Subgroups — C₂₃×#2562 #15670181

Order 2944.

532 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#400	C₂	1	2	yes	maximal, complemented
4	C₂₃×#394	C₂	1	2	yes	maximal
5	C₂₃×#388	C₂	1	4	yes	maximal, complemented
6	C₂₃×#380	C₂	1	2	yes	maximal, complemented
7	C₂₃×#380	C₂	1	2	yes	maximal, complemented
8	C₄×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₂₃	C₂²	1	2	yes	—
11	C₂₃×#125	—	2	4	no	—
12	C₄×D₄×C₂₃	—	2	4	no	—
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×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
22	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
23	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
24	C₂₃×#95	C₂²	1	2	yes	—
25	C₂₃×#95	C₂²	1	4	yes	—
26	C₂₃×#95	C₂²	1	4	yes	—
27	C₂⁴×C₂₃	C₂³	1	2	yes	—
28	C₂×D₄×C₂₃	—	2	4	no	—
29	C₂×D₄×C₂₃	—	2	2	no	—
30	C₂²×C₄×C₂₃	—	2	4	no	—
31	C₂²×C₄×C₂₃	—	2	4	no	—
32	C₂²×C₄×C₂₃	—	2	4	no	—
33	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
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	1	yes	characteristic
40	C₄⋊C₄×C₂₃	—	2	2	no	—
41	C₄⋊C₄×C₂₃	—	2	4	no	—
42	C₄⋊C₄×C₂₃	—	2	4	no	—
43	C₄⋊C₄×C₂₃	—	2	2	no	—
44	C₄⋊C₄×C₂₃	—	2	2	no	—
45	C₄⋊C₄×C₂₃	—	2	4	no	—
46	C₂²⋊C₄×C₂₃	—	2	8	no	—
47	C₂²⋊C₄×C₂₃	—	2	4	no	—
48	C₂²⋊C₄×C₂₃	—	4	4	no	—
49	C₄²×C₂₃	—	4	2	no	—
50	C₄²×C₂₃	D₄	1	4	yes	—
51	C₂³×C₂₃	—	2	2	no	—
52	C₂³×C₂₃	—	2	2	no	—
53	C₂³×C₂₃	—	2	2	no	—
54	C₂³×C₂₃	—	2	2	no	—
55	C₂³×C₂₃	—	2	2	no	—
56	C₂³×C₂₃	—	2	2	no	—
57	C₂³×C₂₃	#41	1	4	yes	—
58	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
59	D₄×C₂₃	—	2	8	no	—
60	C₂×C₄×C₂₃	—	2	2	no	—
61	C₂×C₄×C₂₃	—	2	1	no	—
62	C₂×C₄×C₂₃	—	2	1	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	4	no	—
67	C₂×C₄×C₂₃	—	2	4	no	—
68	C₂×C₄×C₂₃	—	2	4	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	2	no	—
71	C₂×C₄×C₂₃	—	2	4	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	2	2	no	—
74	C₂×C₄×C₂₃	—	4	8	no	—
75	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
76	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₂₃	#41	1	4	yes	—
79	#2562	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂²×C₂₃	—	2	4	no	—
81	C₂²×C₂₃	—	2	4	no	—
82	C₂²×C₂₃	—	2	4	no	—
83	C₂²×C₂₃	—	4	2	no	—
84	C₂²×C₂₃	—	4	2	no	—
85	C₂²×C₂₃	—	4	2	no	—
86	C₂²×C₂₃	—	4	2	no	—
87	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
88	C₂²×C₂₃	C₂×#41	1	2	yes	—
89	C₂²×C₂₃	C₂×#41	1	2	yes	—
90	C₂²×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
91	C₂²×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
92	C₄×C₂₃	—	2	2	no	—
93	C₄×C₂₃	—	2	2	no	—
94	C₄×C₂₃	—	4	2	no	—
95	C₄×C₂₃	—	4	2	no	—
96	C₄×C₂₃	—	4	2	no	—
97	C₄×C₂₃	—	4	4	no	—
98	C₄×C₂₃	#124	1	4	yes	—
99	C₂×#125	C₂×C₂₃	1	1	yes	characteristic, complemented
100	C₂×C₄×D₄	C₂×C₂₃	1	1	yes	characteristic
101	#400	C₂×C₂₃	1	2	yes	complemented
102	#394	C₂×C₂₃	1	2	yes	—
103	#388	C₂×C₂₃	1	4	yes	complemented
104	#380	C₂×C₂₃	1	2	yes	complemented
105	#380	C₂×C₂₃	1	2	yes	complemented
106	C₄×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic, complemented
107	C₂×C₂₃	—	4	4	no	—
108	C₂×C₂₃	#525	1	1	yes	characteristic
109	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
110	C₂×C₂₃	#537	1	1	yes	characteristic
111	C₂×C₂₃	#552	1	2	yes	—
112	C₂×C₂₃	#553	1	2	yes	—
113	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
114	C₂³×C₄	C₂²×C₂₃	1	2	yes	—
115	#125	—	2	4	no	—
116	C₄×D₄	—	2	4	no	—
117	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
119	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
120	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
121	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
122	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
123	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
124	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
125	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
126	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
127	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
128	#95	C₂²×C₂₃	1	2	yes	—
129	#95	C₂²×C₂₃	1	4	yes	—
130	#95	C₂²×C₂₃	1	4	yes	—
131	C₂₃	#2562	1	1	yes	characteristic, direct product factor, Sylow
132	C₂⁴	C₂³×C₂₃	1	2	yes	—
133	C₂×D₄	—	2	4	no	—
134	C₂×D₄	—	2	2	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
139	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
140	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
141	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
142	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
145	C₄⋊C₄	—	2	2	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	4	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	4	no	—
151	C₂²⋊C₄	—	2	8	no	—
152	C₂²⋊C₄	—	2	4	no	—
153	C₂²⋊C₄	—	4	4	no	—
154	C₄²	—	4	2	no	—
155	C₄²	D₄×C₂₃	1	4	yes	—
156	C₂³	—	2	2	no	—
157	C₂³	—	2	2	no	—
158	C₂³	—	2	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	—	2	2	no	—
162	C₂³	#41×C₂₃	1	4	yes	—
163	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
164	D₄	—	2	8	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	1	no	—
167	C₂×C₄	—	2	1	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
181	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
182	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
183	C₂×C₄	#41×C₂₃	1	4	yes	—
184	C₂²	—	2	4	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	4	no	—
187	C₂²	—	4	2	no	—
188	C₂²	—	4	2	no	—
189	C₂²	—	4	2	no	—
190	C₂²	—	4	2	no	—
191	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
192	C₂²	C₂×#41×C₂₃	1	2	yes	—
193	C₂²	C₂×#41×C₂₃	1	2	yes	—
194	C₂²	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
195	C₂²	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	4	4	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	2	no	—
201	C₄	—	4	2	no	—
202	C₄	C₂₃×#124	1	4	yes	—
203	C₂	—	4	4	no	—
204	C₂	C₂₃×#525	1	1	yes	characteristic
205	C₂	C₂×C₂₃×#124	1	1	yes	characteristic
206	C₂	C₂₃×#537	1	1	yes	characteristic
207	C₂	C₂₃×#552	1	2	yes	—
208	C₂	C₂₃×#553	1	2	yes	—

	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. C₂×C₂₃×#125	C₂	semidirect
3. C₂₃×#400	C₂	semidirect
5. C₂₃×#388	C₂	semidirect
6. C₂₃×#380	C₂	semidirect
7. C₂₃×#380	C₂	semidirect
8. C₄×C₄⋊C₄×C₂₃	C₂	semidirect
79. #2562	C₂₃	direct (N × H)
99. C₂×#125	C₂×C₂₃	semidirect
101. #400	C₂×C₂₃	semidirect
103. #388	C₂×C₂₃	semidirect
104. #380	C₂×C₂₃	semidirect
105. #380	C₂×C₂₃	semidirect
106. C₄×C₄⋊C₄	C₂×C₂₃	semidirect
131. C₂₃	#2562	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×#125	2
C₂×C₄×D₄×C₂₃	2
C₂₃×#400	2
C₂₃×#394	2
C₂₃×#388	2
C₂₃×#380	2
C₄×C₄⋊C₄×C₂₃	2
#2562	23

Subgroups — C23×#2562 #15670181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2562 #15670181