Subgroups — C₂×C₁₉×#3502 #18068743

Order 4864.

6826 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₉×#3502	C₂	1	32	yes	maximal, direct product factor
2	C₂×Q₈²×C₁₉	C₂	1	4	yes	maximal, complemented
3	C₂×C₁₉×#557	C₂	1	8	yes	maximal, complemented
4	C₂×C₁₉×#557	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₉×#544	C₂	1	6	yes	maximal
6	C₂×C₁₉×#533	C₂	1	6	yes	maximal
7	C₂×C₁₉×#519	C₂	1	2	yes	maximal
8	C₂²×C₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic
10	Q₈²×C₁₉	C₂²	1	64	yes	complemented
11	C₁₉×#557	C₂²	1	128	yes	complemented
12	C₁₉×#557	C₂²	1	64	yes	complemented
13	C₁₉×#544	C₂²	1	96	yes	—
14	C₁₉×#533	C₂²	1	96	yes	—
15	C₂×C₁₉×C₄⋊Q₈	C₂²	1	6	yes	—
16	C₂×C₁₉×C₄⋊Q₈	C₂²	1	6	yes	—
17	C₂×C₁₉×C₄⋊Q₈	C₂²	1	24	yes	—
18	C₂×C₁₉×#125	C₂²	1	12	yes	—
19	C₂×C₁₉×#125	C₂²	1	24	yes	—
20	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	4	yes	—
21	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	12	yes	—
22	C₁₉×#519	C₂²	1	32	yes	—
23	C₂×C₄×Q₈×C₁₉	C₂²	1	4	yes	—
24	C₂×C₄×Q₈×C₁₉	C₂²	1	4	yes	—
25	C₂×C₄×Q₈×C₁₉	C₂²	1	16	yes	—
26	C₂×C₄×Q₈×C₁₉	C₂²	1	8	yes	—
27	C₂×C₄×Q₈×C₁₉	C₂²	1	8	yes	—
28	C₂×C₄×Q₈×C₁₉	C₂²	1	12	yes	—
29	C₂×C₄×Q₈×C₁₉	C₂²	1	12	yes	—
30	C₂×C₁₉×#117	C₂²	1	12	yes	—
31	C₂²×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
32	C₂²×C₄²×C₁₉	C₂²	1	3	yes	—
33	C₂²×Q₈×C₁₉	Q₈	1	16	yes	complemented
34	C₂²×Q₈×C₁₉	C₂³	1	4	yes	—
35	C₂²×Q₈×C₁₉	C₂³	1	12	yes	—
36	C₂²×Q₈×C₁₉	C₂³	1	4	yes	—
37	C₂²×Q₈×C₁₉	C₂³	1	8	yes	—
38	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
40	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
41	C₁₉×C₄⋊Q₈	C₂³	1	48	yes	—
42	C₁₉×C₄⋊Q₈	C₂³	1	48	yes	—
43	C₁₉×C₄⋊Q₈	C₂³	1	192	yes	—
44	C₁₉×#125	C₂³	1	96	yes	—
45	C₁₉×#125	C₂³	1	192	yes	—
46	C₁₉×C₂²⋊Q₈	C₂³	1	32	yes	—
47	C₁₉×C₂²⋊Q₈	C₂³	1	96	yes	—
48	C₄×Q₈×C₁₉	C₂³	1	32	yes	—
49	C₄×Q₈×C₁₉	C₂³	1	32	yes	—
50	C₄×Q₈×C₁₉	C₂³	1	64	yes	—
51	C₄×Q₈×C₁₉	C₂³	1	64	yes	—
52	C₄×Q₈×C₁₉	C₂³	1	96	yes	—
53	C₄×Q₈×C₁₉	C₂³	1	96	yes	—
54	C₁₉×#117	C₂³	1	96	yes	—
55	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
56	C₂×C₄⋊C₄×C₁₉	C₂³	1	24	yes	—
57	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
58	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
59	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
61	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	yes	—
62	C₂×C₄⋊C₄×C₁₉	C₂³	1	24	yes	—
63	C₂×C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
64	C₂×C₄²×C₁₉	C₂³	1	6	yes	—
65	C₂×C₄²×C₁₉	C₂³	1	6	yes	—
66	C₂×C₄²×C₁₉	C₂³	1	24	yes	—
67	C₂×C₄²×C₁₉	C₂³	1	24	yes	—
68	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
69	C₂×Q₈×C₁₉	—	2	16	no	—
70	C₂×Q₈×C₁₉	—	4	24	no	—
71	C₂×Q₈×C₁₉	C₂×Q₈	1	32	yes	—
72	C₂×Q₈×C₁₉	C₂×Q₈	1	64	yes	complemented
73	C₂×Q₈×C₁₉	C₂⁴	1	16	yes	—
74	C₂×Q₈×C₁₉	C₂⁴	1	48	yes	—
75	C₂×Q₈×C₁₉	C₂⁴	1	16	yes	—
76	C₂²×C₄×C₁₉	—	2	4	no	—
77	C₂²×C₄×C₁₉	—	4	6	no	—
78	C₂²×C₄×C₁₉	C₂×Q₈	1	24	yes	—
79	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
80	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
82	C₂²×C₄×C₁₉	C₂⁴	1	8	yes	—
83	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
84	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
85	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
86	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
87	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
88	C₄⋊C₄×C₁₉	C₂⁴	1	48	yes	—
89	C₄⋊C₄×C₁₉	C₂⁴	1	48	yes	—
90	C₄⋊C₄×C₁₉	C₂⁴	1	48	yes	—
91	C₄⋊C₄×C₁₉	C₂⁴	1	32	yes	—
92	C₄⋊C₄×C₁₉	C₂⁴	1	16	yes	—
93	C₄⋊C₄×C₁₉	C₂⁴	1	48	yes	—
94	C₄⋊C₄×C₁₉	C₂⁴	1	96	yes	—
95	C₂²⋊C₄×C₁₉	C₂⁴	1	32	yes	—
96	C₄²×C₁₉	C₂⁴	1	24	yes	—
97	C₄²×C₁₉	C₂⁴	1	24	yes	—
98	C₄²×C₁₉	C₂⁴	1	96	yes	—
99	C₂×#3502	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₁₉	—	2	2	no	—
101	C₂³×C₁₉	—	2	2	no	—
102	C₂³×C₁₉	C₂²×Q₈	1	4	yes	—
103	C₂³×C₁₉	C₂⁵	1	2	yes	—
104	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
105	Q₈×C₁₉	—	2	32	no	—
106	Q₈×C₁₉	—	4	48	no	—
107	Q₈×C₁₉	C₂²×Q₈	1	64	yes	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	16	no	—
110	C₂×C₄×C₁₉	—	2	12	no	—
111	C₂×C₄×C₁₉	—	2	16	no	—
112	C₂×C₄×C₁₉	—	2	8	no	—
113	C₂×C₄×C₁₉	—	4	12	no	—
114	C₂×C₄×C₁₉	C₂²×Q₈	1	24	yes	—
115	C₂×C₄×C₁₉	C₂²×Q₈	1	48	yes	—
116	C₂×C₄×C₁₉	2₋¹⁺⁴	1	4	yes	—
117	C₂×C₄×C₁₉	C₂⁵	1	2	yes	—
118	C₂×C₄×C₁₉	C₂⁵	1	16	yes	—
119	C₂×C₄×C₁₉	C₂⁵	1	16	yes	—
120	C₂×C₄×C₁₉	C₂⁵	1	12	yes	—
121	C₂×C₄×C₁₉	C₂⁵	1	12	yes	—
122	C₂×C₄×C₁₉	C₂⁵	1	2	yes	—
123	#3502	C₂×C₁₉	1	32	yes	direct product factor
124	C₂×Q₈²	C₂×C₁₉	1	4	yes	complemented
125	C₂×#557	C₂×C₁₉	1	8	yes	complemented
126	C₂×#557	C₂×C₁₉	1	4	yes	complemented
127	C₂×#544	C₂×C₁₉	1	6	yes	—
128	C₂×#533	C₂×C₁₉	1	6	yes	—
129	C₂×#519	C₂×C₁₉	1	2	yes	—
130	C₂²×C₄×Q₈	C₂×C₁₉	1	1	yes	characteristic
131	C₂²×C₁₉	—	2	2	no	—
132	C₂²×C₁₉	—	2	2	no	—
133	C₂²×C₁₉	—	2	8	no	—
134	C₂²×C₁₉	C₂³×Q₈	1	4	yes	—
135	C₂²×C₁₉	C₂³×Q₈	1	2	yes	—
136	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	2	yes	—
137	C₂²×C₁₉	#585	1	2	yes	—
138	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
139	C₄×C₁₉	—	2	2	no	—
140	C₄×C₁₉	—	2	16	no	—
141	C₄×C₁₉	—	2	12	no	—
142	C₄×C₁₉	—	2	16	no	—
143	C₄×C₁₉	C₂³×Q₈	1	24	yes	—
144	C₄×C₁₉	C₂×2₋¹⁺⁴	1	4	yes	—
145	C₂³×Q₈	C₂²×C₁₉	1	1	yes	characteristic
146	Q₈²	C₂²×C₁₉	1	64	yes	complemented
147	#557	C₂²×C₁₉	1	128	yes	complemented
148	#557	C₂²×C₁₉	1	64	yes	complemented
149	#544	C₂²×C₁₉	1	96	yes	—
150	#533	C₂²×C₁₉	1	96	yes	—
151	C₂×C₄⋊Q₈	C₂²×C₁₉	1	6	yes	—
152	C₂×C₄⋊Q₈	C₂²×C₁₉	1	6	yes	—
153	C₂×C₄⋊Q₈	C₂²×C₁₉	1	24	yes	—
154	C₂×#125	C₂²×C₁₉	1	12	yes	—
155	C₂×#125	C₂²×C₁₉	1	24	yes	—
156	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	4	yes	—
157	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	12	yes	—
158	#519	C₂²×C₁₉	1	32	yes	—
159	C₂×C₄×Q₈	C₂²×C₁₉	1	4	yes	—
160	C₂×C₄×Q₈	C₂²×C₁₉	1	4	yes	—
161	C₂×C₄×Q₈	C₂²×C₁₉	1	16	yes	—
162	C₂×C₄×Q₈	C₂²×C₁₉	1	8	yes	—
163	C₂×C₄×Q₈	C₂²×C₁₉	1	8	yes	—
164	C₂×C₄×Q₈	C₂²×C₁₉	1	12	yes	—
165	C₂×C₄×Q₈	C₂²×C₁₉	1	12	yes	—
166	C₂×#117	C₂²×C₁₉	1	12	yes	—
167	C₂²×C₄⋊C₄	C₂²×C₁₉	1	3	yes	—
168	C₂²×C₄²	C₂²×C₁₉	1	3	yes	—
169	C₂×C₁₉	—	2	4	no	—
170	C₂×C₁₉	#3502	1	4	yes	direct product factor
171	C₂×C₁₉	C₂⁴×Q₈	1	1	yes	characteristic
172	C₂×C₁₉	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
173	C₂×C₁₉	C₂×#585	1	1	yes	characteristic
174	C₂²×Q₈	Q₈×C₁₉	1	16	yes	complemented
175	C₂²×Q₈	C₂³×C₁₉	1	4	yes	—
176	C₂²×Q₈	C₂³×C₁₉	1	12	yes	—
177	C₂²×Q₈	C₂³×C₁₉	1	4	yes	—
178	C₂²×Q₈	C₂³×C₁₉	1	8	yes	—
179	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
180	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
181	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
182	C₄⋊Q₈	C₂³×C₁₉	1	48	yes	—
183	C₄⋊Q₈	C₂³×C₁₉	1	48	yes	—
184	C₄⋊Q₈	C₂³×C₁₉	1	192	yes	—
185	#125	C₂³×C₁₉	1	96	yes	—
186	#125	C₂³×C₁₉	1	192	yes	—
187	C₂²⋊Q₈	C₂³×C₁₉	1	32	yes	—
188	C₂²⋊Q₈	C₂³×C₁₉	1	96	yes	—
189	C₄×Q₈	C₂³×C₁₉	1	32	yes	—
190	C₄×Q₈	C₂³×C₁₉	1	32	yes	—
191	C₄×Q₈	C₂³×C₁₉	1	64	yes	—
192	C₄×Q₈	C₂³×C₁₉	1	64	yes	—
193	C₄×Q₈	C₂³×C₁₉	1	96	yes	—
194	C₄×Q₈	C₂³×C₁₉	1	96	yes	—
195	#117	C₂³×C₁₉	1	96	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
197	C₂×C₄⋊C₄	C₂³×C₁₉	1	24	yes	—
198	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
199	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
200	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
202	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
203	C₂×C₄⋊C₄	C₂³×C₁₉	1	24	yes	—
204	C₂×C₂²⋊C₄	C₂³×C₁₉	1	8	yes	—
205	C₂×C₄²	C₂³×C₁₉	1	24	yes	—
206	C₂×C₄²	C₂³×C₁₉	1	6	yes	—
207	C₂×C₄²	C₂³×C₁₉	1	6	yes	—
208	C₂×C₄²	C₂³×C₁₉	1	24	yes	—
209	C₁₉	C₂×#3502	1	1	yes	characteristic, direct product factor, Sylow
210	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
211	C₂×Q₈	—	2	16	no	—
212	C₂×Q₈	—	4	24	no	—
213	C₂×Q₈	C₂×Q₈×C₁₉	1	32	yes	—
214	C₂×Q₈	C₂×Q₈×C₁₉	1	64	yes	complemented
215	C₂×Q₈	C₂⁴×C₁₉	1	16	yes	—
216	C₂×Q₈	C₂⁴×C₁₉	1	48	yes	—
217	C₂×Q₈	C₂⁴×C₁₉	1	16	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	6	no	—
220	C₂²×C₄	C₂×Q₈×C₁₉	1	24	yes	—
221	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
223	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
224	C₂²×C₄	C₂⁴×C₁₉	1	8	yes	—
225	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
226	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
227	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
228	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
229	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
230	C₄⋊C₄	C₂⁴×C₁₉	1	48	yes	—
231	C₄⋊C₄	C₂⁴×C₁₉	1	48	yes	—
232	C₄⋊C₄	C₂⁴×C₁₉	1	48	yes	—
233	C₄⋊C₄	C₂⁴×C₁₉	1	32	yes	—
234	C₄⋊C₄	C₂⁴×C₁₉	1	16	yes	—
235	C₄⋊C₄	C₂⁴×C₁₉	1	48	yes	—
236	C₄⋊C₄	C₂⁴×C₁₉	1	96	yes	—
237	C₂²⋊C₄	C₂⁴×C₁₉	1	32	yes	—
238	C₄²	C₂⁴×C₁₉	1	24	yes	—
239	C₄²	C₂⁴×C₁₉	1	24	yes	—
240	C₄²	C₂⁴×C₁₉	1	96	yes	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₂²×Q₈×C₁₉	1	4	yes	—
244	C₂³	C₂⁵×C₁₉	1	2	yes	—
245	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
246	Q₈	—	2	32	no	—
247	Q₈	—	4	48	no	—
248	Q₈	C₂²×Q₈×C₁₉	1	64	yes	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	16	no	—
251	C₂×C₄	—	2	12	no	—
252	C₂×C₄	—	2	16	no	—
253	C₂×C₄	—	2	8	no	—
254	C₂×C₄	—	4	12	no	—
255	C₂×C₄	C₂²×Q₈×C₁₉	1	24	yes	—
256	C₂×C₄	C₂²×Q₈×C₁₉	1	48	yes	—
257	C₂×C₄	C₁₉×2₋¹⁺⁴	1	4	yes	—
258	C₂×C₄	C₂⁵×C₁₉	1	2	yes	—
259	C₂×C₄	C₂⁵×C₁₉	1	16	yes	—
260	C₂×C₄	C₂⁵×C₁₉	1	16	yes	—
261	C₂×C₄	C₂⁵×C₁₉	1	12	yes	—
262	C₂×C₄	C₂⁵×C₁₉	1	12	yes	—
263	C₂×C₄	C₂⁵×C₁₉	1	2	yes	—
264	C₂²	—	2	8	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	C₂³×Q₈×C₁₉	1	4	yes	—
268	C₂²	C₂³×Q₈×C₁₉	1	2	yes	—
269	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	2	yes	—
270	C₂²	C₁₉×#585	1	2	yes	—
271	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
272	C₄	—	2	2	no	—
273	C₄	—	2	16	no	—
274	C₄	—	2	12	no	—
275	C₄	—	2	16	no	—
276	C₄	C₂³×Q₈×C₁₉	1	24	yes	—
277	C₄	C₂×C₁₉×2₋¹⁺⁴	1	4	yes	—
278	C₂	—	2	4	no	—
279	C₂	C₁₉×#3502	1	4	yes	direct product factor
280	C₂	C₂⁴×Q₈×C₁₉	1	1	yes	characteristic
281	C₂	C₂²×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
282	C₂	C₂×C₁₉×#585	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₁₉×#3502	C₂	direct (N × H)
2. C₂×Q₈²×C₁₉	C₂	semidirect
3. C₂×C₁₉×#557	C₂	semidirect
4. C₂×C₁₉×#557	C₂	semidirect
10. Q₈²×C₁₉	C₂²	semidirect
11. C₁₉×#557	C₂²	semidirect
12. C₁₉×#557	C₂²	semidirect
33. C₂²×Q₈×C₁₉	Q₈	semidirect
72. C₂×Q₈×C₁₉	C₂×Q₈	semidirect
99. C₂×#3502	C₁₉	direct (N × H)
123. #3502	C₂×C₁₉	direct (N × H)
124. C₂×Q₈²	C₂×C₁₉	semidirect
125. C₂×#557	C₂×C₁₉	semidirect
126. C₂×#557	C₂×C₁₉	semidirect
146. Q₈²	C₂²×C₁₉	semidirect
147. #557	C₂²×C₁₉	semidirect
148. #557	C₂²×C₁₉	semidirect
170. C₂×C₁₉	#3502	direct (N × H)
174. C₂²×Q₈	Q₈×C₁₉	semidirect
209. C₁₉	C₂×#3502	direct (N × H)
214. C₂×Q₈	C₂×Q₈×C₁₉	semidirect
279. C₂	C₁₉×#3502	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×#3502	2
C₂×Q₈²×C₁₉	2
C₂×C₁₉×#557	2
C₂×C₁₉×#544	2
C₂×C₁₉×#533	2
C₂×C₁₉×#519	2
C₂²×C₄×Q₈×C₁₉	2
C₂×#3502	19

Subgroups — C2×C19×#3502 #18068743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₉×#3502 #18068743