Subgroups — C₁₉×#2555 #14560412

Order 2432.

696 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#124	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₉×#121	C₂	1	2	yes	maximal, complemented
3	C₁₉×#395	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₉×#394	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₉×#392	C₂	1	2	yes	maximal, complemented
6	C₁₉×#390	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#388	C₂	1	2	yes	maximal, complemented
8	C₁₉×#386	C₂	1	2	yes	maximal, complemented
9	C₄×C₂²⋊C₄×C₁₉	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×D₄×C₁₉	C₂²	1	2	yes	—
12	C₂²×D₄×C₁₉	C₂²	1	2	yes	—
13	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₁₉	C₂²	1	2	yes	—
16	C₁₉×#124	—	2	8	no	—
17	C₁₉×#121	—	2	8	no	—
18	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
27	C₂×C₄²×C₁₉	C₂²	1	2	yes	—
28	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
29	C₁₉×#95	C₂²	1	2	yes	—
30	C₁₉×#95	C₂²	1	2	yes	—
31	C₁₉×#95	C₂²	1	1	yes	characteristic
32	C₁₉×#95	C₂²	1	1	yes	characteristic
33	C₂⁴×C₁₉	C₂³	1	2	yes	—
34	C₂⁴×C₁₉	C₂³	1	2	yes	—
35	C₂×Q₈×C₁₉	—	2	4	no	—
36	C₂×Q₈×C₁₉	—	4	1	no	—
37	C₂×D₄×C₁₉	—	2	4	no	—
38	C₂×D₄×C₁₉	—	2	4	no	—
39	C₂×D₄×C₁₉	—	2	4	no	—
40	C₂×D₄×C₁₉	—	2	4	no	—
41	C₂×D₄×C₁₉	—	2	2	no	—
42	C₂×D₄×C₁₉	—	2	4	no	—
43	C₂×D₄×C₁₉	—	4	1	no	—
44	C₂×D₄×C₁₉	—	4	1	no	—
45	C₂×D₄×C₁₉	—	4	2	no	—
46	C₂×D₄×C₁₉	—	4	2	no	—
47	C₂×D₄×C₁₉	—	4	1	no	—
48	C₂²×C₄×C₁₉	—	2	4	no	—
49	C₂²×C₄×C₁₉	—	2	4	no	—
50	C₂²×C₄×C₁₉	—	4	2	no	—
51	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
58	C₄⋊C₄×C₁₉	—	2	4	no	—
59	C₄⋊C₄×C₁₉	—	4	1	no	—
60	C₂²⋊C₄×C₁₉	—	2	4	no	—
61	C₂²⋊C₄×C₁₉	—	2	4	no	—
62	C₂²⋊C₄×C₁₉	—	2	4	no	—
63	C₂²⋊C₄×C₁₉	—	2	4	no	—
64	C₂²⋊C₄×C₁₉	—	2	4	no	—
65	C₂²⋊C₄×C₁₉	—	2	4	no	—
66	C₂²⋊C₄×C₁₉	D₄	1	8	yes	complemented
67	C₄²×C₁₉	—	2	4	no	—
68	C₄²×C₁₉	—	4	1	no	—
69	C₂³×C₁₉	—	2	2	no	—
70	C₂³×C₁₉	—	2	2	no	—
71	C₂³×C₁₉	—	2	2	no	—
72	C₂³×C₁₉	—	2	2	no	—
73	C₂³×C₁₉	—	2	2	no	—
74	C₂³×C₁₉	—	2	2	no	—
75	C₂³×C₁₉	—	2	2	no	—
76	C₂³×C₁₉	—	2	2	no	—
77	C₂³×C₁₉	—	2	2	no	—
78	C₂³×C₁₉	—	2	2	no	—
79	C₂³×C₁₉	—	2	2	no	—
80	C₂³×C₁₉	—	2	2	no	—
81	C₂³×C₁₉	—	2	2	no	—
82	C₂³×C₁₉	C₂×D₄	1	4	yes	—
83	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
84	Q₈×C₁₉	—	4	4	no	—
85	D₄×C₁₉	—	4	8	no	—
86	D₄×C₁₉	—	4	8	no	—
87	D₄×C₁₉	—	4	4	no	—
88	D₄×C₁₉	—	4	4	no	—
89	C₂×C₄×C₁₉	—	2	2	no	—
90	C₂×C₄×C₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	2	2	no	—
98	C₂×C₄×C₁₉	—	2	2	no	—
99	C₂×C₄×C₁₉	—	2	2	no	—
100	C₂×C₄×C₁₉	—	2	2	no	—
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	4	8	no	—
105	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	—
106	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	—
107	C₂×C₄×C₁₉	#41	1	2	yes	—
108	C₂×C₄×C₁₉	#41	1	4	yes	—
109	#2555	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₁₉	—	2	4	no	—
111	C₂²×C₁₉	—	2	4	no	—
112	C₂²×C₁₉	—	2	4	no	—
113	C₂²×C₁₉	—	4	2	no	—
114	C₂²×C₁₉	—	4	2	no	—
115	C₂²×C₁₉	—	4	2	no	—
116	C₂²×C₁₉	—	4	2	no	—
117	C₂²×C₁₉	—	4	2	no	—
118	C₂²×C₁₉	—	4	2	no	—
119	C₂²×C₁₉	—	4	2	no	—
120	C₂²×C₁₉	—	4	2	no	—
121	C₂²×C₁₉	—	4	2	no	—
122	C₂²×C₁₉	—	4	2	no	—
123	C₂²×C₁₉	—	4	2	no	—
124	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
125	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
126	C₂²×C₁₉	C₂×#41	1	2	yes	—
127	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
128	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
129	C₄×C₁₉	—	2	2	no	—
130	C₄×C₁₉	—	2	4	no	—
131	C₄×C₁₉	—	2	4	no	—
132	C₄×C₁₉	—	2	4	no	—
133	C₄×C₁₉	—	4	2	no	—
134	C₄×C₁₉	—	4	1	no	—
135	C₄×C₁₉	—	4	1	no	—
136	C₂×#124	C₂×C₁₉	1	2	yes	complemented
137	C₂×#121	C₂×C₁₉	1	2	yes	complemented
138	#395	C₂×C₁₉	1	1	yes	characteristic, complemented
139	#394	C₂×C₁₉	1	1	yes	characteristic, complemented
140	#392	C₂×C₁₉	1	2	yes	complemented
141	#390	C₂×C₁₉	1	1	yes	characteristic, complemented
142	#388	C₂×C₁₉	1	2	yes	complemented
143	#386	C₂×C₁₉	1	2	yes	complemented
144	C₄×C₂²⋊C₄	C₂×C₁₉	1	2	yes	complemented
145	C₂×C₁₉	—	4	4	no	—
146	C₂×C₁₉	—	8	2	no	—
147	C₂×C₁₉	#532	1	2	yes	—
148	C₂×C₁₉	#538	1	1	yes	characteristic
149	C₂×C₁₉	D₄²	1	1	yes	characteristic
150	C₂×C₁₉	#546	1	2	yes	—
151	C₂×C₁₉	#555	1	1	yes	characteristic
152	C₂²×Q₈	C₂²×C₁₉	1	1	yes	characteristic
153	C₂²×D₄	C₂²×C₁₉	1	1	yes	characteristic
154	C₂²×D₄	C₂²×C₁₉	1	2	yes	—
155	C₂²×D₄	C₂²×C₁₉	1	2	yes	—
156	C₂²×D₄	C₂²×C₁₉	1	1	yes	characteristic
157	C₂³×C₄	C₂²×C₁₉	1	2	yes	—
158	#124	—	2	8	no	—
159	#121	—	2	8	no	—
160	C₂×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
161	C₂×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
162	C₂×C₂²⋊C₄	C₂²×C₁₉	1	2	yes	—
163	C₂×C₂²⋊C₄	C₂²×C₁₉	1	2	yes	—
164	C₂×C₂²⋊C₄	C₂²×C₁₉	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₁₉	1	2	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₁₉	1	2	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₁₉	1	2	yes	—
168	C₂×C₂²⋊C₄	C₂²×C₁₉	1	2	yes	—
169	C₂×C₄²	C₂²×C₁₉	1	2	yes	—
170	C₂×C₄²	C₂²×C₁₉	1	1	yes	characteristic
171	#95	C₂²×C₁₉	1	2	yes	—
172	#95	C₂²×C₁₉	1	2	yes	—
173	#95	C₂²×C₁₉	1	1	yes	characteristic
174	#95	C₂²×C₁₉	1	1	yes	characteristic
175	C₁₉	#2555	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	C₂³×C₁₉	1	2	yes	—
177	C₂⁴	C₂³×C₁₉	1	2	yes	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	4	1	no	—
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	4	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	4	2	no	—
194	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
195	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
198	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
199	C₂²×C₄	C₂³×C₁₉	1	2	yes	—
200	C₂²×C₄	C₂³×C₁₉	1	1	yes	characteristic
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	4	1	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	2	4	no	—
205	C₂²⋊C₄	—	2	4	no	—
206	C₂²⋊C₄	—	2	4	no	—
207	C₂²⋊C₄	—	2	4	no	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	D₄×C₁₉	1	8	yes	complemented
210	C₄²	—	2	4	no	—
211	C₄²	—	4	1	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	2	2	no	—
217	C₂³	—	2	2	no	—
218	C₂³	—	2	2	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	C₂×D₄×C₁₉	1	4	yes	—
226	C₂³	C₂⁴×C₁₉	1	1	yes	characteristic
227	Q₈	—	4	4	no	—
228	D₄	—	4	8	no	—
229	D₄	—	4	8	no	—
230	D₄	—	4	4	no	—
231	D₄	—	4	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	C₂×D₄×C₁₉	1	4	yes	—
249	C₂×C₄	C₂×D₄×C₁₉	1	4	yes	—
250	C₂×C₄	#41×C₁₉	1	2	yes	—
251	C₂×C₄	#41×C₁₉	1	4	yes	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	2	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	C₂²×D₄×C₁₉	1	2	yes	—
267	C₂²	C₂×#41×C₁₉	1	1	yes	characteristic
268	C₂²	C₂×#41×C₁₉	1	2	yes	—
269	C₂²	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
270	C₂²	C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	2	4	no	—
273	C₄	—	2	4	no	—
274	C₄	—	2	4	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	1	no	—
277	C₄	—	4	1	no	—
278	C₂	—	4	4	no	—
279	C₂	—	8	2	no	—
280	C₂	C₁₉×#532	1	2	yes	—
281	C₂	C₁₉×#538	1	1	yes	characteristic
282	C₂	D₄²×C₁₉	1	1	yes	characteristic
283	C₂	C₁₉×#546	1	2	yes	—
284	C₂	C₁₉×#555	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×#124	C₂	semidirect
2. C₂×C₁₉×#121	C₂	semidirect
3. C₁₉×#395	C₂	semidirect
4. C₁₉×#394	C₂	semidirect
5. C₁₉×#392	C₂	semidirect
6. C₁₉×#390	C₂	semidirect
7. C₁₉×#388	C₂	semidirect
8. C₁₉×#386	C₂	semidirect
9. C₄×C₂²⋊C₄×C₁₉	C₂	semidirect
66. C₂²⋊C₄×C₁₉	D₄	semidirect
109. #2555	C₁₉	direct (N × H)
136. C₂×#124	C₂×C₁₉	semidirect
137. C₂×#121	C₂×C₁₉	semidirect
138. #395	C₂×C₁₉	semidirect
139. #394	C₂×C₁₉	semidirect
140. #392	C₂×C₁₉	semidirect
141. #390	C₂×C₁₉	semidirect
142. #388	C₂×C₁₉	semidirect
143. #386	C₂×C₁₉	semidirect
144. C₄×C₂²⋊C₄	C₂×C₁₉	semidirect
175. C₁₉	#2555	direct (N × H)
209. C₂²⋊C₄	D₄×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#124	2
C₂×C₁₉×#121	2
C₁₉×#395	2
C₁₉×#394	2
C₁₉×#392	2
C₁₉×#390	2
C₁₉×#388	2
C₁₉×#386	2
C₄×C₂²⋊C₄×C₁₉	2
#2555	19

Subgroups — C19×#2555 #14560412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#2555 #14560412