Subgroups — C₂×C₂₃×#2169 #19181057

Order 5888.

742 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	C₂²×C₂₃×M₃₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#2169	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂₃×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄×D₄×C₂₃	C₄	1	1	yes	characteristic
9	C₂×C₂₃×M₃₂	—	2	8	no	—
10	C₂×C₂₃×M₃₂	C₂²	1	8	yes	complemented
11	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
14	C₈×D₄×C₂₃	C₄	1	4	yes	—
15	C₈×D₄×C₂₃	C₂²	1	4	yes	—
16	C₂×C₂₃×C₄⋊C₈	C₄	1	1	yes	characteristic
17	C₂×C₂₃×C₂²⋊C₈	C₄	1	2	yes	—
18	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₂₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
20	C₂₃×C₂²⋊C₁₆	C₂²	1	8	yes	complemented
21	C₄×C₁₆×C₂₃	C₂²	1	4	yes	complemented
22	C₂²×D₄×C₂₃	C₈	1	1	yes	characteristic
23	C₂³×C₄×C₂₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₂₃	—	2	4	no	—
25	C₂²×C₈×C₂₃	—	2	4	no	—
26	C₂²×C₈×C₂₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
30	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
31	C₄×D₄×C₂₃	C₈	1	4	yes	—
32	C₄×D₄×C₂₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₂₃	C₈	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₂₃	C₈	1	2	yes	—
35	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
36	C₂₃×M₃₂	—	2	16	no	—
37	C₂×C₁₆×C₂₃	—	2	4	no	—
38	C₂×C₁₆×C₂₃	—	2	2	no	—
39	C₂×C₁₆×C₂₃	D₄	1	4	yes	complemented
40	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
41	C₂×C₁₆×C₂₃	C₂³	1	4	yes	—
42	C₂×C₁₆×C₂₃	C₂³	1	2	yes	—
43	C₂₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
44	C₂₃×C₂²⋊C₈	C₂×C₄	1	8	yes	—
45	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₂₃	C₂³	1	2	yes	—
47	C₂⁴×C₂₃	C₂×C₈	1	2	yes	—
48	C₂×D₄×C₂₃	—	2	4	no	—
49	C₂×D₄×C₂₃	C₂×C₈	1	4	yes	—
50	C₂²×C₄×C₂₃	—	2	4	no	—
51	C₂²×C₄×C₂₃	—	2	4	no	—
52	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₂₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₂₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂²×C₄	1	4	yes	—
57	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₈×C₂₃	—	2	4	no	—
59	C₂×C₈×C₂₃	—	2	2	no	—
60	C₂×C₈×C₂₃	—	2	8	no	—
61	C₂×C₈×C₂₃	—	2	4	no	—
62	C₂×C₈×C₂₃	—	2	4	no	—
63	C₂×C₈×C₂₃	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₂₃	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₂₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₂₃	#41	1	2	yes	—
68	C₂×C₈×C₂₃	C₂⁴	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	C₂×C₈	1	4	yes	—
70	C₂²⋊C₄×C₂₃	C₂×C₈	1	8	yes	—
71	C₄²×C₂₃	C₂×C₈	1	2	yes	—
72	C₄²×C₂₃	C₂²×C₄	1	2	yes	—
73	C₁₆×C₂₃	—	2	4	no	—
74	C₁₆×C₂₃	—	2	2	no	—
75	C₁₆×C₂₃	C₂×D₄	1	4	yes	complemented
76	C₂×#2169	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₂₃	—	2	4	no	—
78	C₂³×C₂₃	—	2	4	no	—
79	C₂³×C₂₃	—	2	4	no	—
80	C₂³×C₂₃	C₂²×C₈	1	4	yes	—
81	C₂³×C₂₃	C₂²×C₈	1	1	yes	characteristic
82	D₄×C₂₃	—	2	8	no	—
83	C₂×C₄×C₂₃	—	2	4	no	—
84	C₂×C₄×C₂₃	—	2	2	no	—
85	C₂×C₄×C₂₃	—	2	4	no	—
86	C₂×C₄×C₂₃	—	2	4	no	—
87	C₂×C₄×C₂₃	—	2	8	no	—
88	C₂×C₄×C₂₃	M₃₂	1	4	yes	complemented
89	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
91	C₂×C₄×C₂₃	C₂²×C₈	1	4	yes	—
92	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
94	C₂×C₄×C₂₃	#131	1	2	yes	—
95	C₂×C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
96	C₈×C₂₃	—	2	4	no	—
97	C₈×C₂₃	—	2	2	no	—
98	C₈×C₂₃	C₄×D₄	1	2	yes	—
99	C₈×C₂₃	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₂₃	C₂×#41	1	1	yes	characteristic
101	C₂²×M₃₂	C₂×C₂₃	1	2	yes	complemented
102	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
103	#2169	C₂×C₂₃	1	8	yes	direct product factor
104	C₂×C₄⋊C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
105	C₂×C₂²⋊C₁₆	C₂×C₂₃	1	2	yes	complemented
106	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
107	C₂²×C₂₃	—	2	4	no	—
108	C₂²×C₂₃	—	2	4	no	—
109	C₂²×C₂₃	—	2	4	no	—
110	C₂²×C₂₃	—	4	8	no	—
111	C₂²×C₂₃	C₈×D₄	1	2	yes	—
112	C₂²×C₂₃	C₂×M₃₂	1	2	yes	—
113	C₂²×C₂₃	#504	1	2	yes	—
114	C₂²×C₂₃	C₂³×C₈	1	1	yes	characteristic
115	C₄×C₂₃	—	2	2	no	—
116	C₄×C₂₃	—	2	4	no	—
117	C₄×C₂₃	C₈×D₄	1	2	yes	—
118	C₄×C₂₃	C₂×M₃₂	1	4	yes	complemented
119	C₄×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
120	C₄×C₂₃	C₂×#131	1	1	yes	characteristic
121	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
122	C₂×C₄×D₄	C₄×C₂₃	1	1	yes	characteristic
123	C₂×M₃₂	—	2	8	no	—
124	C₂×M₃₂	C₂²×C₂₃	1	8	yes	complemented
125	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
126	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
127	C₂²×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
128	C₈×D₄	C₄×C₂₃	1	4	yes	—
129	C₈×D₄	C₂²×C₂₃	1	4	yes	—
130	C₂×C₄⋊C₈	C₄×C₂₃	1	1	yes	characteristic
131	C₂×C₂²⋊C₈	C₄×C₂₃	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
133	C₄⋊C₁₆	C₂²×C₂₃	1	4	yes	complemented
134	C₂²⋊C₁₆	C₂²×C₂₃	1	8	yes	complemented
135	C₄×C₁₆	C₂²×C₂₃	1	4	yes	complemented
136	C₂×C₂₃	—	4	4	no	—
137	C₂×C₂₃	#2169	1	4	yes	direct product factor
138	C₂×C₂₃	C₂×C₈×D₄	1	1	yes	characteristic
139	C₂×C₂₃	C₂²×M₃₂	1	1	yes	characteristic
140	C₂×C₂₃	C₂×#504	1	1	yes	characteristic
141	C₂²×D₄	C₈×C₂₃	1	1	yes	characteristic
142	C₂³×C₄	C₂×C₄×C₂₃	1	2	yes	—
143	C₂²×C₈	—	2	4	no	—
144	C₂²×C₈	—	2	4	no	—
145	C₂²×C₈	C₂×C₄×C₂₃	1	2	yes	—
146	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
147	C₂²×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
149	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
150	C₄×D₄	C₈×C₂₃	1	4	yes	—
151	C₄×D₄	C₂×C₄×C₂₃	1	4	yes	—
152	C₂×C₄⋊C₄	C₈×C₂₃	1	1	yes	characteristic
153	C₂×C₂²⋊C₄	C₈×C₂₃	1	2	yes	—
154	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
155	M₃₂	—	2	16	no	—
156	C₂×C₁₆	—	2	4	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	D₄×C₂₃	1	4	yes	complemented
159	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
160	C₂×C₁₆	C₂³×C₂₃	1	4	yes	—
161	C₂×C₁₆	C₂³×C₂₃	1	2	yes	—
162	C₄⋊C₈	C₂×C₄×C₂₃	1	4	yes	—
163	C₂²⋊C₈	C₂×C₄×C₂₃	1	8	yes	—
164	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
165	C₄×C₈	C₂³×C₂₃	1	2	yes	—
166	C₂₃	C₂×#2169	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	C₂×C₈×C₂₃	1	2	yes	—
168	C₂×D₄	—	2	4	no	—
169	C₂×D₄	C₂×C₈×C₂₃	1	4	yes	—
170	C₂²×C₄	—	2	4	no	—
171	C₂²×C₄	—	2	4	no	—
172	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₈×C₂₃	1	2	yes	—
174	C₂²×C₄	C₂×C₈×C₂₃	1	4	yes	—
175	C₂²×C₄	C₂×C₈×C₂₃	1	1	yes	characteristic
176	C₂²×C₄	C₂²×C₄×C₂₃	1	4	yes	—
177	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	8	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	C₂²×C₄×C₂₃	1	2	yes	—
184	C₂×C₈	C₂²×C₄×C₂₃	1	4	yes	—
185	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₂₃	1	2	yes	—
187	C₂×C₈	#41×C₂₃	1	2	yes	—
188	C₂×C₈	C₂⁴×C₂₃	1	1	yes	characteristic
189	C₄⋊C₄	C₂×C₈×C₂₃	1	4	yes	—
190	C₂²⋊C₄	C₂×C₈×C₂₃	1	8	yes	—
191	C₄²	C₂×C₈×C₂₃	1	2	yes	—
192	C₄²	C₂²×C₄×C₂₃	1	2	yes	—
193	C₁₆	—	2	4	no	—
194	C₁₆	—	2	2	no	—
195	C₁₆	C₂×D₄×C₂₃	1	4	yes	complemented
196	C₂³	—	2	4	no	—
197	C₂³	—	2	4	no	—
198	C₂³	—	2	4	no	—
199	C₂³	C₂²×C₈×C₂₃	1	4	yes	—
200	C₂³	C₂²×C₈×C₂₃	1	1	yes	characteristic
201	D₄	—	2	8	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	C₂₃×M₃₂	1	4	yes	complemented
208	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
209	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
210	C₂×C₄	C₂²×C₈×C₂₃	1	4	yes	—
211	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
213	C₂×C₄	C₂₃×#131	1	2	yes	—
214	C₂×C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	2	4	no	—
217	C₈	C₄×D₄×C₂₃	1	2	yes	—
218	C₈	C₂²×D₄×C₂₃	1	1	yes	characteristic
219	C₈	C₂×#41×C₂₃	1	1	yes	characteristic
220	C₂²	—	2	4	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	4	8	no	—
224	C₂²	C₈×D₄×C₂₃	1	2	yes	—
225	C₂²	C₂×C₂₃×M₃₂	1	2	yes	—
226	C₂²	C₂₃×#504	1	2	yes	—
227	C₂²	C₂³×C₈×C₂₃	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	C₈×D₄×C₂₃	1	2	yes	—
231	C₄	C₂×C₂₃×M₃₂	1	4	yes	complemented
232	C₄	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
233	C₄	C₂×C₂₃×#131	1	1	yes	characteristic
234	C₂	—	4	4	no	—
235	C₂	C₂₃×#2169	1	4	yes	direct product factor
236	C₂	C₂×C₈×D₄×C₂₃	1	1	yes	characteristic
237	C₂	C₂²×C₂₃×M₃₂	1	1	yes	characteristic
238	C₂	C₂×C₂₃×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈×C₂₃	C₂³
commutator	C₂²	C₂³×C₈×C₂₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₂₃
socle	C₂³×C₂₃	C₂²×C₈
prime radical	C₂²×D₄×C₂₃	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₃×M₃₂	C₂	semidirect
3. C₂₃×#2169	C₂	direct (N × H)
4. C₂×C₂₃×C₄⋊C₁₆	C₂	semidirect
5. C₂×C₂₃×C₂²⋊C₁₆	C₂	semidirect
6. C₂×C₄×C₁₆×C₂₃	C₂	semidirect
10. C₂×C₂₃×M₃₂	C₂²	semidirect
19. C₂₃×C₄⋊C₁₆	C₂²	semidirect
20. C₂₃×C₂²⋊C₁₆	C₂²	semidirect
21. C₄×C₁₆×C₂₃	C₂²	semidirect
39. C₂×C₁₆×C₂₃	D₄	semidirect
75. C₁₆×C₂₃	C₂×D₄	semidirect
76. C₂×#2169	C₂₃	direct (N × H)
88. C₂×C₄×C₂₃	M₃₂	semidirect
101. C₂²×M₃₂	C₂×C₂₃	semidirect
103. #2169	C₂×C₂₃	direct (N × H)
104. C₂×C₄⋊C₁₆	C₂×C₂₃	semidirect
105. C₂×C₂²⋊C₁₆	C₂×C₂₃	semidirect
106. C₂×C₄×C₁₆	C₂×C₂₃	semidirect
118. C₄×C₂₃	C₂×M₃₂	semidirect
124. C₂×M₃₂	C₂²×C₂₃	semidirect
133. C₄⋊C₁₆	C₂²×C₂₃	semidirect
134. C₂²⋊C₁₆	C₂²×C₂₃	semidirect
135. C₄×C₁₆	C₂²×C₂₃	semidirect
137. C₂×C₂₃	#2169	direct (N × H)
158. C₂×C₁₆	D₄×C₂₃	semidirect
166. C₂₃	C₂×#2169	direct (N × H)
195. C₁₆	C₂×D₄×C₂₃	semidirect
207. C₂×C₄	C₂₃×M₃₂	semidirect
231. C₄	C₂×C₂₃×M₃₂	semidirect
235. C₂	C₂₃×#2169	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×M₃₂	2
C₂×C₈×D₄×C₂₃	2
C₂₃×#2169	2
C₂×C₂₃×C₄⋊C₁₆	2
C₂×C₂₃×C₂²⋊C₁₆	2
C₂×C₄×C₁₆×C₂₃	2
C₂×#2169	23

Subgroups — C2×C23×#2169 #19181057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#2169 #19181057