Subgroups — C₂₃×#8606 #19156232

Order 5888.

372 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2922	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1477	C₂	1	2	yes	maximal
5	C₂₃×#1283	C₂	1	2	yes	maximal, complemented
6	C₄×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#433	—	2	1	no	—
8	C₂₃×#433	—	2	1	no	—
9	C₂₃×#431	—	2	2	no	—
10	C₂₃×#431	C₂²	1	2	yes	—
11	C₂₃×#424	C₄	1	2	yes	—
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂₃×#408	C₄	1	2	yes	—
14	C₄×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#335	—	2	2	no	—
17	C₂₃×#335	—	2	1	no	—
18	C₂₃×#334	—	2	1	no	—
19	C₂₃×Q₈⋊C₈	—	2	2	no	—
20	C₂₃×D₄⋊C₈	—	2	2	no	—
21	C₂×#41×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₂₃	—	2	2	no	—
24	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
25	C₄×Q₈×C₂₃	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₂₃	Q₈	1	1	yes	characteristic
27	C₄×D₄×C₂₃	—	2	1	no	—
28	C₄×D₄×C₂₃	—	2	1	no	—
29	C₄×D₄×C₂₃	D₄	1	1	yes	characteristic
30	C₄×D₄×C₂₃	Q₈	1	1	yes	characteristic
31	C₂₃×#117	—	2	1	no	—
32	C₂₃×#117	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	—	2	1	no	—
34	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×C₄⋊C₈	—	2	2	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	—	2	2	no	—
40	C₂₃×C₂²⋊C₈	—	2	2	no	—
41	C₂₃×C₂²⋊C₈	—	2	2	no	—
42	C₂₃×C₂²⋊C₈	—	2	2	no	—
43	C₂₃×#97	—	2	1	no	—
44	C₄×C₈×C₂₃	—	2	1	no	—
45	C₄×C₈×C₂₃	—	2	2	no	—
46	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₂₃	D₄	1	2	yes	—
48	C₄×C₈×C₂₃	D₄	1	1	yes	characteristic
49	#41×C₂₃	—	2	4	no	—
50	C₂×Q₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
51	C₂×D₄×C₂₃	—	2	1	no	—
52	C₂×D₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	—	2	1	no	—
54	C₂²×C₄×C₂₃	—	2	1	no	—
55	C₂²×C₄×C₂₃	—	2	1	no	—
56	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	#41	1	1	yes	characteristic
59	M₁₆×C₂₃	—	2	4	no	—
60	C₂×C₈×C₂₃	—	2	4	no	—
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	—	4	2	no	—
63	C₂×C₈×C₂₃	—	4	2	no	—
64	C₂×C₈×C₂₃	—	8	2	no	—
65	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₂₃	C₂²⋊C₄	1	2	yes	—
67	C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₄⋊C₄×C₂₃	—	2	1	no	—
69	C₄⋊C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₂₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂²⋊C₄×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	—	2	1	no	—
75	C₄²×C₂₃	—	2	1	no	—
76	C₄²×C₂₃	—	2	1	no	—
77	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
81	#8606	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₂₃	—	2	1	no	—
83	C₂³×C₂₃	#117	1	1	yes	characteristic
84	Q₈×C₂₃	—	2	2	no	—
85	D₄×C₂₃	—	2	2	no	—
86	D₄×C₂₃	—	4	2	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	1	no	—
92	C₂×C₄×C₂₃	—	4	1	no	—
93	C₂×C₄×C₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	—	4	1	no	—
96	C₂×C₄×C₂₃	—	4	1	no	—
97	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
100	C₂×C₄×C₂₃	#121	1	1	yes	characteristic
101	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₂₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₂₃	#131	1	4	yes	—
104	C₈×C₂₃	—	2	2	no	—
105	C₈×C₂₃	—	2	2	no	—
106	C₈×C₂₃	—	4	2	no	—
107	C₈×C₂₃	—	8	2	no	—
108	C₈×C₂₃	—	8	2	no	—
109	#2922	C₂×C₂₃	1	1	yes	characteristic
110	#1564	C₂×C₂₃	1	1	yes	characteristic, complemented
111	C₂×#335	C₂×C₂₃	1	1	yes	characteristic, complemented
112	#1477	C₂×C₂₃	1	2	yes	—
113	#1283	C₂×C₂₃	1	2	yes	complemented
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	—	4	1	no	—
117	C₂²×C₂₃	—	4	1	no	—
118	C₂²×C₂₃	—	4	1	no	—
119	C₂²×C₂₃	#380	1	1	yes	characteristic
120	C₂²×C₂₃	#433	1	2	yes	—
121	C₄×C₂₃	—	2	1	no	—
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	—	2	1	no	—
124	C₄×C₂₃	—	4	1	no	—
125	C₄×C₂₃	—	4	1	no	—
126	C₄×C₂₃	—	4	1	no	—
127	C₄×C₂₃	#408	1	2	yes	—
128	C₄×C₂₃	#424	1	2	yes	—
129	C₄×C₂₃	#472	1	1	yes	characteristic
130	C₄×C₂₃	#473	1	1	yes	characteristic
131	C₄×#41	C₂²×C₂₃	1	1	yes	characteristic
132	#433	—	2	1	no	—
133	#433	—	2	1	no	—
134	#431	—	2	2	no	—
135	#431	C₂²×C₂₃	1	2	yes	—
136	#424	C₄×C₂₃	1	2	yes	—
137	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
138	#408	C₄×C₂₃	1	2	yes	—
139	C₄×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
141	#335	—	2	2	no	—
142	#335	—	2	1	no	—
143	#334	—	2	1	no	—
144	Q₈⋊C₈	—	2	2	no	—
145	D₄⋊C₈	—	2	2	no	—
146	C₂×C₂₃	—	2	2	no	—
147	C₂×C₂₃	—	8	1	no	—
148	C₂×C₂₃	#1798	1	1	yes	characteristic
149	C₂×C₂₃	#1814	1	1	yes	characteristic
150	C₂×C₂₃	#1816	1	1	yes	characteristic
151	C₂×#41	C₂×C₄×C₂₃	1	1	yes	characteristic
152	C₂×M₁₆	C₂×C₄×C₂₃	1	2	yes	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
155	C₄×Q₈	D₄×C₂₃	1	1	yes	characteristic
156	C₄×Q₈	Q₈×C₂₃	1	1	yes	characteristic
157	C₄×D₄	—	2	1	no	—
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	D₄×C₂₃	1	1	yes	characteristic
160	C₄×D₄	Q₈×C₂₃	1	1	yes	characteristic
161	#117	—	2	1	no	—
162	#117	C₂×C₄×C₂₃	1	1	yes	characteristic
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₂²⋊C₈	—	2	2	no	—
171	C₂²⋊C₈	—	2	2	no	—
172	C₂²⋊C₈	—	2	2	no	—
173	#97	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
177	C₄×C₈	D₄×C₂₃	1	2	yes	—
178	C₄×C₈	D₄×C₂₃	1	1	yes	characteristic
179	C₂₃	#8606	1	1	yes	characteristic, direct product factor, Sylow
180	#41	—	2	4	no	—
181	C₂×Q₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
188	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	#41×C₂₃	1	1	yes	characteristic
190	M₁₆	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	8	2	no	—
196	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
197	C₂×C₈	C₂²⋊C₄×C₂₃	1	2	yes	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
201	C₄⋊C₄	C₄⋊C₄×C₂₃	1	1	yes	characteristic
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
209	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	C₂³	C₂₃×#117	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	D₄	—	2	2	no	—
216	D₄	—	4	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
228	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
229	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
230	C₂×C₄	C₂₃×#121	1	1	yes	characteristic
231	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
232	C₂×C₄	C₂₃×C₂²⋊Q₈	1	1	yes	characteristic
233	C₂×C₄	C₂₃×#131	1	4	yes	—
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	—	4	2	no	—
237	C₈	—	8	2	no	—
238	C₈	—	8	2	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₂₃×#380	1	1	yes	characteristic
245	C₂²	C₂₃×#433	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	C₂₃×#408	1	2	yes	—
253	C₄	C₂₃×#424	1	2	yes	—
254	C₄	C₂₃×#472	1	1	yes	characteristic
255	C₄	C₂₃×#473	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	8	1	no	—
258	C₂	C₂₃×#1798	1	1	yes	characteristic
259	C₂	C₂₃×#1814	1	1	yes	characteristic
260	C₂	C₂₃×#1816	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	#121
commutator	C₂²×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	#380
prime radical	C₂×#41×C₂₃	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1564	C₂	semidirect
3. C₂×C₂₃×#335	C₂	semidirect
5. C₂₃×#1283	C₂	semidirect
81. #8606	C₂₃	direct (N × H)
110. #1564	C₂×C₂₃	semidirect
111. C₂×#335	C₂×C₂₃	semidirect
113. #1283	C₂×C₂₃	semidirect
179. C₂₃	#8606	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×#2922	2
C₂₃×#1564	2
C₂×C₂₃×#335	2
C₂₃×#1477	2
C₂₃×#1283	2
#8606	23

Subgroups — C23×#8606 #19156232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8606 #19156232