Subgroups — C₂₃×#20523 #19168149

Order 5888.

1450 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₂²⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄×C₂²⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₈×C₂²⋊C₄×C₂₃	C₂	1	8	yes	maximal, complemented
4	C₄×C₂₃×C₂²⋊C₈	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×C₂₃	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₂₃	C₂²	1	4	yes	—
7	C₂²×C₂²⋊C₄×C₂₃	C₄	1	2	yes	—
8	C₂²×C₄²×C₂₃	C₂²	1	2	yes	—
9	C₂×C₂₃×C₂²⋊C₈	C₄	1	16	yes	complemented
10	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
11	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₂₃	C₂²	1	8	yes	complemented
13	C₄×C₂²⋊C₄×C₂₃	C₄	1	4	yes	—
14	C₄×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	—
15	C₂×C₂₃×#95	C₄	1	2	yes	—
16	C₂₃×#336	C₂²	1	8	yes	complemented
17	C₂⁵×C₂₃	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₂₃	—	4	4	no	—
19	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₂₃	C₂×C₄	1	8	yes	—
22	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
23	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
24	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₂₃	—	4	8	no	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	16	yes	complemented
27	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
28	C₂×C₂²⋊C₄×C₂₃	C₈	1	16	yes	complemented
29	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
30	C₂×C₄²×C₂₃	—	4	4	no	—
31	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
33	C₂₃×C₂²⋊C₈	C₂×C₄	1	16	yes	—
34	C₂₃×C₂²⋊C₈	C₂×C₄	1	16	yes	complemented
35	C₄×C₈×C₂₃	—	4	8	no	—
36	C₂₃×#95	C₂×C₄	1	8	yes	—
37	C₂⁴×C₂₃	—	4	2	no	—
38	C₂⁴×C₂₃	—	4	2	no	—
39	C₂⁴×C₂₃	C₄²	1	4	yes	—
40	C₂⁴×C₂₃	C₂×C₈	1	8	yes	—
41	C₂⁴×C₂₃	C₂²×C₄	1	2	yes	—
42	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₂₃	—	2	4	no	—
44	C₂²×C₄×C₂₃	—	2	8	no	—
45	C₂²×C₄×C₂₃	—	4	16	no	—
46	C₂²×C₄×C₂₃	—	4	4	no	—
47	C₂²×C₄×C₂₃	—	4	4	no	—
48	C₂²×C₄×C₂₃	—	4	2	no	—
49	C₂²×C₄×C₂₃	—	4	4	no	—
50	C₂²×C₄×C₂₃	C₄²	1	4	yes	—
51	C₂²×C₄×C₂₃	C₄²	1	8	yes	—
52	C₂²×C₄×C₂₃	C₂×C₈	1	16	yes	complemented
53	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₂₃	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₂₃	—	2	16	no	—
61	C₂×C₈×C₂₃	—	4	16	no	—
62	C₂×C₈×C₂₃	C₂²×C₄	1	16	yes	—
63	C₂²⋊C₄×C₂₃	C₂×C₈	1	32	yes	complemented
64	C₄²×C₂₃	—	2	8	no	—
65	C₄²×C₂₃	—	4	4	no	—
66	#20523	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₂₃	—	2	2	no	—
68	C₂³×C₂₃	—	2	4	no	—
69	C₂³×C₂₃	—	2	2	no	—
70	C₂³×C₂₃	—	2	4	no	—
71	C₂³×C₂₃	—	4	4	no	—
72	C₂³×C₂₃	—	4	4	no	—
73	C₂³×C₂₃	—	4	16	no	—
74	C₂³×C₂₃	C₄×C₈	1	16	yes	complemented
75	C₂³×C₂₃	C₂×C₄²	1	4	yes	—
76	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
77	C₂³×C₂₃	C₂²×C₈	1	8	yes	—
78	C₂³×C₂₃	C₂²×C₈	1	4	yes	—
79	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₃	—	2	8	no	—
81	C₂×C₄×C₂₃	—	2	2	no	—
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	2	8	no	—
84	C₂×C₄×C₂₃	—	2	8	no	—
85	C₂×C₄×C₂₃	—	2	8	no	—
86	C₂×C₄×C₂₃	—	2	8	no	—
87	C₂×C₄×C₂₃	—	4	8	no	—
88	C₂×C₄×C₂₃	—	4	4	no	—
89	C₂×C₄×C₂₃	—	4	16	no	—
90	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₂₃	C₂×C₄²	1	4	yes	—
92	C₂×C₄×C₂₃	C₂²×C₈	1	16	yes	—
93	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
94	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
95	C₈×C₂₃	—	2	16	no	—
96	C₂²×C₂²⋊C₈	C₂×C₂₃	1	2	yes	—
97	C₂×C₄×C₂²⋊C₄	C₂×C₂₃	1	1	yes	characteristic
98	C₈×C₂²⋊C₄	C₂×C₂₃	1	8	yes	complemented
99	C₄×C₂²⋊C₈	C₂×C₂₃	1	4	yes	complemented
100	C₂²×C₂₃	—	2	2	no	—
101	C₂²×C₂₃	—	2	4	no	—
102	C₂²×C₂₃	—	2	8	no	—
103	C₂²×C₂₃	—	2	16	no	—
104	C₂²×C₂₃	—	2	4	no	—
105	C₂²×C₂₃	—	2	2	no	—
106	C₂²×C₂₃	—	4	16	no	—
107	C₂²×C₂₃	C₂×C₄×C₈	1	8	yes	—
108	C₂²×C₂₃	C₂×C₄×C₈	1	4	yes	—
109	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
110	C₂²×C₂₃	#518	1	2	yes	—
111	C₂²×C₂₃	C₂³×C₈	1	2	yes	—
112	C₂²×C₂₃	#568	1	2	yes	—
113	C₄×C₂₃	—	2	16	no	—
114	C₄×C₂₃	—	2	2	no	—
115	C₄×C₂₃	—	2	4	no	—
116	C₄×C₂₃	#518	1	4	yes	—
117	C₂⁴×C₄	C₂²×C₂₃	1	1	yes	characteristic
118	C₂³×C₈	C₂²×C₂₃	1	4	yes	—
119	C₂²×C₂²⋊C₄	C₄×C₂₃	1	2	yes	—
120	C₂²×C₄²	C₂²×C₂₃	1	2	yes	—
121	C₂×C₂²⋊C₈	C₄×C₂₃	1	16	yes	complemented
122	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
123	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
124	C₂×C₄×C₈	C₂²×C₂₃	1	8	yes	complemented
125	C₄×C₂²⋊C₄	C₄×C₂₃	1	4	yes	—
126	C₄×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	—
127	C₂×#95	C₄×C₂₃	1	2	yes	—
128	#336	C₂²×C₂₃	1	8	yes	complemented
129	C₂×C₂₃	—	2	4	no	—
130	C₂×C₂₃	—	2	8	no	—
131	C₂×C₂₃	#2273	1	1	yes	characteristic
132	C₂×C₂₃	C₂²×C₄×C₈	1	1	yes	characteristic
133	C₂×C₂₃	#2873	1	1	yes	characteristic
134	C₂×C₂₃	#2972	1	4	yes	—
135	C₂⁵	C₂×C₄×C₂₃	1	1	yes	characteristic
136	C₂³×C₄	—	4	4	no	—
137	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
138	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
139	C₂³×C₄	C₂×C₄×C₂₃	1	8	yes	—
140	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
141	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
142	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
143	C₂²×C₈	—	4	8	no	—
144	C₂²×C₈	C₂×C₄×C₂₃	1	16	yes	complemented
145	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
146	C₂×C₂²⋊C₄	C₈×C₂₃	1	16	yes	complemented
147	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
148	C₂×C₄²	—	4	4	no	—
149	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
150	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
151	C₂²⋊C₈	C₂×C₄×C₂₃	1	16	yes	—
152	C₂²⋊C₈	C₂×C₄×C₂₃	1	16	yes	complemented
153	C₄×C₈	—	4	8	no	—
154	#95	C₂×C₄×C₂₃	1	8	yes	—
155	C₂₃	#20523	1	1	yes	characteristic, direct product factor, Sylow
156	C₂⁴	—	4	2	no	—
157	C₂⁴	—	4	2	no	—
158	C₂⁴	C₄²×C₂₃	1	4	yes	—
159	C₂⁴	C₂×C₈×C₂₃	1	8	yes	—
160	C₂⁴	C₂²×C₄×C₂₃	1	2	yes	—
161	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	4	16	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	—	4	4	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	—	4	4	no	—
169	C₂²×C₄	C₄²×C₂₃	1	4	yes	—
170	C₂²×C₄	C₄²×C₂₃	1	8	yes	—
171	C₂²×C₄	C₂×C₈×C₂₃	1	16	yes	complemented
172	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
173	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
174	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
175	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
176	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
177	C₂²×C₄	C₂²×C₄×C₂₃	1	8	yes	—
178	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
179	C₂×C₈	—	2	16	no	—
180	C₂×C₈	—	4	16	no	—
181	C₂×C₈	C₂²×C₄×C₂₃	1	16	yes	—
182	C₂²⋊C₄	C₂×C₈×C₂₃	1	32	yes	complemented
183	C₄²	—	2	8	no	—
184	C₄²	—	4	4	no	—
185	C₂³	—	2	2	no	—
186	C₂³	—	2	4	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	4	no	—
189	C₂³	—	4	4	no	—
190	C₂³	—	4	4	no	—
191	C₂³	—	4	16	no	—
192	C₂³	C₄×C₈×C₂₃	1	16	yes	complemented
193	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
194	C₂³	C₂×C₄²×C₂₃	1	4	yes	—
195	C₂³	C₂²×C₈×C₂₃	1	8	yes	—
196	C₂³	C₂²×C₈×C₂₃	1	4	yes	—
197	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	16	no	—
208	C₂×C₄	C₂×C₄²×C₂₃	1	4	yes	—
209	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
210	C₂×C₄	C₂²×C₈×C₂₃	1	16	yes	—
211	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
212	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
213	C₈	—	2	16	no	—
214	C₂²	—	2	8	no	—
215	C₂²	—	2	16	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	4	no	—
220	C₂²	—	4	16	no	—
221	C₂²	C₂×C₄×C₈×C₂₃	1	8	yes	—
222	C₂²	C₂×C₄×C₈×C₂₃	1	4	yes	—
223	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
224	C₂²	C₂₃×#518	1	2	yes	—
225	C₂²	C₂³×C₈×C₂₃	1	2	yes	—
226	C₂²	C₂₃×#568	1	2	yes	—
227	C₄	—	2	16	no	—
228	C₄	—	2	2	no	—
229	C₄	—	2	4	no	—
230	C₄	C₂₃×#518	1	4	yes	—
231	C₂	—	2	4	no	—
232	C₂	—	2	8	no	—
233	C₂	C₂₃×#2273	1	1	yes	characteristic
234	C₂	C₂²×C₄×C₈×C₂₃	1	1	yes	characteristic
235	C₂	C₂₃×#2873	1	1	yes	characteristic
236	C₂	C₂₃×#2972	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₃	C₂⁴
commutator	C₂	C₂²×C₄×C₈×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂³×C₄
prime radical	C₂⁵×C₂₃	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈×C₂²⋊C₄×C₂₃	C₂	semidirect
4. C₄×C₂₃×C₂²⋊C₈	C₂	semidirect
9. C₂×C₂₃×C₂²⋊C₈	C₄	semidirect
12. C₂×C₄×C₈×C₂₃	C₂²	semidirect
16. C₂₃×#336	C₂²	semidirect
26. C₂²×C₈×C₂₃	C₂×C₄	semidirect
28. C₂×C₂²⋊C₄×C₂₃	C₈	semidirect
34. C₂₃×C₂²⋊C₈	C₂×C₄	semidirect
52. C₂²×C₄×C₂₃	C₂×C₈	semidirect
63. C₂²⋊C₄×C₂₃	C₂×C₈	semidirect
66. #20523	C₂₃	direct (N × H)
74. C₂³×C₂₃	C₄×C₈	semidirect
98. C₈×C₂²⋊C₄	C₂×C₂₃	semidirect
99. C₄×C₂²⋊C₈	C₂×C₂₃	semidirect
121. C₂×C₂²⋊C₈	C₄×C₂₃	semidirect
124. C₂×C₄×C₈	C₂²×C₂₃	semidirect
128. #336	C₂²×C₂₃	semidirect
144. C₂²×C₈	C₂×C₄×C₂₃	semidirect
146. C₂×C₂²⋊C₄	C₈×C₂₃	semidirect
152. C₂²⋊C₈	C₂×C₄×C₂₃	semidirect
155. C₂₃	#20523	direct (N × H)
171. C₂²×C₄	C₂×C₈×C₂₃	semidirect
182. C₂²⋊C₄	C₂×C₈×C₂₃	semidirect
192. C₂³	C₄×C₈×C₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₂₃×C₂²⋊C₈	2
C₂×C₄×C₂²⋊C₄×C₂₃	2
C₈×C₂²⋊C₄×C₂₃	2
C₄×C₂₃×C₂²⋊C₈	2
#20523	23

Subgroups — C23×#20523 #19168149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20523 #19168149