Subgroups — C₂×C₁₉×#3531 #18068652

Order 4864.

6554 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3531	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₁₉×#564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₉×#563	C₂	1	10	yes	maximal, complemented
4	C₂×C₁₉×#562	C₂	1	5	yes	maximal, complemented
5	C₂×C₁₉×#533	C₂	1	5	yes	maximal, complemented
6	C₂×C₁₉×#529	C₂	1	10	yes	maximal, complemented
7	C₁₉×#564	C₂²	1	16	yes	complemented
8	C₁₉×#563	C₂²	1	160	yes	complemented
9	C₁₉×#562	C₂²	1	80	yes	complemented
10	C₁₉×#533	C₂²	1	80	yes	complemented
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	10	yes	—
12	C₁₉×#529	C₂²	1	160	yes	complemented
13	C₂×C₁₉×#126	C₂²	1	20	yes	—
14	C₂×C₁₉×#125	C₂²	1	10	yes	—
15	C₂×C₁₉×#125	C₂²	1	5	yes	—
16	C₂×C₁₉×#125	C₂²	1	5	yes	—
17	C₂×C₁₉×#124	C₂²	1	10	yes	—
18	C₂×C₁₉×#123	C₂²	1	20	yes	—
19	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	10	yes	—
20	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	20	yes	—
21	C₂×C₁₉×#121	C₂²	1	10	yes	—
22	C₂×C₄×Q₈×C₁₉	C₂²	1	10	yes	—
23	C₂×C₄×D₄×C₁₉	C₂²	1	10	yes	—
24	C₂×C₁₉×#117	C₂²	1	10	yes	—
25	C₂²×C₄²×C₁₉	C₂²	1	5	yes	—
26	C₂²×Q₈×C₁₉	C₂³	1	10	yes	—
27	C₂²×D₄×C₁₉	C₂³	1	10	yes	—
28	C₂³×C₄×C₁₉	C₂³	1	5	yes	—
29	C₂³×C₄×C₁₉	C₂³	1	10	yes	—
30	C₁₉×C₄⋊Q₈	C₂³	1	80	yes	—
31	C₁₉×#126	C₂³	1	160	yes	—
32	C₁₉×#125	C₂³	1	80	yes	—
33	C₁₉×#125	C₂³	1	40	yes	—
34	C₁₉×#125	C₂³	1	40	yes	—
35	C₁₉×#124	C₂³	1	80	yes	—
36	C₁₉×#123	C₂³	1	160	yes	—
37	C₁₉×C₂²⋊Q₈	C₂³	1	80	yes	—
38	C₁₉×C₂²⋊Q₈	C₂³	1	160	yes	—
39	C₁₉×#121	C₂³	1	80	yes	—
40	C₄×Q₈×C₁₉	C₂³	1	80	yes	—
41	C₄×D₄×C₁₉	C₂³	1	80	yes	—
42	C₁₉×#117	C₂³	1	80	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	10	yes	—
44	C₂×C₄⋊C₄×C₁₉	C₂³	1	10	yes	—
45	C₂×C₄⋊C₄×C₁₉	C₂³	1	10	yes	—
46	C₂×C₄⋊C₄×C₁₉	C₂³	1	10	yes	—
47	C₂×C₄⋊C₄×C₁₉	C₂³	1	20	yes	—
48	C₂×C₂²⋊C₄×C₁₉	C₂³	1	20	yes	—
49	C₂×C₂²⋊C₄×C₁₉	C₂³	1	20	yes	—
50	C₂×C₄²×C₁₉	C₂³	1	5	yes	—
51	C₂×C₄²×C₁₉	C₂³	1	5	yes	—
52	C₂×C₄²×C₁₉	C₂³	1	40	yes	—
53	C₂×C₄²×C₁₉	C₂³	1	10	yes	—
54	C₂⁴×C₁₉	C₂⁴	1	5	yes	—
55	C₂×Q₈×C₁₉	—	4	20	no	—
56	C₂×Q₈×C₁₉	C₂⁴	1	40	yes	—
57	C₂×D₄×C₁₉	—	4	20	no	—
58	C₂×D₄×C₁₉	C₂⁴	1	40	yes	—
59	C₂²×C₄×C₁₉	—	2	20	no	—
60	C₂²×C₄×C₁₉	—	4	20	no	—
61	C₂²×C₄×C₁₉	C₂⁴	1	5	yes	—
62	C₂²×C₄×C₁₉	C₂⁴	1	10	yes	—
63	C₂²×C₄×C₁₉	C₂⁴	1	10	yes	—
64	C₂²×C₄×C₁₉	C₂⁴	1	20	yes	—
65	C₂²×C₄×C₁₉	C₂⁴	1	40	yes	—
66	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₁₉	C₂⁴	1	40	yes	—
68	C₄⋊C₄×C₁₉	C₂⁴	1	40	yes	—
69	C₄⋊C₄×C₁₉	C₂⁴	1	40	yes	—
70	C₄⋊C₄×C₁₉	C₂⁴	1	40	yes	—
71	C₄⋊C₄×C₁₉	C₂⁴	1	80	yes	—
72	C₂²⋊C₄×C₁₉	C₂⁴	1	80	yes	—
73	C₂²⋊C₄×C₁₉	C₂⁴	1	80	yes	—
74	C₄²×C₁₉	C₂⁴	1	40	yes	—
75	C₄²×C₁₉	C₂⁴	1	20	yes	—
76	C₄²×C₁₉	C₂⁴	1	20	yes	—
77	C₂×#3531	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₁₉	—	2	10	no	—
79	C₂³×C₁₉	—	2	20	no	—
80	C₂³×C₁₉	C₂⁵	1	10	yes	—
81	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
82	Q₈×C₁₉	—	4	40	no	—
83	D₄×C₁₉	—	4	40	no	—
84	C₂×C₄×C₁₉	—	2	10	no	—
85	C₂×C₄×C₁₉	—	2	20	no	—
86	C₂×C₄×C₁₉	—	2	20	no	—
87	C₂×C₄×C₁₉	—	2	40	no	—
88	C₂×C₄×C₁₉	—	4	40	no	—
89	C₂×C₄×C₁₉	2₋¹⁺⁴	1	4	yes	—
90	C₂×C₄×C₁₉	C₂⁵	1	10	yes	—
91	C₂×C₄×C₁₉	C₂⁵	1	20	yes	—
92	C₂×C₄×C₁₉	C₂⁵	1	20	yes	—
93	C₂×C₄×C₁₉	C₂⁵	1	2	yes	—
94	#3531	C₂×C₁₉	1	32	yes	direct product factor
95	C₂×#564	C₂×C₁₉	1	1	yes	characteristic, complemented
96	C₂×#563	C₂×C₁₉	1	10	yes	complemented
97	C₂×#562	C₂×C₁₉	1	5	yes	complemented
98	C₂×#533	C₂×C₁₉	1	5	yes	complemented
99	C₂×#529	C₂×C₁₉	1	10	yes	complemented
100	C₂²×C₁₉	—	2	10	no	—
101	C₂²×C₁₉	—	2	20	no	—
102	C₂²×C₁₉	—	4	20	no	—
103	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	2	yes	—
104	C₂²×C₁₉	#585	1	4	yes	—
105	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
106	C₄×C₁₉	—	2	10	no	—
107	C₄×C₁₉	—	2	20	no	—
108	C₄×C₁₉	—	2	20	no	—
109	C₄×C₁₉	C₂×2₋¹⁺⁴	1	4	yes	—
110	#564	C₂²×C₁₉	1	16	yes	complemented
111	#563	C₂²×C₁₉	1	160	yes	complemented
112	#562	C₂²×C₁₉	1	80	yes	complemented
113	#533	C₂²×C₁₉	1	80	yes	complemented
114	C₂×C₄⋊Q₈	C₂²×C₁₉	1	10	yes	—
115	#529	C₂²×C₁₉	1	160	yes	complemented
116	C₂×#126	C₂²×C₁₉	1	20	yes	—
117	C₂×#125	C₂²×C₁₉	1	10	yes	—
118	C₂×#125	C₂²×C₁₉	1	5	yes	—
119	C₂×#125	C₂²×C₁₉	1	5	yes	—
120	C₂×#124	C₂²×C₁₉	1	10	yes	—
121	C₂×#123	C₂²×C₁₉	1	20	yes	—
122	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	10	yes	—
123	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	20	yes	—
124	C₂×#121	C₂²×C₁₉	1	10	yes	—
125	C₂×C₄×Q₈	C₂²×C₁₉	1	10	yes	—
126	C₂×C₄×D₄	C₂²×C₁₉	1	10	yes	—
127	C₂×#117	C₂²×C₁₉	1	10	yes	—
128	C₂²×C₄²	C₂²×C₁₉	1	5	yes	—
129	C₂×C₁₉	—	4	10	no	—
130	C₂×C₁₉	#3531	1	4	yes	direct product factor
131	C₂×C₁₉	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
132	C₂×C₁₉	C₂×#585	1	2	yes	—
133	C₂²×Q₈	C₂³×C₁₉	1	10	yes	—
134	C₂²×D₄	C₂³×C₁₉	1	10	yes	—
135	C₂³×C₄	C₂³×C₁₉	1	5	yes	—
136	C₂³×C₄	C₂³×C₁₉	1	10	yes	—
137	C₄⋊Q₈	C₂³×C₁₉	1	80	yes	—
138	#126	C₂³×C₁₉	1	160	yes	—
139	#125	C₂³×C₁₉	1	80	yes	—
140	#125	C₂³×C₁₉	1	40	yes	—
141	#125	C₂³×C₁₉	1	40	yes	—
142	#124	C₂³×C₁₉	1	80	yes	—
143	#123	C₂³×C₁₉	1	160	yes	—
144	C₂²⋊Q₈	C₂³×C₁₉	1	80	yes	—
145	C₂²⋊Q₈	C₂³×C₁₉	1	160	yes	—
146	#121	C₂³×C₁₉	1	80	yes	—
147	C₄×Q₈	C₂³×C₁₉	1	80	yes	—
148	C₄×D₄	C₂³×C₁₉	1	80	yes	—
149	#117	C₂³×C₁₉	1	80	yes	—
150	C₂×C₄⋊C₄	C₂³×C₁₉	1	10	yes	—
151	C₂×C₄⋊C₄	C₂³×C₁₉	1	10	yes	—
152	C₂×C₄⋊C₄	C₂³×C₁₉	1	10	yes	—
153	C₂×C₄⋊C₄	C₂³×C₁₉	1	10	yes	—
154	C₂×C₄⋊C₄	C₂³×C₁₉	1	20	yes	—
155	C₂×C₂²⋊C₄	C₂³×C₁₉	1	20	yes	—
156	C₂×C₂²⋊C₄	C₂³×C₁₉	1	20	yes	—
157	C₂×C₄²	C₂³×C₁₉	1	10	yes	—
158	C₂×C₄²	C₂³×C₁₉	1	5	yes	—
159	C₂×C₄²	C₂³×C₁₉	1	5	yes	—
160	C₂×C₄²	C₂³×C₁₉	1	40	yes	—
161	C₁₉	C₂×#3531	1	1	yes	characteristic, direct product factor, Sylow
162	C₂⁴	C₂⁴×C₁₉	1	5	yes	—
163	C₂×Q₈	—	4	20	no	—
164	C₂×Q₈	C₂⁴×C₁₉	1	40	yes	—
165	C₂×D₄	—	4	20	no	—
166	C₂×D₄	C₂⁴×C₁₉	1	40	yes	—
167	C₂²×C₄	—	2	20	no	—
168	C₂²×C₄	—	4	20	no	—
169	C₂²×C₄	C₂⁴×C₁₉	1	5	yes	—
170	C₂²×C₄	C₂⁴×C₁₉	1	10	yes	—
171	C₂²×C₄	C₂⁴×C₁₉	1	10	yes	—
172	C₂²×C₄	C₂⁴×C₁₉	1	20	yes	—
173	C₂²×C₄	C₂⁴×C₁₉	1	40	yes	—
174	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
175	C₄⋊C₄	C₂⁴×C₁₉	1	40	yes	—
176	C₄⋊C₄	C₂⁴×C₁₉	1	40	yes	—
177	C₄⋊C₄	C₂⁴×C₁₉	1	40	yes	—
178	C₄⋊C₄	C₂⁴×C₁₉	1	40	yes	—
179	C₄⋊C₄	C₂⁴×C₁₉	1	80	yes	—
180	C₂²⋊C₄	C₂⁴×C₁₉	1	80	yes	—
181	C₂²⋊C₄	C₂⁴×C₁₉	1	80	yes	—
182	C₄²	C₂⁴×C₁₉	1	40	yes	—
183	C₄²	C₂⁴×C₁₉	1	20	yes	—
184	C₄²	C₂⁴×C₁₉	1	20	yes	—
185	C₂³	—	2	10	no	—
186	C₂³	—	2	20	no	—
187	C₂³	C₂⁵×C₁₉	1	10	yes	—
188	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
189	Q₈	—	4	40	no	—
190	D₄	—	4	40	no	—
191	C₂×C₄	—	2	10	no	—
192	C₂×C₄	—	2	20	no	—
193	C₂×C₄	—	2	20	no	—
194	C₂×C₄	—	2	40	no	—
195	C₂×C₄	—	4	40	no	—
196	C₂×C₄	C₁₉×2₋¹⁺⁴	1	4	yes	—
197	C₂×C₄	C₂⁵×C₁₉	1	10	yes	—
198	C₂×C₄	C₂⁵×C₁₉	1	20	yes	—
199	C₂×C₄	C₂⁵×C₁₉	1	20	yes	—
200	C₂×C₄	C₂⁵×C₁₉	1	2	yes	—
201	C₂²	—	2	10	no	—
202	C₂²	—	2	20	no	—
203	C₂²	—	4	20	no	—
204	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	2	yes	—
205	C₂²	C₁₉×#585	1	4	yes	—
206	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
207	C₄	—	2	10	no	—
208	C₄	—	2	20	no	—
209	C₄	—	2	20	no	—
210	C₄	C₂×C₁₉×2₋¹⁺⁴	1	4	yes	—
211	C₂	—	4	10	no	—
212	C₂	C₁₉×#3531	1	4	yes	direct product factor
213	C₂	C₂²×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
214	C₂	C₂×C₁₉×#585	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₉
socle	C₂³×C₁₉	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#3531	C₂	direct (N × H)
2. C₂×C₁₉×#564	C₂	semidirect
3. C₂×C₁₉×#563	C₂	semidirect
4. C₂×C₁₉×#562	C₂	semidirect
5. C₂×C₁₉×#533	C₂	semidirect
6. C₂×C₁₉×#529	C₂	semidirect
7. C₁₉×#564	C₂²	semidirect
8. C₁₉×#563	C₂²	semidirect
9. C₁₉×#562	C₂²	semidirect
10. C₁₉×#533	C₂²	semidirect
12. C₁₉×#529	C₂²	semidirect
77. C₂×#3531	C₁₉	direct (N × H)
94. #3531	C₂×C₁₉	direct (N × H)
95. C₂×#564	C₂×C₁₉	semidirect
96. C₂×#563	C₂×C₁₉	semidirect
97. C₂×#562	C₂×C₁₉	semidirect
98. C₂×#533	C₂×C₁₉	semidirect
99. C₂×#529	C₂×C₁₉	semidirect
110. #564	C₂²×C₁₉	semidirect
111. #563	C₂²×C₁₉	semidirect
112. #562	C₂²×C₁₉	semidirect
113. #533	C₂²×C₁₉	semidirect
115. #529	C₂²×C₁₉	semidirect
130. C₂×C₁₉	#3531	direct (N × H)
161. C₁₉	C₂×#3531	direct (N × H)
212. C₂	C₁₉×#3531	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₉×#3531	2
C₂×C₁₉×#564	2
C₂×C₁₉×#563	2
C₂×C₁₉×#562	2
C₂×C₁₉×#533	2
C₂×C₁₉×#529	2
C₂×#3531	19

Subgroups — C2×C19×#3531 #18068652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#3531 #18068652