Subgroups — C₂₃×#2782 #15670401

Order 2944.

448 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#400	C₂	1	2	yes	maximal, complemented
3	C₂₃×#397	C₂	1	2	yes	maximal, complemented
4	C₂₃×#394	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#388	C₂	1	2	yes	maximal, complemented
6	C₂₃×#386	C₂	1	2	yes	maximal
7	C₂₃×#385	C₂	1	2	yes	maximal, complemented
8	C₂₃×#385	C₂	1	2	yes	maximal, complemented
9	C₂₃×#382	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
23	C₂₃×#95	C₂²	1	2	yes	—
24	C₂₃×#95	C₂²	1	2	yes	—
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	1	yes	characteristic
30	C₂₃×#95	C₂²	1	1	yes	characteristic
31	C₂⁴×C₂₃	C₂³	1	2	yes	—
32	C₂×D₄×C₂₃	—	2	4	no	—
33	C₂×D₄×C₂₃	—	4	1	no	—
34	C₂²×C₄×C₂₃	—	4	2	no	—
35	C₂²×C₄×C₂₃	—	4	2	no	—
36	C₂²×C₄×C₂₃	—	4	2	no	—
37	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
42	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₂₃	—	4	1	no	—
46	C₄⋊C₄×C₂₃	—	4	1	no	—
47	C₄⋊C₄×C₂₃	—	4	2	no	—
48	C₄⋊C₄×C₂₃	—	4	1	no	—
49	C₄⋊C₄×C₂₃	—	4	2	no	—
50	C₂²⋊C₄×C₂₃	—	2	4	no	—
51	C₂²⋊C₄×C₂₃	—	2	4	no	—
52	C₂²⋊C₄×C₂₃	—	2	4	no	—
53	C₂²⋊C₄×C₂₃	—	4	2	no	—
54	C₂²⋊C₄×C₂₃	—	4	2	no	—
55	C₄²×C₂₃	—	2	2	no	—
56	C₂³×C₂₃	—	2	2	no	—
57	C₂³×C₂₃	—	2	2	no	—
58	C₂³×C₂₃	—	2	2	no	—
59	C₂³×C₂₃	—	2	2	no	—
60	C₂³×C₂₃	—	2	2	no	—
61	C₂³×C₂₃	—	2	2	no	—
62	C₂³×C₂₃	#41	1	4	yes	—
63	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
64	D₄×C₂₃	—	4	4	no	—
65	C₂×C₄×C₂₃	—	2	1	no	—
66	C₂×C₄×C₂₃	—	2	1	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	2	no	—
77	C₂×C₄×C₂₃	—	2	2	no	—
78	C₂×C₄×C₂₃	—	2	1	no	—
79	C₂×C₄×C₂₃	—	2	2	no	—
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	1	no	—
85	C₂×C₄×C₂₃	—	2	2	no	—
86	C₂×C₄×C₂₃	—	4	8	no	—
87	C₂×C₄×C₂₃	#41	1	2	yes	—
88	#2782	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂²×C₂₃	—	2	4	no	—
90	C₂²×C₂₃	—	2	4	no	—
91	C₂²×C₂₃	—	2	4	no	—
92	C₂²×C₂₃	—	4	2	no	—
93	C₂²×C₂₃	—	4	2	no	—
94	C₂²×C₂₃	—	4	2	no	—
95	C₂²×C₂₃	—	4	2	no	—
96	C₂²×C₂₃	C₂×#41	1	2	yes	—
97	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
98	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
100	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
101	C₄×C₂₃	—	2	2	no	—
102	C₄×C₂₃	—	4	2	no	—
103	C₄×C₂₃	—	4	1	no	—
104	C₄×C₂₃	—	4	1	no	—
105	C₄×C₂₃	—	4	2	no	—
106	C₄×C₂₃	—	4	2	no	—
107	C₄×C₂₃	—	4	2	no	—
108	C₄×C₂₃	—	4	2	no	—
109	#400	C₂×C₂₃	1	2	yes	complemented
110	#400	C₂×C₂₃	1	1	yes	characteristic, complemented
111	#397	C₂×C₂₃	1	2	yes	complemented
112	#394	C₂×C₂₃	1	1	yes	characteristic
113	#388	C₂×C₂₃	1	2	yes	complemented
114	#386	C₂×C₂₃	1	2	yes	—
115	#385	C₂×C₂₃	1	2	yes	complemented
116	#385	C₂×C₂₃	1	2	yes	complemented
117	#382	C₂×C₂₃	1	1	yes	characteristic, complemented
118	C₂×C₂₃	—	4	4	no	—
119	C₂×C₂₃	#539	1	1	yes	characteristic
120	C₂×C₂₃	#539	1	2	yes	—
121	C₂×C₂₃	#551	1	1	yes	characteristic
122	C₂×C₂₃	#553	1	2	yes	—
123	C₂×C₂₃	H(3,𝔽₄)	1	1	yes	characteristic
124	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
125	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
126	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
127	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
128	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
129	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
130	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
131	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
132	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
133	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
134	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
135	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
136	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
137	#95	C₂²×C₂₃	1	2	yes	—
138	#95	C₂²×C₂₃	1	2	yes	—
139	#95	C₂²×C₂₃	1	2	yes	—
140	#95	C₂²×C₂₃	1	2	yes	—
141	#95	C₂²×C₂₃	1	2	yes	—
142	#95	C₂²×C₂₃	1	2	yes	—
143	#95	C₂²×C₂₃	1	1	yes	characteristic
144	#95	C₂²×C₂₃	1	1	yes	characteristic
145	C₂₃	#2782	1	1	yes	characteristic, direct product factor, Sylow
146	C₂⁴	C₂³×C₂₃	1	2	yes	—
147	C₂×D₄	—	2	4	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂²×C₄	—	4	2	no	—
150	C₂²×C₄	—	4	2	no	—
151	C₂²×C₄	—	4	2	no	—
152	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
153	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
154	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
155	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
156	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
157	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
158	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
159	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
160	C₄⋊C₄	—	4	1	no	—
161	C₄⋊C₄	—	4	1	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	1	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₂²⋊C₄	—	2	4	no	—
166	C₂²⋊C₄	—	2	4	no	—
167	C₂²⋊C₄	—	2	4	no	—
168	C₂²⋊C₄	—	4	2	no	—
169	C₂²⋊C₄	—	4	2	no	—
170	C₄²	—	2	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	2	2	no	—
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	C₂³	—	2	2	no	—
177	C₂³	#41×C₂₃	1	4	yes	—
178	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
179	D₄	—	4	4	no	—
180	C₂×C₄	—	2	1	no	—
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	#41×C₂₃	1	2	yes	—
203	C₂²	—	2	4	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	4	2	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	C₂×#41×C₂₃	1	2	yes	—
211	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
212	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
213	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
214	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	4	1	no	—
217	C₄	—	4	1	no	—
218	C₄	—	4	2	no	—
219	C₄	—	4	2	no	—
220	C₄	—	4	2	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₂	—	4	4	no	—
224	C₂	C₂₃×#539	1	1	yes	characteristic
225	C₂	C₂₃×#539	1	2	yes	—
226	C₂	C₂₃×#551	1	1	yes	characteristic
227	C₂	C₂₃×#553	1	2	yes	—
228	C₂	C₂₃×H(3,𝔽₄)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₃
prime radical	C₂²×D₄×C₂₃	C₂²
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
1. C₂₃×#400	C₂	semidirect
2. C₂₃×#400	C₂	semidirect
3. C₂₃×#397	C₂	semidirect
5. C₂₃×#388	C₂	semidirect
7. C₂₃×#385	C₂	semidirect
8. C₂₃×#385	C₂	semidirect
9. C₂₃×#382	C₂	semidirect
88. #2782	C₂₃	direct (N × H)
109. #400	C₂×C₂₃	semidirect
110. #400	C₂×C₂₃	semidirect
111. #397	C₂×C₂₃	semidirect
113. #388	C₂×C₂₃	semidirect
115. #385	C₂×C₂₃	semidirect
116. #385	C₂×C₂₃	semidirect
117. #382	C₂×C₂₃	semidirect
145. C₂₃	#2782	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×#400	2
C₂₃×#397	2
C₂₃×#394	2
C₂₃×#388	2
C₂₃×#386	2
C₂₃×#385	2
C₂₃×#382	2
#2782	23

Subgroups — C23×#2782 #15670401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2782 #15670401