Subgroups — C₁₉×#62848 #18068807

Order 4864.

10962 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×2₋¹⁺⁴	C₂	1	2	yes	maximal, complemented
2	C₁₉×#3497	C₂	1	16	yes	maximal, complemented
3	C₁₉×#3487	C₂	1	16	yes	maximal, complemented
4	C₁₉×#3486	C₂	1	16	yes	maximal, complemented
5	C₁₉×#3449	C₂	1	12	yes	maximal, complemented
6	C₂×C₁₉×#519	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×2₋¹⁺⁴	—	2	32	no	—
8	C₂×C₁₉×2₋¹⁺⁴	C₂²	1	16	yes	—
9	C₂×C₁₉×2₋¹⁺⁴	C₂²	1	16	yes	complemented
10	C₂²×#41×C₁₉	C₂²	1	12	yes	—
11	C₂²×#41×C₁₉	C₂²	1	8	yes	complemented
12	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₁₉	C₂²	1	8	yes	complemented
14	C₁₉×#550	C₂²	1	64	yes	complemented
15	D₄×Q₈×C₁₉	C₂²	1	64	yes	complemented
16	C₁₉×#548	C₂²	1	128	yes	complemented
17	C₁₉×#537	C₂²	1	48	yes	complemented
18	C₁₉×#536	C₂²	1	48	yes	complemented
19	C₁₉×#535	C₂²	1	48	yes	complemented
20	C₁₉×#534	C₂²	1	16	yes	—
21	C₁₉×#532	C₂²	1	48	yes	complemented
22	C₁₉×#525	C₂²	1	96	yes	complemented
23	C₁₉×#519	C₂²	1	16	yes	complemented
24	C₂×C₄×Q₈×C₁₉	C₂²	1	8	yes	complemented
25	C₂×C₁₉×#117	C₂²	1	6	yes	—
26	C₁₉×2₋¹⁺⁴	—	2	128	no	—
27	C₁₉×2₋¹⁺⁴	—	2	128	no	—
28	C₂×#41×C₁₉	—	2	64	no	—
29	C₂×#41×C₁₉	—	2	96	no	—
30	C₂×#41×C₁₉	C₂³	1	48	yes	—
31	C₂×#41×C₁₉	C₂³	1	48	yes	—
32	C₂×#41×C₁₉	C₂³	1	64	yes	—
33	C₂²×Q₈×C₁₉	—	2	64	no	—
34	C₂²×Q₈×C₁₉	D₄	1	16	yes	complemented
35	C₂²×Q₈×C₁₉	C₂³	1	12	yes	—
36	C₂²×Q₈×C₁₉	C₂³	1	64	yes	—
37	C₂²×Q₈×C₁₉	C₂³	1	4	yes	—
38	C₂²×Q₈×C₁₉	C₂³	1	8	yes	—
39	C₂²×D₄×C₁₉	C₂³	1	8	yes	—
40	C₂²×D₄×C₁₉	C₂³	1	12	yes	—
41	C₂³×C₄×C₁₉	C₂³	1	24	yes	—
42	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
43	C₁₉×C₄⋊Q₈	C₂³	1	48	yes	—
44	C₁₉×#127	C₂³	1	48	yes	—
45	C₁₉×#124	C₂³	1	96	yes	—
46	C₁₉×#123	C₂³	1	64	yes	—
47	C₁₉×C₂²⋊Q₈	C₂³	1	192	yes	—
48	C₁₉×#121	C₂³	1	192	yes	—
49	C₁₉×#121	C₂³	1	64	yes	—
50	C₁₉×#120	C₂³	1	64	yes	—
51	C₄×Q₈×C₁₉	C₂³	1	64	yes	—
52	C₄×D₄×C₁₉	C₂³	1	192	yes	—
53	C₁₉×#117	C₂³	1	48	yes	—
54	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
55	C₂×C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
56	C₂×C₄²×C₁₉	C₂³	1	12	yes	—
57	C₂⁴×C₁₉	C₂⁴	1	8	yes	—
58	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
59	#41×C₁₉	—	2	256	no	—
60	#41×C₁₉	—	2	192	no	—
61	#41×C₁₉	—	2	192	no	—
62	C₂×Q₈×C₁₉	—	2	256	no	—
63	C₂×Q₈×C₁₉	—	2	48	no	—
64	C₂×Q₈×C₁₉	C₂×D₄	1	64	yes	—
65	C₂×Q₈×C₁₉	C₂×D₄	1	32	yes	—
66	C₂×Q₈×C₁₉	C₂⁴	1	48	yes	—
67	C₂×Q₈×C₁₉	C₂⁴	1	16	yes	—
68	C₂×D₄×C₁₉	—	2	32	no	—
69	C₂×D₄×C₁₉	—	2	48	no	—
70	C₂×D₄×C₁₉	C₂⁴	1	16	yes	—
71	C₂×D₄×C₁₉	C₂⁴	1	16	yes	—
72	C₂×D₄×C₁₉	C₂⁴	1	48	yes	—
73	C₂×D₄×C₁₉	C₂⁴	1	64	yes	—
74	C₂²×C₄×C₁₉	—	2	96	no	—
75	C₂²×C₄×C₁₉	C₂×D₄	1	24	yes	—
76	C₂²×C₄×C₁₉	C₂⁴	1	96	yes	—
77	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
78	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
79	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
80	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
81	C₄⋊C₄×C₁₉	C₂⁴	1	48	yes	—
82	C₄⋊C₄×C₁₉	C₂⁴	1	64	yes	—
83	C₂²⋊C₄×C₁₉	C₂⁴	1	16	yes	—
84	C₂²⋊C₄×C₁₉	C₂⁴	1	128	yes	—
85	C₄²×C₁₉	C₂⁴	1	48	yes	—
86	#62848	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₁₉	—	2	16	no	—
88	C₂³×C₁₉	—	4	2	no	—
89	C₂³×C₁₉	—	4	16	no	—
90	C₂³×C₁₉	C₂²×D₄	1	4	yes	—
91	C₂³×C₁₉	C₂⁵	1	16	yes	—
92	C₂³×C₁₉	C₂⁵	1	2	yes	—
93	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
94	Q₈×C₁₉	—	2	96	no	—
95	Q₈×C₁₉	C₂²×D₄	1	64	yes	—
96	D₄×C₁₉	—	2	32	no	—
97	D₄×C₁₉	—	2	32	no	—
98	D₄×C₁₉	—	2	96	no	—
99	D₄×C₁₉	—	4	64	no	—
100	C₂×C₄×C₁₉	—	2	16	no	—
101	C₂×C₄×C₁₉	—	2	192	no	—
102	C₂×C₄×C₁₉	—	4	8	no	—
103	C₂×C₄×C₁₉	C₂²×D₄	1	48	yes	—
104	C₂×C₄×C₁₉	C₂²×D₄	1	24	yes	—
105	C₂×C₄×C₁₉	C₂⁵	1	16	yes	—
106	C₂×C₄×C₁₉	C₂⁵	1	16	yes	—
107	C₂×C₄×C₁₉	C₂⁵	1	12	yes	—
108	C₂²×2₋¹⁺⁴	C₂×C₁₉	1	2	yes	complemented
109	#3497	C₂×C₁₉	1	16	yes	complemented
110	#3487	C₂×C₁₉	1	16	yes	complemented
111	#3486	C₂×C₁₉	1	16	yes	complemented
112	#3449	C₂×C₁₉	1	12	yes	complemented
113	C₂×#519	C₂×C₁₉	1	1	yes	characteristic, complemented
114	C₂²×C₁₉	—	2	2	no	—
115	C₂²×C₁₉	—	2	4	no	—
116	C₂²×C₁₉	—	2	16	no	—
117	C₂²×C₁₉	—	2	32	no	—
118	C₂²×C₁₉	—	4	4	no	—
119	C₂²×C₁₉	—	4	32	no	—
120	C₂²×C₁₉	C₂³×D₄	1	4	yes	—
121	C₂²×C₁₉	C₂³×D₄	1	2	yes	—
122	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
123	C₄×C₁₉	—	2	16	no	—
124	C₄×C₁₉	—	2	16	no	—
125	C₄×C₁₉	C₂³×D₄	1	24	yes	—
126	C₂×2₋¹⁺⁴	—	2	32	no	—
127	C₂×2₋¹⁺⁴	C₂²×C₁₉	1	16	yes	—
128	C₂×2₋¹⁺⁴	C₂²×C₁₉	1	16	yes	complemented
129	C₂²×#41	C₂²×C₁₉	1	12	yes	—
130	C₂²×#41	C₂²×C₁₉	1	8	yes	complemented
131	C₂³×Q₈	C₂²×C₁₉	1	1	yes	characteristic
132	C₂³×Q₈	C₂²×C₁₉	1	8	yes	complemented
133	#550	C₂²×C₁₉	1	64	yes	complemented
134	D₄×Q₈	C₂²×C₁₉	1	64	yes	complemented
135	#548	C₂²×C₁₉	1	128	yes	complemented
136	#537	C₂²×C₁₉	1	48	yes	complemented
137	#536	C₂²×C₁₉	1	48	yes	complemented
138	#535	C₂²×C₁₉	1	48	yes	complemented
139	#534	C₂²×C₁₉	1	16	yes	—
140	#532	C₂²×C₁₉	1	48	yes	complemented
141	#525	C₂²×C₁₉	1	96	yes	complemented
142	#519	C₂²×C₁₉	1	16	yes	complemented
143	C₂×C₄×Q₈	C₂²×C₁₉	1	8	yes	complemented
144	C₂×#117	C₂²×C₁₉	1	6	yes	—
145	C₂×C₁₉	—	2	4	no	—
146	C₂×C₁₉	—	2	2	no	—
147	C₂×C₁₉	—	4	16	no	—
148	C₂×C₁₉	C₂⁴×D₄	1	1	yes	characteristic
149	C₂×C₁₉	2₋¹⁺⁶	1	2	yes	—
150	2₋¹⁺⁴	—	2	128	no	—
151	2₋¹⁺⁴	—	2	128	no	—
152	C₂×#41	—	2	64	no	—
153	C₂×#41	—	2	96	no	—
154	C₂×#41	C₂³×C₁₉	1	48	yes	—
155	C₂×#41	C₂³×C₁₉	1	48	yes	—
156	C₂×#41	C₂³×C₁₉	1	64	yes	—
157	C₂²×Q₈	—	2	64	no	—
158	C₂²×Q₈	D₄×C₁₉	1	16	yes	complemented
159	C₂²×Q₈	C₂³×C₁₉	1	12	yes	—
160	C₂²×Q₈	C₂³×C₁₉	1	64	yes	—
161	C₂²×Q₈	C₂³×C₁₉	1	4	yes	—
162	C₂²×Q₈	C₂³×C₁₉	1	8	yes	—
163	C₂²×D₄	C₂³×C₁₉	1	8	yes	—
164	C₂²×D₄	C₂³×C₁₉	1	12	yes	—
165	C₂³×C₄	C₂³×C₁₉	1	24	yes	—
166	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
167	C₄⋊Q₈	C₂³×C₁₉	1	48	yes	—
168	#127	C₂³×C₁₉	1	48	yes	—
169	#124	C₂³×C₁₉	1	96	yes	—
170	#123	C₂³×C₁₉	1	64	yes	—
171	C₂²⋊Q₈	C₂³×C₁₉	1	192	yes	—
172	#121	C₂³×C₁₉	1	192	yes	—
173	#121	C₂³×C₁₉	1	64	yes	—
174	#120	C₂³×C₁₉	1	64	yes	—
175	C₄×Q₈	C₂³×C₁₉	1	64	yes	—
176	C₄×D₄	C₂³×C₁₉	1	192	yes	—
177	#117	C₂³×C₁₉	1	48	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
179	C₂×C₂²⋊C₄	C₂³×C₁₉	1	4	yes	—
180	C₂×C₄²	C₂³×C₁₉	1	12	yes	—
181	C₁₉	#62848	1	1	yes	characteristic, direct product factor, Sylow
182	C₂⁴	C₂⁴×C₁₉	1	8	yes	—
183	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
184	#41	—	2	256	no	—
185	#41	—	2	192	no	—
186	#41	—	2	192	no	—
187	C₂×Q₈	—	2	256	no	—
188	C₂×Q₈	—	2	48	no	—
189	C₂×Q₈	C₂×D₄×C₁₉	1	64	yes	—
190	C₂×Q₈	C₂×D₄×C₁₉	1	32	yes	—
191	C₂×Q₈	C₂⁴×C₁₉	1	48	yes	—
192	C₂×Q₈	C₂⁴×C₁₉	1	16	yes	—
193	C₂×D₄	—	2	32	no	—
194	C₂×D₄	—	2	48	no	—
195	C₂×D₄	C₂⁴×C₁₉	1	16	yes	—
196	C₂×D₄	C₂⁴×C₁₉	1	16	yes	—
197	C₂×D₄	C₂⁴×C₁₉	1	48	yes	—
198	C₂×D₄	C₂⁴×C₁₉	1	64	yes	—
199	C₂²×C₄	—	2	96	no	—
200	C₂²×C₄	C₂×D₄×C₁₉	1	24	yes	—
201	C₂²×C₄	C₂⁴×C₁₉	1	96	yes	—
202	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
203	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
204	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
205	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
206	C₄⋊C₄	C₂⁴×C₁₉	1	48	yes	—
207	C₄⋊C₄	C₂⁴×C₁₉	1	64	yes	—
208	C₂²⋊C₄	C₂⁴×C₁₉	1	16	yes	—
209	C₂²⋊C₄	C₂⁴×C₁₉	1	128	yes	—
210	C₄²	C₂⁴×C₁₉	1	48	yes	—
211	C₂³	—	2	16	no	—
212	C₂³	—	4	2	no	—
213	C₂³	—	4	16	no	—
214	C₂³	C₂²×D₄×C₁₉	1	4	yes	—
215	C₂³	C₂⁵×C₁₉	1	16	yes	—
216	C₂³	C₂⁵×C₁₉	1	2	yes	—
217	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
218	Q₈	—	2	96	no	—
219	Q₈	C₂²×D₄×C₁₉	1	64	yes	—
220	D₄	—	2	32	no	—
221	D₄	—	2	32	no	—
222	D₄	—	2	96	no	—
223	D₄	—	4	64	no	—
224	C₂×C₄	—	2	16	no	—
225	C₂×C₄	—	2	192	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	C₂²×D₄×C₁₉	1	48	yes	—
228	C₂×C₄	C₂²×D₄×C₁₉	1	24	yes	—
229	C₂×C₄	C₂⁵×C₁₉	1	16	yes	—
230	C₂×C₄	C₂⁵×C₁₉	1	16	yes	—
231	C₂×C₄	C₂⁵×C₁₉	1	12	yes	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	4	no	—
234	C₂²	—	2	16	no	—
235	C₂²	—	2	32	no	—
236	C₂²	—	4	4	no	—
237	C₂²	—	4	32	no	—
238	C₂²	C₂³×D₄×C₁₉	1	4	yes	—
239	C₂²	C₂³×D₄×C₁₉	1	2	yes	—
240	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
241	C₄	—	2	16	no	—
242	C₄	—	2	16	no	—
243	C₄	C₂³×D₄×C₁₉	1	24	yes	—
244	C₂	—	2	4	no	—
245	C₂	—	2	2	no	—
246	C₂	—	4	16	no	—
247	C₂	C₂⁴×D₄×C₁₉	1	1	yes	characteristic
248	C₂	C₁₉×2₋¹⁺⁶	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₉×2₋¹⁺⁴	C₂	semidirect
2. C₁₉×#3497	C₂	semidirect
3. C₁₉×#3487	C₂	semidirect
4. C₁₉×#3486	C₂	semidirect
5. C₁₉×#3449	C₂	semidirect
6. C₂×C₁₉×#519	C₂	semidirect
9. C₂×C₁₉×2₋¹⁺⁴	C₂²	semidirect
11. C₂²×#41×C₁₉	C₂²	semidirect
13. C₂³×Q₈×C₁₉	C₂²	semidirect
14. C₁₉×#550	C₂²	semidirect
15. D₄×Q₈×C₁₉	C₂²	semidirect
16. C₁₉×#548	C₂²	semidirect
17. C₁₉×#537	C₂²	semidirect
18. C₁₉×#536	C₂²	semidirect
19. C₁₉×#535	C₂²	semidirect
21. C₁₉×#532	C₂²	semidirect
22. C₁₉×#525	C₂²	semidirect
23. C₁₉×#519	C₂²	semidirect
24. C₂×C₄×Q₈×C₁₉	C₂²	semidirect
34. C₂²×Q₈×C₁₉	D₄	semidirect
86. #62848	C₁₉	direct (N × H)
108. C₂²×2₋¹⁺⁴	C₂×C₁₉	semidirect
109. #3497	C₂×C₁₉	semidirect
110. #3487	C₂×C₁₉	semidirect
111. #3486	C₂×C₁₉	semidirect
112. #3449	C₂×C₁₉	semidirect
113. C₂×#519	C₂×C₁₉	semidirect
128. C₂×2₋¹⁺⁴	C₂²×C₁₉	semidirect
130. C₂²×#41	C₂²×C₁₉	semidirect
132. C₂³×Q₈	C₂²×C₁₉	semidirect
133. #550	C₂²×C₁₉	semidirect
134. D₄×Q₈	C₂²×C₁₉	semidirect
135. #548	C₂²×C₁₉	semidirect
136. #537	C₂²×C₁₉	semidirect
137. #536	C₂²×C₁₉	semidirect
138. #535	C₂²×C₁₉	semidirect
140. #532	C₂²×C₁₉	semidirect
141. #525	C₂²×C₁₉	semidirect
142. #519	C₂²×C₁₉	semidirect
143. C₂×C₄×Q₈	C₂²×C₁₉	semidirect
158. C₂²×Q₈	D₄×C₁₉	semidirect
181. C₁₉	#62848	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×C₁₉×2₋¹⁺⁴	2
C₁₉×#3497	2
C₁₉×#3487	2
C₁₉×#3486	2
C₁₉×#3449	2
C₂×C₁₉×#519	2
#62848	19

Subgroups — C19×#62848 #18068807

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#62848 #18068807