Subgroups — C₂₃×#7567 #19155193

Order 5888.

528 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1744	C₂	1	2	yes	maximal, complemented
3	C₂₃×#1272	C₂	1	2	yes	maximal, complemented
4	C₂₃×#1270	C₂	1	2	yes	maximal
5	C₂×C₂₃×#124	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₂₃	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₂₃	—	2	2	no	—
9	C₂₃×#391	C₂²	1	1	yes	characteristic
10	C₂₃×#390	C₂²	1	1	yes	characteristic
11	C₂₃×#388	—	2	2	no	—
12	C₂₃×#382	C₄	1	1	yes	characteristic
13	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
14	C₄³×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#336	—	2	2	no	—
16	C₂₃×#336	C₂²	1	4	yes	—
17	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₂₃	—	2	1	no	—
20	C₂²×C₈×C₂₃	—	2	4	no	—
21	C₂₃×#124	—	2	2	no	—
22	C₂₃×#124	C₈	1	4	yes	complemented
23	C₄×Q₈×C₂₃	—	4	2	no	—
24	C₄×D₄×C₂₃	—	4	2	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
26	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
29	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
30	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
31	C₂×C₄²×C₂₃	—	2	1	no	—
32	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	D₄	1	2	yes	—
35	C₂×C₄²×C₂₃	D₄	1	2	yes	—
36	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
37	C₄×C₈×C₂₃	—	2	8	no	—
38	C₂₃×#95	—	2	2	no	—
39	C₂₃×#95	—	2	2	no	—
40	C₂⁴×C₂₃	—	2	1	no	—
41	C₂×Q₈×C₂₃	—	2	1	no	—
42	C₂×Q₈×C₂₃	—	4	1	no	—
43	C₂×Q₈×C₂₃	—	4	1	no	—
44	C₂×Q₈×C₂₃	C₂×C₈	1	2	yes	—
45	C₂×D₄×C₂₃	—	2	1	no	—
46	C₂×D₄×C₂₃	—	4	1	no	—
47	C₂×D₄×C₂₃	—	4	1	no	—
48	C₂×D₄×C₂₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₂₃	—	2	1	no	—
50	C₂²×C₄×C₂₃	—	2	1	no	—
51	C₂²×C₄×C₂₃	—	2	1	no	—
52	C₂²×C₄×C₂₃	—	4	2	no	—
53	C₂²×C₄×C₂₃	—	4	4	no	—
54	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
59	C₂×C₈×C₂₃	—	2	8	no	—
60	C₂×C₈×C₂₃	—	4	4	no	—
61	C₂×C₈×C₂₃	—	4	4	no	—
62	C₄⋊C₄×C₂₃	—	4	1	no	—
63	C₄⋊C₄×C₂₃	—	4	1	no	—
64	C₄⋊C₄×C₂₃	—	4	2	no	—
65	C₂²⋊C₄×C₂₃	—	2	8	no	—
66	C₂²⋊C₄×C₂₃	—	4	2	no	—
67	C₂²⋊C₄×C₂₃	—	4	4	no	—
68	C₄²×C₂₃	—	2	2	no	—
69	C₄²×C₂₃	—	2	4	no	—
70	C₄²×C₂₃	—	2	4	no	—
71	C₄²×C₂₃	—	4	1	no	—
72	C₄²×C₂₃	—	4	2	no	—
73	C₄²×C₂₃	C₂×C₈	1	2	yes	—
74	C₄²×C₂₃	M₁₆	1	2	yes	—
75	#7567	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₂₃	—	2	2	no	—
77	C₂³×C₂₃	—	4	1	no	—
78	C₂³×C₂₃	—	4	1	no	—
79	C₂³×C₂₃	—	4	2	no	—
80	C₂³×C₂₃	—	4	2	no	—
81	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	Q₈×C₂₃	—	4	2	no	—
83	Q₈×C₂₃	—	4	2	no	—
84	D₄×C₂₃	—	4	2	no	—
85	D₄×C₂₃	—	4	2	no	—
86	C₂×C₄×C₂₃	—	2	2	no	—
87	C₂×C₄×C₂₃	—	2	1	no	—
88	C₂×C₄×C₂₃	—	2	1	no	—
89	C₂×C₄×C₂₃	—	2	2	no	—
90	C₂×C₄×C₂₃	—	2	2	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	4	1	no	—
96	C₂×C₄×C₂₃	—	4	1	no	—
97	C₂×C₄×C₂₃	—	4	1	no	—
98	C₂×C₄×C₂₃	—	4	1	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	8	4	no	—
102	C₂×C₄×C₂₃	C₂²⋊C₈	1	4	yes	—
103	C₂×C₄×C₂₃	#100	1	2	yes	—
104	C₂×C₄×C₂₃	#104	1	4	yes	—
105	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#135	1	4	yes	—
108	C₈×C₂₃	—	4	8	no	—
109	C₄×#124	C₂×C₂₃	1	1	yes	characteristic
110	#1744	C₂×C₂₃	1	2	yes	complemented
111	#1272	C₂×C₂₃	1	2	yes	complemented
112	#1270	C₂×C₂₃	1	2	yes	—
113	C₂²×C₂₃	—	4	2	no	—
114	C₂²×C₂₃	—	4	4	no	—
115	C₂²×C₂₃	—	8	1	no	—
116	C₂²×C₂₃	—	8	2	no	—
117	C₂²×C₂₃	—	8	1	no	—
118	C₂²×C₂₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
119	C₂²×C₂₃	#410	1	1	yes	characteristic
120	C₂²×C₂₃	C₂×#100	1	1	yes	characteristic
121	C₂²×C₂₃	#416	1	2	yes	—
122	C₂²×C₂₃	C₂×#104	1	2	yes	—
123	C₄×C₂₃	—	2	1	no	—
124	C₄×C₂₃	—	2	1	no	—
125	C₄×C₂₃	—	2	4	no	—
126	C₄×C₂₃	—	2	4	no	—
127	C₄×C₂₃	—	4	1	no	—
128	C₄×C₂₃	—	4	2	no	—
129	C₄×C₂₃	—	8	1	no	—
130	C₄×C₂₃	—	8	1	no	—
131	C₄×C₂₃	#329	1	4	yes	—
132	C₂×#124	C₄×C₂₃	1	1	yes	characteristic
133	C₂×C₄×Q₈	C₄×C₂₃	1	1	yes	characteristic
134	C₂×C₄×D₄	C₄×C₂₃	1	1	yes	characteristic
135	C₂×C₄×C₈	—	2	2	no	—
136	#391	C₂²×C₂₃	1	1	yes	characteristic
137	#390	C₂²×C₂₃	1	1	yes	characteristic
138	#388	—	2	2	no	—
139	#382	C₄×C₂₃	1	1	yes	characteristic
140	C₄×C₂²⋊C₄	—	2	2	no	—
141	C₄³	C₂²×C₂₃	1	1	yes	characteristic
142	#336	—	2	2	no	—
143	#336	C₂²×C₂₃	1	4	yes	—
144	C₂×C₂₃	—	8	2	no	—
145	C₂×C₂₃	#1460	1	1	yes	characteristic
146	C₂×C₂₃	#1476	1	2	yes	—
147	C₂×C₂₃	#1500	1	2	yes	—
148	C₂×C₂₃	C₂×#329	1	1	yes	characteristic
149	C₂×C₂₃	#1543	1	1	yes	characteristic
150	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
151	C₂²×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
152	C₂³×C₄	—	2	1	no	—
153	C₂²×C₈	—	2	4	no	—
154	#124	—	2	2	no	—
155	#124	C₈×C₂₃	1	4	yes	complemented
156	C₄×Q₈	—	4	2	no	—
157	C₄×D₄	—	4	2	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
161	C₂×C₂²⋊C₄	—	2	2	no	—
162	C₂×C₂²⋊C₄	—	2	1	no	—
163	C₂×C₂²⋊C₄	—	2	2	no	—
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
166	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
167	C₂×C₄²	D₄×C₂₃	1	2	yes	—
168	C₂×C₄²	D₄×C₂₃	1	2	yes	—
169	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
170	C₄×C₈	—	2	8	no	—
171	#95	—	2	2	no	—
172	#95	—	2	2	no	—
173	C₂₃	#7567	1	1	yes	characteristic, direct product factor, Sylow
174	C₂⁴	—	2	1	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	—	4	1	no	—
177	C₂×Q₈	—	4	1	no	—
178	C₂×Q₈	C₂×C₈×C₂₃	1	2	yes	—
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂×D₄	C₂×C₈×C₂₃	1	2	yes	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	4	4	no	—
188	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
189	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
190	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
193	C₂×C₈	—	2	8	no	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₄⋊C₄	—	4	1	no	—
197	C₄⋊C₄	—	4	1	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	2	8	no	—
200	C₂²⋊C₄	—	4	2	no	—
201	C₂²⋊C₄	—	4	4	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	4	1	no	—
206	C₄²	—	4	2	no	—
207	C₄²	C₂×C₈×C₂₃	1	2	yes	—
208	C₄²	M₁₆×C₂₃	1	2	yes	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	1	no	—
211	C₂³	—	4	1	no	—
212	C₂³	—	4	2	no	—
213	C₂³	—	4	2	no	—
214	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
215	Q₈	—	4	2	no	—
216	Q₈	—	4	2	no	—
217	D₄	—	4	2	no	—
218	D₄	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	8	4	no	—
235	C₂×C₄	C₂₃×C₂²⋊C₈	1	4	yes	—
236	C₂×C₄	C₂₃×#100	1	2	yes	—
237	C₂×C₄	C₂₃×#104	1	4	yes	—
238	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
239	C₂×C₄	C₂×M₁₆×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂₃×#135	1	4	yes	—
241	C₈	—	4	8	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	4	no	—
244	C₂²	—	8	2	no	—
245	C₂²	—	8	1	no	—
246	C₂²	—	8	1	no	—
247	C₂²	C₂×C₂₃×C₂²⋊C₈	1	1	yes	characteristic
248	C₂²	C₂₃×#410	1	1	yes	characteristic
249	C₂²	C₂×C₂₃×#100	1	1	yes	characteristic
250	C₂²	C₂₃×#416	1	2	yes	—
251	C₂²	C₂×C₂₃×#104	1	2	yes	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	2	4	no	—
255	C₄	—	2	4	no	—
256	C₄	—	4	2	no	—
257	C₄	—	4	1	no	—
258	C₄	—	8	1	no	—
259	C₄	—	8	1	no	—
260	C₄	C₂₃×#329	1	4	yes	—
261	C₂	—	8	2	no	—
262	C₂	C₂₃×#1460	1	1	yes	characteristic
263	C₂	C₂₃×#1476	1	2	yes	—
264	C₂	C₂₃×#1500	1	2	yes	—
265	C₂	C₂×C₂₃×#329	1	1	yes	characteristic
266	C₂	C₂₃×#1543	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₂₃	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1744	C₂	semidirect
3. C₂₃×#1272	C₂	semidirect
22. C₂₃×#124	C₈	semidirect
75. #7567	C₂₃	direct (N × H)
110. #1744	C₂×C₂₃	semidirect
111. #1272	C₂×C₂₃	semidirect
155. #124	C₈×C₂₃	semidirect
173. C₂₃	#7567	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₂₃×#124	2
C₂₃×#1744	2
C₂₃×#1272	2
C₂₃×#1270	2
#7567	23

Subgroups — C23×#7567 #19155193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7567 #19155193