Subgroups — C₁₉×#36675 #18042634

Order 4864.

2070 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	C₁₉×#2840	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₉×#2835	C₂	1	3	yes	maximal, complemented
3	C₁₉×#2824	C₂	1	9	yes	maximal, complemented
4	C₁₉×#2812	C₂	1	3	yes	maximal, complemented
5	C₁₉×#2398	C₂	1	9	yes	maximal, complemented
6	C₁₉×#2388	C₂	1	3	yes	maximal, complemented
7	C₄×C₁₉×#120	C₂	1	3	yes	maximal, complemented
8	C₂²×#41×C₁₉	C₂²	1	1	yes	characteristic, complemented
9	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₂×C₁₉×#120	C₂²	1	3	yes	—
11	C₂×C₄×Q₈×C₁₉	C₂²	1	3	yes	complemented
12	C₂×C₄×D₄×C₁₉	C₂²	1	9	yes	—
13	C₁₉×#400	C₂²	1	18	yes	complemented
14	C₁₉×#397	C₂²	1	9	yes	—
15	C₁₉×#397	C₂²	1	9	yes	complemented
16	C₁₉×#395	C₂²	1	3	yes	complemented
17	C₁₉×#394	C₂²	1	18	yes	—
18	C₁₉×#394	C₂²	1	9	yes	complemented
19	C₁₉×#393	C₂²	1	9	yes	—
20	C₁₉×#393	C₂²	1	3	yes	—
21	C₁₉×#393	C₂²	1	9	yes	complemented
22	C₁₉×#392	C₂²	1	9	yes	—
23	C₁₉×#386	C₂²	1	9	yes	—
24	C₁₉×#385	C₂²	1	9	yes	—
25	C₁₉×#379	C₂²	1	3	yes	—
26	C₄×C₂²⋊C₄×C₁₉	C₂²	1	9	yes	—
27	C₄×C₂²⋊C₄×C₁₉	C₂²	1	9	yes	complemented
28	C₁₉×#376	C₂²	1	3	yes	complemented
29	C₂⁵×C₁₉	C₂³	1	1	yes	characteristic
30	C₂×#41×C₁₉	—	4	6	no	—
31	C₂²×Q₈×C₁₉	C₂³	1	3	yes	—
32	C₂²×Q₈×C₁₉	C₂³	1	1	yes	characteristic
33	C₂²×D₄×C₁₉	C₂³	1	9	yes	—
34	C₂²×D₄×C₁₉	C₂³	1	3	yes	—
35	C₂³×C₄×C₁₉	—	4	6	no	—
36	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
37	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
38	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
39	C₁₉×#120	—	2	12	no	—
40	C₄×Q₈×C₁₉	—	4	6	no	—
41	C₄×D₄×C₁₉	—	4	18	no	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	9	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	9	yes	—
44	C₂×C₂²⋊C₄×C₁₉	C₂³	1	9	yes	—
45	C₂×C₂²⋊C₄×C₁₉	C₂³	1	9	yes	—
46	C₂×C₂²⋊C₄×C₁₉	C₂³	1	18	yes	—
47	C₂×C₂²⋊C₄×C₁₉	C₂³	1	9	yes	—
48	C₂×C₂²⋊C₄×C₁₉	C₂³	1	9	yes	—
49	C₂×C₄²×C₁₉	C₂³	1	9	yes	—
50	C₁₉×#95	C₂³	1	3	yes	—
51	C₁₉×#95	C₂³	1	9	yes	—
52	C₁₉×#95	C₂³	1	18	yes	—
53	C₁₉×#95	C₂³	1	9	yes	—
54	C₁₉×#95	C₂³	1	9	yes	—
55	C₂⁴×C₁₉	—	4	3	no	—
56	C₂⁴×C₁₉	—	4	3	no	—
57	C₂⁴×C₁₉	#41	1	4	yes	complemented
58	C₂⁴×C₁₉	C₂⁴	1	3	yes	—
59	C₂⁴×C₁₉	C₂⁴	1	3	yes	—
60	#41×C₁₉	—	4	16	no	—
61	C₂×Q₈×C₁₉	—	4	3	no	—
62	C₂×Q₈×C₁₉	—	4	6	no	—
63	C₂×Q₈×C₁₉	—	4	3	no	—
64	C₂×D₄×C₁₉	—	2	18	no	—
65	C₂×D₄×C₁₉	—	4	9	no	—
66	C₂×D₄×C₁₉	—	4	9	no	—
67	C₂×D₄×C₁₉	—	4	9	no	—
68	C₂²×C₄×C₁₉	—	2	18	no	—
69	C₂²×C₄×C₁₉	—	2	18	no	—
70	C₂²×C₄×C₁₉	—	4	9	no	—
71	C₂²×C₄×C₁₉	—	4	16	no	—
72	C₂²×C₄×C₁₉	C₂⁴	1	9	yes	—
73	C₂²×C₄×C₁₉	C₂⁴	1	9	yes	—
74	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
75	C₂²×C₄×C₁₉	C₂⁴	1	3	yes	—
76	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
77	C₄⋊C₄×C₁₉	—	4	9	no	—
78	C₄⋊C₄×C₁₉	—	4	9	no	—
79	C₂²⋊C₄×C₁₉	—	2	18	no	—
80	C₂²⋊C₄×C₁₉	—	4	18	no	—
81	C₂²⋊C₄×C₁₉	—	4	9	no	—
82	C₂²⋊C₄×C₁₉	—	4	9	no	—
83	C₂²⋊C₄×C₁₉	—	4	9	no	—
84	C₄²×C₁₉	—	2	18	no	—
85	#36675	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₁₉	—	2	9	no	—
87	C₂³×C₁₉	—	2	9	no	—
88	C₂³×C₁₉	—	2	3	no	—
89	C₂³×C₁₉	—	2	9	no	—
90	C₂³×C₁₉	—	2	9	no	—
91	C₂³×C₁₉	—	4	12	no	—
92	C₂³×C₁₉	—	4	4	no	—
93	C₂³×C₁₉	—	4	12	no	—
94	C₂³×C₁₉	C₂×#41	1	6	yes	—
95	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
96	Q₈×C₁₉	—	4	12	no	—
97	Q₈×C₁₉	—	4	4	no	—
98	D₄×C₁₉	—	4	36	no	—
99	D₄×C₁₉	—	4	12	no	—
100	C₂×C₄×C₁₉	—	2	9	no	—
101	C₂×C₄×C₁₉	—	2	9	no	—
102	C₂×C₄×C₁₉	—	2	9	no	—
103	C₂×C₄×C₁₉	—	2	9	no	—
104	C₂×C₄×C₁₉	—	2	9	no	—
105	C₂×C₄×C₁₉	—	2	9	no	—
106	C₂×C₄×C₁₉	—	2	18	no	—
107	C₂×C₄×C₁₉	—	4	12	no	—
108	C₂×C₄×C₁₉	—	4	12	no	—
109	C₂×C₄×C₁₉	—	4	12	no	—
110	C₂×C₄×C₁₉	H(5,𝔽₂)	1	6	yes	—
111	#2840	C₂×C₁₉	1	1	yes	characteristic, complemented
112	#2835	C₂×C₁₉	1	3	yes	complemented
113	#2824	C₂×C₁₉	1	9	yes	complemented
114	#2812	C₂×C₁₉	1	3	yes	complemented
115	#2398	C₂×C₁₉	1	9	yes	complemented
116	#2388	C₂×C₁₉	1	3	yes	complemented
117	C₄×#120	C₂×C₁₉	1	3	yes	complemented
118	C₂²×C₁₉	—	2	18	no	—
119	C₂²×C₁₉	—	4	16	no	—
120	C₂²×C₁₉	—	4	3	no	—
121	C₂²×C₁₉	—	4	9	no	—
122	C₂²×C₁₉	—	4	9	no	—
123	C₂²×C₁₉	—	4	3	no	—
124	C₂²×C₁₉	—	4	9	no	—
125	C₂²×C₁₉	C₂²×#41	1	1	yes	characteristic
126	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	3	yes	—
127	C₂²×C₁₉	#585	1	3	yes	—
128	C₄×C₁₉	—	4	9	no	—
129	C₄×C₁₉	—	4	3	no	—
130	C₄×C₁₉	—	4	3	no	—
131	C₄×C₁₉	—	4	9	no	—
132	C₄×C₁₉	H(3,𝔽₄)	1	4	yes	—
133	C₂²×#41	C₂²×C₁₉	1	1	yes	characteristic, complemented
134	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
135	C₂×#120	C₂²×C₁₉	1	3	yes	—
136	C₂×C₄×Q₈	C₂²×C₁₉	1	3	yes	complemented
137	C₂×C₄×D₄	C₂²×C₁₉	1	9	yes	—
138	#400	C₂²×C₁₉	1	18	yes	complemented
139	#397	C₂²×C₁₉	1	9	yes	—
140	#397	C₂²×C₁₉	1	9	yes	complemented
141	#395	C₂²×C₁₉	1	3	yes	complemented
142	#394	C₂²×C₁₉	1	18	yes	—
143	#394	C₂²×C₁₉	1	9	yes	complemented
144	#393	C₂²×C₁₉	1	9	yes	—
145	#393	C₂²×C₁₉	1	3	yes	—
146	#393	C₂²×C₁₉	1	9	yes	complemented
147	#392	C₂²×C₁₉	1	9	yes	—
148	#386	C₂²×C₁₉	1	9	yes	—
149	#385	C₂²×C₁₉	1	9	yes	—
150	#379	C₂²×C₁₉	1	3	yes	—
151	C₄×C₂²⋊C₄	C₂²×C₁₉	1	9	yes	—
152	C₄×C₂²⋊C₄	C₂²×C₁₉	1	9	yes	complemented
153	#376	C₂²×C₁₉	1	3	yes	complemented
154	C₂×C₁₉	—	4	6	no	—
155	C₂×C₁₉	—	8	3	no	—
156	C₂×C₁₉	#3459	1	3	yes	—
157	C₂×C₁₉	C₂×H(3,𝔽₄)	1	1	yes	characteristic
158	C₂×C₁₉	#3529	1	3	yes	—
159	C₂⁵	C₂³×C₁₉	1	1	yes	characteristic
160	C₂×#41	—	4	6	no	—
161	C₂²×Q₈	C₂³×C₁₉	1	3	yes	—
162	C₂²×Q₈	C₂³×C₁₉	1	1	yes	characteristic
163	C₂²×D₄	C₂³×C₁₉	1	9	yes	—
164	C₂²×D₄	C₂³×C₁₉	1	3	yes	—
165	C₂³×C₄	—	4	6	no	—
166	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
167	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
168	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
169	#120	—	2	12	no	—
170	C₄×Q₈	—	4	6	no	—
171	C₄×D₄	—	4	18	no	—
172	C₂×C₄⋊C₄	C₂³×C₁₉	1	9	yes	—
173	C₂×C₄⋊C₄	C₂³×C₁₉	1	9	yes	—
174	C₂×C₂²⋊C₄	C₂³×C₁₉	1	9	yes	—
175	C₂×C₂²⋊C₄	C₂³×C₁₉	1	9	yes	—
176	C₂×C₂²⋊C₄	C₂³×C₁₉	1	9	yes	—
177	C₂×C₂²⋊C₄	C₂³×C₁₉	1	9	yes	—
178	C₂×C₂²⋊C₄	C₂³×C₁₉	1	18	yes	—
179	C₂×C₄²	C₂³×C₁₉	1	9	yes	—
180	#95	C₂³×C₁₉	1	3	yes	—
181	#95	C₂³×C₁₉	1	9	yes	—
182	#95	C₂³×C₁₉	1	18	yes	—
183	#95	C₂³×C₁₉	1	9	yes	—
184	#95	C₂³×C₁₉	1	9	yes	—
185	C₁₉	#36675	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	—	4	3	no	—
187	C₂⁴	—	4	3	no	—
188	C₂⁴	#41×C₁₉	1	4	yes	complemented
189	C₂⁴	C₂⁴×C₁₉	1	3	yes	—
190	C₂⁴	C₂⁴×C₁₉	1	3	yes	—
191	#41	—	4	16	no	—
192	C₂×Q₈	—	4	3	no	—
193	C₂×Q₈	—	4	6	no	—
194	C₂×Q₈	—	4	3	no	—
195	C₂×D₄	—	2	18	no	—
196	C₂×D₄	—	4	9	no	—
197	C₂×D₄	—	4	9	no	—
198	C₂×D₄	—	4	9	no	—
199	C₂²×C₄	—	2	18	no	—
200	C₂²×C₄	—	2	18	no	—
201	C₂²×C₄	—	4	9	no	—
202	C₂²×C₄	—	4	16	no	—
203	C₂²×C₄	C₂⁴×C₁₉	1	9	yes	—
204	C₂²×C₄	C₂⁴×C₁₉	1	9	yes	—
205	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
206	C₂²×C₄	C₂⁴×C₁₉	1	3	yes	—
207	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
208	C₄⋊C₄	—	4	9	no	—
209	C₄⋊C₄	—	4	9	no	—
210	C₂²⋊C₄	—	2	18	no	—
211	C₂²⋊C₄	—	4	18	no	—
212	C₂²⋊C₄	—	4	9	no	—
213	C₂²⋊C₄	—	4	9	no	—
214	C₂²⋊C₄	—	4	9	no	—
215	C₄²	—	2	18	no	—
216	C₂³	—	2	9	no	—
217	C₂³	—	2	9	no	—
218	C₂³	—	2	3	no	—
219	C₂³	—	2	9	no	—
220	C₂³	—	2	9	no	—
221	C₂³	—	4	12	no	—
222	C₂³	—	4	4	no	—
223	C₂³	—	4	12	no	—
224	C₂³	C₂×#41×C₁₉	1	6	yes	—
225	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
226	Q₈	—	4	4	no	—
227	Q₈	—	4	12	no	—
228	D₄	—	4	36	no	—
229	D₄	—	4	12	no	—
230	C₂×C₄	—	2	9	no	—
231	C₂×C₄	—	2	9	no	—
232	C₂×C₄	—	2	9	no	—
233	C₂×C₄	—	2	9	no	—
234	C₂×C₄	—	2	9	no	—
235	C₂×C₄	—	2	9	no	—
236	C₂×C₄	—	2	18	no	—
237	C₂×C₄	—	4	12	no	—
238	C₂×C₄	—	4	12	no	—
239	C₂×C₄	—	4	12	no	—
240	C₂×C₄	C₁₉×H(5,𝔽₂)	1	6	yes	—
241	C₂²	—	2	18	no	—
242	C₂²	—	4	16	no	—
243	C₂²	—	4	3	no	—
244	C₂²	—	4	9	no	—
245	C₂²	—	4	9	no	—
246	C₂²	—	4	3	no	—
247	C₂²	—	4	9	no	—
248	C₂²	C₂²×#41×C₁₉	1	1	yes	characteristic
249	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	3	yes	—
250	C₂²	C₁₉×#585	1	3	yes	—
251	C₄	—	4	9	no	—
252	C₄	—	4	3	no	—
253	C₄	—	4	3	no	—
254	C₄	—	4	9	no	—
255	C₄	C₁₉×H(3,𝔽₄)	1	4	yes	—
256	C₂	—	4	6	no	—
257	C₂	—	8	3	no	—
258	C₂	C₁₉×#3459	1	3	yes	—
259	C₂	C₂×C₁₉×H(3,𝔽₄)	1	1	yes	characteristic
260	C₂	C₁₉×#3529	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₉
socle	C₂³×C₁₉	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₉×#2840	C₂	semidirect
2. C₁₉×#2835	C₂	semidirect
3. C₁₉×#2824	C₂	semidirect
4. C₁₉×#2812	C₂	semidirect
5. C₁₉×#2398	C₂	semidirect
6. C₁₉×#2388	C₂	semidirect
7. C₄×C₁₉×#120	C₂	semidirect
8. C₂²×#41×C₁₉	C₂²	semidirect
11. C₂×C₄×Q₈×C₁₉	C₂²	semidirect
13. C₁₉×#400	C₂²	semidirect
15. C₁₉×#397	C₂²	semidirect
16. C₁₉×#395	C₂²	semidirect
18. C₁₉×#394	C₂²	semidirect
21. C₁₉×#393	C₂²	semidirect
27. C₄×C₂²⋊C₄×C₁₉	C₂²	semidirect
28. C₁₉×#376	C₂²	semidirect
57. C₂⁴×C₁₉	#41	semidirect
85. #36675	C₁₉	direct (N × H)
111. #2840	C₂×C₁₉	semidirect
112. #2835	C₂×C₁₉	semidirect
113. #2824	C₂×C₁₉	semidirect
114. #2812	C₂×C₁₉	semidirect
115. #2398	C₂×C₁₉	semidirect
116. #2388	C₂×C₁₉	semidirect
117. C₄×#120	C₂×C₁₉	semidirect
133. C₂²×#41	C₂²×C₁₉	semidirect
136. C₂×C₄×Q₈	C₂²×C₁₉	semidirect
138. #400	C₂²×C₁₉	semidirect
140. #397	C₂²×C₁₉	semidirect
141. #395	C₂²×C₁₉	semidirect
143. #394	C₂²×C₁₉	semidirect
146. #393	C₂²×C₁₉	semidirect
152. C₄×C₂²⋊C₄	C₂²×C₁₉	semidirect
153. #376	C₂²×C₁₉	semidirect
185. C₁₉	#36675	direct (N × H)
188. C₂⁴	#41×C₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₉×#2840	2
C₁₉×#2835	2
C₁₉×#2824	2
C₁₉×#2812	2
C₁₉×#2398	2
C₁₉×#2388	2
C₄×C₁₉×#120	2
#36675	19

Subgroups — C19×#36675 #18042634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#36675 #18042634