Subgroups — C₂×C₄×C₁₉×#107 #18026407

Order 4864.

1358 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×#107	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₁₉	C₂	1	2	yes	maximal
4	C₄×C₁₉×#107	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₉×#340	C₂	1	2	yes	maximal
6	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
9	C₂²×C₄²×C₁₉	—	2	2	no	—
10	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×#107	C₄	1	16	yes	direct product factor
12	C₂×C₁₉×#107	C₂²	1	8	yes	—
13	C₂×C₄×C₈×C₁₉	C₄	1	8	yes	complemented
14	C₂×C₄×C₈×C₁₉	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄×C₁₉	—	2	4	no	—
16	C₄×C₄⋊C₄×C₁₉	C₂²	1	8	yes	—
17	C₂×C₁₉×#95	—	2	2	no	—
18	C₁₉×#340	C₂²	1	8	yes	—
19	C₂³×C₄×C₁₉	—	2	4	no	—
20	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
21	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	—	2	4	no	—
24	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₉	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₉	C₂³	1	2	yes	—
29	C₂×C₄⋊C₄×C₁₉	—	2	4	no	—
30	C₂×C₄⋊C₄×C₁₉	—	2	8	no	—
31	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	16	yes	—
32	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
33	C₂×C₄²×C₁₉	—	2	8	no	—
34	C₂×C₄²×C₁₉	—	4	8	no	—
35	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₉	D₄	1	2	yes	—
37	C₂×C₄²×C₁₉	Q₈	1	4	yes	—
38	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
39	C₁₉×#107	C₂×C₄	1	32	yes	direct product factor
40	C₄×C₈×C₁₉	C₂×C₄	1	16	yes	complemented
41	C₁₉×#95	—	2	8	no	—
42	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	—	2	4	no	—
44	C₂²×C₄×C₁₉	—	2	16	no	—
45	C₂²×C₄×C₁₉	—	2	8	no	—
46	C₂²×C₄×C₁₉	—	4	8	no	—
47	C₂²×C₄×C₁₉	—	4	8	no	—
48	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₁₉	D₈	1	4	yes	—
51	C₂²×C₄×C₁₉	Q₁₆	1	4	yes	—
52	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₁₉	#41	1	4	yes	—
59	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
60	C₂×C₈×C₁₉	—	2	8	no	—
61	C₂×C₈×C₁₉	—	2	4	no	—
62	C₂×C₈×C₁₉	C₄²	1	16	yes	complemented
63	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₁₉	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₁₉	C₂²×C₄	1	8	yes	—
66	C₄⋊C₄×C₁₉	—	2	16	no	—
67	C₄⋊C₄×C₁₉	—	2	8	no	—
68	C₄⋊C₄×C₁₉	C₂²×C₄	1	16	yes	—
69	C₄²×C₁₉	—	4	16	no	—
70	C₄²×C₁₉	C₄⋊C₄	1	8	yes	—
71	C₄²×C₁₉	C₂²×C₄	1	4	yes	—
72	C₄²×C₁₉	C₂×Q₈	1	4	yes	—
73	C₂×C₄×#107	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
75	C₂³×C₁₉	C₄×D₄	1	4	yes	—
76	C₂³×C₁₉	C₂×D₈	1	2	yes	—
77	C₂³×C₁₉	C₂×Q₁₆	1	2	yes	—
78	C₂³×C₁₉	#135	1	4	yes	—
79	C₂³×C₁₉	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉	—	2	4	no	—
81	C₂×C₄×C₁₉	—	2	8	no	—
82	C₂×C₄×C₁₉	—	2	16	no	—
83	C₂×C₄×C₁₉	—	4	8	no	—
84	C₂×C₄×C₁₉	—	4	8	no	—
85	C₂×C₄×C₁₉	—	4	32	no	—
86	C₂×C₄×C₁₉	#107	1	16	yes	direct product factor
87	C₂×C₄×C₁₉	C₂×C₄²	1	4	yes	—
88	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₁₉	#117	1	4	yes	—
92	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
93	C₂×C₄×C₁₉	C₄×Q₈	1	8	yes	—
94	C₂×C₄×C₁₉	C₂×D₈	1	4	yes	—
95	C₂×C₄×C₁₉	C₂×Q₁₆	1	4	yes	—
96	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂³×C₄	1	2	yes	—
98	C₂×C₄×C₁₉	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂×#41	1	2	yes	—
100	C₈×C₁₉	—	2	4	no	—
101	C₈×C₁₉	C₂×C₄²	1	8	yes	complemented
102	C₂²×#107	C₂×C₁₉	1	2	yes	—
103	C₂²×C₄×C₈	C₂×C₁₉	1	1	yes	characteristic
104	C₂×C₄×C₄⋊C₄	C₂×C₁₉	1	2	yes	—
105	C₄×#107	C₂×C₁₉	1	8	yes	direct product factor
106	C₂×#340	C₂×C₁₉	1	2	yes	—
107	C₂²×C₁₉	C₄×C₄⋊C₄	1	4	yes	—
108	C₂²×C₁₉	C₂×#107	1	4	yes	—
109	C₂²×C₁₉	#427	1	4	yes	—
110	C₂²×C₁₉	C₄×D₈	1	8	yes	—
111	C₂²×C₁₉	C₄×Q₁₆	1	8	yes	—
112	C₂²×C₁₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₁₉	C₂×C₄×D₄	1	2	yes	—
114	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
115	C₂²×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
116	C₂²×C₁₉	C₂×#135	1	2	yes	—
117	C₄×C₁₉	—	2	4	no	—
118	C₄×C₁₉	—	4	16	no	—
119	C₄×C₁₉	C₄×C₄⋊C₄	1	4	yes	—
120	C₄×C₁₉	C₂×#107	1	8	yes	direct product factor
121	C₄×C₁₉	C₂²×C₄²	1	1	yes	characteristic
122	C₄×C₁₉	C₂×#117	1	1	yes	characteristic
123	C₄×C₁₉	C₂×C₄×Q₈	1	2	yes	—
124	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
125	C₂³×C₈	C₂²×C₁₉	1	1	yes	characteristic
126	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
127	C₂²×C₄²	—	2	2	no	—
128	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
129	C₂×#107	C₄×C₁₉	1	16	yes	direct product factor
130	C₂×#107	C₂²×C₁₉	1	8	yes	—
131	C₂×C₄×C₈	C₄×C₁₉	1	8	yes	complemented
132	C₂×C₄×C₈	C₂²×C₁₉	1	4	yes	—
133	C₄×C₄⋊C₄	—	2	4	no	—
134	C₄×C₄⋊C₄	C₂²×C₁₉	1	8	yes	—
135	C₂×#95	—	2	2	no	—
136	#340	C₂²×C₁₉	1	8	yes	—
137	C₂×C₁₉	C₄×#107	1	8	yes	direct product factor
138	C₂×C₁₉	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₁₉	C₂²×#107	1	1	yes	characteristic
140	C₂×C₁₉	C₂×#427	1	1	yes	characteristic
141	C₂×C₁₉	C₂×C₄×D₈	1	2	yes	—
142	C₂×C₁₉	C₂×C₄×Q₁₆	1	2	yes	—
143	C₂³×C₄	—	2	4	no	—
144	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
145	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
146	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
149	C₂²×C₈	C₂×C₄×C₁₉	1	4	yes	—
150	C₂²×C₈	C₂×C₄×C₁₉	1	8	yes	—
151	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
152	C₂²×C₈	C₂³×C₁₉	1	2	yes	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	—	2	8	no	—
155	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	16	yes	—
156	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
157	C₂×C₄²	—	2	8	no	—
158	C₂×C₄²	—	4	8	no	—
159	C₂×C₄²	C₂×C₄×C₁₉	1	4	yes	—
160	C₂×C₄²	D₄×C₁₉	1	2	yes	—
161	C₂×C₄²	Q₈×C₁₉	1	4	yes	—
162	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
163	#107	C₂×C₄×C₁₉	1	32	yes	direct product factor
164	C₄×C₈	C₂×C₄×C₁₉	1	16	yes	complemented
165	#95	—	2	8	no	—
166	C₁₉	C₂×C₄×#107	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	C₂×D₄×C₁₉	1	1	yes	characteristic
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	2	16	no	—
170	C₂²×C₄	—	2	8	no	—
171	C₂²×C₄	—	4	8	no	—
172	C₂²×C₄	—	4	8	no	—
173	C₂²×C₄	C₄⋊C₄×C₁₉	1	4	yes	—
174	C₂²×C₄	C₄⋊C₄×C₁₉	1	4	yes	—
175	C₂²×C₄	D₈×C₁₉	1	4	yes	—
176	C₂²×C₄	Q₁₆×C₁₉	1	4	yes	—
177	C₂²×C₄	C₂²×C₄×C₁₉	1	2	yes	—
178	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
179	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
180	C₂²×C₄	C₂×D₄×C₁₉	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₁₉	1	2	yes	—
182	C₂²×C₄	C₂×Q₈×C₁₉	1	2	yes	—
183	C₂²×C₄	#41×C₁₉	1	4	yes	—
184	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
185	C₂×C₈	—	2	8	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	C₄²×C₁₉	1	16	yes	complemented
188	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
189	C₂×C₈	C₂²×C₄×C₁₉	1	4	yes	—
190	C₂×C₈	C₂²×C₄×C₁₉	1	8	yes	—
191	C₄⋊C₄	—	2	16	no	—
192	C₄⋊C₄	—	2	8	no	—
193	C₄⋊C₄	C₂²×C₄×C₁₉	1	16	yes	—
194	C₄²	—	4	16	no	—
195	C₄²	C₄⋊C₄×C₁₉	1	8	yes	—
196	C₄²	C₂²×C₄×C₁₉	1	4	yes	—
197	C₄²	C₂×Q₈×C₁₉	1	4	yes	—
198	C₂³	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
199	C₂³	C₄×D₄×C₁₉	1	4	yes	—
200	C₂³	C₂×D₈×C₁₉	1	2	yes	—
201	C₂³	C₂×Q₁₆×C₁₉	1	2	yes	—
202	C₂³	C₁₉×#135	1	4	yes	—
203	C₂³	C₂²×D₄×C₁₉	1	1	yes	characteristic
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	16	no	—
207	C₂×C₄	—	4	32	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	—	4	8	no	—
210	C₂×C₄	C₁₉×#107	1	16	yes	direct product factor
211	C₂×C₄	C₂×C₄²×C₁₉	1	4	yes	—
212	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
213	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	2	yes	—
214	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	4	yes	—
215	C₂×C₄	C₁₉×#117	1	4	yes	—
216	C₂×C₄	C₄×D₄×C₁₉	1	4	yes	—
217	C₂×C₄	C₄×Q₈×C₁₉	1	8	yes	—
218	C₂×C₄	C₂×D₈×C₁₉	1	4	yes	—
219	C₂×C₄	C₂×Q₁₆×C₁₉	1	4	yes	—
220	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
221	C₂×C₄	C₂³×C₄×C₁₉	1	2	yes	—
222	C₂×C₄	C₂²×Q₈×C₁₉	1	1	yes	characteristic
223	C₂×C₄	C₂×#41×C₁₉	1	2	yes	—
224	C₈	—	2	4	no	—
225	C₈	C₂×C₄²×C₁₉	1	8	yes	complemented
226	C₂²	C₄×C₄⋊C₄×C₁₉	1	4	yes	—
227	C₂²	C₂×C₁₉×#107	1	4	yes	—
228	C₂²	C₁₉×#427	1	4	yes	—
229	C₂²	C₄×D₈×C₁₉	1	8	yes	—
230	C₂²	C₄×Q₁₆×C₁₉	1	8	yes	—
231	C₂²	C₂²×C₄⋊C₄×C₁₉	1	1	yes	characteristic
232	C₂²	C₂×C₄×D₄×C₁₉	1	2	yes	—
233	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
234	C₂²	C₂²×Q₁₆×C₁₉	1	1	yes	characteristic
235	C₂²	C₂×C₁₉×#135	1	2	yes	—
236	C₄	—	2	4	no	—
237	C₄	—	4	16	no	—
238	C₄	C₄×C₄⋊C₄×C₁₉	1	4	yes	—
239	C₄	C₂×C₁₉×#107	1	8	yes	direct product factor
240	C₄	C₂²×C₄²×C₁₉	1	1	yes	characteristic
241	C₄	C₂×C₁₉×#117	1	1	yes	characteristic
242	C₄	C₂×C₄×Q₈×C₁₉	1	2	yes	—
243	C₂	C₄×C₁₉×#107	1	8	yes	direct product factor
244	C₂	C₂×C₄×C₄⋊C₄×C₁₉	1	1	yes	characteristic
245	C₂	C₂²×C₁₉×#107	1	1	yes	characteristic
246	C₂	C₂×C₁₉×#427	1	1	yes	characteristic
247	C₂	C₂×C₄×D₈×C₁₉	1	2	yes	—
248	C₂	C₂×C₄×Q₁₆×C₁₉	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₉	D₄
commutator	C₄	C₂²×C₄²×C₁₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle, prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×C₁₉×#107	C₂	direct (N × H)
11. C₂×C₁₉×#107	C₄	direct (N × H)
13. C₂×C₄×C₈×C₁₉	C₄	semidirect
39. C₁₉×#107	C₂×C₄	direct (N × H)
40. C₄×C₈×C₁₉	C₂×C₄	semidirect
62. C₂×C₈×C₁₉	C₄²	semidirect
73. C₂×C₄×#107	C₁₉	direct (N × H)
86. C₂×C₄×C₁₉	#107	direct (N × H)
101. C₈×C₁₉	C₂×C₄²	semidirect
105. C₄×#107	C₂×C₁₉	direct (N × H)
120. C₄×C₁₉	C₂×#107	direct (N × H)
129. C₂×#107	C₄×C₁₉	direct (N × H)
131. C₂×C₄×C₈	C₄×C₁₉	semidirect
137. C₂×C₁₉	C₄×#107	direct (N × H)
163. #107	C₂×C₄×C₁₉	direct (N × H)
164. C₄×C₈	C₂×C₄×C₁₉	semidirect
166. C₁₉	C₂×C₄×#107	direct (N × H)
187. C₂×C₈	C₄²×C₁₉	semidirect
210. C₂×C₄	C₁₉×#107	direct (N × H)
225. C₈	C₂×C₄²×C₁₉	semidirect
239. C₄	C₂×C₁₉×#107	direct (N × H)
243. C₂	C₄×C₁₉×#107	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×#107	2
C₂²×C₄×C₈×C₁₉	2
C₂×C₄×C₄⋊C₄×C₁₉	2
C₄×C₁₉×#107	2
C₂×C₁₉×#340	2
C₂×C₄×#107	19

Subgroups — C2×C4×C19×#107 #18026407

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₁₉×#107 #18026407