Subgroups — C₂₃×#21960 #19169586

Order 5888.

818 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#2283	C₂	1	2	yes	maximal
3	C₂₃×#1949	C₂	1	8	yes	maximal, complemented
4	C₂₃×#1808	C₂	1	2	yes	maximal
5	C₂₃×#1807	C₂	1	2	yes	maximal
6	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
9	C₂×C₂₃×#107	C₂²	1	4	yes	—
10	C₂×C₂₃×#106	C₂²	1	4	yes	—
11	C₂×C₂₃×C₄⋊C₈	C₄	1	8	yes	complemented
12	C₂×C₂₃×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂₃×#383	—	2	2	no	—
14	C₂₃×#382	C₂²	1	4	yes	—
15	C₂₃×#381	—	2	2	no	—
16	C₂₃×#380	C₂²	1	4	yes	—
17	C₄×C₄⋊C₄×C₂₃	C₂²	1	4	yes	—
18	C₄×C₂²⋊C₄×C₂₃	—	2	2	no	—
19	C₂₃×#376	—	2	2	no	—
20	C₂₃×#340	C₂²	1	4	yes	—
21	C₂₃×#340	C₂²	1	4	yes	—
22	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₂₃	—	4	2	no	—
26	C₂²×C₈×C₂₃	—	4	2	no	—
27	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
29	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
31	C₂×C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
32	C₂×C₂²⋊C₄×C₂₃	—	2	4	no	—
33	C₂×C₄²×C₂₃	—	2	4	no	—
34	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₂₃	D₄	1	2	yes	—
36	C₂×C₄²×C₂₃	Q₈	1	4	yes	—
37	C₂×C₄²×C₂₃	C₂³	1	2	yes	—
38	C₂₃×#107	—	4	4	no	—
39	C₂₃×#106	—	4	4	no	—
40	C₂₃×C₄⋊C₈	C₂×C₄	1	16	yes	—
41	C₂₃×#95	—	2	4	no	—
42	C₂₃×#95	—	2	4	no	—
43	C₂₃×#95	—	2	4	no	—
44	C₂₃×#95	—	2	4	no	—
45	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₂₃	—	2	4	no	—
47	C₂²×C₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	—	2	2	no	—
49	C₂²×C₄×C₂₃	—	2	8	no	—
50	C₂²×C₄×C₂₃	—	4	2	no	—
51	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₂₃	C₂×Q₈	1	2	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	—	4	4	no	—
62	C₂×C₈×C₂₃	—	4	4	no	—
63	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
65	C₄⋊C₄×C₂₃	—	2	8	no	—
66	C₄⋊C₄×C₂₃	—	2	8	no	—
67	C₂²⋊C₄×C₂₃	—	4	8	no	—
68	C₄²×C₂₃	—	4	8	no	—
69	C₄²×C₂₃	C₄⋊C₄	1	8	yes	—
70	C₄²×C₂₃	C₂²×C₄	1	4	yes	—
71	C₄²×C₂₃	C₂×Q₈	1	4	yes	—
72	#21960	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₂₃	—	2	2	no	—
74	C₂³×C₂₃	—	2	2	no	—
75	C₂³×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂³×C₂₃	#135	1	4	yes	—
77	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	—	2	2	no	—
79	C₂×C₄×C₂₃	—	2	4	no	—
80	C₂×C₄×C₂₃	—	2	4	no	—
81	C₂×C₄×C₂₃	—	2	8	no	—
82	C₂×C₄×C₂₃	—	2	2	no	—
83	C₂×C₄×C₂₃	—	2	2	no	—
84	C₂×C₄×C₂₃	—	2	4	no	—
85	C₂×C₄×C₂₃	—	4	4	no	—
86	C₂×C₄×C₂₃	—	4	8	no	—
87	C₂×C₄×C₂₃	—	4	8	no	—
88	C₂×C₄×C₂₃	—	4	8	no	—
89	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
95	C₂×C₄×C₂₃	2₋¹⁺⁴	1	2	yes	—
96	C₈×C₂₃	—	2	4	no	—
97	C₈×C₂₃	—	2	4	no	—
98	C₂²×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
99	#2283	C₂×C₂₃	1	2	yes	—
100	#1949	C₂×C₂₃	1	8	yes	complemented
101	#1808	C₂×C₂₃	1	2	yes	—
102	#1807	C₂×C₂₃	1	2	yes	—
103	C₂²×C₂₃	—	2	2	no	—
104	C₂²×C₂₃	—	2	2	no	—
105	C₂²×C₂₃	—	2	8	no	—
106	C₂²×C₂₃	#427	1	4	yes	—
107	C₂²×C₂₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₂₃	#537	1	2	yes	—
109	C₂²×C₂₃	C₂×#135	1	2	yes	—
110	C₂²×C₂₃	#575	1	2	yes	—
111	C₄×C₂₃	—	2	2	no	—
112	C₄×C₂₃	—	2	4	no	—
113	C₄×C₂₃	—	2	4	no	—
114	C₄×C₂₃	—	8	8	no	—
115	C₄×C₂₃	#518	1	1	yes	characteristic
116	C₄×C₂₃	#519	1	1	yes	characteristic
117	C₄×C₂₃	#544	1	2	yes	—
118	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
119	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
120	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
121	C₂×#107	C₂²×C₂₃	1	4	yes	—
122	C₂×#106	C₂²×C₂₃	1	4	yes	—
123	C₂×C₄⋊C₈	C₄×C₂₃	1	8	yes	complemented
124	C₂×C₄⋊C₈	C₂²×C₂₃	1	4	yes	—
125	#383	—	2	2	no	—
126	#382	C₂²×C₂₃	1	4	yes	—
127	#381	—	2	2	no	—
128	#380	C₂²×C₂₃	1	4	yes	—
129	C₄×C₄⋊C₄	C₂²×C₂₃	1	4	yes	—
130	C₄×C₂²⋊C₄	—	2	2	no	—
131	#376	—	2	2	no	—
132	#340	C₂²×C₂₃	1	4	yes	—
133	#340	C₂²×C₂₃	1	4	yes	—
134	C₂×C₂₃	—	2	4	no	—
135	C₂×C₂₃	#2316	1	1	yes	characteristic
136	C₂×C₂₃	C₂×#427	1	1	yes	characteristic
137	C₂×C₂₃	#2911	1	1	yes	characteristic
138	C₂×C₂₃	#3250	1	4	yes	—
139	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
140	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
141	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
142	C₂²×C₈	—	4	2	no	—
143	C₂²×C₈	—	4	2	no	—
144	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
145	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
146	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
147	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
148	C₂×C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
149	C₂×C₂²⋊C₄	—	2	4	no	—
150	C₂×C₄²	—	2	4	no	—
151	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
152	C₂×C₄²	D₄×C₂₃	1	2	yes	—
153	C₂×C₄²	Q₈×C₂₃	1	4	yes	—
154	C₂×C₄²	C₂³×C₂₃	1	2	yes	—
155	#107	—	4	4	no	—
156	#106	—	4	4	no	—
157	C₄⋊C₈	C₂×C₄×C₂₃	1	16	yes	—
158	#95	—	2	4	no	—
159	#95	—	2	4	no	—
160	#95	—	2	4	no	—
161	#95	—	2	4	no	—
162	C₂₃	#21960	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	—	2	2	no	—
167	C₂²×C₄	—	2	8	no	—
168	C₂²×C₄	—	4	2	no	—
169	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
170	C₂²×C₄	C₄⋊C₄×C₂₃	1	4	yes	—
171	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
172	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
173	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
174	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
175	C₂²×C₄	C₂×Q₈×C₂₃	1	2	yes	—
176	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	4	4	no	—
181	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
182	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
183	C₄⋊C₄	—	2	8	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₂²⋊C₄	—	4	8	no	—
186	C₄²	—	4	8	no	—
187	C₄²	C₄⋊C₄×C₂₃	1	8	yes	—
188	C₄²	C₂²×C₄×C₂₃	1	4	yes	—
189	C₄²	C₂×Q₈×C₂₃	1	4	yes	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
193	C₂³	C₂₃×#135	1	4	yes	—
194	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	8	no	—
204	C₂×C₄	—	4	8	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	4	yes	—
209	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×Q₈×C₂₃	1	1	yes	characteristic
211	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
212	C₂×C₄	C₂₃×2₋¹⁺⁴	1	2	yes	—
213	C₈	—	2	4	no	—
214	C₈	—	2	4	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	C₂₃×#427	1	4	yes	—
219	C₂²	C₂²×C₄⋊C₄×C₂₃	1	1	yes	characteristic
220	C₂²	C₂₃×#537	1	2	yes	—
221	C₂²	C₂×C₂₃×#135	1	2	yes	—
222	C₂²	C₂₃×#575	1	2	yes	—
223	C₄	—	2	4	no	—
224	C₄	—	2	4	no	—
225	C₄	—	2	2	no	—
226	C₄	—	8	8	no	—
227	C₄	C₂₃×#518	1	1	yes	characteristic
228	C₄	C₂₃×#519	1	1	yes	characteristic
229	C₄	C₂₃×#544	1	2	yes	—
230	C₂	—	2	4	no	—
231	C₂	C₂₃×#2316	1	1	yes	characteristic
232	C₂	C₂×C₂₃×#427	1	1	yes	characteristic
233	C₂	C₂₃×#2911	1	1	yes	characteristic
234	C₂	C₂₃×#3250	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₂₃
prime radical	C₂⁴×C₂₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂₃×#1949	C₂	semidirect
11. C₂×C₂₃×C₄⋊C₈	C₄	semidirect
72. #21960	C₂₃	direct (N × H)
100. #1949	C₂×C₂₃	semidirect
123. C₂×C₄⋊C₈	C₄×C₂₃	semidirect
162. C₂₃	#21960	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×C₄⋊C₈	2
C₂₃×#2283	2
C₂₃×#1949	2
C₂₃×#1808	2
C₂₃×#1807	2
#21960	23

Subgroups — C23×#21960 #19169586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#21960 #19169586