Subgroups — C₂₃×#20538 #19168164

Order 5888.

922 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#431	C₂	1	2	yes	maximal
2	C₂₃×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1770	C₂	1	4	yes	maximal, complemented
4	C₈×C₄⋊C₄×C₂₃	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1750	C₂	1	4	yes	maximal
6	C₂×C₂₃×#117	C₂²	1	2	yes	—
7	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂₃×#431	C₄	1	16	yes	complemented
9	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
12	C₂×C₂₃×#97	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
15	C₂₃×#384	C₄	1	2	yes	—
16	C₂₃×#382	C₄	1	2	yes	—
17	C₂₃×#380	C₄	1	4	yes	—
18	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
19	C₂₃×#336	C₂²	1	8	yes	—
20	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
23	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
24	C₂₃×#117	—	4	2	no	—
25	C₂₃×#117	—	4	2	no	—
26	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
27	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₂₃	C₂×C₄	1	8	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
33	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
34	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
35	C₂₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
36	C₂₃×C₂²⋊C₈	C₂×C₄	1	16	yes	—
37	C₂₃×#97	—	4	4	no	—
38	C₄×C₈×C₂₃	—	4	4	no	—
39	C₄×C₈×C₂₃	C₂×C₄	1	16	yes	—
40	C₂₃×#95	C₂×C₄	1	8	yes	—
41	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₂₃	—	2	8	no	—
43	C₂²×C₄×C₂₃	—	2	1	no	—
44	C₂²×C₄×C₂₃	—	2	1	no	—
45	C₂²×C₄×C₂₃	—	2	2	no	—
46	C₂²×C₄×C₂₃	C₄²	1	8	yes	—
47	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
48	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
50	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
53	C₂×C₈×C₂₃	—	2	4	no	—
54	C₂×C₈×C₂₃	—	2	4	no	—
55	C₂×C₈×C₂₃	—	2	4	no	—
56	C₂×C₈×C₂₃	—	2	4	no	—
57	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
58	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
59	C₄⋊C₄×C₂₃	—	2	16	no	—
60	C₄⋊C₄×C₂₃	—	4	4	no	—
61	C₂²⋊C₄×C₂₃	—	4	4	no	—
62	C₄²×C₂₃	—	2	4	no	—
63	C₄²×C₂₃	—	2	4	no	—
64	C₄²×C₂₃	C₄²	1	4	yes	—
65	C₄²×C₂₃	C₄²	1	4	yes	—
66	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
67	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
68	#20538	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₂₃	—	2	1	no	—
70	C₂³×C₂₃	—	2	1	no	—
71	C₂³×C₂₃	—	2	4	no	—
72	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
73	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₃	—	2	1	no	—
75	C₂×C₄×C₂₃	—	2	1	no	—
76	C₂×C₄×C₂₃	—	2	4	no	—
77	C₂×C₄×C₂₃	—	2	4	no	—
78	C₂×C₄×C₂₃	—	2	4	no	—
79	C₂×C₄×C₂₃	—	2	16	no	—
80	C₂×C₄×C₂₃	—	2	8	no	—
81	C₂×C₄×C₂₃	—	2	16	no	—
82	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
83	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
84	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
85	C₂×C₄×C₂₃	#131	1	8	yes	—
86	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
87	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
88	C₂×C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
89	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
90	C₈×C₂₃	—	2	8	no	—
91	C₈×C₂₃	—	2	8	no	—
92	C₂×#431	C₂×C₂₃	1	2	yes	—
93	#2285	C₂×C₂₃	1	1	yes	characteristic
94	#1770	C₂×C₂₃	1	4	yes	complemented
95	C₈×C₄⋊C₄	C₂×C₂₃	1	4	yes	complemented
96	#1750	C₂×C₂₃	1	4	yes	—
97	C₂²×C₂₃	—	2	4	no	—
98	C₂²×C₂₃	—	2	2	no	—
99	C₂²×C₂₃	—	4	2	no	—
100	C₂²×C₂₃	—	4	2	no	—
101	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
102	C₂²×C₂₃	#518	1	1	yes	characteristic
103	C₂²×C₂₃	#519	1	1	yes	characteristic
104	C₂²×C₂₃	C₂×#131	1	4	yes	—
105	C₄×C₂₃	—	2	4	no	—
106	C₄×C₂₃	—	2	2	no	—
107	C₄×C₂₃	—	4	8	no	—
108	C₄×C₂₃	#405	1	8	yes	—
109	C₄×C₂₃	#520	1	2	yes	—
110	C₄×C₂₃	#520	1	2	yes	—
111	C₂×#117	C₂²×C₂₃	1	2	yes	—
112	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
113	#431	C₄×C₂₃	1	16	yes	complemented
114	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
115	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
116	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
117	C₂×#97	C₂²×C₂₃	1	4	yes	—
118	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
119	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
120	#384	C₄×C₂₃	1	2	yes	—
121	#382	C₄×C₂₃	1	2	yes	—
122	#380	C₄×C₂₃	1	4	yes	—
123	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
124	#336	C₂²×C₂₃	1	8	yes	—
125	C₂×C₂₃	—	4	2	no	—
126	C₂×C₂₃	#2274	1	1	yes	characteristic
127	C₂×C₂₃	C₂×#405	1	2	yes	—
128	C₂×C₂₃	#2979	1	2	yes	—
129	C₂×C₂₃	#2986	1	2	yes	—
130	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
131	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
133	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
134	#117	—	4	2	no	—
135	#117	—	4	2	no	—
136	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
137	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
138	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
139	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
140	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
141	C₂×C₄²	C₂×C₄×C₂₃	1	8	yes	—
142	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
143	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
144	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
145	C₄⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
146	C₂²⋊C₈	C₂×C₄×C₂₃	1	16	yes	—
147	#97	—	4	4	no	—
148	C₄×C₈	—	4	4	no	—
149	C₄×C₈	C₂×C₄×C₂₃	1	16	yes	—
150	#95	C₂×C₄×C₂₃	1	8	yes	—
151	C₂₃	#20538	1	1	yes	characteristic, direct product factor, Sylow
152	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
153	C₂²×C₄	—	2	8	no	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	2	no	—
157	C₂²×C₄	C₄²×C₂₃	1	8	yes	—
158	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
159	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
161	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
162	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
163	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
169	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
170	C₄⋊C₄	—	2	16	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₂²⋊C₄	—	4	4	no	—
173	C₄²	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	C₄²×C₂₃	1	4	yes	—
176	C₄²	C₄²×C₂₃	1	4	yes	—
177	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
178	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
179	C₂³	—	2	4	no	—
180	C₂³	—	2	1	no	—
181	C₂³	—	2	1	no	—
182	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
183	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	16	no	—
188	C₂×C₄	—	2	4	no	—
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	2	16	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
193	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
194	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
195	C₂×C₄	C₂₃×#131	1	8	yes	—
196	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
197	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
198	C₂×C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
199	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
200	C₈	—	2	8	no	—
201	C₈	—	2	8	no	—
202	C₂²	—	2	2	no	—
203	C₂²	—	2	4	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
207	C₂²	C₂₃×#518	1	1	yes	characteristic
208	C₂²	C₂₃×#519	1	1	yes	characteristic
209	C₂²	C₂×C₂₃×#131	1	4	yes	—
210	C₄	—	2	2	no	—
211	C₄	—	2	4	no	—
212	C₄	—	4	8	no	—
213	C₄	C₂₃×#405	1	8	yes	—
214	C₄	C₂₃×#520	1	2	yes	—
215	C₄	C₂₃×#520	1	2	yes	—
216	C₂	—	4	2	no	—
217	C₂	C₂₃×#2274	1	1	yes	characteristic
218	C₂	C₂×C₂₃×#405	1	2	yes	—
219	C₂	C₂₃×#2979	1	2	yes	—
220	C₂	C₂₃×#2986	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1770	C₂	semidirect
4. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
8. C₂₃×#431	C₄	semidirect
68. #20538	C₂₃	direct (N × H)
94. #1770	C₂×C₂₃	semidirect
95. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
113. #431	C₄×C₂₃	semidirect
151. C₂₃	#20538	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×#431	2
C₂₃×#2285	2
C₂₃×#1770	2
C₈×C₄⋊C₄×C₂₃	2
C₂₃×#1750	2
#20538	23

Subgroups — C23×#20538 #19168164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20538 #19168164