Subgroups — C₂₃×#2517 #15670136

Order 2944.

412 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#399	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#397	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#397	C₂	1	2	yes	maximal
4	C₂₃×#387	C₂	1	2	yes	maximal, complemented
5	C₂₃×#387	C₂	1	2	yes	maximal, complemented
6	C₂₃×#387	C₂	1	2	yes	maximal, complemented
7	C₂₃×#385	C₂	1	1	yes	maximal, characteristic
8	C₂₃×#382	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₂₃	C₂	1	2	yes	maximal
10	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
22	C₂₃×#95	C₂²	1	2	yes	—
23	C₂₃×#95	C₂²	1	2	yes	—
24	C₂₃×#95	C₂²	1	1	yes	characteristic
25	C₂₃×#95	C₂²	1	2	yes	—
26	C₂₃×#95	C₂²	1	2	yes	—
27	C₂₃×#95	C₂²	1	2	yes	—
28	C₂₃×#95	C₂²	1	2	yes	—
29	C₂₃×#95	C₂²	1	2	yes	—
30	C₂₃×#95	C₂²	1	1	yes	characteristic
31	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
32	C₂²×C₄×C₂₃	—	2	4	no	—
33	C₂²×C₄×C₂₃	—	4	1	no	—
34	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
35	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
36	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
38	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₄⋊C₄×C₂₃	—	2	4	no	—
43	C₄⋊C₄×C₂₃	—	2	4	no	—
44	C₄⋊C₄×C₂₃	—	2	4	no	—
45	C₄⋊C₄×C₂₃	—	4	1	no	—
46	C₄⋊C₄×C₂₃	—	4	1	no	—
47	C₂²⋊C₄×C₂₃	—	2	4	no	—
48	C₂²⋊C₄×C₂₃	—	2	2	no	—
49	C₂²⋊C₄×C₂₃	—	2	4	no	—
50	C₂²⋊C₄×C₂₃	—	4	1	no	—
51	C₄²×C₂₃	—	2	4	no	—
52	C₄²×C₂₃	—	4	2	no	—
53	C₂³×C₂₃	—	2	2	no	—
54	C₂³×C₂₃	—	2	2	no	—
55	C₂³×C₂₃	—	2	1	no	—
56	C₂³×C₂₃	—	2	1	no	—
57	C₂³×C₂₃	#41	1	2	yes	—
58	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
59	C₂×C₄×C₂₃	—	2	2	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	2	2	no	—
62	C₂×C₄×C₂₃	—	2	2	no	—
63	C₂×C₄×C₂₃	—	2	2	no	—
64	C₂×C₄×C₂₃	—	2	2	no	—
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	2	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	2	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	2	2	no	—
75	C₂×C₄×C₂₃	—	2	2	no	—
76	C₂×C₄×C₂₃	—	2	1	no	—
77	C₂×C₄×C₂₃	—	2	1	no	—
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	4	4	no	—
80	C₂×C₄×C₂₃	#41	1	4	yes	—
81	C₂×C₄×C₂₃	#41	1	4	yes	—
82	#2517	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂²×C₂₃	—	2	4	no	—
84	C₂²×C₂₃	—	2	2	no	—
85	C₂²×C₂₃	—	4	1	no	—
86	C₂²×C₂₃	—	4	2	no	—
87	C₂²×C₂₃	—	4	1	no	—
88	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
89	C₂²×C₂₃	C₂×#41	1	2	yes	—
90	C₂²×C₂₃	C₂×#41	1	2	yes	—
91	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
93	C₄×C₂₃	—	2	4	no	—
94	C₄×C₂₃	—	2	4	no	—
95	C₄×C₂₃	—	4	2	no	—
96	C₄×C₂₃	—	4	2	no	—
97	C₄×C₂₃	—	4	2	no	—
98	C₄×C₂₃	—	4	1	no	—
99	C₄×C₂₃	—	4	2	no	—
100	C₄×C₂₃	—	4	1	no	—
101	#399	C₂×C₂₃	1	1	yes	characteristic
102	#397	C₂×C₂₃	1	1	yes	characteristic
103	#397	C₂×C₂₃	1	2	yes	—
104	#387	C₂×C₂₃	1	2	yes	complemented
105	#387	C₂×C₂₃	1	2	yes	complemented
106	#387	C₂×C₂₃	1	2	yes	complemented
107	#385	C₂×C₂₃	1	1	yes	characteristic
108	#382	C₂×C₂₃	1	2	yes	complemented
109	C₄×C₂²⋊C₄	C₂×C₂₃	1	2	yes	—
110	C₂×C₂₃	—	4	2	no	—
111	C₂×C₂₃	#529	1	2	yes	—
112	C₂×C₂₃	#539	1	1	yes	characteristic
113	C₂×C₂₃	#551	1	1	yes	characteristic
114	C₂×C₂₃	#552	1	2	yes	—
115	C₂×C₂₃	#559	1	1	yes	characteristic
116	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
117	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
118	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
120	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
121	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
122	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
123	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
124	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
125	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
126	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
127	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
128	#95	C₂²×C₂₃	1	2	yes	—
129	#95	C₂²×C₂₃	1	2	yes	—
130	#95	C₂²×C₂₃	1	1	yes	characteristic
131	#95	C₂²×C₂₃	1	2	yes	—
132	#95	C₂²×C₂₃	1	2	yes	—
133	#95	C₂²×C₂₃	1	2	yes	—
134	#95	C₂²×C₂₃	1	2	yes	—
135	#95	C₂²×C₂₃	1	2	yes	—
136	#95	C₂²×C₂₃	1	1	yes	characteristic
137	C₂₃	#2517	1	1	yes	characteristic, direct product factor, Sylow
138	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	4	1	no	—
141	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
142	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
143	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
144	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
145	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
147	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
148	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	4	1	no	—
153	C₄⋊C₄	—	4	1	no	—
154	C₂²⋊C₄	—	2	4	no	—
155	C₂²⋊C₄	—	2	2	no	—
156	C₂²⋊C₄	—	2	4	no	—
157	C₂²⋊C₄	—	4	1	no	—
158	C₄²	—	2	4	no	—
159	C₄²	—	4	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	—	2	2	no	—
162	C₂³	—	2	1	no	—
163	C₂³	—	2	1	no	—
164	C₂³	#41×C₂₃	1	2	yes	—
165	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	#41×C₂₃	1	4	yes	—
188	C₂×C₄	#41×C₂₃	1	4	yes	—
189	C₂²	—	2	4	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	4	1	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	1	no	—
194	C₂²	C₂×#41×C₂₃	1	2	yes	—
195	C₂²	C₂×#41×C₂₃	1	2	yes	—
196	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
197	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
198	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	2	4	no	—
201	C₄	—	4	1	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	1	no	—
204	C₄	—	4	2	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	2	no	—
207	C₂	—	4	2	no	—
208	C₂	C₂₃×#529	1	2	yes	—
209	C₂	C₂₃×#539	1	1	yes	characteristic
210	C₂	C₂₃×#551	1	1	yes	characteristic
211	C₂	C₂₃×#552	1	2	yes	—
212	C₂	C₂₃×#559	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₂₃×#387	C₂	semidirect
5. C₂₃×#387	C₂	semidirect
6. C₂₃×#387	C₂	semidirect
8. C₂₃×#382	C₂	semidirect
82. #2517	C₂₃	direct (N × H)
104. #387	C₂×C₂₃	semidirect
105. #387	C₂×C₂₃	semidirect
106. #387	C₂×C₂₃	semidirect
108. #382	C₂×C₂₃	semidirect
137. C₂₃	#2517	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×#399	2
C₂₃×#397	2
C₂₃×#387	2
C₂₃×#385	2
C₂₃×#382	2
C₄×C₂²⋊C₄×C₂₃	2
#2517	23

Subgroups — C23×#2517 #15670136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2517 #15670136