Subgroups — C₂₃×#7898 #19155524

Order 5888.

710 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2283	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1279	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₂₃	—	2	1	no	—
6	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#432	—	2	2	no	—
8	C₂₃×#432	C₄	1	4	yes	complemented
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
11	C₂×C₂₃×#97	—	2	1	no	—
12	C₂×C₄×C₈×C₂₃	—	2	1	no	—
13	C₂×C₄×C₈×C₂₃	—	2	2	no	—
14	C₂×C₄×C₈×C₂₃	C₄	1	4	yes	complemented
15	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
16	C₂₃×#383	—	2	1	no	—
17	C₂₃×#382	C₄	1	2	yes	—
18	C₂₃×#381	—	2	1	no	—
19	C₂₃×#380	C₄	1	2	yes	—
20	C₄×C₄⋊C₄×C₂₃	C₂²	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	C₂²×C₈×C₂₃	—	2	8	no	—
27	C₂²×C₈×C₂₃	—	2	4	no	—
28	C₂²×C₈×C₂₃	—	2	2	no	—
29	C₂²×C₈×C₂₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
31	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
32	C₂×C₄²×C₂₃	—	2	2	no	—
33	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₂₃	D₄	1	2	yes	—
39	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
40	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂₃×C₄⋊C₈	—	2	2	no	—
42	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
43	C₂₃×C₂²⋊C₈	—	2	4	no	—
44	C₂₃×#97	—	2	4	no	—
45	C₄×C₈×C₂₃	—	2	4	no	—
46	C₄×C₈×C₂₃	—	2	2	no	—
47	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
48	C₂₃×#95	—	2	2	no	—
49	C₂₃×#95	—	2	2	no	—
50	C₂₃×#95	—	2	2	no	—
51	C₂₃×#95	—	2	2	no	—
52	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	—	2	4	no	—
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	C₄²	1	2	yes	—
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	2	yes	—
62	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₂₃	—	2	16	no	—
70	C₂×C₈×C₂₃	—	2	8	no	—
71	C₂×C₈×C₂₃	—	2	4	no	—
72	C₂×C₈×C₂₃	—	2	4	no	—
73	C₂×C₈×C₂₃	—	4	4	no	—
74	C₄⋊C₄×C₂₃	—	2	8	no	—
75	C₂²⋊C₄×C₂₃	—	4	4	no	—
76	C₄²×C₂₃	—	4	4	no	—
77	C₄²×C₂₃	C₄²	1	2	yes	—
78	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
79	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
80	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
82	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
83	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
87	#7898	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₂₃	—	2	2	no	—
89	C₂³×C₂₃	—	2	2	no	—
90	C₂³×C₂₃	#95	1	2	yes	—
91	C₂³×C₂₃	#104	1	4	yes	—
92	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	—	2	4	no	—
94	C₂×C₄×C₂₃	—	2	4	no	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	2	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	4	no	—
100	C₂×C₄×C₂₃	—	2	4	no	—
101	C₂×C₄×C₂₃	—	2	4	no	—
102	C₂×C₄×C₂₃	—	2	4	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	—	4	4	no	—
105	C₂×C₄×C₂₃	—	4	4	no	—
106	C₂×C₄×C₂₃	#95	1	2	yes	—
107	C₂×C₄×C₂₃	#95	1	2	yes	—
108	C₂×C₄×C₂₃	#95	1	2	yes	—
109	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	#131	1	4	yes	—
116	C₂×C₄×C₂₃	#135	1	4	yes	—
117	C₈×C₂₃	—	2	8	no	—
118	C₈×C₂₃	—	4	4	no	—
119	C₂×#432	C₂×C₂₃	1	1	yes	characteristic
120	C₂²×C₄×C₈	C₂×C₂₃	1	1	yes	characteristic
121	#2283	C₂×C₂₃	1	1	yes	characteristic
122	#1279	C₂×C₂₃	1	4	yes	complemented
123	C₂²×C₂₃	—	2	2	no	—
124	C₂²×C₂₃	—	2	2	no	—
125	C₂²×C₂₃	—	2	8	no	—
126	C₂²×C₂₃	#339	1	4	yes	—
127	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
128	C₂²×C₂₃	#408	1	2	yes	—
129	C₂²×C₂₃	#416	1	2	yes	—
130	C₂²×C₂₃	C₂×#104	1	2	yes	—
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	2	1	no	—
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	8	4	no	—
137	C₄×C₂₃	#405	1	2	yes	—
138	C₄×C₂₃	#416	1	2	yes	—
139	C₄×C₂₃	#424	1	2	yes	—
140	C₄×C₂₃	#427	1	2	yes	—
141	C₂³×C₈	—	2	1	no	—
142	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
143	#432	—	2	2	no	—
144	#432	C₄×C₂₃	1	4	yes	complemented
145	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
146	C₂×C₂²⋊C₈	—	2	1	no	—
147	C₂×#97	—	2	1	no	—
148	C₂×C₄×C₈	—	2	1	no	—
149	C₂×C₄×C₈	—	2	2	no	—
150	C₂×C₄×C₈	C₄×C₂₃	1	4	yes	complemented
151	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
152	#383	—	2	1	no	—
153	#382	C₄×C₂₃	1	2	yes	—
154	#381	—	2	1	no	—
155	#380	C₄×C₂₃	1	2	yes	—
156	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
157	C₄×C₂²⋊C₄	—	2	1	no	—
158	#376	—	2	1	no	—
159	C₂×C₂₃	—	2	4	no	—
160	C₂×C₂₃	#1494	1	4	yes	—
161	C₂×C₂₃	#1729	1	1	yes	characteristic
162	C₂×C₂₃	C₂×#339	1	1	yes	characteristic
163	C₂×C₂₃	#1735	1	1	yes	characteristic
164	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
165	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
166	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
167	C₂²×C₈	—	2	8	no	—
168	C₂²×C₈	—	2	4	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₄²	—	2	2	no	—
174	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
176	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
177	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
178	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
179	C₂×C₄²	D₄×C₂₃	1	2	yes	—
180	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
181	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
184	C₂²⋊C₈	—	2	4	no	—
185	#97	—	2	4	no	—
186	C₄×C₈	—	2	4	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
189	#95	—	2	2	no	—
190	#95	—	2	2	no	—
191	#95	—	2	2	no	—
192	#95	—	2	2	no	—
193	C₂₃	#7898	1	1	yes	characteristic, direct product factor, Sylow
194	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	4	no	—
199	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
200	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
201	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
202	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
203	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
205	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
206	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
207	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
208	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
211	C₂×C₈	—	2	16	no	—
212	C₂×C₈	—	2	8	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	4	4	no	—
216	C₄⋊C₄	—	2	8	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₄²	—	4	4	no	—
219	C₄²	C₄²×C₂₃	1	2	yes	—
220	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
221	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
222	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
223	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
224	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
225	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
226	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
227	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
228	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	C₂₃×#95	1	2	yes	—
232	C₂³	C₂₃×#104	1	4	yes	—
233	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	4	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	C₂₃×#95	1	2	yes	—
248	C₂×C₄	C₂₃×#95	1	2	yes	—
249	C₂×C₄	C₂₃×#95	1	2	yes	—
250	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
256	C₂×C₄	C₂₃×#131	1	4	yes	—
257	C₂×C₄	C₂₃×#135	1	4	yes	—
258	C₈	—	2	8	no	—
259	C₈	—	4	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	8	no	—
263	C₂²	C₂₃×#339	1	4	yes	—
264	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
265	C₂²	C₂₃×#408	1	2	yes	—
266	C₂²	C₂₃×#416	1	2	yes	—
267	C₂²	C₂×C₂₃×#104	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	8	4	no	—
274	C₄	C₂₃×#405	1	2	yes	—
275	C₄	C₂₃×#416	1	2	yes	—
276	C₄	C₂₃×#424	1	2	yes	—
277	C₄	C₂₃×#427	1	2	yes	—
278	C₂	—	2	4	no	—
279	C₂	C₂₃×#1494	1	4	yes	—
280	C₂	C₂₃×#1729	1	1	yes	characteristic
281	C₂	C₂×C₂₃×#339	1	1	yes	characteristic
282	C₂	C₂₃×#1735	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂×D₄
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 ▷ C₂ ▷ 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×#1279	C₂	semidirect
8. C₂₃×#432	C₄	semidirect
14. C₂×C₄×C₈×C₂₃	C₄	semidirect
87. #7898	C₂₃	direct (N × H)
122. #1279	C₂×C₂₃	semidirect
144. #432	C₄×C₂₃	semidirect
150. C₂×C₄×C₈	C₄×C₂₃	semidirect
193. C₂₃	#7898	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×#432	2
C₂²×C₄×C₈×C₂₃	2
C₂₃×#2283	2
C₂₃×#1279	2
#7898	23

Subgroups — C23×#7898 #19155524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7898 #19155524