Subgroups — C₂₃×#8279 #19155905

Order 5888.

394 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#1767	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#1277	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×#97	—	2	1	no	—
13	C₂×C₄×C₈×C₂₃	—	2	1	no	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#336	—	2	1	no	—
18	C₂₃×#336	—	2	1	no	—
19	C₂₃×#336	C₄	1	4	yes	—
20	C₂₃×#335	C₄	1	4	yes	complemented
21	C₂₃×#334	C₄	1	4	yes	complemented
22	C₂²×C₈×C₂₃	—	2	1	no	—
23	C₂²×C₈×C₂₃	—	2	1	no	—
24	C₂²×C₈×C₂₃	—	2	1	no	—
25	C₂²×C₈×C₂₃	—	2	1	no	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
33	C₂₃×C₄⋊C₈	—	2	1	no	—
34	C₂₃×C₄⋊C₈	—	2	1	no	—
35	C₂₃×C₄⋊C₈	—	2	1	no	—
36	C₂₃×C₄⋊C₈	—	2	1	no	—
37	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
38	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₂₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₂₃×#97	—	4	2	no	—
42	C₄×C₈×C₂₃	—	2	1	no	—
43	C₄×C₈×C₂₃	—	2	1	no	—
44	C₄×C₈×C₂₃	—	4	2	no	—
45	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₂₃	—	2	4	no	—
53	C₂×C₈×C₂₃	—	2	2	no	—
54	C₂×C₈×C₂₃	—	2	2	no	—
55	C₂×C₈×C₂₃	—	2	2	no	—
56	C₂×C₈×C₂₃	—	2	2	no	—
57	C₂×C₈×C₂₃	—	2	2	no	—
58	C₂×C₈×C₂₃	—	2	4	no	—
59	C₂×C₈×C₂₃	—	4	1	no	—
60	C₂×C₈×C₂₃	—	4	1	no	—
61	C₂×C₈×C₂₃	—	4	1	no	—
62	C₂×C₈×C₂₃	—	4	1	no	—
63	C₂×C₈×C₂₃	—	4	1	no	—
64	C₂×C₈×C₂₃	—	4	1	no	—
65	C₂×C₈×C₂₃	—	4	1	no	—
66	C₂×C₈×C₂₃	—	4	1	no	—
67	C₂×C₈×C₂₃	C₄²	1	4	yes	—
68	C₄²×C₂₃	—	2	4	no	—
69	C₄²×C₂₃	—	2	4	no	—
70	C₄²×C₂₃	—	2	1	no	—
71	C₄²×C₂₃	—	2	1	no	—
72	C₄²×C₂₃	—	2	1	no	—
73	C₄²×C₂₃	—	2	1	no	—
74	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₂₃	#41	1	1	yes	characteristic
77	C₄²×C₂₃	#41	1	1	yes	characteristic
78	#8279	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	—	2	2	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	#108	1	4	yes	—
87	C₂×C₄×C₂₃	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄×C₂₃	#117	1	1	yes	characteristic
89	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
91	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₂₃	C₄×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₂₃	#131	1	4	yes	—
94	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
96	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
97	C₂×C₄×C₂₃	#137	1	1	yes	characteristic
98	C₈×C₂₃	—	2	4	no	—
99	C₈×C₂₃	—	4	2	no	—
100	C₈×C₂₃	—	4	2	no	—
101	C₈×C₂₃	—	4	2	no	—
102	C₈×C₂₃	—	4	2	no	—
103	C₈×C₂₃	—	4	2	no	—
104	#1767	C₂×C₂₃	1	1	yes	characteristic
105	C₄×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
106	#1744	C₂×C₂₃	1	1	yes	characteristic
107	C₂×#335	C₂×C₂₃	1	1	yes	characteristic
108	C₂×#334	C₂×C₂₃	1	1	yes	characteristic
109	#1277	C₂×C₂₃	1	1	yes	characteristic
110	#1277	C₂×C₂₃	1	1	yes	characteristic
111	C₂²×C₂₃	C₄×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₂₃	#424	1	2	yes	—
113	C₂²×C₂₃	#428	1	1	yes	characteristic
114	C₂²×C₂₃	#428	1	1	yes	characteristic
115	C₂²×C₂₃	C₂×#108	1	2	yes	—
116	C₄×C₂₃	—	2	4	no	—
117	C₄×C₂₃	—	2	1	no	—
118	C₄×C₂₃	—	2	1	no	—
119	C₄×C₂₃	—	4	2	no	—
120	C₄×C₂₃	#405	1	2	yes	—
121	C₄×C₂₃	#433	1	2	yes	—
122	C₄×C₂₃	#440	1	1	yes	characteristic
123	C₄×C₂₃	#440	1	1	yes	characteristic
124	C₄×C₂₃	#441	1	1	yes	characteristic
125	C₄×C₂₃	#442	1	1	yes	characteristic
126	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
127	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
128	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
129	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
130	C₂×#97	—	2	1	no	—
131	C₂×C₄×C₈	—	2	1	no	—
132	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
134	C₄³	C₂²×C₂₃	1	1	yes	characteristic
135	#336	—	2	1	no	—
136	#336	—	2	1	no	—
137	#336	C₄×C₂₃	1	4	yes	—
138	#335	C₄×C₂₃	1	4	yes	complemented
139	#334	C₄×C₂₃	1	4	yes	complemented
140	C₂×C₂₃	#1573	1	2	yes	—
141	C₂×C₂₃	#1574	1	2	yes	—
142	C₂×C₂₃	#1768	1	1	yes	characteristic
143	C₂×C₂₃	#1776	1	1	yes	characteristic
144	C₂×C₂₃	C₄×#108	1	1	yes	characteristic
145	C₂²×C₈	—	2	1	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
150	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
151	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
152	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
153	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
154	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
155	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
161	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
162	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
163	C₄⋊C₈	C₂×C₄×C₂₃	1	2	yes	—
164	#97	—	4	2	no	—
165	C₄×C₈	—	2	1	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	4	2	no	—
168	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
169	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
170	C₂₃	#8279	1	1	yes	characteristic, direct product factor, Sylow
171	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
172	C₂²×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
173	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
175	C₂²×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	C₄²×C₂₃	1	4	yes	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
199	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
200	C₄²	#41×C₂₃	1	1	yes	characteristic
201	C₄²	#41×C₂₃	1	1	yes	characteristic
202	C₂³	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	C₂₃×#108	1	4	yes	—
210	C₂×C₄	C₂×C₄²×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂₃×#117	1	1	yes	characteristic
212	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
213	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
214	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
215	C₂×C₄	C₄×Q₈×C₂₃	1	1	yes	characteristic
216	C₂×C₄	C₂₃×#131	1	4	yes	—
217	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
218	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
219	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
220	C₂×C₄	C₂₃×#137	1	1	yes	characteristic
221	C₈	—	2	4	no	—
222	C₈	—	4	2	no	—
223	C₈	—	4	2	no	—
224	C₈	—	4	2	no	—
225	C₈	—	4	2	no	—
226	C₈	—	4	2	no	—
227	C₂²	C₄×C₄⋊C₄×C₂₃	1	1	yes	characteristic
228	C₂²	C₂₃×#424	1	2	yes	—
229	C₂²	C₂₃×#428	1	1	yes	characteristic
230	C₂²	C₂₃×#428	1	1	yes	characteristic
231	C₂²	C₂×C₂₃×#108	1	2	yes	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	C₂₃×#405	1	2	yes	—
237	C₄	C₂₃×#433	1	2	yes	—
238	C₄	C₂₃×#440	1	1	yes	characteristic
239	C₄	C₂₃×#440	1	1	yes	characteristic
240	C₄	C₂₃×#441	1	1	yes	characteristic
241	C₄	C₂₃×#442	1	1	yes	characteristic
242	C₂	C₂₃×#1573	1	2	yes	—
243	C₂	C₂₃×#1574	1	2	yes	—
244	C₂	C₂₃×#1768	1	1	yes	characteristic
245	C₂	C₂₃×#1776	1	1	yes	characteristic
246	C₂	C₄×C₂₃×#108	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, prime radical	C₂³×C₂₃	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
20. C₂₃×#335	C₄	semidirect
21. C₂₃×#334	C₄	semidirect
78. #8279	C₂₃	direct (N × H)
138. #335	C₄×C₂₃	semidirect
139. #334	C₄×C₂₃	semidirect
170. C₂₃	#8279	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×#1767	2
C₄×C₂₃×C₄⋊C₈	2
C₂₃×#1744	2
C₂×C₂₃×#335	2
C₂×C₂₃×#334	2
C₂₃×#1277	2
#8279	23

Subgroups — C23×#8279 #19155905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8279 #19155905