Subgroups — C₂₃×#7894 #19155520

Order 5888.

614 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×#431	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	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂₃×#431	—	2	2	no	—
7	C₂₃×#431	—	2	2	no	—
8	C₂₃×#431	C₄	1	4	yes	complemented
9	C₂₃×#431	C₄	1	4	yes	complemented
10	C₂×C₂₃×C₄⋊C₈	—	2	1	no	—
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
13	C₂×C₂₃×C₂²⋊C₈	—	2	1	no	—
14	C₂×C₄×C₈×C₂₃	—	2	1	no	—
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	2	no	—
27	C₂²×C₈×C₂₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
30	C₂×C₄²×C₂₃	—	2	2	no	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₂₃	D₄	1	2	yes	—
37	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
38	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
39	C₂₃×C₄⋊C₈	—	2	4	no	—
40	C₂₃×C₄⋊C₈	—	2	2	no	—
41	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
42	C₂₃×C₂²⋊C₈	—	2	4	no	—
43	C₂₃×C₂²⋊C₈	—	2	4	no	—
44	C₄×C₈×C₂₃	—	2	4	no	—
45	C₄×C₈×C₂₃	—	2	2	no	—
46	C₄×C₈×C₂₃	C₂×C₄	1	4	yes	—
47	C₂₃×#95	—	2	2	no	—
48	C₂₃×#95	—	2	2	no	—
49	C₂₃×#95	—	2	2	no	—
50	C₂₃×#95	—	2	2	no	—
51	C₂⁴×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	—	2	4	no	—
53	C₂²×C₄×C₂₃	—	2	4	no	—
54	C₂²×C₄×C₂₃	—	2	4	no	—
55	C₂²×C₄×C₂₃	—	4	2	no	—
56	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
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	2	yes	—
61	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₂₃	C₄⋊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	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₂₃	—	2	4	no	—
69	C₂×C₈×C₂₃	—	2	4	no	—
70	C₂×C₈×C₂₃	—	4	4	no	—
71	C₂×C₈×C₂₃	—	4	4	no	—
72	C₄⋊C₄×C₂₃	—	2	8	no	—
73	C₂²⋊C₄×C₂₃	—	4	4	no	—
74	C₄²×C₂₃	—	4	4	no	—
75	C₄²×C₂₃	C₄²	1	2	yes	—
76	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
77	C₄²×C₂₃	C₂²⋊C₄	1	2	yes	—
78	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
79	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
80	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
81	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
85	#7894	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₂₃	—	2	2	no	—
87	C₂³×C₂₃	—	2	2	no	—
88	C₂³×C₂₃	—	2	2	no	—
89	C₂³×C₂₃	#95	1	2	yes	—
90	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	2	2	no	—
93	C₂×C₄×C₂₃	—	2	2	no	—
94	C₂×C₄×C₂₃	—	2	2	no	—
95	C₂×C₄×C₂₃	—	2	4	no	—
96	C₂×C₄×C₂₃	—	2	4	no	—
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	—	2	4	no	—
99	C₂×C₄×C₂₃	—	4	2	no	—
100	C₂×C₄×C₂₃	—	4	2	no	—
101	C₂×C₄×C₂₃	—	4	4	no	—
102	C₂×C₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	#95	1	2	yes	—
105	C₂×C₄×C₂₃	#95	1	2	yes	—
106	C₂×C₄×C₂₃	#95	1	2	yes	—
107	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	#131	1	4	yes	—
114	C₂×C₄×C₂₃	#135	1	4	yes	—
115	C₈×C₂₃	—	4	4	no	—
116	C₈×C₂₃	—	4	4	no	—
117	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
118	C₂×#431	C₂×C₂₃	1	1	yes	characteristic
119	#2283	C₂×C₂₃	1	1	yes	characteristic
120	#1279	C₂×C₂₃	1	4	yes	complemented
121	C₂²×C₂₃	—	2	2	no	—
122	C₂²×C₂₃	—	2	2	no	—
123	C₂²×C₂₃	—	2	2	no	—
124	C₂²×C₂₃	—	4	4	no	—
125	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
126	C₂²×C₂₃	#408	1	2	yes	—
127	C₂²×C₂₃	#416	1	2	yes	—
128	C₂²×C₂₃	#421	1	2	yes	—
129	C₄×C₂₃	—	2	2	no	—
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	2	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	2	2	no	—
134	C₄×C₂₃	—	8	4	no	—
135	C₄×C₂₃	#405	1	2	yes	—
136	C₄×C₂₃	#416	1	2	yes	—
137	C₄×C₂₃	#424	1	2	yes	—
138	C₄×C₂₃	#427	1	2	yes	—
139	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
140	#431	—	2	2	no	—
141	#431	—	2	2	no	—
142	#431	C₄×C₂₃	1	4	yes	complemented
143	#431	C₄×C₂₃	1	4	yes	complemented
144	C₂×C₄⋊C₈	—	2	1	no	—
145	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
146	C₂×C₂²⋊C₈	—	2	1	no	—
147	C₂×C₂²⋊C₈	—	2	1	no	—
148	C₂×C₄×C₈	—	2	1	no	—
149	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
150	#383	—	2	1	no	—
151	#382	C₄×C₂₃	1	2	yes	—
152	#381	—	2	1	no	—
153	#380	C₄×C₂₃	1	2	yes	—
154	C₄×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
155	C₄×C₂²⋊C₄	—	2	1	no	—
156	#376	—	2	1	no	—
157	C₂×C₂₃	—	4	2	no	—
158	C₂×C₂₃	#1482	1	4	yes	—
159	C₂×C₂₃	#1729	1	1	yes	characteristic
160	C₂×C₂₃	#1733	1	1	yes	characteristic
161	C₂×C₂₃	#1735	1	1	yes	characteristic
162	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
163	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
164	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
168	C₂×C₂²⋊C₄	—	2	2	no	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
171	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
172	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
173	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
174	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
175	C₂×C₄²	D₄×C₂₃	1	2	yes	—
176	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
177	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
178	C₄⋊C₈	—	2	4	no	—
179	C₄⋊C₈	—	2	2	no	—
180	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	2	4	no	—
183	C₄×C₈	—	2	4	no	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	C₂×C₄×C₂₃	1	4	yes	—
186	#95	—	2	2	no	—
187	#95	—	2	2	no	—
188	#95	—	2	2	no	—
189	#95	—	2	2	no	—
190	C₂₃	#7894	1	1	yes	characteristic, direct product factor, Sylow
191	C₂⁴	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
197	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
198	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊C₄×C₂₃	1	1	yes	characteristic
201	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
202	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
203	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
204	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	4	4	no	—
211	C₂×C₈	—	4	4	no	—
212	C₄⋊C₄	—	2	8	no	—
213	C₂²⋊C₄	—	4	4	no	—
214	C₄²	—	4	4	no	—
215	C₄²	C₄²×C₂₃	1	2	yes	—
216	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
217	C₄²	C₂²⋊C₄×C₂₃	1	2	yes	—
218	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
219	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
220	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
221	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
222	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
223	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
224	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	2	no	—
228	C₂³	C₂₃×#95	1	2	yes	—
229	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	C₂₃×#95	1	2	yes	—
244	C₂×C₄	C₂₃×#95	1	2	yes	—
245	C₂×C₄	C₂₃×#95	1	2	yes	—
246	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
247	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
249	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂₃×#131	1	4	yes	—
253	C₂×C₄	C₂₃×#135	1	4	yes	—
254	C₈	—	4	4	no	—
255	C₈	—	4	4	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	4	no	—
260	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
261	C₂²	C₂₃×#408	1	2	yes	—
262	C₂²	C₂₃×#416	1	2	yes	—
263	C₂²	C₂₃×#421	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	8	4	no	—
270	C₄	C₂₃×#405	1	2	yes	—
271	C₄	C₂₃×#416	1	2	yes	—
272	C₄	C₂₃×#424	1	2	yes	—
273	C₄	C₂₃×#427	1	2	yes	—
274	C₂	—	4	2	no	—
275	C₂	C₂₃×#1482	1	4	yes	—
276	C₂	C₂₃×#1729	1	1	yes	characteristic
277	C₂	C₂₃×#1733	1	1	yes	characteristic
278	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×#431	C₄	semidirect
9. C₂₃×#431	C₄	semidirect
85. #7894	C₂₃	direct (N × H)
120. #1279	C₂×C₂₃	semidirect
142. #431	C₄×C₂₃	semidirect
143. #431	C₄×C₂₃	semidirect
190. C₂₃	#7894	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×#431	2
C₂₃×#2283	2
C₂₃×#1279	2
#7894	23

Subgroups — C23×#7894 #19155520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#7894 #19155520