Subgroups — #18081092

Order 4864.

382 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564354	C₂	1	2	yes	maximal
2	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926265	C₂	1	1	yes	maximal, characteristic
4	#14561476	C₂	1	2	yes	maximal
5	C₄×C₁₉×#106	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
10	#11926299	—	2	2	no	—
11	#11926297	—	2	2	no	—
12	#11926265	—	2	2	no	—
13	C₂×C₁₉×#106	C₄	1	1	yes	characteristic
14	C₂×C₁₉×#106	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
17	C₁₉×#340	C₄	1	2	yes	—
18	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
21	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₉	—	2	2	no	—
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂²×C₁₉⋊C₈	—	2	1	no	—
25	C₂²×C₁₉⋊C₈	—	2	2	no	—
26	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
27	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
28	C₁₉×#106	—	2	2	no	—
29	C₁₉×#106	C₈	1	4	yes	complemented
30	C₄×C₈×C₁₉	D₄	1	2	yes	—
31	C₄×C₈×C₁₉	Q₈	1	2	yes	—
32	C₁₉×#95	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
34	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
35	C₄⋊(C₁₉⋊C₈)	—	4	4	no	—
36	C₄×C₁₉⋊C₈	—	2	1	no	—
37	C₄×C₁₉⋊C₈	—	2	1	no	—
38	C₄×C₁₉⋊C₈	—	2	1	no	—
39	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
40	C₂²×C₄×C₁₉	—	2	2	no	—
41	C₂²×C₄×C₁₉	—	2	2	no	—
42	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
45	C₂×C₈×C₁₉	—	2	1	no	—
46	C₂×C₈×C₁₉	—	2	1	no	—
47	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
51	C₄⋊C₄×C₁₉	—	2	2	no	—
52	C₄⋊C₄×C₁₉	—	2	2	no	—
53	C₄⋊C₄×C₁₉	—	2	2	no	—
54	C₄⋊C₄×C₁₉	C₂×C₈	1	4	yes	—
55	C₄²×C₁₉	—	4	2	no	—
56	C₄²×C₁₉	—	4	2	no	—
57	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₁₉	#41	1	1	yes	characteristic
61	C₂×C₁₉⋊C₈	—	2	1	no	—
62	C₂×C₁₉⋊C₈	—	2	1	no	—
63	C₂×C₁₉⋊C₈	—	2	2	no	—
64	C₂×C₁₉⋊C₈	—	2	2	no	—
65	C₂×C₁₉⋊C₈	—	2	2	no	—
66	C₂×C₁₉⋊C₈	—	4	2	no	—
67	C₂×C₁₉⋊C₈	—	4	2	no	—
68	C₂×C₁₉⋊C₈	—	4	2	no	—
69	C₂×C₁₉⋊C₈	—	4	2	no	—
70	#9596	—	19	1	no	maximal, Sylow
71	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	—	2	4	no	—
74	C₂×C₄×C₁₉	—	4	2	no	—
75	C₂×C₄×C₁₉	—	4	2	no	—
76	C₂×C₄×C₁₉	—	4	2	no	—
77	C₂×C₄×C₁₉	—	4	2	no	—
78	C₂×C₄×C₁₉	—	4	2	no	—
79	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
84	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
88	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
89	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
90	C₂×C₄×C₁₉	#137	1	2	yes	—
91	C₈×C₁₉	—	2	2	no	—
92	C₈×C₁₉	C₄⋊C₈	1	4	yes	complemented
93	C₁₉⋊C₈	—	4	2	no	—
94	C₁₉⋊C₈	—	4	1	no	—
95	C₁₉⋊C₈	—	4	1	no	—
96	C₁₉⋊C₈	—	8	2	no	—
97	#1939	—	19	2	no	—
98	C₄×#106	D₁₉	1	1	yes	characteristic
99	C₂×#335	—	19	1	no	—
100	C₂×#322	—	19	1	no	—
101	#1279	—	19	2	no	—
102	C₂²×C₁₉	#389	1	1	yes	characteristic
103	C₂²×C₁₉	C₈×D₄	1	1	yes	characteristic
104	C₂²×C₁₉	#436	1	1	yes	characteristic
105	C₂²×C₁₉	#440	1	2	yes	—
106	C₂²×C₁₉	#444	1	2	yes	—
107	C₄×C₁₉	—	2	1	no	—
108	C₄×C₁₉	—	2	1	no	—
109	C₄×C₁₉	—	4	4	no	—
110	C₄×C₁₉	—	8	2	no	—
111	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₄×C₁₉	#424	1	1	yes	characteristic
113	C₄×C₁₉	C₈×Q₈	1	1	yes	characteristic
114	C₄×C₁₉	#446	1	1	yes	characteristic
115	C₄×C₁₉	#469	1	1	yes	characteristic
116	C₄×C₁₉	#470	1	1	yes	characteristic
117	C₄×C₁₉	#501	1	2	yes	—
118	C₂×#106	Dic₁₉	1	1	yes	characteristic
119	C₂×#106	Dic₁₉	1	1	yes	characteristic
120	C₂×C₄⋊C₈	—	19	2	no	—
121	C₂×C₄⋊C₈	—	38	2	no	—
122	C₂×C₄×C₈	—	19	1	no	—
123	C₂×C₄×C₈	—	19	1	no	—
124	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
126	#340	Dic₁₉	1	2	yes	—
127	#336	—	38	2	no	—
128	#335	—	38	2	no	—
129	#322	—	38	2	no	—
130	C₂×C₁₉	SD₁₆⋊C₈	1	2	yes	—
131	C₂×C₁₉	#1592	1	2	yes	—
132	C₂×C₁₉	#1939	1	1	yes	characteristic
133	C₂×C₁₉	#1944	1	1	yes	characteristic
134	C₂×C₁₉	#1946	1	1	yes	characteristic
135	C₂²×C₈	—	19	1	no	—
136	C₂²×C₈	—	19	1	no	—
137	C₂²×C₈	—	38	1	no	—
138	C₂²×C₈	—	38	2	no	—
139	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
140	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
141	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
142	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
143	C₂×C₄²	—	2	2	no	—
144	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
145	#106	—	2	2	no	—
146	#106	C₁₉⋊C₈	1	4	yes	complemented
147	C₄⋊C₈	—	38	2	no	—
148	C₄⋊C₈	—	38	2	no	—
149	C₄⋊C₈	—	76	4	no	—
150	C₄×C₈	—	19	2	no	—
151	C₄×C₈	—	38	1	no	—
152	C₄×C₈	—	38	1	no	—
153	C₄×C₈	—	38	1	no	—
154	C₄×C₈	Dic₃₈	1	2	yes	—
155	C₄×C₈	D₇₆	1	2	yes	—
156	#95	—	2	2	no	—
157	C₁₉	#9596	1	1	yes	characteristic, complemented, Sylow
158	C₂²×C₄	—	2	2	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
161	C₂²×C₄	#64543	1	1	yes	characteristic
162	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	38	1	no	—
166	C₂×C₈	—	38	1	no	—
167	C₂×C₈	—	38	2	no	—
168	C₂×C₈	—	38	2	no	—
169	C₂×C₈	—	38	2	no	—
170	C₂×C₈	—	76	2	no	—
171	C₂×C₈	—	76	2	no	—
172	C₂×C₈	—	76	2	no	—
173	C₂×C₈	—	76	2	no	—
174	C₂×C₈	C₂×C₁₉⋊C₈	1	2	yes	—
175	C₂×C₈	#64520	1	2	yes	—
176	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
177	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	C₂×C₁₉⋊C₈	1	4	yes	—
182	C₄²	—	4	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
185	C₄²	C₂×D₇₆	1	1	yes	characteristic
186	C₄²	Q₈×D₁₉	1	1	yes	characteristic
187	C₄²	#64545	1	1	yes	characteristic
188	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	4	2	no	—
192	C₂×C₄	—	4	2	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	#10597334	1	1	yes	characteristic
197	C₂×C₄	#10597336	1	1	yes	characteristic
198	C₂×C₄	#10597341	1	1	yes	characteristic
199	C₂×C₄	#10597343	1	1	yes	characteristic
200	C₂×C₄	#10597360	1	2	yes	—
201	C₂×C₄	#10597361	1	2	yes	—
202	C₂×C₄	C₂²×C₁₉⋊C₈	1	1	yes	characteristic
203	C₂×C₄	C₂×#64520	1	1	yes	characteristic
204	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
205	C₂×C₄	Q₈×Dic₁₉	1	1	yes	characteristic
206	C₂×C₄	#10597394	1	1	yes	characteristic
207	C₂×C₄	#10597394	1	1	yes	characteristic
208	C₈	—	2	2	no	—
209	C₈	—	76	2	no	—
210	C₈	—	76	1	no	—
211	C₈	—	76	1	no	—
212	C₈	—	152	2	no	—
213	C₈	C₄⋊(C₁₉⋊C₈)	1	4	yes	complemented
214	C₂²	#11926443	1	1	yes	characteristic
215	C₂²	D₄×C₁₉⋊C₈	1	1	yes	characteristic
216	C₂²	#11926622	1	1	yes	characteristic
217	C₂²	#11926645	1	2	yes	—
218	C₂²	#11926666	1	2	yes	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	4	4	no	—
222	C₄	—	8	2	no	—
223	C₄	C₂×C₄⋊(C₁₉⋊C₈)	1	1	yes	characteristic
224	C₄	#11926571	1	1	yes	characteristic
225	C₄	Q₈×C₁₉⋊C₈	1	1	yes	characteristic
226	C₄	#11926675	1	1	yes	characteristic
227	C₄	#11926798	1	1	yes	characteristic
228	C₄	#11926803	1	1	yes	characteristic
229	C₄	#11926951	1	2	yes	—
230	C₂	SD₁₆⋊(C₁₉⋊C₈)	1	2	yes	—
231	C₂	#14562538	1	2	yes	—
232	C₂	#14564354	1	1	yes	characteristic
233	C₂	#14564379	1	1	yes	characteristic
234	C₂	#14564389	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#106	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#106	D₁₉
hypocenter	C₁₉	#9596
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#106

Complements

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

N	H ≅ G/N	type
29. C₁₉×#106	C₈	semidirect
92. C₈×C₁₉	C₄⋊C₈	semidirect
146. #106	C₁₉⋊C₈	semidirect
157. C₁₉	#9596	semidirect
213. C₈	C₄⋊(C₁₉⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#14564354	2
C₂×#11926297	2
C₂×#11926265	2
#14561476	2
C₄×C₁₉×#106	2
#9596	19