Subgroups — C₂₃×#10543 #19158169

Order 5888.

422 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#500	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1987	C₂	1	2	yes	maximal
3	C₂₃×#1939	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#333	C₂	1	2	yes	maximal
5	C₂₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂₃×#500	—	2	4	no	—
8	C₂×C₂₃×#107	—	2	1	no	—
9	C₂×C₂₃×#107	—	2	1	no	—
10	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#340	—	2	2	no	—
16	C₂₃×#340	—	2	2	no	—
17	C₂₃×#336	—	2	1	no	—
18	C₂₃×#333	—	2	4	no	—
19	C₂²×Q₈×C₂₃	—	2	2	no	—
20	C₂²×C₈×C₂₃	—	2	1	no	—
21	C₂²×C₈×C₂₃	—	2	2	no	—
22	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
24	C₂₃×C₄⋊Q₈	—	2	2	no	—
25	C₂₃×C₄⋊Q₈	—	2	2	no	—
26	C₂₃×C₄⋊Q₈	—	2	2	no	—
27	C₂₃×C₄⋊Q₈	D₄	1	4	yes	—
28	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	—	2	1	no	—
33	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
34	C₂₃×#107	—	2	4	no	—
35	C₂₃×#107	—	2	4	no	—
36	C₂₃×C₄⋊C₈	—	2	2	no	—
37	C₂₃×C₄⋊C₈	—	2	2	no	—
38	C₂₃×C₄⋊C₈	—	2	2	no	—
39	C₂₃×C₄⋊C₈	—	4	2	no	—
40	C₄×C₈×C₂₃	—	2	2	no	—
41	C₂₃×#95	—	2	1	no	—
42	C₂×Q₈×C₂₃	—	2	4	no	—
43	C₂×Q₈×C₂₃	—	4	2	no	—
44	C₂×Q₈×C₂₃	—	8	2	no	—
45	C₂×Q₈×C₂₃	—	8	2	no	—
46	C₂²×C₄×C₂₃	—	2	1	no	—
47	C₂²×C₄×C₂₃	—	2	1	no	—
48	C₂²×C₄×C₂₃	—	4	2	no	—
49	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
52	C₂×C₈×C₂₃	—	2	1	no	—
53	C₂×C₈×C₂₃	—	2	1	no	—
54	C₂×C₈×C₂₃	—	2	1	no	—
55	C₂×C₈×C₂₃	—	2	1	no	—
56	C₂×C₈×C₂₃	—	4	1	no	—
57	C₂×C₈×C₂₃	—	4	2	no	—
58	C₂×C₈×C₂₃	—	4	1	no	—
59	C₂×C₈×C₂₃	—	4	2	no	—
60	C₂×C₈×C₂₃	—	4	1	no	—
61	C₂×C₈×C₂₃	—	4	2	no	—
62	C₂×C₈×C₂₃	SD₁₆	1	2	yes	—
63	C₂×C₈×C₂₃	Q₁₆	1	2	yes	—
64	C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₄⋊C₄×C₂₃	—	2	4	no	—
66	C₄⋊C₄×C₂₃	—	2	4	no	—
67	C₄⋊C₄×C₂₃	—	2	2	no	—
68	C₄⋊C₄×C₂₃	—	4	2	no	—
69	C₄⋊C₄×C₂₃	—	4	2	no	—
70	C₄⋊C₄×C₂₃	Q₁₆	1	2	yes	—
71	C₄²×C₂₃	—	4	1	no	—
72	C₄²×C₂₃	—	4	1	no	—
73	C₄²×C₂₃	C₂×D₄	1	2	yes	—
74	C₄²×C₂₃	#41	1	2	yes	—
75	#10543	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	#121	1	1	yes	characteristic
77	Q₈×C₂₃	—	8	4	no	—
78	Q₈×C₂₃	—	8	4	no	—
79	C₂×C₄×C₂₃	—	2	1	no	—
80	C₂×C₄×C₂₃	—	2	1	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	4	1	no	—
83	C₂×C₄×C₂₃	—	4	1	no	—
84	C₂×C₄×C₂₃	—	4	4	no	—
85	C₂×C₄×C₂₃	—	4	1	no	—
86	C₂×C₄×C₂₃	—	4	1	no	—
87	C₂×C₄×C₂₃	—	4	1	no	—
88	C₂×C₄×C₂₃	—	8	2	no	—
89	C₂×C₄×C₂₃	—	8	2	no	—
90	C₂×C₄×C₂₃	#120	1	1	yes	characteristic
91	C₂×C₄×C₂₃	#121	1	2	yes	—
92	C₂×C₄×C₂₃	#123	1	2	yes	—
93	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
94	C₂×C₄×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
96	C₂×C₄×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#135	1	1	yes	characteristic
98	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
99	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
100	C₂×C₄×C₂₃	#137	1	2	yes	—
101	C₈×C₂₃	—	2	2	no	—
102	C₈×C₂₃	—	2	2	no	—
103	C₈×C₂₃	—	8	1	no	—
104	C₈×C₂₃	—	8	2	no	—
105	C₂×#500	C₂×C₂₃	1	1	yes	characteristic
106	#1987	C₂×C₂₃	1	2	yes	—
107	#1939	C₂×C₂₃	1	1	yes	characteristic
108	C₂×#333	C₂×C₂₃	1	2	yes	—
109	#1277	C₂×C₂₃	1	1	yes	characteristic
110	C₂²×C₂₃	#394	1	1	yes	characteristic
111	C₂²×C₂₃	#461	1	1	yes	characteristic
112	C₂²×C₂₃	#462	1	1	yes	characteristic
113	C₂²×C₂₃	#467	1	1	yes	characteristic
114	C₂²×C₂₃	#470	1	1	yes	characteristic
115	C₂²×C₂₃	#472	1	1	yes	characteristic
116	C₂²×C₂₃	#472	1	1	yes	characteristic
117	C₄×C₂₃	—	2	2	no	—
118	C₄×C₂₃	—	4	2	no	—
119	C₄×C₂₃	—	8	1	no	—
120	C₄×C₂₃	—	8	4	no	—
121	C₄×C₂₃	#448	1	1	yes	characteristic
122	C₄×C₂₃	#451	1	1	yes	characteristic
123	C₄×C₂₃	#462	1	2	yes	—
124	C₄×C₂₃	#464	1	2	yes	—
125	C₄×C₂₃	#487	1	1	yes	characteristic
126	C₄×C₂₃	#491	1	1	yes	characteristic
127	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
128	#500	—	2	4	no	—
129	C₂×#107	—	2	1	no	—
130	C₂×#107	—	2	1	no	—
131	C₂×C₄⋊C₈	—	2	1	no	—
132	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
133	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
134	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
135	C₄×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
136	#340	—	2	2	no	—
137	#340	—	2	2	no	—
138	#336	—	2	1	no	—
139	#333	—	2	4	no	—
140	C₂×C₂₃	#1675	1	1	yes	characteristic
141	C₂×C₂₃	#1676	1	1	yes	characteristic
142	C₂×C₂₃	#1695	1	1	yes	characteristic
143	C₂×C₂₃	#1696	1	1	yes	characteristic
144	C₂×C₂₃	#2043	1	1	yes	characteristic
145	C₂×C₂₃	#2045	1	1	yes	characteristic
146	C₂×C₂₃	#2056	1	1	yes	characteristic
147	C₂²×Q₈	—	2	2	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
151	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
152	C₄⋊Q₈	—	2	2	no	—
153	C₄⋊Q₈	—	2	2	no	—
154	C₄⋊Q₈	—	2	2	no	—
155	C₄⋊Q₈	D₄×C₂₃	1	4	yes	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
159	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
160	C₂×C₄²	—	2	1	no	—
161	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
162	#107	—	2	4	no	—
163	#107	—	2	4	no	—
164	C₄⋊C₈	—	2	2	no	—
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	4	2	no	—
168	C₄×C₈	—	2	2	no	—
169	#95	—	2	1	no	—
170	C₂₃	#10543	1	1	yes	characteristic, direct product factor, Sylow
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	8	2	no	—
174	C₂×Q₈	—	8	2	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
180	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	SD₁₆×C₂₃	1	2	yes	—
192	C₂×C₈	Q₁₆×C₂₃	1	2	yes	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	Q₁₆×C₂₃	1	2	yes	—
200	C₄²	—	4	1	no	—
201	C₄²	—	4	1	no	—
202	C₄²	C₂×D₄×C₂₃	1	2	yes	—
203	C₄²	#41×C₂₃	1	2	yes	—
204	C₂³	C₂₃×#121	1	1	yes	characteristic
205	Q₈	—	8	4	no	—
206	Q₈	—	8	4	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	1	no	—
214	C₂×C₄	—	4	1	no	—
215	C₂×C₄	—	4	1	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	C₂₃×#120	1	1	yes	characteristic
219	C₂×C₄	C₂₃×#121	1	2	yes	—
220	C₂×C₄	C₂₃×#123	1	2	yes	—
221	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
222	C₂×C₄	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
223	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
224	C₂×C₄	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
225	C₂×C₄	C₂₃×#135	1	1	yes	characteristic
226	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
227	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
228	C₂×C₄	C₂₃×#137	1	2	yes	—
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	8	1	no	—
232	C₈	—	8	2	no	—
233	C₂²	C₂₃×#394	1	1	yes	characteristic
234	C₂²	C₂₃×#461	1	1	yes	characteristic
235	C₂²	C₂₃×#462	1	1	yes	characteristic
236	C₂²	C₂₃×#467	1	1	yes	characteristic
237	C₂²	C₂₃×#470	1	1	yes	characteristic
238	C₂²	C₂₃×#472	1	1	yes	characteristic
239	C₂²	C₂₃×#472	1	1	yes	characteristic
240	C₄	—	2	2	no	—
241	C₄	—	4	2	no	—
242	C₄	—	8	1	no	—
243	C₄	—	8	4	no	—
244	C₄	C₂₃×#448	1	1	yes	characteristic
245	C₄	C₂₃×#451	1	1	yes	characteristic
246	C₄	C₂₃×#462	1	2	yes	—
247	C₄	C₂₃×#464	1	2	yes	—
248	C₄	C₂₃×#487	1	1	yes	characteristic
249	C₄	C₂₃×#491	1	1	yes	characteristic
250	C₂	C₂₃×#1675	1	1	yes	characteristic
251	C₂	C₂₃×#1676	1	1	yes	characteristic
252	C₂	C₂₃×#1695	1	1	yes	characteristic
253	C₂	C₂₃×#1696	1	1	yes	characteristic
254	C₂	C₂₃×#2043	1	1	yes	characteristic
255	C₂	C₂₃×#2045	1	1	yes	characteristic
256	C₂	C₂₃×#2056	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#121
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
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
75. #10543	C₂₃	direct (N × H)
170. C₂₃	#10543	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#500	2
C₂₃×#1987	2
C₂₃×#1939	2
C₂×C₂₃×#333	2
C₂₃×#1277	2
#10543	23

Subgroups — C23×#10543 #19158169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10543 #19158169