Subgroups — C₂₃×#32850 #19180476

Order 5888.

604 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×#2859	C₂	1	2	yes	maximal
2	C₂₃×#2100	C₂	1	4	yes	maximal
3	C₂₃×#2087	C₂	1	4	yes	maximal
4	C₂₃×#2083	C₂	1	4	yes	maximal
5	C₂₃×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×#125	—	2	2	no	—
7	C₂×C₂₃×#107	C₂²	1	4	yes	—
8	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂₃×#389	—	2	2	no	—
10	C₂₃×#389	—	2	2	no	—
11	C₂₃×#389	C₂²	1	4	yes	—
12	C₂₃×#389	C₂²	1	4	yes	—
13	C₂₃×#384	C₂²	1	2	yes	—
14	C₂₃×#384	C₂²	1	2	yes	—
15	C₂₃×#382	—	2	2	no	—
16	C₄³×C₂₃	C₂²	1	1	yes	characteristic
17	C₂₃×#340	C₂²	1	8	yes	—
18	C₂₃×#340	C₂²	1	4	yes	—
19	C₂₃×#336	C₂²	1	4	yes	—
20	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
21	C₂₃×#125	—	4	8	no	—
22	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
23	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
29	C₂×C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
30	C₂×C₄²×C₂₃	D₄	1	2	yes	—
31	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
34	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×#107	—	4	4	no	—
36	C₂₃×C₄⋊C₈	—	4	2	no	—
37	C₂₃×#95	—	2	4	no	—
38	C₂₃×#95	—	2	4	no	—
39	C₂²×C₄×C₂₃	—	2	4	no	—
40	C₂²×C₄×C₂₃	—	2	4	no	—
41	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
43	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
46	C₂×C₈×C₂₃	—	2	4	no	—
47	C₂×C₈×C₂₃	—	2	8	no	—
48	C₄⋊C₄×C₂₃	—	2	4	no	—
49	C₄⋊C₄×C₂₃	—	2	8	no	—
50	C₄⋊C₄×C₂₃	—	2	4	no	—
51	C₄⋊C₄×C₂₃	—	4	4	no	—
52	C₄⋊C₄×C₂₃	—	4	4	no	—
53	C₄⋊C₄×C₂₃	—	4	4	no	—
54	C₄⋊C₄×C₂₃	—	4	8	no	—
55	C₄⋊C₄×C₂₃	—	4	4	no	—
56	C₄⋊C₄×C₂₃	—	4	8	no	—
57	C₄²×C₂₃	—	2	4	no	—
58	C₄²×C₂₃	—	2	1	no	—
59	C₄²×C₂₃	—	2	4	no	—
60	C₄²×C₂₃	—	2	1	no	—
61	C₄²×C₂₃	—	4	1	no	—
62	C₄²×C₂₃	Q₁₆	1	4	yes	—
63	#32850	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
65	C₂×C₄×C₂₃	—	2	2	no	—
66	C₂×C₄×C₂₃	—	2	2	no	—
67	C₂×C₄×C₂₃	—	2	1	no	—
68	C₂×C₄×C₂₃	—	2	2	no	—
69	C₂×C₄×C₂₃	—	2	2	no	—
70	C₂×C₄×C₂₃	—	2	1	no	—
71	C₂×C₄×C₂₃	—	2	1	no	—
72	C₂×C₄×C₂₃	—	2	2	no	—
73	C₂×C₄×C₂₃	—	4	4	no	—
74	C₂×C₄×C₂₃	—	4	4	no	—
75	C₂×C₄×C₂₃	—	4	4	no	—
76	C₂×C₄×C₂₃	—	4	4	no	—
77	C₂×C₄×C₂₃	—	4	4	no	—
78	C₂×C₄×C₂₃	—	4	4	no	—
79	C₂×C₄×C₂₃	C₂×Q₁₆	1	4	yes	—
80	C₂×C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
82	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
83	C₂×C₄×C₂₃	2₋¹⁺⁴	1	4	yes	—
84	C₂×C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
85	C₈×C₂₃	—	4	4	no	—
86	#2859	C₂×C₂₃	1	2	yes	—
87	#2100	C₂×C₂₃	1	4	yes	—
88	#2087	C₂×C₂₃	1	4	yes	—
89	#2083	C₂×C₂₃	1	4	yes	—
90	#1831	C₂×C₂₃	1	1	yes	characteristic
91	C₂²×C₂₃	#536	1	2	yes	—
92	C₂²×C₂₃	#537	1	1	yes	characteristic
93	C₂²×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
94	C₂²×C₂₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
95	C₂²×C₂₃	#575	1	1	yes	characteristic
96	C₄×C₂₃	—	2	4	no	—
97	C₄×C₂₃	—	2	4	no	—
98	C₄×C₂₃	—	2	2	no	—
99	C₄×C₂₃	—	4	1	no	—
100	C₄×C₂₃	—	8	4	no	—
101	C₄×C₂₃	—	8	4	no	—
102	C₄×C₂₃	#563	1	2	yes	—
103	C₄×C₂₃	#564	1	2	yes	—
104	C₂×#125	—	2	2	no	—
105	C₂×#107	C₂²×C₂₃	1	4	yes	—
106	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
107	#389	—	2	2	no	—
108	#389	—	2	2	no	—
109	#389	C₂²×C₂₃	1	4	yes	—
110	#389	C₂²×C₂₃	1	4	yes	—
111	#384	C₂²×C₂₃	1	2	yes	—
112	#384	C₂²×C₂₃	1	2	yes	—
113	#382	—	2	2	no	—
114	C₄³	C₂²×C₂₃	1	1	yes	characteristic
115	#340	C₂²×C₂₃	1	8	yes	—
116	#340	C₂²×C₂₃	1	4	yes	—
117	#336	C₂²×C₂₃	1	4	yes	—
118	C₂×C₂₃	#2822	1	1	yes	characteristic
119	C₂×C₂₃	#3230	1	2	yes	—
120	C₂×C₂₃	#3234	1	2	yes	—
121	C₂×C₂₃	#3249	1	1	yes	characteristic
122	C₂×C₂₃	#3253	1	1	yes	characteristic
123	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
124	#125	—	4	8	no	—
125	C₂×C₄⋊C₄	—	2	2	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₄⋊C₄	—	2	4	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
132	C₂×C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
133	C₂×C₄²	D₄×C₂₃	1	2	yes	—
134	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
135	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
136	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
137	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
138	#107	—	4	4	no	—
139	C₄⋊C₈	—	4	2	no	—
140	#95	—	2	4	no	—
141	#95	—	2	4	no	—
142	C₂₃	#32850	1	1	yes	characteristic, direct product factor, Sylow
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
146	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
147	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
148	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
149	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	2	8	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	8	no	—
154	C₄⋊C₄	—	2	4	no	—
155	C₄⋊C₄	—	4	4	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	—	4	8	no	—
158	C₄⋊C₄	—	4	4	no	—
159	C₄⋊C₄	—	4	8	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄²	—	2	4	no	—
162	C₄²	—	2	1	no	—
163	C₄²	—	2	4	no	—
164	C₄²	—	2	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	Q₁₆×C₂₃	1	4	yes	—
167	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	—	4	4	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	C₂×Q₁₆×C₂₃	1	4	yes	—
183	C₂×C₄	C₂×Q₁₆×C₂₃	1	2	yes	—
184	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
185	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
186	C₂×C₄	C₂₃×2₋¹⁺⁴	1	4	yes	—
187	C₂×C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
188	C₈	—	4	4	no	—
189	C₂²	C₂₃×#536	1	2	yes	—
190	C₂²	C₂₃×#537	1	1	yes	characteristic
191	C₂²	C₂²×Q₁₆×C₂₃	1	1	yes	characteristic
192	C₂²	C₂×C₂₃×AGL(1,ℤ/8)	1	2	yes	—
193	C₂²	C₂₃×#575	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	4	1	no	—
198	C₄	—	8	4	no	—
199	C₄	—	8	4	no	—
200	C₄	C₂₃×#563	1	2	yes	—
201	C₄	C₂₃×#564	1	2	yes	—
202	C₂	C₂₃×#2822	1	1	yes	characteristic
203	C₂	C₂₃×#3230	1	2	yes	—
204	C₂	C₂₃×#3234	1	2	yes	—
205	C₂	C₂₃×#3249	1	1	yes	characteristic
206	C₂	C₂₃×#3253	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₄³×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
63. #32850	C₂₃	direct (N × H)
142. C₂₃	#32850	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×#2859	2
C₂₃×#2100	2
C₂₃×#2087	2
C₂₃×#2083	2
C₂₃×#1831	2
#32850	23

Subgroups — C23×#32850 #19180476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#32850 #19180476