Subgroups — C₂₃×#7693 #19155319

Order 5888.

390 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×#2863	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1745	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1272	C₂	1	2	yes	maximal
5	C₂₃×#1271	C₂	1	2	yes	maximal
6	C₂×C₂₃×#97	—	2	1	no	—
7	C₂×C₄×C₈×C₂₃	—	2	1	no	—
8	C₂₃×#391	C₄	1	1	yes	characteristic
9	C₂₃×#391	C₄	1	1	yes	characteristic
10	C₂₃×#391	C₂²	1	1	yes	characteristic
11	C₂₃×#389	C₄	1	1	yes	characteristic
12	C₂₃×#389	C₄	1	1	yes	characteristic
13	C₂₃×#389	C₂²	1	1	yes	characteristic
14	C₂₃×#387	—	2	1	no	—
15	C₂₃×#387	—	2	1	no	—
16	C₂₃×#387	—	2	1	no	—
17	C₂₃×#383	—	2	1	no	—
18	C₄³×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×#336	—	2	1	no	—
20	C₂₃×#336	—	2	1	no	—
21	C₂₃×#336	C₂²	1	2	yes	—
22	C₂₃×#336	C₂²	1	2	yes	—
23	C₂²×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₂₃	—	2	2	no	—
25	C₂²×C₈×C₂₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
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	1	yes	characteristic
35	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×#97	—	2	4	no	—
40	C₄×C₈×C₂₃	—	2	4	no	—
41	C₂₃×#95	—	2	1	no	—
42	C₂₃×#95	—	2	1	no	—
43	C₂₃×#95	—	2	1	no	—
44	C₂₃×#95	—	2	1	no	—
45	C₂₃×#95	—	2	1	no	—
46	C₂₃×#95	—	2	1	no	—
47	C₂₃×#95	—	2	1	no	—
48	C₂₃×#95	—	2	1	no	—
49	C₂₃×#95	—	2	1	no	—
50	C₂×Q₈×C₂₃	—	2	2	no	—
51	C₂×Q₈×C₂₃	—	4	1	no	—
52	C₂×Q₈×C₂₃	—	4	1	no	—
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	—	2	1	no	—
57	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₂₃	—	2	4	no	—
65	C₂×C₈×C₂₃	—	2	4	no	—
66	C₂×C₈×C₂₃	—	4	2	no	—
67	C₂×C₈×C₂₃	—	4	2	no	—
68	C₂×C₈×C₂₃	—	4	2	no	—
69	C₂×C₈×C₂₃	—	4	2	no	—
70	C₄⋊C₄×C₂₃	—	2	2	no	—
71	C₄⋊C₄×C₂₃	—	4	1	no	—
72	C₄⋊C₄×C₂₃	—	4	1	no	—
73	C₄⋊C₄×C₂₃	—	4	2	no	—
74	C₄⋊C₄×C₂₃	—	4	2	no	—
75	C₄⋊C₄×C₂₃	—	4	2	no	—
76	C₄²×C₂₃	—	2	2	no	—
77	C₄²×C₂₃	—	2	2	no	—
78	C₄²×C₂₃	—	2	2	no	—
79	C₄²×C₂₃	—	2	2	no	—
80	C₄²×C₂₃	—	2	2	no	—
81	C₄²×C₂₃	—	4	1	no	—
82	C₄²×C₂₃	—	4	1	no	—
83	#7693	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₂₃	—	8	2	no	—
86	C₂×C₄×C₂₃	—	2	1	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	—	2	1	no	—
93	C₂×C₄×C₂₃	—	2	1	no	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	1	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	4	1	no	—
103	C₂×C₄×C₂₃	—	4	1	no	—
104	C₂×C₄×C₂₃	—	4	1	no	—
105	C₂×C₄×C₂₃	—	4	1	no	—
106	C₂×C₄×C₂₃	—	4	1	no	—
107	C₂×C₄×C₂₃	—	4	1	no	—
108	C₂×C₄×C₂₃	—	4	1	no	—
109	C₂×C₄×C₂₃	—	4	1	no	—
110	C₂×C₄×C₂₃	—	4	1	no	—
111	C₂×C₄×C₂₃	—	4	1	no	—
112	C₂×C₄×C₂₃	—	4	1	no	—
113	C₂×C₄×C₂₃	—	4	1	no	—
114	C₂×C₄×C₂₃	#104	1	2	yes	—
115	C₂×C₄×C₂₃	#104	1	2	yes	—
116	C₂×C₄×C₂₃	#131	1	2	yes	—
117	C₂×C₄×C₂₃	#135	1	2	yes	—
118	C₂×C₄×C₂₃	#137	1	2	yes	—
119	C₈×C₂₃	—	4	4	no	—
120	C₈×C₂₃	—	4	4	no	—
121	#2863	C₂×C₂₃	1	1	yes	characteristic
122	#1745	C₂×C₂₃	1	1	yes	characteristic
123	#1744	C₂×C₂₃	1	1	yes	characteristic
124	#1272	C₂×C₂₃	1	2	yes	—
125	#1271	C₂×C₂₃	1	2	yes	—
126	C₂²×C₂₃	#408	1	1	yes	characteristic
127	C₂²×C₂₃	#410	1	1	yes	characteristic
128	C₂²×C₂₃	#413	1	1	yes	characteristic
129	C₂²×C₂₃	#416	1	1	yes	characteristic
130	C₂²×C₂₃	#419	1	1	yes	characteristic
131	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
132	C₂²×C₂₃	C₂×#104	1	1	yes	characteristic
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	2	2	no	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	2	2	no	—
137	C₄×C₂₃	—	2	2	no	—
138	C₄×C₂₃	—	4	1	no	—
139	C₄×C₂₃	—	4	1	no	—
140	C₄×C₂₃	—	8	1	no	—
141	C₄×C₂₃	—	8	1	no	—
142	C₄×C₂₃	—	8	1	no	—
143	C₄×C₂₃	—	8	1	no	—
144	C₂×#97	—	2	1	no	—
145	C₂×C₄×C₈	—	2	1	no	—
146	#391	C₄×C₂₃	1	1	yes	characteristic
147	#391	C₄×C₂₃	1	1	yes	characteristic
148	#391	C₂²×C₂₃	1	1	yes	characteristic
149	#389	C₄×C₂₃	1	1	yes	characteristic
150	#389	C₄×C₂₃	1	1	yes	characteristic
151	#389	C₂²×C₂₃	1	1	yes	characteristic
152	#387	—	2	1	no	—
153	#387	—	2	1	no	—
154	#387	—	2	1	no	—
155	#383	—	2	1	no	—
156	C₄³	C₂²×C₂₃	1	1	yes	characteristic
157	#336	—	2	1	no	—
158	#336	—	2	1	no	—
159	#336	C₂²×C₂₃	1	2	yes	—
160	#336	C₂²×C₂₃	1	2	yes	—
161	C₂×C₂₃	#1473	1	1	yes	characteristic
162	C₂×C₂₃	#1485	1	1	yes	characteristic
163	C₂×C₂₃	#1494	1	1	yes	characteristic
164	C₂×C₂₃	#1500	1	1	yes	characteristic
165	C₂×C₂₃	#1506	1	1	yes	characteristic
166	C₂×C₂₃	#1524	1	1	yes	characteristic
167	C₂×C₂₃	#1554	1	1	yes	characteristic
168	C₂²×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	1	no	—
174	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
176	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
178	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
180	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
181	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
184	#97	—	2	4	no	—
185	C₄×C₈	—	2	4	no	—
186	#95	—	2	1	no	—
187	#95	—	2	1	no	—
188	#95	—	2	1	no	—
189	#95	—	2	1	no	—
190	#95	—	2	1	no	—
191	#95	—	2	1	no	—
192	#95	—	2	1	no	—
193	#95	—	2	1	no	—
194	#95	—	2	1	no	—
195	C₂₃	#7693	1	1	yes	characteristic, direct product factor, Sylow
196	C₂×Q₈	—	2	2	no	—
197	C₂×Q₈	—	4	1	no	—
198	C₂×Q₈	—	4	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	4	2	no	—
213	C₂×C₈	—	4	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	1	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	2	2	no	—
224	C₄²	—	2	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	4	1	no	—
228	C₄²	—	4	1	no	—
229	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
230	Q₈	—	8	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	—	4	1	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	C₂₃×#104	1	2	yes	—
260	C₂×C₄	C₂₃×#104	1	2	yes	—
261	C₂×C₄	C₂₃×#131	1	2	yes	—
262	C₂×C₄	C₂₃×#135	1	2	yes	—
263	C₂×C₄	C₂₃×#137	1	2	yes	—
264	C₈	—	4	4	no	—
265	C₈	—	4	4	no	—
266	C₂²	C₂₃×#408	1	1	yes	characteristic
267	C₂²	C₂₃×#410	1	1	yes	characteristic
268	C₂²	C₂₃×#413	1	1	yes	characteristic
269	C₂²	C₂₃×#416	1	1	yes	characteristic
270	C₂²	C₂₃×#419	1	1	yes	characteristic
271	C₂²	C₂×C₂₃×#104	1	1	yes	characteristic
272	C₂²	C₂×C₂₃×#104	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	2	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	8	1	no	—
281	C₄	—	8	1	no	—
282	C₄	—	8	1	no	—
283	C₄	—	8	1	no	—
284	C₂	C₂₃×#1473	1	1	yes	characteristic
285	C₂	C₂₃×#1485	1	1	yes	characteristic
286	C₂	C₂₃×#1494	1	1	yes	characteristic
287	C₂	C₂₃×#1500	1	1	yes	characteristic
288	C₂	C₂₃×#1506	1	1	yes	characteristic
289	C₂	C₂₃×#1524	1	1	yes	characteristic
290	C₂	C₂₃×#1554	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
83. #7693	C₂₃	direct (N × H)
195. C₂₃	#7693	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×#2863	2
C₂₃×#1745	2
C₂₃×#1744	2
C₂₃×#1272	2
C₂₃×#1271	2
#7693	23

Subgroups — C23×#7693 #19155319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7693 #19155319