Subgroups — C₂₃×#10615 #19158241

Order 5888.

418 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	C₂₃×#1954	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#1802	C₂	1	2	yes	maximal
4	C₂₃×#1296	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1277	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₂₃×#107	—	2	1	no	—
8	C₂×C₂₃×#107	—	2	1	no	—
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₂₃×#102	—	2	2	no	—
12	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
13	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#384	C₂²	1	2	yes	—
16	C₂₃×#340	—	2	2	no	—
17	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
18	C₂³×C₄×C₂₃	—	2	1	no	—
19	C₂²×C₈×C₂₃	—	2	2	no	—
20	C₂²×C₈×C₂₃	—	2	1	no	—
21	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₄×D₄×C₂₃	—	2	2	no	—
24	C₄×D₄×C₂₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
30	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
31	C₂₃×#107	—	4	2	no	—
32	C₂₃×#107	—	4	2	no	—
33	C₂₃×C₄⋊C₈	—	2	2	no	—
34	C₂₃×C₄⋊C₈	—	2	2	no	—
35	C₂₃×C₄⋊C₈	—	2	2	no	—
36	C₂₃×#102	—	4	4	no	—
37	C₂₃×C₂²⋊C₈	—	4	2	no	—
38	C₂₃×C₂²⋊C₈	—	4	2	no	—
39	C₄×C₈×C₂₃	—	2	2	no	—
40	C₂⁴×C₂₃	—	2	1	no	—
41	C₂×D₄×C₂₃	—	2	2	no	—
42	C₂×D₄×C₂₃	—	2	1	no	—
43	C₂×D₄×C₂₃	—	2	2	no	—
44	C₂×D₄×C₂₃	D₈	1	2	yes	—
45	C₂²×C₄×C₂₃	—	2	1	no	—
46	C₂²×C₄×C₂₃	—	4	1	no	—
47	C₂²×C₄×C₂₃	—	4	1	no	—
48	C₂²×C₄×C₂₃	—	4	1	no	—
49	C₂²×C₄×C₂₃	—	4	1	no	—
50	C₂²×C₄×C₂₃	—	4	1	no	—
51	C₂²×C₄×C₂₃	—	4	1	no	—
52	C₂²×C₄×C₂₃	—	4	2	no	—
53	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₂₃	—	2	1	no	—
57	C₂×C₈×C₂₃	—	2	1	no	—
58	C₂×C₈×C₂₃	—	2	1	no	—
59	C₂×C₈×C₂₃	—	2	1	no	—
60	C₂×C₈×C₂₃	—	4	2	no	—
61	C₂×C₈×C₂₃	—	4	1	no	—
62	C₂×C₈×C₂₃	—	4	2	no	—
63	C₂×C₈×C₂₃	—	4	1	no	—
64	C₂×C₈×C₂₃	—	4	1	no	—
65	C₂×C₈×C₂₃	—	4	2	no	—
66	C₂×C₈×C₂₃	D₈	1	2	yes	—
67	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
68	C₄⋊C₄×C₂₃	—	2	2	no	—
69	C₄⋊C₄×C₂₃	—	4	2	no	—
70	C₄⋊C₄×C₂₃	—	4	2	no	—
71	C₄⋊C₄×C₂₃	—	4	2	no	—
72	C₄⋊C₄×C₂₃	—	4	2	no	—
73	C₄⋊C₄×C₂₃	—	8	2	no	—
74	C₂²⋊C₄×C₂₃	—	4	2	no	—
75	C₄²×C₂₃	#41	1	2	yes	—
76	C₄²×C₂₃	#41	1	2	yes	—
77	#10615	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₂₃	—	2	2	no	—
79	C₂³×C₂₃	—	4	1	no	—
80	C₂³×C₂₃	—	4	1	no	—
81	C₂³×C₂₃	—	4	1	no	—
82	C₂³×C₂₃	—	4	1	no	—
83	C₂³×C₂₃	—	4	1	no	—
84	C₂³×C₂₃	—	4	1	no	—
85	C₂³×C₂₃	#121	1	1	yes	characteristic
86	D₄×C₂₃	—	2	4	no	—
87	D₄×C₂₃	—	4	2	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	4	1	no	—
91	C₂×C₄×C₂₃	—	4	1	no	—
92	C₂×C₄×C₂₃	—	4	1	no	—
93	C₂×C₄×C₂₃	—	8	1	no	—
94	C₂×C₄×C₂₃	—	8	1	no	—
95	C₂×C₄×C₂₃	—	8	1	no	—
96	C₂×C₄×C₂₃	—	8	1	no	—
97	C₂×C₄×C₂₃	—	8	2	no	—
98	C₂×C₄×C₂₃	—	8	2	no	—
99	C₂×C₄×C₂₃	—	8	2	no	—
100	C₂×C₄×C₂₃	#123	1	1	yes	characteristic
101	C₂×C₄×C₂₃	#123	1	2	yes	—
102	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
103	C₂×C₄×C₂₃	#126	1	2	yes	—
104	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
108	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₄×C₂₃	#137	1	2	yes	—
110	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
111	C₈×C₂₃	—	2	2	no	—
112	C₈×C₂₃	—	2	2	no	—
113	C₈×C₂₃	—	8	2	no	—
114	C₈×C₂₃	—	8	1	no	—
115	#1954	C₂×C₂₃	1	1	yes	characteristic
116	#1845	C₂×C₂₃	1	1	yes	characteristic, complemented
117	#1802	C₂×C₂₃	1	2	yes	—
118	#1296	C₂×C₂₃	1	2	yes	complemented
119	#1277	C₂×C₂₃	1	1	yes	characteristic, complemented
120	C₂²×C₂₃	—	4	2	no	—
121	C₂²×C₂₃	—	4	2	no	—
122	C₂²×C₂₃	—	4	2	no	—
123	C₂²×C₂₃	—	8	1	no	—
124	C₂²×C₂₃	—	8	1	no	—
125	C₂²×C₂₃	—	8	1	no	—
126	C₂²×C₂₃	—	8	1	no	—
127	C₂²×C₂₃	#397	1	1	yes	characteristic
128	C₂²×C₂₃	#459	1	1	yes	characteristic
129	C₂²×C₂₃	#461	1	1	yes	characteristic
130	C₂²×C₂₃	#466	1	1	yes	characteristic
131	C₂²×C₂₃	#470	1	1	yes	characteristic
132	C₂²×C₂₃	#472	1	1	yes	characteristic
133	C₂²×C₂₃	#473	1	1	yes	characteristic
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	—	8	1	no	—
136	C₄×C₂₃	—	16	2	no	—
137	C₄×C₂₃	#480	1	1	yes	characteristic
138	C₄×C₂₃	#480	1	2	yes	—
139	C₄×C₂₃	#481	1	1	yes	characteristic
140	C₄×C₂₃	#485	1	2	yes	—
141	C₄×C₂₃	#486	1	1	yes	characteristic
142	C₄×C₂₃	#489	1	1	yes	characteristic
143	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
144	C₂×#107	—	2	1	no	—
145	C₂×#107	—	2	1	no	—
146	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
147	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
148	C₂×#102	—	2	2	no	—
149	C₂×C₂²⋊C₈	—	2	1	no	—
150	C₂×C₂²⋊C₈	—	2	1	no	—
151	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
152	#384	C₂²×C₂₃	1	2	yes	—
153	#340	—	2	2	no	—
154	C₂×C₂₃	—	8	2	no	—
155	C₂×C₂₃	#1667	1	1	yes	characteristic
156	C₂×C₂₃	#1673	1	1	yes	characteristic
157	C₂×C₂₃	#1689	1	1	yes	characteristic
158	C₂×C₂₃	#1695	1	1	yes	characteristic
159	C₂×C₂₃	#2071	1	1	yes	characteristic
160	C₂×C₂₃	#2075	1	1	yes	characteristic
161	C₂×C₂₃	#2078	1	1	yes	characteristic
162	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
163	C₂³×C₄	—	2	1	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
168	C₄×D₄	—	2	2	no	—
169	C₄×D₄	—	2	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	—	2	1	no	—
175	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
176	#107	—	4	2	no	—
177	#107	—	4	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	—	2	2	no	—
181	#102	—	4	4	no	—
182	C₂²⋊C₈	—	4	2	no	—
183	C₂²⋊C₈	—	4	2	no	—
184	C₄×C₈	—	2	2	no	—
185	C₂₃	#10615	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	—	2	1	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	D₈×C₂₃	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	4	1	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	1	no	—
209	C₂×C₈	—	4	2	no	—
210	C₂×C₈	—	4	1	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	D₈×C₂₃	1	2	yes	—
213	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₄⋊C₄	—	4	2	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	8	2	no	—
220	C₂²⋊C₄	—	4	2	no	—
221	C₄²	#41×C₂₃	1	2	yes	—
222	C₄²	#41×C₂₃	1	2	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	4	1	no	—
227	C₂³	—	4	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	—	4	1	no	—
230	C₂³	C₂₃×#121	1	1	yes	characteristic
231	D₄	—	2	4	no	—
232	D₄	—	4	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	4	1	no	—
236	C₂×C₄	—	4	1	no	—
237	C₂×C₄	—	4	1	no	—
238	C₂×C₄	—	8	1	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	—	8	1	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	—	8	2	no	—
243	C₂×C₄	—	8	2	no	—
244	C₂×C₄	—	8	2	no	—
245	C₂×C₄	C₂₃×#123	1	1	yes	characteristic
246	C₂×C₄	C₂₃×#123	1	2	yes	—
247	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
248	C₂×C₄	C₂₃×#126	1	2	yes	—
249	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
253	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₂×C₄	C₂₃×#137	1	2	yes	—
255	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	—	8	2	no	—
259	C₈	—	8	1	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	8	1	no	—
264	C₂²	—	8	1	no	—
265	C₂²	—	8	1	no	—
266	C₂²	—	8	1	no	—
267	C₂²	C₂₃×#397	1	1	yes	characteristic
268	C₂²	C₂₃×#459	1	1	yes	characteristic
269	C₂²	C₂₃×#461	1	1	yes	characteristic
270	C₂²	C₂₃×#466	1	1	yes	characteristic
271	C₂²	C₂₃×#470	1	1	yes	characteristic
272	C₂²	C₂₃×#472	1	1	yes	characteristic
273	C₂²	C₂₃×#473	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	8	1	no	—
276	C₄	—	16	2	no	—
277	C₄	C₂₃×#480	1	1	yes	characteristic
278	C₄	C₂₃×#480	1	2	yes	—
279	C₄	C₂₃×#481	1	1	yes	characteristic
280	C₄	C₂₃×#485	1	2	yes	—
281	C₄	C₂₃×#486	1	1	yes	characteristic
282	C₄	C₂₃×#489	1	1	yes	characteristic
283	C₂	—	8	2	no	—
284	C₂	C₂₃×#1667	1	1	yes	characteristic
285	C₂	C₂₃×#1673	1	1	yes	characteristic
286	C₂	C₂₃×#1689	1	1	yes	characteristic
287	C₂	C₂₃×#1695	1	1	yes	characteristic
288	C₂	C₂₃×#2071	1	1	yes	characteristic
289	C₂	C₂₃×#2075	1	1	yes	characteristic
290	C₂	C₂₃×#2078	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂²×D₄×C₂₃	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1845	C₂	semidirect
4. C₂₃×#1296	C₂	semidirect
5. C₂₃×#1277	C₂	semidirect
77. #10615	C₂₃	direct (N × H)
116. #1845	C₂×C₂₃	semidirect
118. #1296	C₂×C₂₃	semidirect
119. #1277	C₂×C₂₃	semidirect
185. C₂₃	#10615	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×#1954	2
C₂₃×#1845	2
C₂₃×#1802	2
C₂₃×#1296	2
C₂₃×#1277	2
#10615	23

Subgroups — C23×#10615 #19158241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10615 #19158241