Subgroups — C₂₃×#3256 #15670875

Order 2944.

354 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#529	C₂	1	2	yes	maximal, complemented
2	C₂₃×#479	C₂	1	2	yes	maximal, complemented
3	C₂₃×#478	C₂	1	2	yes	maximal, complemented
4	C₂₃×#470	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#469	C₂	1	1	yes	maximal, characteristic
6	C₂₃×#468	C₂	1	2	yes	maximal, complemented
7	C₂₃×#464	C₂	1	2	yes	maximal, complemented
8	C₂₃×#463	C₂	1	2	yes	maximal, complemented
9	C₂₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂₃×#126	—	2	2	no	—
16	C₂₃×#125	C₂²	1	2	yes	—
17	C₂₃×#124	C₂²	1	2	yes	—
18	C₂₃×#123	—	2	2	no	—
19	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂₃×#121	C₂²	1	2	yes	—
21	C₄×Q₈×C₂₃	C₂²	1	2	yes	—
22	C₄×D₄×C₂₃	—	2	2	no	—
23	C₄×D₄×C₂₃	C₂²	1	2	yes	—
24	C₂₃×#117	—	2	2	no	—
25	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
26	C₂₃×#107	C₂²	1	2	yes	—
27	C₂₃×#106	C₂²	1	1	yes	characteristic
28	C₂₃×#106	C₂²	1	1	yes	characteristic
29	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
31	C₂₃×#103	C₂²	1	2	yes	—
32	C₂₃×#103	C₂²	1	2	yes	—
33	C₂₃×#102	C₂²	1	2	yes	—
34	C₂₃×#102	C₂²	1	2	yes	—
35	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
36	C₂×D₄×C₂₃	—	2	2	no	—
37	C₂×D₄×C₂₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₂₃	—	2	2	no	—
39	C₂²×C₄×C₂₃	D₄	1	2	yes	—
40	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
41	Q₁₆×C₂₃	—	4	1	no	—
42	SD₁₆×C₂₃	—	4	2	no	—
43	D₈×C₂₃	—	4	1	no	—
44	M₁₆×C₂₃	—	4	1	no	—
45	M₁₆×C₂₃	—	4	1	no	—
46	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₂₃	C₂³	1	2	yes	—
48	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₄×C₂₃	—	2	2	no	—
50	C₄⋊C₄×C₂₃	—	2	2	no	—
51	C₄⋊C₄×C₂₃	—	2	2	no	—
52	C₄⋊C₄×C₂₃	—	2	2	no	—
53	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
54	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₂₃	—	2	2	no	—
56	C₂²⋊C₄×C₂₃	—	2	2	no	—
57	C₂²⋊C₄×C₂₃	—	2	2	no	—
58	C₂²⋊C₄×C₂₃	—	2	2	no	—
59	C₂²⋊C₄×C₂₃	—	2	2	no	—
60	C₄²×C₂₃	—	2	2	no	—
61	C₄²×C₂₃	D₄	1	1	yes	characteristic
62	C₄²×C₂₃	D₄	1	1	yes	characteristic
63	C₄²×C₂₃	C₂³	1	2	yes	—
64	C₂³×C₂₃	—	2	2	no	—
65	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
66	Q₈×C₂₃	—	2	2	no	—
67	Q₈×C₂₃	—	2	2	no	—
68	D₄×C₂₃	—	2	2	no	—
69	D₄×C₂₃	—	2	2	no	—
70	D₄×C₂₃	—	4	4	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	1	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	—	4	2	no	—
77	C₂×C₄×C₂₃	—	4	4	no	—
78	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
82	C₈×C₂₃	—	2	1	no	—
83	C₈×C₂₃	—	2	2	no	—
84	C₈×C₂₃	—	2	1	no	—
85	#3256	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₂₃	—	2	1	no	—
87	C₂²×C₂₃	—	2	2	no	—
88	C₂²×C₂₃	—	4	2	no	—
89	C₂²×C₂₃	—	4	2	no	—
90	C₂²×C₂₃	—	4	2	no	—
91	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
92	C₄×C₂₃	—	2	1	no	—
93	C₄×C₂₃	—	2	2	no	—
94	C₄×C₂₃	—	2	2	no	—
95	C₄×C₂₃	—	4	2	no	—
96	C₄×C₂₃	—	4	2	no	—
97	C₄×C₂₃	—	4	2	no	—
98	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
99	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
100	#529	C₂×C₂₃	1	2	yes	complemented
101	#479	C₂×C₂₃	1	2	yes	complemented
102	#478	C₂×C₂₃	1	2	yes	complemented
103	#470	C₂×C₂₃	1	1	yes	characteristic, complemented
104	#469	C₂×C₂₃	1	1	yes	characteristic
105	#468	C₂×C₂₃	1	2	yes	complemented
106	#464	C₂×C₂₃	1	2	yes	complemented
107	#463	C₂×C₂₃	1	2	yes	complemented
108	#423	C₂×C₂₃	1	1	yes	characteristic, complemented
109	C₂×C₂₃	—	4	1	no	—
110	C₂×C₂₃	—	8	2	no	—
111	C₂×C₂₃	#537	1	1	yes	characteristic
112	C₂×C₂₃	#575	1	2	yes	—
113	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
114	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
115	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
116	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
117	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
118	#126	—	2	2	no	—
119	#125	C₂²×C₂₃	1	2	yes	—
120	#124	C₂²×C₂₃	1	2	yes	—
121	#123	—	2	2	no	—
122	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
123	#121	C₂²×C₂₃	1	2	yes	—
124	C₄×Q₈	C₂²×C₂₃	1	2	yes	—
125	C₄×D₄	—	2	2	no	—
126	C₄×D₄	C₂²×C₂₃	1	2	yes	—
127	#117	—	2	2	no	—
128	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
129	#107	C₂²×C₂₃	1	2	yes	—
130	#106	C₂²×C₂₃	1	1	yes	characteristic
131	#106	C₂²×C₂₃	1	1	yes	characteristic
132	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
133	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
134	#103	C₂²×C₂₃	1	2	yes	—
135	#103	C₂²×C₂₃	1	2	yes	—
136	#102	C₂²×C₂₃	1	2	yes	—
137	#102	C₂²×C₂₃	1	2	yes	—
138	C₂₃	#3256	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
140	C₂×D₄	—	2	2	no	—
141	C₂×D₄	C₂³×C₂₃	1	2	yes	—
142	C₂²×C₄	—	2	2	no	—
143	C₂²×C₄	D₄×C₂₃	1	2	yes	—
144	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
145	Q₁₆	—	4	1	no	—
146	SD₁₆	—	4	2	no	—
147	D₈	—	4	1	no	—
148	M₁₆	—	4	1	no	—
149	M₁₆	—	4	1	no	—
150	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
151	C₂×C₈	C₂³×C₂₃	1	2	yes	—
152	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
153	C₄⋊C₄	—	2	2	no	—
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	2	2	no	—
156	C₄⋊C₄	—	2	2	no	—
157	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
158	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
159	C₂²⋊C₄	—	2	2	no	—
160	C₂²⋊C₄	—	2	2	no	—
161	C₂²⋊C₄	—	2	2	no	—
162	C₂²⋊C₄	—	2	2	no	—
163	C₂²⋊C₄	—	2	2	no	—
164	C₄²	—	2	2	no	—
165	C₄²	D₄×C₂₃	1	1	yes	characteristic
166	C₄²	D₄×C₂₃	1	1	yes	characteristic
167	C₄²	C₂³×C₂₃	1	2	yes	—
168	C₂³	—	2	2	no	—
169	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	D₄	—	2	2	no	—
173	D₄	—	2	2	no	—
174	D₄	—	4	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	4	2	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
183	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
184	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
185	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
186	C₈	—	2	1	no	—
187	C₈	—	2	2	no	—
188	C₈	—	2	1	no	—
189	C₂²	—	2	2	no	—
190	C₂²	—	2	1	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	2	no	—
194	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	1	no	—
198	C₄	—	4	2	no	—
199	C₄	—	4	2	no	—
200	C₄	—	4	2	no	—
201	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
202	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
203	C₂	—	4	1	no	—
204	C₂	—	8	2	no	—
205	C₂	C₂₃×#537	1	1	yes	characteristic
206	C₂	C₂₃×#575	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂₃×#469	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#529	C₂	semidirect
2. C₂₃×#479	C₂	semidirect
3. C₂₃×#478	C₂	semidirect
4. C₂₃×#470	C₂	semidirect
6. C₂₃×#468	C₂	semidirect
7. C₂₃×#464	C₂	semidirect
8. C₂₃×#463	C₂	semidirect
9. C₂₃×#423	C₂	semidirect
85. #3256	C₂₃	direct (N × H)
100. #529	C₂×C₂₃	semidirect
101. #479	C₂×C₂₃	semidirect
102. #478	C₂×C₂₃	semidirect
103. #470	C₂×C₂₃	semidirect
105. #468	C₂×C₂₃	semidirect
106. #464	C₂×C₂₃	semidirect
107. #463	C₂×C₂₃	semidirect
108. #423	C₂×C₂₃	semidirect
138. C₂₃	#3256	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×#529	2
C₂₃×#479	2
C₂₃×#478	2
C₂₃×#470	2
C₂₃×#469	2
C₂₃×#468	2
C₂₃×#464	2
C₂₃×#463	2
C₂₃×#423	2
#3256	23

Subgroups — C23×#3256 #15670875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3256 #15670875