	2	4	no	—
42	C₂₃×M₃₂	—	2	2	no	—
43	C₂₃×M₃₂	—	2	2	no	—
44	C₂₃×M₃₂	—	2	2	no	—
45	C₂₃×M₃₂	—	2	2	no	—
46	C₂×C₁₆×C₂₃	—	2	4	no	—
47	C₂×C₁₆×C₂₃	—	2	2	no	—
48	C₂×C₁₆×C₂₃	D₄	1	4	yes	complemented
49	C₂₃×C₄⋊C₈	—	2	2	no	—
50	C₂₃×C₂²⋊C₈	—	2	2	no	—
51	C₂₃×C₂²⋊C₈	—	2	2	no	—
52	C₂₃×#97	—	2	1	no	—
53	C₄×C₈×C₂₃	—	2	1	no	—
54	C₂⁴×C₂₃	C₂×C₈	1	1	yes	characteristic
55	#41×C₂₃	—	2	2	no	—
56	#41×C₂₃	—	2	2	no	—
57	C₂×Q₈×C₂₃	M₁₆	1	1	yes	characteristic
58	C₂×D₄×C₂₃	—	2	2	no	—
59	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
60	C₂×D₄×C₂₃	M₁₆	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	—	2	1	no	—
62	C₂²×C₄×C₂₃	—	2	2	no	—
63	C₂²×C₄×C₂₃	—	2	1	no	—
64	C₂²×C₄×C₂₃	—	2	2	no	—
65	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	M₁₆×C₂₃	—	2	2	no	—
69	M₁₆×C₂₃	—	2	4	no	—
70	M₁₆×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	2	no	—
72	C₂×C₈×C₂₃	—	2	2	no	—
73	C₂×C₈×C₂₃	—	2	1	no	—
74	C₂×C₈×C₂₃	—	2	1	no	—
75	C₂×C₈×C₂₃	—	2	2	no	—
76	C₂×C₈×C₂₃	—	4	1	no	—
77	C₂×C₈×C₂₃	—	4	2	no	—
78	C₂×C₈×C₂₃	—	4	1	no	—
79	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₂₃	#41	1	2	yes	—
84	C₄⋊C₄×C₂₃	—	2	2	no	—
85	C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂²⋊C₄×C₂₃	—	2	2	no	—
87	C₂²⋊C₄×C₂₃	—	2	2	no	—
88	C₂²⋊C₄×C₂₃	C₂×C₈	1	2	yes	—
89	C₄²×C₂₃	—	2	1	no	—
90	C₁₆×C₂₃	—	2	4	no	—
91	C₁₆×C₂₃	—	4	2	no	—
92	#12981	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₂₃	—	2	1	no	—
94	C₂³×C₂₃	—	2	2	no	—
95	C₂³×C₂₃	—	4	2	no	—
96	C₂³×C₂₃	C₂²⋊C₈	1	2	yes	—
97	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
98	Q₈×C₂₃	—	2	2	no	—
99	D₄×C₂₃	—	2	4	no	—
100	D₄×C₂₃	—	2	2	no	—
101	D₄×C₂₃	—	4	4	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	1	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	—	4	1	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
111	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₂₃	#131	1	2	yes	—
115	C₈×C₂₃	—	2	2	no	—
116	C₈×C₂₃	—	4	1	no	—
117	C₈×C₂₃	—	4	1	no	—
118	C₈×C₂₃	#120	1	1	yes	characteristic
119	C₈×C₂₃	#121	1	2	yes	—
120	C₈×C₂₃	#123	1	1	yes	characteristic
121	#3407	C₂×C₂₃	1	1	yes	characteristic, complemented
122	#2929	C₂×C₂₃	1	1	yes	characteristic
123	#2115	C₂×C₂₃	1	1	yes	characteristic, complemented
124	#2112	C₂×C₂₃	1	2	yes	complemented
125	#1378	C₂×C₂₃	1	2	yes	complemented
126	C₂²×C₂₃	—	2	2	no	—
127	C₂²×C₂₃	—	2	4	no	—
128	C₂²×C₂₃	—	2	2	no	—
129	C₂²×C₂₃	—	4	1	no	—
130	C₂²×C₂₃	—	4	2	no	—
131	C₂²×C₂₃	—	4	1	no	—
132	C₂²×C₂₃	—	4	2	no	—
133	C₂²×C₂₃	—	4	1	no	—
134	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₂²×C₂₃	C₈×D₄	1	2	yes	—
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	—	4	2	no	—
138	C₄×C₂₃	—	4	1	no	—
139	C₄×C₂₃	#386	1	1	yes	characteristic
140	C₄×C₂₃	#408	1	1	yes	characteristic
141	C₄×C₂₃	#435	1	2	yes	—
142	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
143	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
144	#525	C₄×C₂₃	1	1	yes	characteristic
145	#504	—	2	4	no	—
146	C₂×M₃₂	—	2	2	no	—
147	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
148	C₂×M₃₂	C₂²×C₂₃	1	2	yes	—
149	#435	—	2	2	no	—
150	C₈×D₄	—	2	2	no	—
151	#424	C₄×C₂₃	1	1	yes	characteristic
152	#408	C₄×C₂₃	1	1	yes	characteristic
153	#407	C₄×C₂₃	1	1	yes	characteristic
154	#405	C₂²×C₂₃	1	1	yes	characteristic
155	#349	—	2	2	no	—
156	C₂²⋊C₁₆	—	2	4	no	—
157	C₂×C₂₃	—	2	1	no	—
158	C₂×C₂₃	—	2	2	no	—
159	C₂×C₂₃	—	4	2	no	—
160	C₂×C₂₃	—	4	2	no	—
161	C₂×C₂₃	#1869	1	1	yes	characteristic
162	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
163	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
164	#131	—	2	2	no	—
165	#131	—	2	2	no	—
166	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
167	C₂×M₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
168	C₂×M₁₆	D₄×C₂₃	1	2	yes	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	—	2	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
174	C₂²×C₈	D₄×C₂₃	1	2	yes	—
175	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
176	#123	C₈×C₂₃	1	2	yes	—
177	C₂²⋊Q₈	—	2	1	no	—
178	#121	—	2	1	no	—
179	#120	C₈×C₂₃	1	2	yes	—
180	C₄×D₄	—	2	2	no	—
181	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
182	M₃₂	—	2	4	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	2	2	no	—
185	M₃₂	—	2	2	no	—
186	M₃₂	—	2	2	no	—
187	C₂×C₁₆	—	2	4	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	D₄×C₂₃	1	4	yes	complemented
190	C₄⋊C₈	—	2	2	no	—
191	C₂²⋊C₈	—	2	2	no	—
192	C₂²⋊C₈	—	2	2	no	—
193	#97	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₂₃	#12981	1	1	yes	characteristic, direct product factor, Sylow
196	C₂⁴	C₂×C₈×C₂₃	1	1	yes	characteristic
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	C₂×Q₈	M₁₆×C₂₃	1	1	yes	characteristic
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
202	C₂×D₄	M₁₆×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	1	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
208	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
210	M₁₆	—	2	2	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	4	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
223	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
225	C₂×C₈	#41×C₂₃	1	2	yes	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	C₂×C₈×C₂₃	1	2	yes	—
231	C₄²	—	2	1	no	—
232	C₁₆	—	2	4	no	—
233	C₁₆	—	4	2	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	C₂₃×C₂²⋊C₈	1	2	yes	—
238	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
239	Q₈	—	2	2	no	—
240	D₄	—	2	4	no	—
241	D₄	—	2	2	no	—
242	D₄	—	4	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	C₂₃×C₂²⋊C₈	1	2	yes	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
254	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂₃×#131	1	2	yes	—
256	C₈	—	2	2	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	C₂₃×#120	1	1	yes	characteristic
260	C₈	C₂₃×#121	1	2	yes	—
261	C₈	C₂₃×#123	1	1	yes	characteristic
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
271	C₂²	C₈×D₄×C₂₃	1	2	yes	—
272	C₄	—	2	2	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	1	no	—
275	C₄	C₂₃×#386	1	1	yes	characteristic
276	C₄	C₂₃×#408	1	1	yes	characteristic
277	C₄	C₂₃×#435	1	2	yes	—
278	C₂	—	2	1	no	—
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	—	4	2	no	—
282	C₂	C₂₃×#1869	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₂₃	#120
commutator	C₂³	C₂²×C₈×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂×C₂₃	#1869
prime radical	C₂₃×#525	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₂₃ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#3407	C₂	semidirect
3. C₂₃×#2115	C₂	semidirect
4. C₂₃×#2112	C₂	semidirect
5. C₂₃×#1378	C₂	semidirect
48. C₂×C₁₆×C₂₃	D₄	semidirect
92. #12981	C₂₃	direct (N × H)
121. #3407	C₂×C₂₃	semidirect
123. #2115	C₂×C₂₃	semidirect
124. #2112	C₂×C₂₃	semidirect
125. #1378	C₂×C₂₃	semidirect
189. C₂×C₁₆	D₄×C₂₃	semidirect
195. C₂₃	#12981	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#3407	2
C₂₃×#2929	2
C₂₃×#2115	2
C₂₃×#2112	2
C₂₃×#1378	2
#12981	23

Subgroups — C23×#12981 #19160607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12981 #19160607