Subgroups — C₂×C₂₃×#2242 #19181322

Order 5888.

610 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#495	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2242	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×#358	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#357	C₂	1	2	yes	maximal
6	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	C₂²×SD₁₆×C₂₃	—	2	1	no	—
9	C₂²×D₈×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂₃×#125	—	2	1	no	—
11	C₂×C₂₃×#124	—	2	1	no	—
12	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
13	C₂₃×#499	—	2	2	no	—
14	C₂₃×#499	C₂²	1	4	yes	complemented
15	C₂₃×#495	—	2	2	no	—
16	C₂₃×#495	C₂²	1	4	yes	—
17	C₂×C₂₃×#107	C₂²	1	2	yes	—
18	C₂×C₂₃×#106	—	2	1	no	—
19	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#358	C₂²	1	8	yes	complemented
21	C₂₃×#357	C₂²	1	8	yes	—
22	C₄×C₁₆×C₂₃	C₂²	1	4	yes	complemented
23	C₂²×Q₈×C₂₃	—	2	1	no	—
24	C₂²×D₄×C₂₃	—	2	1	no	—
25	C₂×Q₁₆×C₂₃	—	2	1	no	—
26	C₂×Q₁₆×C₂₃	—	2	4	no	—
27	C₂×Q₁₆×C₂₃	C₂³	1	2	yes	—
28	C₂×SD₁₆×C₂₃	—	2	4	no	—
29	C₂×SD₁₆×C₂₃	—	4	4	no	—
30	C₂×D₈×C₂₃	—	2	1	no	—
31	C₂×D₈×C₂₃	—	2	4	no	—
32	C₂×D₈×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
35	C₂₃×#125	—	2	4	no	—
36	C₂₃×#125	—	4	2	no	—
37	C₂₃×#124	—	2	4	no	—
38	C₂₃×#124	—	4	2	no	—
39	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₂₃	—	4	2	no	—
41	C₂×C₂²⋊C₄×C₂₃	—	4	2	no	—
42	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₂₃	—	2	4	no	—
44	C₂×C₁₆×C₂₃	C₂³	1	4	yes	—
45	C₂₃×#107	—	2	2	no	—
46	C₂₃×#107	C₂³	1	4	yes	—
47	C₂₃×#106	—	2	4	no	—
48	C₄×C₈×C₂₃	D₄	1	2	yes	—
49	C₄×C₈×C₂₃	C₂³	1	2	yes	—
50	C₂⁴×C₂₃	—	4	1	no	—
51	C₂×Q₈×C₂₃	—	2	2	no	—
52	C₂×Q₈×C₂₃	—	4	1	no	—
53	C₂×Q₈×C₂₃	—	4	2	no	—
54	C₂×Q₈×C₂₃	—	4	2	no	—
55	C₂×D₄×C₂₃	—	2	2	no	—
56	C₂×D₄×C₂₃	—	4	2	no	—
57	C₂×D₄×C₂₃	—	4	1	no	—
58	C₂×D₄×C₂₃	—	4	2	no	—
59	C₂²×C₄×C₂₃	—	4	1	no	—
60	C₂²×C₄×C₂₃	—	4	2	no	—
61	C₂²×C₄×C₂₃	D₈	1	2	yes	—
62	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₂₃	—	2	4	no	—
64	Q₁₆×C₂₃	—	4	2	no	—
65	SD₁₆×C₂₃	—	4	8	no	—
66	D₈×C₂₃	—	2	4	no	—
67	D₈×C₂₃	—	4	2	no	—
68	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
69	C₂×C₈×C₂₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₂₃	#41	1	4	yes	—
72	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
73	C₄⋊C₄×C₂₃	—	2	4	no	—
74	C₄⋊C₄×C₂₃	—	4	2	no	—
75	C₄⋊C₄×C₂₃	—	4	8	no	—
76	C₂²⋊C₄×C₂₃	—	4	8	no	—
77	C₄²×C₂₃	D₈	1	2	yes	—
78	C₄²×C₂₃	C₂×D₄	1	2	yes	—
79	C₁₆×C₂₃	—	2	4	no	—
80	C₂×#2242	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₂₃	—	4	2	no	—
82	C₂³×C₂₃	—	8	2	no	—
83	C₂³×C₂₃	—	8	4	no	—
84	C₂³×C₂₃	C₂×D₈	1	1	yes	characteristic
85	Q₈×C₂₃	—	4	2	no	—
86	Q₈×C₂₃	—	4	2	no	—
87	Q₈×C₂₃	—	8	2	no	—
88	D₄×C₂₃	—	4	2	no	—
89	D₄×C₂₃	—	4	2	no	—
90	D₄×C₂₃	—	8	2	no	—
91	C₂×C₄×C₂₃	—	2	4	no	—
92	C₂×C₄×C₂₃	—	4	2	no	—
93	C₂×C₄×C₂₃	—	4	4	no	—
94	C₂×C₄×C₂₃	—	8	2	no	—
95	C₂×C₄×C₂₃	—	8	4	no	—
96	C₂×C₄×C₂₃	#124	1	2	yes	—
97	C₂×C₄×C₂₃	C₂×D₈	1	4	yes	—
98	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
100	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
101	C₈×C₂₃	#124	1	2	yes	—
102	C₈×C₂₃	C₂×SD₁₆	1	4	yes	—
103	C₈×C₂₃	C₂×#41	1	2	yes	—
104	C₂×#499	C₂×C₂₃	1	1	yes	characteristic, complemented
105	C₂×#495	C₂×C₂₃	1	1	yes	characteristic
106	#2242	C₂×C₂₃	1	8	yes	direct product factor
107	C₂×#358	C₂×C₂₃	1	2	yes	complemented
108	C₂×#357	C₂×C₂₃	1	2	yes	—
109	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
110	C₂²×C₂₃	—	8	2	no	—
111	C₂²×C₂₃	—	8	4	no	—
112	C₂²×C₂₃	—	16	4	no	—
113	C₂²×C₂₃	#486	1	2	yes	—
114	C₂²×C₂₃	#508	1	4	yes	—
115	C₂²×C₂₃	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₂₃	—	2	4	no	—
117	C₄×C₂₃	—	8	2	no	—
118	C₄×C₂₃	—	8	4	no	—
119	C₄×C₂₃	#486	1	2	yes	—
120	C₄×C₂₃	C₂×#124	1	1	yes	characteristic
121	C₄×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
122	C₂²×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic
123	C₂²×SD₁₆	—	2	1	no	—
124	C₂²×D₈	C₂²×C₂₃	1	1	yes	characteristic
125	C₂×#125	—	2	1	no	—
126	C₂×#124	—	2	1	no	—
127	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
128	#499	—	2	2	no	—
129	#499	C₂²×C₂₃	1	4	yes	complemented
130	#495	—	2	2	no	—
131	#495	C₂²×C₂₃	1	4	yes	—
132	C₂×#107	C₂²×C₂₃	1	2	yes	—
133	C₂×#106	—	2	1	no	—
134	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
135	#358	C₂²×C₂₃	1	8	yes	complemented
136	#357	C₂²×C₂₃	1	8	yes	—
137	C₄×C₁₆	C₂²×C₂₃	1	4	yes	complemented
138	C₂×C₂₃	—	16	2	no	—
139	C₂×C₂₃	#2242	1	4	yes	direct product factor
140	C₂×C₂₃	C₂×#486	1	1	yes	characteristic
141	C₂×C₂₃	C₂×#508	1	2	yes	—
142	C₂²×Q₈	—	2	1	no	—
143	C₂²×D₄	—	2	1	no	—
144	C₂×Q₁₆	—	2	1	no	—
145	C₂×Q₁₆	—	2	4	no	—
146	C₂×Q₁₆	C₂³×C₂₃	1	2	yes	—
147	C₂×SD₁₆	—	2	4	no	—
148	C₂×SD₁₆	—	4	4	no	—
149	C₂×D₈	—	2	1	no	—
150	C₂×D₈	—	2	4	no	—
151	C₂×D₈	C₂³×C₂₃	1	2	yes	—
152	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
154	#125	—	2	4	no	—
155	#125	—	4	2	no	—
156	#124	—	2	4	no	—
157	#124	—	4	2	no	—
158	C₂×C₄⋊C₄	—	2	2	no	—
159	C₂×C₄⋊C₄	—	4	2	no	—
160	C₂×C₂²⋊C₄	—	4	2	no	—
161	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
162	C₂×C₁₆	—	2	4	no	—
163	C₂×C₁₆	C₂³×C₂₃	1	4	yes	—
164	#107	—	2	2	no	—
165	#107	C₂³×C₂₃	1	4	yes	—
166	#106	—	2	4	no	—
167	C₄×C₈	D₄×C₂₃	1	2	yes	—
168	C₄×C₈	C₂³×C₂₃	1	2	yes	—
169	C₂₃	C₂×#2242	1	1	yes	characteristic, direct product factor, Sylow
170	C₂⁴	—	4	1	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	D₈×C₂₃	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
183	Q₁₆	—	2	4	no	—
184	Q₁₆	—	4	2	no	—
185	SD₁₆	—	4	8	no	—
186	D₈	—	2	4	no	—
187	D₈	—	4	2	no	—
188	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
189	C₂×C₈	C₂×D₄×C₂₃	1	1	yes	characteristic
190	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
191	C₂×C₈	#41×C₂₃	1	4	yes	—
192	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	4	2	no	—
195	C₄⋊C₄	—	4	8	no	—
196	C₂²⋊C₄	—	4	8	no	—
197	C₄²	D₈×C₂₃	1	2	yes	—
198	C₄²	C₂×D₄×C₂₃	1	2	yes	—
199	C₁₆	—	2	4	no	—
200	C₂³	—	4	2	no	—
201	C₂³	—	8	2	no	—
202	C₂³	—	8	4	no	—
203	C₂³	C₂×D₈×C₂₃	1	1	yes	characteristic
204	Q₈	—	4	2	no	—
205	Q₈	—	4	2	no	—
206	Q₈	—	8	2	no	—
207	D₄	—	4	2	no	—
208	D₄	—	4	2	no	—
209	D₄	—	8	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	4	no	—
215	C₂×C₄	C₂₃×#124	1	2	yes	—
216	C₂×C₄	C₂×D₈×C₂₃	1	4	yes	—
217	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
218	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
219	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
220	C₈	C₂₃×#124	1	2	yes	—
221	C₈	C₂×SD₁₆×C₂₃	1	4	yes	—
222	C₈	C₂×#41×C₂₃	1	2	yes	—
223	C₂²	—	8	2	no	—
224	C₂²	—	8	4	no	—
225	C₂²	—	16	4	no	—
226	C₂²	C₂₃×#486	1	2	yes	—
227	C₂²	C₂₃×#508	1	4	yes	—
228	C₂²	C₂²×D₈×C₂₃	1	1	yes	characteristic
229	C₄	—	2	4	no	—
230	C₄	—	8	2	no	—
231	C₄	—	8	4	no	—
232	C₄	C₂₃×#486	1	2	yes	—
233	C₄	C₂×C₂₃×#124	1	1	yes	characteristic
234	C₄	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
235	C₂	—	16	2	no	—
236	C₂	C₂₃×#2242	1	4	yes	direct product factor
237	C₂	C₂×C₂₃×#486	1	1	yes	characteristic
238	C₂	C₂×C₂₃×#508	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
prime radical	C₂²×D₈×C₂₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₂₃×#499	C₂	semidirect
3. C₂₃×#2242	C₂	direct (N × H)
4. C₂×C₂₃×#358	C₂	semidirect
6. C₂×C₄×C₁₆×C₂₃	C₂	semidirect
14. C₂₃×#499	C₂²	semidirect
20. C₂₃×#358	C₂²	semidirect
22. C₄×C₁₆×C₂₃	C₂²	semidirect
80. C₂×#2242	C₂₃	direct (N × H)
104. C₂×#499	C₂×C₂₃	semidirect
106. #2242	C₂×C₂₃	direct (N × H)
107. C₂×#358	C₂×C₂₃	semidirect
109. C₂×C₄×C₁₆	C₂×C₂₃	semidirect
129. #499	C₂²×C₂₃	semidirect
135. #358	C₂²×C₂₃	semidirect
137. C₄×C₁₆	C₂²×C₂₃	semidirect
139. C₂×C₂₃	#2242	direct (N × H)
169. C₂₃	C₂×#2242	direct (N × H)
236. C₂	C₂₃×#2242	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×#499	2
C₂×C₂₃×#495	2
C₂₃×#2242	2
C₂×C₂₃×#358	2
C₂×C₂₃×#357	2
C₂×C₄×C₁₆×C₂₃	2
C₂×#2242	23

Subgroups — C2×C23×#2242 #19181322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2242 #19181322