Subgroups — C₁₉×#36645 #18042604

Order 4864.

2502 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#533	C₂	1	2	yes	maximal
2	C₁₉×#2868	C₂	1	2	yes	maximal, complemented
3	C₁₉×#2862	C₂	1	2	yes	maximal, complemented
4	C₁₉×#2859	C₂	1	4	yes	maximal, complemented
5	C₁₉×#2857	C₂	1	8	yes	maximal
6	C₄×C₁₉×C₄⋊Q₈	C₂	1	8	yes	maximal, complemented
7	C₁₉×#2285	C₂	1	4	yes	maximal
8	C₂×C₄³×C₁₉	C₂	1	1	yes	maximal, characteristic
9	C₁₉×#533	—	2	16	no	—
10	C₂×C₁₉×C₄⋊Q₈	C₂²	1	4	yes	—
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	8	yes	—
12	C₂×C₁₉×#125	C₂²	1	4	yes	—
13	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	8	yes	—
14	C₂×C₄×Q₈×C₁₉	C₂²	1	8	yes	—
15	C₂×C₁₉×#117	C₂²	1	4	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	4	yes	—
17	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₁₉	C₂²	1	2	yes	—
19	C₁₉×#391	C₂²	1	16	yes	—
20	C₁₉×#389	C₂²	1	32	yes	—
21	C₁₉×#384	C₂²	1	8	yes	—
22	C₁₉×#384	C₂²	1	8	yes	—
23	C₁₉×#382	C₂²	1	8	yes	—
24	C₁₉×#380	C₂²	1	16	yes	—
25	C₄×C₄⋊C₄×C₁₉	C₂²	1	16	yes	—
26	C₄³×C₁₉	C₂²	1	4	yes	—
27	C₄³×C₁₉	C₂²	1	2	yes	—
28	C₄³×C₁₉	C₂²	1	1	yes	characteristic
29	C₄³×C₁₉	C₂²	1	1	yes	characteristic
30	C₂²×Q₈×C₁₉	C₂³	1	8	yes	—
31	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
32	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
33	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
34	C₁₉×C₄⋊Q₈	—	2	16	no	—
35	C₁₉×C₄⋊Q₈	—	2	32	no	—
36	C₁₉×#125	—	2	16	no	—
37	C₁₉×C₂²⋊Q₈	—	2	32	no	—
38	C₄×Q₈×C₁₉	—	2	32	no	—
39	C₁₉×#117	—	2	16	no	—
40	C₂×C₄⋊C₄×C₁₉	C₂³	1	32	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	8	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	16	yes	—
44	C₂×C₂²⋊C₄×C₁₉	C₂³	1	8	yes	—
45	C₂×C₄²×C₁₉	—	2	16	no	—
46	C₂×C₄²×C₁₉	D₄	1	8	yes	—
47	C₂×C₄²×C₁₉	Q₈	1	16	yes	complemented
48	C₂×C₄²×C₁₉	C₂³	1	8	yes	—
49	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
50	C₂×C₄²×C₁₉	C₂³	1	8	yes	—
51	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
54	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₉	C₂³	1	2	yes	—
56	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
57	C₂×C₄²×C₁₉	C₂³	1	4	yes	—
58	C₁₉×#95	C₂³	1	32	yes	—
59	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
60	C₂×Q₈×C₁₉	—	2	16	no	—
61	C₂×Q₈×C₁₉	—	2	32	no	—
62	C₂²×C₄×C₁₉	—	2	8	no	—
63	C₂²×C₄×C₁₉	—	2	16	no	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	8	yes	—
65	C₂²×C₄×C₁₉	C₂×D₄	1	4	yes	—
66	C₂²×C₄×C₁₉	C₂×Q₈	1	16	yes	—
67	C₂²×C₄×C₁₉	C₂×Q₈	1	8	yes	—
68	C₂²×C₄×C₁₉	C₂⁴	1	16	yes	—
69	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
70	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
71	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
73	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
74	C₂²×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₁₉	—	2	16	no	—
76	C₄⋊C₄×C₁₉	—	2	32	no	—
77	C₄⋊C₄×C₁₉	—	2	16	no	—
78	C₄⋊C₄×C₁₉	—	2	64	no	—
79	C₂²⋊C₄×C₁₉	—	2	16	no	—
80	C₄²×C₁₉	—	2	16	no	—
81	C₄²×C₁₉	—	2	8	no	—
82	C₄²×C₁₉	—	2	16	no	—
83	C₄²×C₁₉	C₂×D₄	1	8	yes	—
84	C₄²×C₁₉	C₂×D₄	1	4	yes	—
85	C₄²×C₁₉	C₂×D₄	1	4	yes	—
86	C₄²×C₁₉	C₂×Q₈	1	8	yes	—
87	C₄²×C₁₉	C₂×Q₈	1	16	yes	—
88	C₄²×C₁₉	C₂×Q₈	1	8	yes	—
89	C₄²×C₁₉	#41	1	16	yes	—
90	#36645	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₁₉	—	2	4	no	—
92	C₂³×C₁₉	C₂²×D₄	1	2	yes	—
93	C₂³×C₁₉	C₂²×Q₈	1	4	yes	—
94	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
95	Q₈×C₁₉	—	4	32	no	—
96	C₂×C₄×C₁₉	—	2	4	no	—
97	C₂×C₄×C₁₉	—	2	8	no	—
98	C₂×C₄×C₁₉	—	2	16	no	—
99	C₂×C₄×C₁₉	—	2	32	no	—
100	C₂×C₄×C₁₉	—	2	16	no	—
101	C₂×C₄×C₁₉	—	2	32	no	—
102	C₂×C₄×C₁₉	C₄⋊Q₈	1	16	yes	complemented
103	C₂×C₄×C₁₉	C₂²×D₄	1	4	yes	—
104	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
105	C₂×C₄×C₁₉	C₂²×D₄	1	2	yes	—
106	C₂×C₄×C₁₉	C₂²×D₄	1	4	yes	—
107	C₂×C₄×C₁₉	C₂²×Q₈	1	8	yes	—
108	C₂×C₄×C₁₉	C₂²×Q₈	1	4	yes	—
109	C₂×C₄×C₁₉	C₂²×Q₈	1	4	yes	—
110	C₂×C₄×C₁₉	C₂²×Q₈	1	8	yes	—
111	C₂×C₄×C₁₉	C₂×#41	1	8	yes	—
112	C₂×C₄×C₁₉	C₂×#41	1	16	yes	—
113	C₂×#533	C₂×C₁₉	1	2	yes	—
114	#2868	C₂×C₁₉	1	2	yes	complemented
115	#2862	C₂×C₁₉	1	2	yes	complemented
116	#2859	C₂×C₁₉	1	4	yes	complemented
117	#2857	C₂×C₁₉	1	8	yes	—
118	C₄×C₄⋊Q₈	C₂×C₁₉	1	8	yes	complemented
119	#2285	C₂×C₁₉	1	4	yes	—
120	C₂×C₄³	C₂×C₁₉	1	1	yes	characteristic
121	C₂²×C₁₉	—	2	4	no	—
122	C₂²×C₁₉	—	2	8	no	—
123	C₂²×C₁₉	C₂×C₄⋊Q₈	1	4	yes	—
124	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
125	C₂²×C₁₉	C₂³×Q₈	1	2	yes	—
126	C₂²×C₁₉	C₂²×#41	1	4	yes	—
127	C₄×C₁₉	—	2	4	no	—
128	C₄×C₁₉	—	2	8	no	—
129	C₄×C₁₉	—	4	16	no	—
130	C₄×C₁₉	C₂×C₄⋊Q₈	1	4	yes	—
131	C₄×C₁₉	C₂×C₄⋊Q₈	1	4	yes	—
132	C₄×C₁₉	#532	1	8	yes	—
133	C₄×C₁₉	#533	1	16	yes	complemented
134	#533	—	2	16	no	—
135	C₂×C₄⋊Q₈	C₂²×C₁₉	1	8	yes	—
136	C₂×C₄⋊Q₈	C₂²×C₁₉	1	4	yes	—
137	C₂×#125	C₂²×C₁₉	1	4	yes	—
138	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	8	yes	—
139	C₂×C₄×Q₈	C₂²×C₁₉	1	8	yes	—
140	C₂×#117	C₂²×C₁₉	1	4	yes	—
141	C₂²×C₄²	C₂²×C₁₉	1	4	yes	—
142	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
143	C₂²×C₄²	C₂²×C₁₉	1	2	yes	—
144	#391	C₂²×C₁₉	1	16	yes	—
145	#389	C₂²×C₁₉	1	32	yes	—
146	#384	C₂²×C₁₉	1	8	yes	—
147	#384	C₂²×C₁₉	1	8	yes	—
148	#382	C₂²×C₁₉	1	8	yes	—
149	#380	C₂²×C₁₉	1	16	yes	—
150	C₄×C₄⋊C₄	C₂²×C₁₉	1	16	yes	—
151	C₄³	C₂²×C₁₉	1	4	yes	—
152	C₄³	C₂²×C₁₉	1	2	yes	—
153	C₄³	C₂²×C₁₉	1	1	yes	characteristic
154	C₄³	C₂²×C₁₉	1	1	yes	characteristic
155	C₂×C₁₉	—	2	4	no	—
156	C₂×C₁₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
157	C₂×C₁₉	C₂×#532	1	2	yes	—
158	C₂×C₁₉	C₂×#533	1	4	yes	—
159	C₂²×Q₈	C₂³×C₁₉	1	8	yes	—
160	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
161	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
162	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
163	C₄⋊Q₈	—	2	16	no	—
164	C₄⋊Q₈	—	2	32	no	—
165	#125	—	2	16	no	—
166	C₂²⋊Q₈	—	2	32	no	—
167	C₄×Q₈	—	2	32	no	—
168	#117	—	2	16	no	—
169	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
170	C₂×C₄⋊C₄	C₂³×C₁₉	1	16	yes	—
171	C₂×C₄⋊C₄	C₂³×C₁₉	1	8	yes	—
172	C₂×C₄⋊C₄	C₂³×C₁₉	1	32	yes	—
173	C₂×C₂²⋊C₄	C₂³×C₁₉	1	8	yes	—
174	C₂×C₄²	—	2	16	no	—
175	C₂×C₄²	D₄×C₁₉	1	8	yes	—
176	C₂×C₄²	Q₈×C₁₉	1	16	yes	complemented
177	C₂×C₄²	C₂³×C₁₉	1	8	yes	—
178	C₂×C₄²	C₂³×C₁₉	1	8	yes	—
179	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
180	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
181	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
182	C₂×C₄²	C₂³×C₁₉	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
184	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
185	C₂×C₄²	C₂³×C₁₉	1	2	yes	—
186	C₂×C₄²	C₂³×C₁₉	1	4	yes	—
187	#95	C₂³×C₁₉	1	32	yes	—
188	C₁₉	#36645	1	1	yes	characteristic, direct product factor, Sylow
189	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
190	C₂×Q₈	—	2	16	no	—
191	C₂×Q₈	—	2	32	no	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	16	no	—
194	C₂²×C₄	C₂×D₄×C₁₉	1	8	yes	—
195	C₂²×C₄	C₂×D₄×C₁₉	1	4	yes	—
196	C₂²×C₄	C₂×Q₈×C₁₉	1	16	yes	—
197	C₂²×C₄	C₂×Q₈×C₁₉	1	8	yes	—
198	C₂²×C₄	C₂⁴×C₁₉	1	16	yes	—
199	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
200	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
201	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
202	C₂²×C₄	C₂⁴×C₁₉	1	1	yes	characteristic
203	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
204	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
205	C₄⋊C₄	—	2	16	no	—
206	C₄⋊C₄	—	2	32	no	—
207	C₄⋊C₄	—	2	16	no	—
208	C₄⋊C₄	—	2	64	no	—
209	C₂²⋊C₄	—	2	16	no	—
210	C₄²	—	2	16	no	—
211	C₄²	—	2	8	no	—
212	C₄²	—	2	16	no	—
213	C₄²	C₂×D₄×C₁₉	1	8	yes	—
214	C₄²	C₂×D₄×C₁₉	1	4	yes	—
215	C₄²	C₂×D₄×C₁₉	1	4	yes	—
216	C₄²	C₂×Q₈×C₁₉	1	8	yes	—
217	C₄²	C₂×Q₈×C₁₉	1	16	yes	—
218	C₄²	C₂×Q₈×C₁₉	1	8	yes	—
219	C₄²	#41×C₁₉	1	16	yes	—
220	C₂³	—	2	4	no	—
221	C₂³	C₂²×D₄×C₁₉	1	2	yes	—
222	C₂³	C₂²×Q₈×C₁₉	1	4	yes	—
223	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
224	Q₈	—	4	32	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	2	8	no	—
227	C₂×C₄	—	2	16	no	—
228	C₂×C₄	—	2	32	no	—
229	C₂×C₄	—	2	16	no	—
230	C₂×C₄	—	2	32	no	—
231	C₂×C₄	C₁₉×C₄⋊Q₈	1	16	yes	complemented
232	C₂×C₄	C₂²×D₄×C₁₉	1	4	yes	—
233	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
234	C₂×C₄	C₂²×D₄×C₁₉	1	2	yes	—
235	C₂×C₄	C₂²×D₄×C₁₉	1	4	yes	—
236	C₂×C₄	C₂²×Q₈×C₁₉	1	8	yes	—
237	C₂×C₄	C₂²×Q₈×C₁₉	1	4	yes	—
238	C₂×C₄	C₂²×Q₈×C₁₉	1	4	yes	—
239	C₂×C₄	C₂²×Q₈×C₁₉	1	8	yes	—
240	C₂×C₄	C₂×#41×C₁₉	1	8	yes	—
241	C₂×C₄	C₂×#41×C₁₉	1	16	yes	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	8	no	—
244	C₂²	C₂×C₁₉×C₄⋊Q₈	1	4	yes	—
245	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
246	C₂²	C₂³×Q₈×C₁₉	1	2	yes	—
247	C₂²	C₂²×#41×C₁₉	1	4	yes	—
248	C₄	—	2	4	no	—
249	C₄	—	2	8	no	—
250	C₄	—	4	16	no	—
251	C₄	C₂×C₁₉×C₄⋊Q₈	1	4	yes	—
252	C₄	C₂×C₁₉×C₄⋊Q₈	1	4	yes	—
253	C₄	C₁₉×#532	1	8	yes	—
254	C₄	C₁₉×#533	1	16	yes	complemented
255	C₂	—	2	4	no	—
256	C₂	C₂²×C₁₉×C₄⋊Q₈	1	1	yes	characteristic
257	C₂	C₂×C₁₉×#532	1	2	yes	—
258	C₂	C₂×C₁₉×#533	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₉
socle	C₂³×C₁₉	C₂⁵
prime radical	C₂⁴×C₁₉	C₂⁴
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
2. C₁₉×#2868	C₂	semidirect
3. C₁₉×#2862	C₂	semidirect
4. C₁₉×#2859	C₂	semidirect
6. C₄×C₁₉×C₄⋊Q₈	C₂	semidirect
47. C₂×C₄²×C₁₉	Q₈	semidirect
90. #36645	C₁₉	direct (N × H)
102. C₂×C₄×C₁₉	C₄⋊Q₈	semidirect
114. #2868	C₂×C₁₉	semidirect
115. #2862	C₂×C₁₉	semidirect
116. #2859	C₂×C₁₉	semidirect
118. C₄×C₄⋊Q₈	C₂×C₁₉	semidirect
133. C₄×C₁₉	#533	semidirect
176. C₂×C₄²	Q₈×C₁₉	semidirect
188. C₁₉	#36645	direct (N × H)
231. C₂×C₄	C₁₉×C₄⋊Q₈	semidirect
254. C₄	C₁₉×#533	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#533	2
C₁₉×#2868	2
C₁₉×#2862	2
C₁₉×#2859	2
C₁₉×#2857	2
C₄×C₁₉×C₄⋊Q₈	2
C₁₉×#2285	2
C₂×C₄³×C₁₉	2
#36645	19

Subgroups — C19×#36645 #18042604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₉×#36645 #18042604