Subgroups — C₂₃×#20540 #19168166

Order 5888.

906 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#433	C₂	1	2	yes	maximal
2	C₂₃×#2296	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1771	C₂	1	4	yes	maximal, complemented
4	C₈×C₄⋊C₄×C₂₃	C₂	1	4	yes	maximal, complemented
5	C₂₃×#1750	C₂	1	2	yes	maximal
6	C₄×C₂₃×C₂²⋊C₈	C₂	1	2	yes	maximal
7	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
8	C₂×C₂₃×#117	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
10	C₂₃×#433	C₄	1	16	yes	complemented
11	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₂₃×#97	C₂²	1	2	yes	—
14	C₂×C₂₃×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
18	C₂₃×#389	C₄	1	4	yes	—
19	C₂₃×#385	C₄	1	4	yes	—
20	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
21	C₂₃×#336	C₂²	1	2	yes	—
22	C₂₃×#336	C₂²	1	2	yes	—
23	C₂₃×#336	C₂²	1	4	yes	—
24	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
27	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
28	C₂₃×#117	—	4	2	no	—
29	C₂₃×#117	C₂×C₄	1	8	yes	—
30	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₂₃	—	4	2	no	—
36	C₂×C₄²×C₂₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
38	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
39	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
40	C₂₃×C₄⋊C₈	C₂×C₄	1	16	yes	complemented
41	C₂₃×C₂²⋊C₈	C₂×C₄	1	16	yes	—
42	C₂₃×#97	—	4	2	no	—
43	C₂₃×#97	C₂×C₄	1	8	yes	complemented
44	C₄×C₈×C₂₃	—	4	4	no	—
45	C₄×C₈×C₂₃	—	4	2	no	—
46	C₄×C₈×C₂₃	C₂×C₄	1	8	yes	complemented
47	C₂₃×#95	C₂×C₄	1	8	yes	—
48	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₂₃	—	2	2	no	—
50	C₂²×C₄×C₂₃	—	2	2	no	—
51	C₂²×C₄×C₂₃	—	2	8	no	—
52	C₂²×C₄×C₂₃	—	4	2	no	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
59	C₂×C₈×C₂₃	—	2	4	no	—
60	C₂×C₈×C₂₃	—	2	4	no	—
61	C₂×C₈×C₂₃	—	2	4	no	—
62	C₂×C₈×C₂₃	—	2	4	no	—
63	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
64	C₂×C₈×C₂₃	C₂²×C₄	1	8	yes	—
65	C₄⋊C₄×C₂₃	—	2	16	no	—
66	C₄⋊C₄×C₂₃	—	4	2	no	—
67	C₄⋊C₄×C₂₃	C₄²	1	8	yes	—
68	C₂²⋊C₄×C₂₃	—	4	2	no	—
69	C₂²⋊C₄×C₂₃	C₄²	1	8	yes	—
70	C₄²×C₂₃	—	2	4	no	—
71	C₄²×C₂₃	—	2	2	no	—
72	C₄²×C₂₃	—	2	2	no	—
73	C₄²×C₂₃	—	4	2	no	—
74	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
75	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
76	#20540	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₂₃	—	2	1	no	—
78	C₂³×C₂₃	—	2	1	no	—
79	C₂³×C₂₃	—	2	4	no	—
80	C₂³×C₂₃	C₂×C₄²	1	2	yes	—
81	C₂³×C₂₃	C₂³×C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₃	—	2	4	no	—
83	C₂×C₄×C₂₃	—	2	1	no	—
84	C₂×C₄×C₂₃	—	2	1	no	—
85	C₂×C₄×C₂₃	—	2	8	no	—
86	C₂×C₄×C₂₃	—	2	4	no	—
87	C₂×C₄×C₂₃	—	2	8	no	—
88	C₂×C₄×C₂₃	—	2	4	no	—
89	C₂×C₄×C₂₃	—	2	8	no	—
90	C₂×C₄×C₂₃	—	4	8	no	—
91	C₂×C₄×C₂₃	C₂×C₄²	1	8	yes	—
92	C₂×C₄×C₂₃	C₂×C₄²	1	2	yes	—
93	C₂×C₄×C₂₃	#131	1	8	yes	—
94	C₂×C₄×C₂₃	C₂³×C₄	1	2	yes	—
95	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
96	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
97	C₈×C₂₃	—	2	8	no	—
98	C₈×C₂₃	—	2	8	no	—
99	C₂×#433	C₂×C₂₃	1	2	yes	—
100	#2296	C₂×C₂₃	1	1	yes	characteristic
101	#1771	C₂×C₂₃	1	4	yes	complemented
102	C₈×C₄⋊C₄	C₂×C₂₃	1	4	yes	complemented
103	#1750	C₂×C₂₃	1	2	yes	—
104	C₄×C₂²⋊C₈	C₂×C₂₃	1	2	yes	—
105	C₂²×C₂₃	—	2	4	no	—
106	C₂²×C₂₃	—	2	2	no	—
107	C₂²×C₂₃	—	4	2	no	—
108	C₂²×C₂₃	—	4	2	no	—
109	C₂²×C₂₃	C₂²×C₄²	1	1	yes	characteristic
110	C₂²×C₂₃	#518	1	1	yes	characteristic
111	C₂²×C₂₃	#519	1	1	yes	characteristic
112	C₂²×C₂₃	C₂×#131	1	4	yes	—
113	C₄×C₂₃	—	2	8	no	—
114	C₄×C₂₃	—	2	8	no	—
115	C₄×C₂₃	—	2	2	no	—
116	C₄×C₂₃	—	4	4	no	—
117	C₄×C₂₃	#520	1	4	yes	—
118	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
119	C₂×#117	C₂²×C₂₃	1	1	yes	characteristic
120	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
121	#433	C₄×C₂₃	1	16	yes	complemented
122	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
123	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
124	C₂×#97	C₂²×C₂₃	1	2	yes	—
125	C₂×#97	C₂²×C₂₃	1	2	yes	—
126	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
127	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
128	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
129	#389	C₄×C₂₃	1	4	yes	—
130	#385	C₄×C₂₃	1	4	yes	—
131	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
132	#336	C₂²×C₂₃	1	2	yes	—
133	#336	C₂²×C₂₃	1	2	yes	—
134	#336	C₂²×C₂₃	1	4	yes	—
135	C₂×C₂₃	—	4	2	no	—
136	C₂×C₂₃	#2274	1	1	yes	characteristic
137	C₂×C₂₃	C₄×#131	1	2	yes	—
138	C₂×C₂₃	#2977	1	2	yes	—
139	C₂×C₂₃	#2986	1	2	yes	—
140	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
141	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
142	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
143	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
144	#117	—	4	2	no	—
145	#117	C₂×C₄×C₂₃	1	8	yes	—
146	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	8	yes	—
147	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
148	C₂×C₄⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
149	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
150	C₂×C₂²⋊C₄	C₂×C₄×C₂₃	1	2	yes	—
151	C₂×C₄²	—	4	2	no	—
152	C₂×C₄²	C₂×C₄×C₂₃	1	2	yes	—
153	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
154	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
155	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
156	C₄⋊C₈	C₂×C₄×C₂₃	1	16	yes	complemented
157	C₂²⋊C₈	C₂×C₄×C₂₃	1	16	yes	—
158	#97	—	4	2	no	—
159	#97	C₂×C₄×C₂₃	1	8	yes	complemented
160	C₄×C₈	—	4	4	no	—
161	C₄×C₈	—	4	2	no	—
162	C₄×C₈	C₂×C₄×C₂₃	1	8	yes	complemented
163	#95	C₂×C₄×C₂₃	1	8	yes	—
164	C₂₃	#20540	1	1	yes	characteristic, direct product factor, Sylow
165	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	4	2	no	—
170	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
171	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
172	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
173	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
174	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
175	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
181	C₂×C₈	C₂²×C₄×C₂₃	1	8	yes	—
182	C₄⋊C₄	—	2	16	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	C₄²×C₂₃	1	8	yes	—
185	C₂²⋊C₄	—	4	2	no	—
186	C₂²⋊C₄	C₄²×C₂₃	1	8	yes	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
192	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
193	C₂³	—	2	4	no	—
194	C₂³	—	2	1	no	—
195	C₂³	—	2	1	no	—
196	C₂³	C₂×C₄²×C₂₃	1	2	yes	—
197	C₂³	C₂³×C₄×C₂₃	1	1	yes	characteristic
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	8	no	—
207	C₂×C₄	C₂×C₄²×C₂₃	1	8	yes	—
208	C₂×C₄	C₂×C₄²×C₂₃	1	2	yes	—
209	C₂×C₄	C₂₃×#131	1	8	yes	—
210	C₂×C₄	C₂³×C₄×C₂₃	1	2	yes	—
211	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
212	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
213	C₈	—	2	8	no	—
214	C₈	—	2	8	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	C₂²×C₄²×C₂₃	1	1	yes	characteristic
220	C₂²	C₂₃×#518	1	1	yes	characteristic
221	C₂²	C₂₃×#519	1	1	yes	characteristic
222	C₂²	C₂×C₂₃×#131	1	4	yes	—
223	C₄	—	2	8	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	8	no	—
226	C₄	—	4	4	no	—
227	C₄	C₂₃×#520	1	4	yes	—
228	C₂	—	4	2	no	—
229	C₂	C₂₃×#2274	1	1	yes	characteristic
230	C₂	C₄×C₂₃×#131	1	2	yes	—
231	C₂	C₂₃×#2977	1	2	yes	—
232	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×#1771	C₂	semidirect
4. C₈×C₄⋊C₄×C₂₃	C₂	semidirect
10. C₂₃×#433	C₄	semidirect
40. C₂₃×C₄⋊C₈	C₂×C₄	semidirect
43. C₂₃×#97	C₂×C₄	semidirect
46. C₄×C₈×C₂₃	C₂×C₄	semidirect
76. #20540	C₂₃	direct (N × H)
101. #1771	C₂×C₂₃	semidirect
102. C₈×C₄⋊C₄	C₂×C₂₃	semidirect
121. #433	C₄×C₂₃	semidirect
156. C₄⋊C₈	C₂×C₄×C₂₃	semidirect
159. #97	C₂×C₄×C₂₃	semidirect
162. C₄×C₈	C₂×C₄×C₂₃	semidirect
164. C₂₃	#20540	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×#433	2
C₂₃×#2296	2
C₂₃×#1771	2
C₈×C₄⋊C₄×C₂₃	2
C₂₃×#1750	2
C₄×C₂₃×C₂²⋊C₈	2
#20540	23

Subgroups — C23×#20540 #19168166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#20540 #19168166