Subgroups — C₁₉×#60648 #18066607

Order 4864.

3114 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#578	C₂	1	4	yes	maximal, complemented
2	C₁₉×#3499	C₂	1	2	yes	maximal, complemented
3	C₁₉×#3156	C₂	1	12	yes	maximal, complemented
4	C₁₉×#3152	C₂	1	6	yes	maximal, complemented
5	C₁₉×#3147	C₂	1	3	yes	maximal, complemented
6	C₂×C₁₉×#494	C₂	1	3	yes	maximal, complemented
7	C₄×C₁₉×#131	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₉×2₋¹⁺⁴	C₂²	1	4	yes	—
9	C₁₉×#578	—	2	32	no	—
10	C₂×C₁₉×#137	C₂²	1	24	yes	—
11	C₂×C₁₉×#135	C₂²	1	12	yes	—
12	C₂²×Q₁₆×C₁₉	C₂²	1	12	yes	—
13	C₂×C₁₉×#131	C₂²	1	2	yes	—
14	D₄×Q₈×C₁₉	—	2	4	no	—
15	C₁₉×#547	—	2	12	no	—
16	C₁₉×#536	C₂²	1	12	yes	—
17	C₁₉×#532	C₂²	1	6	yes	—
18	C₂×C₁₉×C₄⋊Q₈	C₂²	1	6	yes	—
19	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#497	C₂²	1	24	yes	—
21	C₁₉×#495	C₂²	1	12	yes	—
22	C₁₉×#494	C₂²	1	3	yes	—
23	C₁₉×#494	C₂²	1	6	yes	—
24	C₁₉×#494	C₂²	1	12	yes	—
25	C₁₉×#493	C₂²	1	1	yes	characteristic
26	C₁₉×#492	C₂²	1	6	yes	—
27	C₁₉×#405	C₂²	1	6	yes	—
28	C₄×M₁₆×C₁₉	C₂²	1	3	yes	—
29	C₂×C₄×C₈×C₁₉	C₂²	1	3	yes	—
30	C₁₉×2₋¹⁺⁴	—	2	32	no	—
31	C₂×#41×C₁₉	—	2	12	no	—
32	C₂×#41×C₁₉	C₂³	1	12	yes	—
33	C₂×#41×C₁₉	C₂³	1	1	yes	characteristic
34	C₂²×Q₈×C₁₉	—	2	4	no	—
35	C₂²×Q₈×C₁₉	C₂³	1	12	yes	—
36	C₁₉×#137	—	2	96	no	—
37	C₁₉×#135	—	2	48	no	—
38	C₂×Q₁₆×C₁₉	—	2	48	no	—
39	C₂×Q₁₆×C₁₉	C₂³	1	12	yes	—
40	C₂×Q₁₆×C₁₉	C₂³	1	24	yes	—
41	C₂×SD₁₆×C₁₉	C₂³	1	24	yes	—
42	C₂×D₈×C₁₉	C₂³	1	4	yes	—
43	C₁₉×#131	D₄	1	16	yes	complemented
44	C₂×M₁₆×C₁₉	C₂³	1	6	yes	—
45	C₂²×C₈×C₁₉	C₂³	1	6	yes	—
46	C₁₉×C₄⋊Q₈	—	2	12	no	—
47	C₁₉×C₄⋊Q₈	C₂³	1	6	yes	—
48	C₁₉×C₄⋊Q₈	C₂³	1	12	yes	—
49	C₁₉×#127	C₂³	1	2	yes	—
50	C₁₉×#124	C₂³	1	12	yes	—
51	C₁₉×#123	—	2	24	no	—
52	C₁₉×C₂²⋊Q₈	—	2	24	no	—
53	C₁₉×#121	—	2	12	no	—
54	C₄×Q₈×C₁₉	D₄	1	2	yes	—
55	C₄×D₄×C₁₉	—	2	12	no	—
56	C₄×D₄×C₁₉	D₄	1	6	yes	—
57	C₁₉×#117	C₂³	1	3	yes	—
58	C₂×C₄⋊C₄×C₁₉	—	2	12	no	—
59	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
60	C₁₉×#97	C₂³	1	6	yes	—
61	C₄×C₈×C₁₉	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₁₉	C₂³	1	3	yes	—
63	C₄×C₈×C₁₉	C₂³	1	6	yes	—
64	#41×C₁₉	—	2	48	no	—
65	#41×C₁₉	—	4	48	no	—
66	#41×C₁₉	C₂×D₄	1	8	yes	—
67	C₂×Q₈×C₁₉	—	2	12	no	—
68	C₂×Q₈×C₁₉	—	2	48	no	—
69	C₂×Q₈×C₁₉	—	4	16	no	—
70	C₂×Q₈×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₁₉	C₂⁴	1	12	yes	—
72	C₂×D₄×C₁₉	—	2	12	no	—
73	C₂×D₄×C₁₉	—	2	4	no	—
74	C₂×D₄×C₁₉	C₂×D₄	1	3	yes	—
75	C₂×D₄×C₁₉	C₂⁴	1	4	yes	—
76	C₂²×C₄×C₁₉	—	2	12	no	—
77	C₂²×C₄×C₁₉	—	2	12	no	—
78	C₂²×C₄×C₁₉	C₂×D₄	1	6	yes	—
79	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
80	Q₁₆×C₁₉	—	2	24	no	—
81	Q₁₆×C₁₉	—	2	48	no	—
82	SD₁₆×C₁₉	—	2	48	no	—
83	D₈×C₁₉	—	2	8	no	—
84	M₁₆×C₁₉	C₂×D₄	1	24	yes	—
85	C₂×C₈×C₁₉	C₂×D₄	1	24	yes	—
86	C₂×C₈×C₁₉	C₂⁴	1	2	yes	—
87	C₂×C₈×C₁₉	C₂⁴	1	6	yes	—
88	C₄⋊C₄×C₁₉	—	2	12	no	—
89	C₄⋊C₄×C₁₉	—	2	24	no	—
90	C₄⋊C₄×C₁₉	C₂×D₄	1	6	yes	—
91	C₂²⋊C₄×C₁₉	—	2	24	no	—
92	C₂²⋊C₄×C₁₉	C₂×D₄	1	6	yes	—
93	C₄²×C₁₉	C₂×D₄	1	6	yes	—
94	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
95	C₄²×C₁₉	C₂⁴	1	3	yes	—
96	#60648	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₁₉	—	2	4	no	—
98	C₂³×C₁₉	C₂²×D₄	1	3	yes	—
99	Q₈×C₁₉	—	2	24	no	—
100	Q₈×C₁₉	—	4	24	no	—
101	Q₈×C₁₉	#127	1	4	yes	—
102	D₄×C₁₉	—	2	8	no	—
103	D₄×C₁₉	—	4	24	no	—
104	D₄×C₁₉	—	4	8	no	—
105	D₄×C₁₉	#127	1	12	yes	—
106	C₂×C₄×C₁₉	—	2	12	no	—
107	C₂×C₄×C₁₉	—	2	12	no	—
108	C₂×C₄×C₁₉	—	4	24	no	—
109	C₂×C₄×C₁₉	—	4	24	no	—
110	C₂×C₄×C₁₉	C₂²×D₄	1	6	yes	—
111	C₂×C₄×C₁₉	C₂²×D₄	1	3	yes	—
112	C₂×C₄×C₁₉	C₂²×D₄	1	12	yes	—
113	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
114	C₂×C₄×C₁₉	C₂⁵	1	1	yes	characteristic
115	C₈×C₁₉	C₂²×D₄	1	4	yes	—
116	C₈×C₁₉	C₂²×D₄	1	12	yes	—
117	C₂×#578	C₂×C₁₉	1	4	yes	complemented
118	#3499	C₂×C₁₉	1	2	yes	complemented
119	#3156	C₂×C₁₉	1	12	yes	complemented
120	#3152	C₂×C₁₉	1	6	yes	complemented
121	#3147	C₂×C₁₉	1	3	yes	complemented
122	C₂×#494	C₂×C₁₉	1	3	yes	complemented
123	C₄×#131	C₂×C₁₉	1	1	yes	characteristic, complemented
124	C₂²×C₁₉	—	2	6	no	—
125	C₂²×C₁₉	—	4	4	no	—
126	C₂²×C₁₉	—	4	8	no	—
127	C₂²×C₁₉	C₂×#127	1	6	yes	—
128	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
129	C₄×C₁₉	—	2	6	no	—
130	C₄×C₁₉	—	4	12	no	—
131	C₄×C₁₉	C₂×#127	1	6	yes	—
132	C₄×C₁₉	#578	1	4	yes	complemented
133	C₄×C₁₉	C₂³×D₄	1	2	yes	—
134	C₂×2₋¹⁺⁴	C₂²×C₁₉	1	4	yes	—
135	#578	—	2	32	no	—
136	C₂×#137	C₂²×C₁₉	1	24	yes	—
137	C₂×#135	C₂²×C₁₉	1	12	yes	—
138	C₂²×Q₁₆	C₂²×C₁₉	1	12	yes	—
139	C₂×#131	C₂²×C₁₉	1	2	yes	—
140	D₄×Q₈	—	2	4	no	—
141	#547	—	2	12	no	—
142	#536	C₂²×C₁₉	1	12	yes	—
143	#532	C₂²×C₁₉	1	6	yes	—
144	C₂×C₄⋊Q₈	C₂²×C₁₉	1	6	yes	—
145	C₄×#41	C₂²×C₁₉	1	1	yes	characteristic
146	#497	C₂²×C₁₉	1	24	yes	—
147	#495	C₂²×C₁₉	1	12	yes	—
148	#494	C₂²×C₁₉	1	3	yes	—
149	#494	C₂²×C₁₉	1	6	yes	—
150	#494	C₂²×C₁₉	1	12	yes	—
151	#493	C₂²×C₁₉	1	1	yes	characteristic
152	#492	C₂²×C₁₉	1	6	yes	—
153	#405	C₂²×C₁₉	1	6	yes	—
154	C₄×M₁₆	C₂²×C₁₉	1	3	yes	—
155	C₂×C₄×C₈	C₂²×C₁₉	1	3	yes	—
156	C₂×C₁₉	—	2	6	no	—
157	C₂×C₁₉	—	8	4	no	—
158	C₂×C₁₉	C₂²×#127	1	1	yes	characteristic
159	C₂×C₁₉	C₂×#578	1	2	yes	—
160	2₋¹⁺⁴	—	2	32	no	—
161	C₂×#41	—	2	12	no	—
162	C₂×#41	C₂³×C₁₉	1	12	yes	—
163	C₂×#41	C₂³×C₁₉	1	1	yes	characteristic
164	C₂²×Q₈	—	2	4	no	—
165	C₂²×Q₈	C₂³×C₁₉	1	12	yes	—
166	#137	—	2	96	no	—
167	#135	—	2	48	no	—
168	C₂×Q₁₆	—	2	48	no	—
169	C₂×Q₁₆	C₂³×C₁₉	1	12	yes	—
170	C₂×Q₁₆	C₂³×C₁₉	1	24	yes	—
171	C₂×SD₁₆	C₂³×C₁₉	1	24	yes	—
172	C₂×D₈	C₂³×C₁₉	1	4	yes	—
173	#131	D₄×C₁₉	1	16	yes	complemented
174	C₂×M₁₆	C₂³×C₁₉	1	6	yes	—
175	C₂²×C₈	C₂³×C₁₉	1	6	yes	—
176	C₄⋊Q₈	—	2	12	no	—
177	C₄⋊Q₈	C₂³×C₁₉	1	6	yes	—
178	C₄⋊Q₈	C₂³×C₁₉	1	12	yes	—
179	#127	C₂³×C₁₉	1	2	yes	—
180	#124	C₂³×C₁₉	1	12	yes	—
181	#123	—	2	24	no	—
182	C₂²⋊Q₈	—	2	24	no	—
183	#121	—	2	12	no	—
184	C₄×Q₈	D₄×C₁₉	1	2	yes	—
185	C₄×D₄	—	2	12	no	—
186	C₄×D₄	D₄×C₁₉	1	6	yes	—
187	#117	C₂³×C₁₉	1	3	yes	—
188	C₂×C₄⋊C₄	—	2	12	no	—
189	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
190	#97	C₂³×C₁₉	1	6	yes	—
191	C₄×C₈	C₂³×C₁₉	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₁₉	1	3	yes	—
193	C₄×C₈	C₂³×C₁₉	1	6	yes	—
194	C₁₉	#60648	1	1	yes	characteristic, direct product factor, Sylow
195	#41	—	2	48	no	—
196	#41	—	4	48	no	—
197	#41	C₂×D₄×C₁₉	1	8	yes	—
198	C₂×Q₈	—	2	12	no	—
199	C₂×Q₈	—	2	48	no	—
200	C₂×Q₈	—	4	16	no	—
201	C₂×Q₈	C₂×D₄×C₁₉	1	1	yes	characteristic
202	C₂×Q₈	C₂⁴×C₁₉	1	12	yes	—
203	C₂×D₄	—	2	12	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	C₂×D₄×C₁₉	1	3	yes	—
206	C₂×D₄	C₂⁴×C₁₉	1	4	yes	—
207	C₂²×C₄	—	2	12	no	—
208	C₂²×C₄	—	2	12	no	—
209	C₂²×C₄	C₂×D₄×C₁₉	1	6	yes	—
210	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
211	Q₁₆	—	2	24	no	—
212	Q₁₆	—	2	48	no	—
213	SD₁₆	—	2	48	no	—
214	D₈	—	2	8	no	—
215	M₁₆	C₂×D₄×C₁₉	1	24	yes	—
216	C₂×C₈	C₂×D₄×C₁₉	1	24	yes	—
217	C₂×C₈	C₂⁴×C₁₉	1	2	yes	—
218	C₂×C₈	C₂⁴×C₁₉	1	6	yes	—
219	C₄⋊C₄	—	2	12	no	—
220	C₄⋊C₄	—	2	24	no	—
221	C₄⋊C₄	C₂×D₄×C₁₉	1	6	yes	—
222	C₂²⋊C₄	—	2	24	no	—
223	C₂²⋊C₄	C₂×D₄×C₁₉	1	6	yes	—
224	C₄²	C₂×D₄×C₁₉	1	6	yes	—
225	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
226	C₄²	C₂⁴×C₁₉	1	3	yes	—
227	C₂³	—	2	4	no	—
228	C₂³	C₂²×D₄×C₁₉	1	3	yes	—
229	Q₈	—	2	24	no	—
230	Q₈	—	4	24	no	—
231	Q₈	C₁₉×#127	1	4	yes	—
232	D₄	—	2	8	no	—
233	D₄	—	4	24	no	—
234	D₄	—	4	8	no	—
235	D₄	C₁₉×#127	1	12	yes	—
236	C₂×C₄	—	2	12	no	—
237	C₂×C₄	—	2	12	no	—
238	C₂×C₄	—	4	24	no	—
239	C₂×C₄	—	4	24	no	—
240	C₂×C₄	C₂²×D₄×C₁₉	1	6	yes	—
241	C₂×C₄	C₂²×D₄×C₁₉	1	3	yes	—
242	C₂×C₄	C₂²×D₄×C₁₉	1	12	yes	—
243	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
244	C₂×C₄	C₂⁵×C₁₉	1	1	yes	characteristic
245	C₈	C₂²×D₄×C₁₉	1	4	yes	—
246	C₈	C₂²×D₄×C₁₉	1	12	yes	—
247	C₂²	—	2	6	no	—
248	C₂²	—	4	4	no	—
249	C₂²	—	4	8	no	—
250	C₂²	C₂×C₁₉×#127	1	6	yes	—
251	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
252	C₄	—	2	6	no	—
253	C₄	—	4	12	no	—
254	C₄	C₂×C₁₉×#127	1	6	yes	—
255	C₄	C₁₉×#578	1	4	yes	complemented
256	C₄	C₂³×D₄×C₁₉	1	2	yes	—
257	C₂	—	2	6	no	—
258	C₂	—	8	4	no	—
259	C₂	C₂²×C₁₉×#127	1	1	yes	characteristic
260	C₂	C₂×C₁₉×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×#41×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×#578	C₂	semidirect
2. C₁₉×#3499	C₂	semidirect
3. C₁₉×#3156	C₂	semidirect
4. C₁₉×#3152	C₂	semidirect
5. C₁₉×#3147	C₂	semidirect
6. C₂×C₁₉×#494	C₂	semidirect
7. C₄×C₁₉×#131	C₂	semidirect
43. C₁₉×#131	D₄	semidirect
96. #60648	C₁₉	direct (N × H)
117. C₂×#578	C₂×C₁₉	semidirect
118. #3499	C₂×C₁₉	semidirect
119. #3156	C₂×C₁₉	semidirect
120. #3152	C₂×C₁₉	semidirect
121. #3147	C₂×C₁₉	semidirect
122. C₂×#494	C₂×C₁₉	semidirect
123. C₄×#131	C₂×C₁₉	semidirect
132. C₄×C₁₉	#578	semidirect
173. #131	D₄×C₁₉	semidirect
194. C₁₉	#60648	direct (N × H)
255. C₄	C₁₉×#578	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#578	2
C₁₉×#3499	2
C₁₉×#3156	2
C₁₉×#3152	2
C₁₉×#3147	2
C₂×C₁₉×#494	2
C₄×C₁₉×#131	2
#60648	19

Subgroups — C19×#60648 #18066607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#60648 #18066607