Subgroups — C₂₃×#12807 #19160433

Order 5888.

428 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2931	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2126	C₂	1	2	yes	maximal
3	C₂₃×#1812	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1391	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#131	C₄	1	1	yes	characteristic, complemented
6	C₂×C₂₃×#131	C₂²	1	1	yes	characteristic
7	C₂₃×#532	C₂²	1	1	yes	characteristic
8	C₂₃×#436	—	2	2	no	—
9	C₈×D₄×C₂₃	—	2	2	no	—
10	C₂₃×#427	—	2	2	no	—
11	C₂₃×#423	C₄	1	1	yes	characteristic, complemented
12	C₂₃×#410	C₂²	1	4	yes	—
13	C₂₃×#408	C₄	1	1	yes	characteristic, complemented
14	C₂₃×#408	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₂₃	C₄	1	1	yes	characteristic, complemented
16	C₂₃×#354	—	2	2	no	—
17	C₂₃×#353	—	2	2	no	—
18	C₂₃×H(3,ℤ/4)	—	2	2	no	—
19	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂×#41×C₂₃	C₂³	1	1	yes	characteristic
21	C₂₃×#131	—	4	1	no	—
22	C₂₃×#131	—	4	1	no	—
23	C₂₃×#131	—	4	1	no	—
24	C₂₃×#131	—	4	1	no	—
25	C₂×M₁₆×C₂₃	—	2	1	no	—
26	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₂₃	D₄	1	2	yes	—
29	C₂²×C₈×C₂₃	—	2	1	no	—
30	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₂₃	D₄	1	2	yes	—
33	C₂₃×C₄⋊Q₈	Q₈	1	1	yes	characteristic
34	C₂₃×#127	D₄	1	1	yes	characteristic
35	C₂₃×#124	D₄	1	1	yes	characteristic
36	C₂₃×#124	Q₈	1	1	yes	characteristic
37	C₂₃×#121	—	2	2	no	—
38	C₄×D₄×C₂₃	—	2	2	no	—
39	C₂₃×#117	—	2	4	no	—
40	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
41	C₂₃×#107	—	4	2	no	—
42	C₂₃×#106	—	4	2	no	—
43	C₂₃×C₄⋊C₈	—	2	2	no	—
44	C₂₃×#99	—	2	4	no	—
45	C₂₃×#99	—	2	4	no	—
46	C₂₃×C₂²⋊C₈	—	2	2	no	—
47	C₂₃×C₂²⋊C₈	—	2	2	no	—
48	C₄×C₈×C₂₃	—	2	1	no	—
49	C₄×C₈×C₂₃	—	2	1	no	—
50	#41×C₂₃	—	4	1	no	—
51	#41×C₂₃	—	4	1	no	—
52	#41×C₂₃	—	4	1	no	—
53	#41×C₂₃	—	4	1	no	—
54	C₂×Q₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
55	C₂×Q₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
56	C₂×D₄×C₂₃	—	2	1	no	—
57	C₂×D₄×C₂₃	—	2	2	no	—
58	C₂×D₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×D₄×C₂₃	C₂×D₄	1	1	yes	characteristic
60	C₂×D₄×C₂₃	#41	1	2	yes	—
61	C₂²×C₄×C₂₃	—	2	1	no	—
62	C₂²×C₄×C₂₃	—	2	1	no	—
63	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
65	M₁₆×C₂₃	—	4	1	no	—
66	M₁₆×C₂₃	—	4	2	no	—
67	M₁₆×C₂₃	—	4	2	no	—
68	M₁₆×C₂₃	—	4	1	no	—
69	C₂×C₈×C₂₃	—	2	2	no	—
70	C₂×C₈×C₂₃	—	2	2	no	—
71	C₂×C₈×C₂₃	—	2	1	no	—
72	C₂×C₈×C₂₃	—	2	1	no	—
73	C₂×C₈×C₂₃	—	2	1	no	—
74	C₂×C₈×C₂₃	—	2	1	no	—
75	C₂×C₈×C₂₃	—	4	1	no	—
76	C₂×C₈×C₂₃	—	4	1	no	—
77	C₂×C₈×C₂₃	—	4	1	no	—
78	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₂₃	—	2	2	no	—
82	C₄⋊C₄×C₂₃	—	4	4	no	—
83	C₂²⋊C₄×C₂₃	—	2	2	no	—
84	C₂²⋊C₄×C₂₃	—	2	2	no	—
85	C₂²⋊C₄×C₂₃	—	4	4	no	—
86	C₄²×C₂₃	—	2	1	no	—
87	C₄²×C₂₃	—	4	4	no	—
88	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
89	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
90	#12807	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₂₃	—	2	1	no	—
92	C₂³×C₂₃	#117	1	1	yes	characteristic
93	C₂³×C₂₃	#121	1	2	yes	—
94	Q₈×C₂₃	—	4	1	no	—
95	Q₈×C₂₃	—	4	1	no	—
96	D₄×C₂₃	—	4	1	no	—
97	D₄×C₂₃	—	4	2	no	—
98	D₄×C₂₃	—	4	1	no	—
99	D₄×C₂₃	—	4	4	no	—
100	D₄×C₂₃	—	4	2	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	1	no	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	4	2	no	—
109	C₂×C₄×C₂₃	—	4	1	no	—
110	C₂×C₄×C₂₃	—	4	1	no	—
111	C₂×C₄×C₂₃	—	8	4	no	—
112	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
115	C₈×C₂₃	—	2	2	no	—
116	C₈×C₂₃	—	2	1	no	—
117	C₈×C₂₃	—	4	1	no	—
118	C₈×C₂₃	—	4	1	no	—
119	C₈×C₂₃	#99	1	1	yes	characteristic
120	C₈×C₂₃	#99	1	1	yes	characteristic
121	#2931	C₂×C₂₃	1	1	yes	characteristic
122	#2126	C₂×C₂₃	1	2	yes	—
123	#1812	C₂×C₂₃	1	2	yes	complemented
124	#1391	C₂×C₂₃	1	2	yes	complemented
125	C₂²×C₂₃	—	2	2	no	—
126	C₂²×C₂₃	—	2	1	no	—
127	C₂²×C₂₃	—	4	1	no	—
128	C₂²×C₂₃	—	4	2	no	—
129	C₂²×C₂₃	—	4	2	no	—
130	C₂²×C₂₃	—	4	1	no	—
131	C₂²×C₂₃	#380	1	1	yes	characteristic
132	C₄×C₂₃	—	2	2	no	—
133	C₄×C₂₃	—	2	1	no	—
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	—	4	1	no	—
136	C₄×C₂₃	—	4	1	no	—
137	C₄×C₂₃	—	16	4	no	—
138	C₄×C₂₃	C₂×#99	1	1	yes	characteristic
139	C₄×C₂₃	#410	1	1	yes	characteristic
140	C₂×#131	C₄×C₂₃	1	1	yes	characteristic, complemented
141	C₂×#131	C₂²×C₂₃	1	1	yes	characteristic
142	#532	C₂²×C₂₃	1	1	yes	characteristic
143	#436	—	2	2	no	—
144	C₈×D₄	—	2	2	no	—
145	#427	—	2	2	no	—
146	#423	C₄×C₂₃	1	1	yes	characteristic, complemented
147	#410	C₂²×C₂₃	1	4	yes	—
148	#408	C₄×C₂₃	1	1	yes	characteristic, complemented
149	#408	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×C₄×C₈	C₄×C₂₃	1	1	yes	characteristic, complemented
151	#354	—	2	2	no	—
152	#353	—	2	2	no	—
153	H(3,ℤ/4)	—	2	2	no	—
154	C₂×C₂₃	—	2	1	no	—
155	C₂×C₂₃	—	4	2	no	—
156	C₂×C₂₃	—	4	1	no	—
157	C₂×C₂₃	—	8	1	no	—
158	C₂×C₂₃	#1799	1	1	yes	characteristic
159	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
160	C₂×#41	C₂³×C₂₃	1	1	yes	characteristic
161	#131	—	4	1	no	—
162	#131	—	4	1	no	—
163	#131	—	4	1	no	—
164	#131	—	4	1	no	—
165	C₂×M₁₆	—	2	1	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	1	no	—
170	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
172	C₂²×C₈	D₄×C₂₃	1	2	yes	—
173	C₄⋊Q₈	Q₈×C₂₃	1	1	yes	characteristic
174	#127	D₄×C₂₃	1	1	yes	characteristic
175	#124	D₄×C₂₃	1	1	yes	characteristic
176	#124	Q₈×C₂₃	1	1	yes	characteristic
177	#121	—	2	2	no	—
178	C₄×D₄	—	2	2	no	—
179	#117	—	2	4	no	—
180	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
181	#107	—	4	2	no	—
182	#106	—	4	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	#99	—	2	4	no	—
185	#99	—	2	4	no	—
186	C₂²⋊C₈	—	2	2	no	—
187	C₂²⋊C₈	—	2	2	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₂₃	#12807	1	1	yes	characteristic, direct product factor, Sylow
191	#41	—	4	1	no	—
192	#41	—	4	1	no	—
193	#41	—	4	1	no	—
194	#41	—	4	1	no	—
195	C₂×Q₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
196	C₂×Q₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
200	C₂×D₄	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₂×D₄	#41×C₂₃	1	2	yes	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	—	2	1	no	—
204	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
206	M₁₆	—	4	1	no	—
207	M₁₆	—	4	2	no	—
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	1	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
220	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
221	C₂×C₈	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	4	4	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	4	4	no	—
229	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
230	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
231	C₂³	—	2	1	no	—
232	C₂³	C₂₃×#117	1	1	yes	characteristic
233	C₂³	C₂₃×#121	1	2	yes	—
234	Q₈	—	4	1	no	—
235	Q₈	—	4	1	no	—
236	D₄	—	4	1	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	1	no	—
239	D₄	—	4	4	no	—
240	D₄	—	4	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	8	4	no	—
252	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
255	C₈	—	2	1	no	—
256	C₈	—	2	2	no	—
257	C₈	—	4	1	no	—
258	C₈	—	4	1	no	—
259	C₈	C₂₃×#99	1	1	yes	characteristic
260	C₈	C₂₃×#99	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	C₂₃×#380	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	16	4	no	—
274	C₄	C₂×C₂₃×#99	1	1	yes	characteristic
275	C₄	C₂₃×#410	1	1	yes	characteristic
276	C₂	—	2	1	no	—
277	C₂	—	4	2	no	—
278	C₂	—	4	1	no	—
279	C₂	—	8	1	no	—
280	C₂	C₂₃×#1799	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₃	#410
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×#41	C₂³×C₂₃
socle	C₂×C₂₃	#1799
prime radical	C₂₃×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂×C₈×C₂₃ ◁ C₂×C₂₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1812	C₂	semidirect
4. C₂₃×#1391	C₂	semidirect
5. C₂×C₂₃×#131	C₄	semidirect
11. C₂₃×#423	C₄	semidirect
13. C₂₃×#408	C₄	semidirect
15. C₂×C₄×C₈×C₂₃	C₄	semidirect
90. #12807	C₂₃	direct (N × H)
123. #1812	C₂×C₂₃	semidirect
124. #1391	C₂×C₂₃	semidirect
140. C₂×#131	C₄×C₂₃	semidirect
146. #423	C₄×C₂₃	semidirect
148. #408	C₄×C₂₃	semidirect
150. C₂×C₄×C₈	C₄×C₂₃	semidirect
190. C₂₃	#12807	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×#2931	2
C₂₃×#2126	2
C₂₃×#1812	2
C₂₃×#1391	2
#12807	23

Subgroups — C23×#12807 #19160433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12807 #19160433