Subgroups — C₂₃×#23380 #19171006

Order 5888.

612 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#2401	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1940	C₂	1	3	yes	maximal, complemented
3	C₂₃×#1924	C₂	1	3	yes	maximal
4	C₂₃×#1917	C₂	1	3	yes	maximal, complemented
5	C₂₃×#1850	C₂	1	3	yes	maximal
6	C₂₃×#1843	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#1840	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	—
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	3	yes	—
12	C₂×C₂₃×C₂²⋊C₈	C₂²	1	3	yes	—
13	C₂×C₂₃×#97	C₂²	1	3	yes	—
14	C₂₃×#401	C₄	1	1	yes	characteristic
15	C₂₃×#400	C₄	1	3	yes	—
16	C₂₃×#397	C₄	1	3	yes	—
17	C₂₃×#396	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₂₃	C₂²	1	3	yes	—
19	C₄×C₂²⋊C₄×C₂₃	C₂²	1	3	yes	—
20	C₂₃×#376	C₂²	1	1	yes	characteristic
21	C₂₃×#336	C₂²	1	3	yes	—
22	C₂₃×#336	C₂²	1	3	yes	—
23	C₂₃×#336	C₂²	1	3	yes	—
24	C₂₃×#336	C₂²	1	3	yes	—
25	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	—	4	3	no	—
27	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
30	C₂²×C₈×C₂₃	C₂³	1	3	yes	—
31	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	3	yes	—
32	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	3	yes	—
33	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	3	yes	—
34	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	3	yes	—
35	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
36	C₂×C₄²×C₂₃	C₂³	1	3	yes	—
37	C₂₃×C₄⋊C₈	—	4	3	no	—
38	C₂₃×C₄⋊C₈	—	4	3	no	—
39	C₂₃×C₄⋊C₈	—	4	3	no	—
40	C₂₃×C₂²⋊C₈	—	4	3	no	—
41	C₂₃×#97	—	4	3	no	—
42	C₂₃×#95	C₂×C₄	1	1	yes	characteristic
43	C₂₃×#95	C₂×C₄	1	3	yes	—
44	C₂₃×#95	C₂×C₄	1	3	yes	—
45	C₂₃×#95	C₂×C₄	1	3	yes	—
46	C₂₃×#95	C₂×C₄	1	3	yes	—
47	C₂₃×#95	C₂×C₄	1	3	yes	—
48	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	—	2	3	no	—
50	C₂²×C₄×C₂₃	—	2	3	no	—
51	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
52	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
57	C₂×C₈×C₂₃	—	2	3	no	—
58	C₂×C₈×C₂₃	—	2	3	no	—
59	C₂×C₈×C₂₃	—	2	3	no	—
60	C₂×C₈×C₂₃	—	2	3	no	—
61	C₂×C₈×C₂₃	—	2	3	no	—
62	C₂×C₈×C₂₃	—	2	3	no	—
63	C₂×C₈×C₂₃	—	2	6	no	—
64	C₂×C₈×C₂₃	—	4	4	no	—
65	C₄⋊C₄×C₂₃	—	4	3	no	—
66	C₄⋊C₄×C₂₃	—	4	3	no	—
67	C₂²⋊C₄×C₂₃	—	2	6	no	—
68	C₂²⋊C₄×C₂₃	—	4	3	no	—
69	C₄²×C₂₃	—	2	3	no	—
70	C₄²×C₂₃	—	2	3	no	—
71	C₄²×C₂₃	—	2	3	no	—
72	C₄²×C₂₃	—	2	3	no	—
73	#23380	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₂₃	—	2	3	no	—
75	C₂³×C₂₃	—	2	3	no	—
76	C₂³×C₂₃	#131	1	2	yes	—
77	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	—	2	3	no	—
79	C₂×C₄×C₂₃	—	2	3	no	—
80	C₂×C₄×C₂₃	—	2	3	no	—
81	C₂×C₄×C₂₃	—	2	3	no	—
82	C₂×C₄×C₂₃	—	2	3	no	—
83	C₂×C₄×C₂₃	—	2	3	no	—
84	C₂×C₄×C₂₃	—	2	3	no	—
85	C₂×C₄×C₂₃	—	2	3	no	—
86	C₂×C₄×C₂₃	—	2	3	no	—
87	C₂×C₄×C₂₃	—	2	3	no	—
88	C₂×C₄×C₂₃	—	2	3	no	—
89	C₂×C₄×C₂₃	—	2	3	no	—
90	C₂×C₄×C₂₃	—	2	3	no	—
91	C₂×C₄×C₂₃	—	4	1	no	—
92	C₂×C₄×C₂₃	—	4	3	no	—
93	C₂×C₄×C₂₃	H(5,𝔽₂)	1	3	yes	—
94	C₂×C₄×C₂₃	2₋¹⁺⁴	1	3	yes	—
95	C₈×C₂₃	—	4	3	no	—
96	C₈×C₂₃	—	4	1	no	—
97	C₈×C₂₃	—	4	1	no	—
98	C₈×C₂₃	—	4	3	no	—
99	#2401	C₂×C₂₃	1	1	yes	characteristic
100	#1940	C₂×C₂₃	1	3	yes	complemented
101	#1924	C₂×C₂₃	1	3	yes	—
102	#1917	C₂×C₂₃	1	3	yes	complemented
103	#1850	C₂×C₂₃	1	3	yes	—
104	#1843	C₂×C₂₃	1	1	yes	characteristic, complemented
105	#1840	C₂×C₂₃	1	1	yes	characteristic, complemented
106	C₂²×C₂₃	—	2	6	no	—
107	C₂²×C₂₃	—	4	1	no	—
108	C₂²×C₂₃	—	4	3	no	—
109	C₂²×C₂₃	#520	1	3	yes	—
110	C₂²×C₂₃	C₂×#131	1	1	yes	characteristic
111	C₂²×C₂₃	#568	1	3	yes	—
112	C₄×C₂₃	—	4	1	no	—
113	C₄×C₂₃	—	4	3	no	—
114	C₄×C₂₃	—	4	3	no	—
115	C₄×C₂₃	—	4	3	no	—
116	C₄×C₂₃	—	4	3	no	—
117	C₄×C₂₃	#560	1	1	yes	characteristic
118	C₄×C₂₃	#563	1	3	yes	—
119	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
120	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	—
121	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	—
122	C₂×C₄⋊C₈	C₂²×C₂₃	1	3	yes	—
123	C₂×C₂²⋊C₈	C₂²×C₂₃	1	3	yes	—
124	C₂×#97	C₂²×C₂₃	1	3	yes	—
125	#401	C₄×C₂₃	1	1	yes	characteristic
126	#400	C₄×C₂₃	1	3	yes	—
127	#397	C₄×C₂₃	1	3	yes	—
128	#396	C₄×C₂₃	1	1	yes	characteristic
129	C₄×C₄⋊C₄	C₂²×C₂₃	1	3	yes	—
130	C₄×C₂²⋊C₄	C₂²×C₂₃	1	3	yes	—
131	#376	C₂²×C₂₃	1	1	yes	characteristic
132	#336	C₂²×C₂₃	1	3	yes	—
133	#336	C₂²×C₂₃	1	3	yes	—
134	#336	C₂²×C₂₃	1	3	yes	—
135	#336	C₂²×C₂₃	1	3	yes	—
136	C₂×C₂₃	—	4	2	no	—
137	C₂×C₂₃	#2383	1	1	yes	characteristic
138	C₂×C₂₃	#2992	1	3	yes	—
139	C₂×C₂₃	#2995	1	3	yes	—
140	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
141	C₂²×C₈	—	4	3	no	—
142	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
145	C₂²×C₈	C₂³×C₂₃	1	3	yes	—
146	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	3	yes	—
147	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	3	yes	—
148	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	3	yes	—
149	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	3	yes	—
150	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
151	C₂×C₄²	C₂³×C₂₃	1	3	yes	—
152	C₄⋊C₈	—	4	3	no	—
153	C₄⋊C₈	—	4	3	no	—
154	C₄⋊C₈	—	4	3	no	—
155	C₂²⋊C₈	—	4	3	no	—
156	#97	—	4	3	no	—
157	#95	C₂×C₄×C₂₃	1	1	yes	characteristic
158	#95	C₂×C₄×C₂₃	1	3	yes	—
159	#95	C₂×C₄×C₂₃	1	3	yes	—
160	#95	C₂×C₄×C₂₃	1	3	yes	—
161	#95	C₂×C₄×C₂₃	1	3	yes	—
162	#95	C₂×C₄×C₂₃	1	3	yes	—
163	C₂₃	#23380	1	1	yes	characteristic, direct product factor, Sylow
164	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
165	C₂²×C₄	—	2	3	no	—
166	C₂²×C₄	—	2	3	no	—
167	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
169	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
170	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
171	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
172	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
173	C₂×C₈	—	2	3	no	—
174	C₂×C₈	—	2	3	no	—
175	C₂×C₈	—	2	3	no	—
176	C₂×C₈	—	2	3	no	—
177	C₂×C₈	—	2	3	no	—
178	C₂×C₈	—	2	3	no	—
179	C₂×C₈	—	2	6	no	—
180	C₂×C₈	—	4	4	no	—
181	C₄⋊C₄	—	4	3	no	—
182	C₄⋊C₄	—	4	3	no	—
183	C₂²⋊C₄	—	2	6	no	—
184	C₂²⋊C₄	—	4	3	no	—
185	C₄²	—	2	3	no	—
186	C₄²	—	2	3	no	—
187	C₄²	—	2	3	no	—
188	C₄²	—	2	3	no	—
189	C₂³	—	2	3	no	—
190	C₂³	—	2	3	no	—
191	C₂³	C₂₃×#131	1	2	yes	—
192	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	—	2	3	no	—
199	C₂×C₄	—	2	3	no	—
200	C₂×C₄	—	2	3	no	—
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	3	no	—
208	C₂×C₄	C₂₃×H(5,𝔽₂)	1	3	yes	—
209	C₂×C₄	C₂₃×2₋¹⁺⁴	1	3	yes	—
210	C₈	—	4	1	no	—
211	C₈	—	4	1	no	—
212	C₈	—	4	3	no	—
213	C₈	—	4	3	no	—
214	C₂²	—	2	6	no	—
215	C₂²	—	4	3	no	—
216	C₂²	—	4	1	no	—
217	C₂²	C₂₃×#520	1	3	yes	—
218	C₂²	C₂×C₂₃×#131	1	1	yes	characteristic
219	C₂²	C₂₃×#568	1	3	yes	—
220	C₄	—	4	3	no	—
221	C₄	—	4	3	no	—
222	C₄	—	4	3	no	—
223	C₄	—	4	3	no	—
224	C₄	—	4	1	no	—
225	C₄	C₂₃×#560	1	1	yes	characteristic
226	C₄	C₂₃×#563	1	3	yes	—
227	C₂	—	4	2	no	—
228	C₂	C₂₃×#2383	1	1	yes	characteristic
229	C₂	C₂₃×#2992	1	3	yes	—
230	C₂	C₂₃×#2995	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂₃×#1940	C₂	semidirect
4. C₂₃×#1917	C₂	semidirect
6. C₂₃×#1843	C₂	semidirect
7. C₂₃×#1840	C₂	semidirect
73. #23380	C₂₃	direct (N × H)
100. #1940	C₂×C₂₃	semidirect
102. #1917	C₂×C₂₃	semidirect
104. #1843	C₂×C₂₃	semidirect
105. #1840	C₂×C₂₃	semidirect
163. C₂₃	#23380	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×#2401	2
C₂₃×#1940	2
C₂₃×#1924	2
C₂₃×#1917	2
C₂₃×#1850	2
C₂₃×#1843	2
C₂₃×#1840	2
#23380	23

Subgroups — C23×#23380 #19171006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#23380 #19171006