Subgroups — C₂₃×#8029 #19155655

Order 5888.

746 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂₃×#97	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1295	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₂₃	—	2	1	no	—
6	C₂³×C₈×C₂₃	—	2	1	no	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₄×M₁₆×C₂₃	—	2	2	no	—
9	C₄×M₁₆×C₂₃	C₄	1	4	yes	complemented
10	C₂×C₂₃×#97	—	2	1	no	—
11	C₂×C₂₃×#97	—	2	2	no	—
12	C₂×C₂₃×#97	C₄	1	4	yes	complemented
13	C₂×C₂₃×#97	C₂²	1	2	yes	—
14	C₂×C₂₃×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₂₃	—	2	1	no	—
16	C₂₃×#383	—	2	1	no	—
17	C₂₃×#382	C₂²	1	2	yes	—
18	C₂₃×#381	—	2	1	no	—
19	C₂₃×#380	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₂₃	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₂₃	—	2	1	no	—
22	C₂₃×#376	—	2	1	no	—
23	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₂₃	—	2	8	no	—
27	C₂×M₁₆×C₂₃	—	2	4	no	—
28	C₂²×C₈×C₂₃	—	2	8	no	—
29	C₂²×C₈×C₂₃	—	2	4	no	—
30	C₂²×C₈×C₂₃	—	2	2	no	—
31	C₂²×C₈×C₂₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
33	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
34	C₂×C₄²×C₂₃	—	2	2	no	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₂₃	D₄	1	2	yes	—
41	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
42	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
43	C₂₃×#97	—	2	4	no	—
44	C₂₃×#97	—	2	2	no	—
45	C₂₃×#97	—	2	2	no	—
46	C₂₃×#97	C₂×C₄	1	4	yes	—
47	C₂₃×#97	C₂×C₄	1	4	yes	—
48	C₄×C₈×C₂₃	—	2	4	no	—
49	C₂₃×#95	—	2	2	no	—
50	C₂₃×#95	—	2	2	no	—
51	C₂₃×#95	—	2	2	no	—
52	C₂₃×#95	—	2	2	no	—
53	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	2	4	no	—
56	C₂²×C₄×C₂₃	—	2	4	no	—
57	C₂²×C₄×C₂₃	—	4	2	no	—
58	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
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	2	yes	—
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
70	M₁₆×C₂₃	—	2	16	no	—
71	C₂×C₈×C₂₃	—	2	8	no	—
72	C₂×C₈×C₂₃	—	2	16	no	—
73	C₂×C₈×C₂₃	—	2	8	no	—
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	—	2	4	no	—
76	C₄⋊C₄×C₂₃	—	4	4	no	—
77	C₂²⋊C₄×C₂₃	—	4	4	no	—
78	C₄²×C₂₃	—	4	4	no	—
79	C₄²×C₂₃	C₄²	1	2	yes	—
80	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
84	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
85	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
89	#8029	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₂₃	—	2	2	no	—
91	C₂³×C₂₃	—	2	2	no	—
92	C₂³×C₂₃	#95	1	2	yes	—
93	C₂³×C₂₃	#104	1	4	yes	—
94	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₂₃	—	2	4	no	—
96	C₂×C₄×C₂₃	—	2	4	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	2	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	2	4	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	—	4	4	no	—
107	C₂×C₄×C₂₃	—	4	4	no	—
108	C₂×C₄×C₂₃	—	4	4	no	—
109	C₂×C₄×C₂₃	#95	1	2	yes	—
110	C₂×C₄×C₂₃	#95	1	2	yes	—
111	C₂×C₄×C₂₃	#95	1	2	yes	—
112	C₂×C₄×C₂₃	#104	1	4	yes	—
113	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₈×C₂₃	—	2	8	no	—
120	C₈×C₂₃	—	2	8	no	—
121	C₂×C₄×M₁₆	C₂×C₂₃	1	1	yes	characteristic
122	C₂²×#97	C₂×C₂₃	1	1	yes	characteristic
123	#2283	C₂×C₂₃	1	1	yes	characteristic
124	#1295	C₂×C₂₃	1	4	yes	complemented
125	C₂²×C₂₃	—	2	2	no	—
126	C₂²×C₂₃	—	2	2	no	—
127	C₂²×C₂₃	—	2	8	no	—
128	C₂²×C₂₃	#339	1	4	yes	—
129	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
130	C₂²×C₂₃	#413	1	2	yes	—
131	C₂²×C₂₃	C₂×#104	1	2	yes	—
132	C₂²×C₂₃	C₂×#104	1	2	yes	—
133	C₄×C₂₃	—	2	1	no	—
134	C₄×C₂₃	—	2	1	no	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	2	1	no	—
137	C₄×C₂₃	—	2	1	no	—
138	C₄×C₂₃	—	2	2	no	—
139	C₄×C₂₃	—	2	1	no	—
140	C₄×C₂₃	—	2	1	no	—
141	C₄×C₂₃	—	8	4	no	—
142	C₄×C₂₃	#339	1	4	yes	—
143	C₂²×M₁₆	—	2	1	no	—
144	C₂³×C₈	—	2	1	no	—
145	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
146	C₄×M₁₆	—	2	2	no	—
147	C₄×M₁₆	C₄×C₂₃	1	4	yes	complemented
148	C₂×#97	—	2	1	no	—
149	C₂×#97	—	2	2	no	—
150	C₂×#97	C₄×C₂₃	1	4	yes	complemented
151	C₂×#97	C₂²×C₂₃	1	2	yes	—
152	C₂×#97	C₂²×C₂₃	1	2	yes	—
153	C₂×C₄×C₈	—	2	1	no	—
154	#383	—	2	1	no	—
155	#382	C₂²×C₂₃	1	2	yes	—
156	#381	—	2	1	no	—
157	#380	C₄×C₂₃	1	2	yes	—
158	C₄×C₄⋊C₄	C₄×C₂₃	1	2	yes	—
159	C₄×C₂²⋊C₄	—	2	1	no	—
160	#376	—	2	1	no	—
161	C₂×C₂₃	—	2	4	no	—
162	C₂×C₂₃	#1524	1	4	yes	—
163	C₂×C₂₃	C₂×#339	1	1	yes	characteristic
164	C₂×C₂₃	C₂×#339	1	1	yes	characteristic
165	C₂×C₂₃	#1742	1	1	yes	characteristic
166	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
167	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
168	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
169	C₂×M₁₆	—	2	8	no	—
170	C₂×M₁₆	—	2	4	no	—
171	C₂²×C₈	—	2	8	no	—
172	C₂²×C₈	—	2	4	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
176	C₂×C₂²⋊C₄	—	2	2	no	—
177	C₂×C₄²	—	2	2	no	—
178	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
179	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
180	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
181	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
182	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
183	C₂×C₄²	D₄×C₂₃	1	2	yes	—
184	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
185	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
186	#97	—	2	4	no	—
187	#97	—	2	2	no	—
188	#97	—	2	2	no	—
189	#97	C₂×C₄×C₂₃	1	4	yes	—
190	#97	C₂×C₄×C₂₃	1	4	yes	—
191	C₄×C₈	—	2	4	no	—
192	#95	—	2	2	no	—
193	#95	—	2	2	no	—
194	#95	—	2	2	no	—
195	#95	—	2	2	no	—
196	C₂₃	#8029	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	4	2	no	—
202	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
203	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
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	C₄⋊C₄×C₂₃	1	2	yes	—
209	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
210	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
214	M₁₆	—	2	16	no	—
215	C₂×C₈	—	2	8	no	—
216	C₂×C₈	—	2	16	no	—
217	C₂×C₈	—	2	8	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	—	4	4	no	—
223	C₄²	C₄²×C₂₃	1	2	yes	—
224	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
225	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
226	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
227	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
228	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
229	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
230	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
231	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
232	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	C₂₃×#95	1	2	yes	—
236	C₂³	C₂₃×#104	1	4	yes	—
237	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	C₂₃×#95	1	2	yes	—
253	C₂×C₄	C₂₃×#95	1	2	yes	—
254	C₂×C₄	C₂₃×#95	1	2	yes	—
255	C₂×C₄	C₂₃×#104	1	4	yes	—
256	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
257	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
258	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
259	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
260	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
261	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
262	C₈	—	2	8	no	—
263	C₈	—	2	8	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	8	no	—
267	C₂²	C₂₃×#339	1	4	yes	—
268	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
269	C₂²	C₂₃×#413	1	2	yes	—
270	C₂²	C₂×C₂₃×#104	1	2	yes	—
271	C₂²	C₂×C₂₃×#104	1	2	yes	—
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	2	no	—
280	C₄	—	8	4	no	—
281	C₄	C₂₃×#339	1	4	yes	—
282	C₂	—	2	4	no	—
283	C₂	C₂₃×#1524	1	4	yes	—
284	C₂	C₂×C₂₃×#339	1	1	yes	characteristic
285	C₂	C₂×C₂₃×#339	1	1	yes	characteristic
286	C₂	C₂₃×#1742	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
socle	C₂³×C₂₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₃	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂₃×#1295	C₂	semidirect
9. C₄×M₁₆×C₂₃	C₄	semidirect
12. C₂×C₂₃×#97	C₄	semidirect
89. #8029	C₂₃	direct (N × H)
124. #1295	C₂×C₂₃	semidirect
147. C₄×M₁₆	C₄×C₂₃	semidirect
150. C₂×#97	C₄×C₂₃	semidirect
196. C₂₃	#8029	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×C₄×M₁₆×C₂₃	2
C₂²×C₂₃×#97	2
C₂₃×#2283	2
C₂₃×#1295	2
#8029	23

Subgroups — C23×#8029 #19155655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8029 #19155655