Subgroups — #18076282

Order 4864.

422 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563752	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926296	C₂	1	2	yes	maximal
4	C₂×#11926294	C₂	1	2	yes	maximal
5	C₁₉×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₁₉⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
13	#11926299	—	2	2	no	—
14	#11926296	—	2	4	no	—
15	#11926294	—	2	4	no	—
16	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
17	C₄³×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#336	—	2	2	no	—
19	C₂²×C₈×C₁₉	—	2	2	no	—
20	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	D₄	1	2	yes	—
23	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
24	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉⋊C₈	—	2	2	no	—
26	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₁₉⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₁₉×C₄⋊C₈	—	2	2	no	—
31	C₁₉×C₄⋊C₈	—	2	2	no	—
32	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
34	C₄×C₁₉⋊C₈	—	2	2	no	—
35	C₄×C₁₉⋊C₈	—	2	2	no	—
36	C₄×C₁₉⋊C₈	—	2	2	no	—
37	C₄×C₁₉⋊C₈	C₈	1	4	yes	complemented
38	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
39	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
40	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
41	C₄×C₁₉⋊C₈	Q₈	1	2	yes	—
42	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₉	—	4	2	no	—
48	C₂×C₈×C₁₉	—	4	4	no	—
49	C₄²×C₁₉	—	2	4	no	—
50	C₄²×C₁₉	—	2	1	no	—
51	C₄²×C₁₉	—	2	1	no	—
52	C₄²×C₁₉	—	2	4	no	—
53	C₄²×C₁₉	C₂×C₈	1	2	yes	—
54	C₄²×C₁₉	M₁₆	1	2	yes	—
55	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₁₉⋊C₈	—	2	1	no	—
60	C₂×C₁₉⋊C₈	—	2	1	no	—
61	C₂×C₁₉⋊C₈	—	2	1	no	—
62	C₂×C₁₉⋊C₈	—	2	1	no	—
63	C₂×C₁₉⋊C₈	—	4	2	no	—
64	C₂×C₁₉⋊C₈	—	4	4	no	—
65	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
66	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
67	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
68	C₂×C₁₉⋊C₈	C₄⋊C₄	1	2	yes	—
69	C₂×C₁₉⋊C₈	C₂×C₈	1	2	yes	—
70	C₂×C₁₉⋊C₈	C₂×C₈	1	2	yes	—
71	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	—
72	C₂×C₁₉⋊C₈	M₁₆	1	2	yes	—
73	#8807	—	19	1	no	maximal, Sylow
74	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₁₉	—	2	2	no	—
76	C₂×C₄×C₁₉	—	2	2	no	—
77	C₂×C₄×C₁₉	—	2	2	no	—
78	C₂×C₄×C₁₉	—	2	2	no	—
79	C₂×C₄×C₁₉	—	2	2	no	—
80	C₂×C₄×C₁₉	C₄⋊C₈	1	4	yes	—
81	C₂×C₄×C₁₉	#108	1	4	yes	—
82	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
85	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₁₉	#135	1	4	yes	—
93	C₈×C₁₉	—	8	2	no	—
94	C₁₉⋊C₈	—	2	2	no	—
95	C₁₉⋊C₈	—	2	2	no	—
96	C₁₉⋊C₈	—	8	2	no	—
97	C₁₉⋊C₈	C₄⋊C₈	1	4	yes	—
98	C₁₉⋊C₈	C₄⋊C₈	1	4	yes	—
99	#1831	—	19	1	no	—
100	#1831	D₁₉	1	1	yes	characteristic
101	C₄²×C₈	—	19	1	no	—
102	C₂×#335	—	19	2	no	—
103	C₂×#334	—	19	2	no	—
104	C₂²×C₁₉	#384	1	1	yes	characteristic
105	C₂²×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₂²×C₁₉	#423	1	1	yes	characteristic
107	C₂²×C₁₉	#427	1	2	yes	—
108	C₂²×C₁₉	C₂×#108	1	2	yes	—
109	C₄×C₁₉	—	2	4	no	—
110	C₄×C₁₉	—	2	4	no	—
111	C₄×C₁₉	—	2	1	no	—
112	C₄×C₁₉	—	2	1	no	—
113	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₄×C₁₉	#423	1	1	yes	characteristic
116	C₄×C₁₉	#423	1	1	yes	characteristic
117	C₄×C₁₉	#495	1	2	yes	—
118	C₄×C₁₉	#499	1	2	yes	—
119	C₂×C₄⋊C₈	—	19	2	no	—
120	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
121	C₂×C₄×C₈	—	19	1	no	—
122	C₂×C₄×C₈	—	19	1	no	—
123	C₂×C₄×C₈	—	19	1	no	—
124	C₂×C₄×C₈	—	19	1	no	—
125	C₂×C₄×C₈	—	19	1	no	—
126	C₂×C₄×C₈	—	19	1	no	—
127	C₄³	C₂×D₁₉	1	1	yes	characteristic
128	#336	—	2	2	no	—
129	#336	—	38	2	no	—
130	#335	—	38	4	no	—
131	#334	—	38	4	no	—
132	C₂×C₁₉	#1564	1	2	yes	—
133	C₂×C₁₉	#1568	1	2	yes	—
134	C₂×C₁₉	#1842	1	1	yes	characteristic
135	C₂×C₁₉	#1846	1	1	yes	characteristic
136	C₂×C₁₉	#1848	1	1	yes	characteristic
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	19	1	no	—
139	C₂²×C₈	—	19	1	no	—
140	C₂²×C₈	—	19	1	no	—
141	C₂²×C₈	—	19	1	no	—
142	C₂²×C₈	—	38	2	no	—
143	C₂×C₄²	Dic₃₈	1	2	yes	—
144	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
145	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
147	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	—	2	2	no	—
150	C₄⋊C₈	—	38	2	no	—
151	C₄⋊C₈	—	38	2	no	—
152	C₄×C₈	—	19	4	no	—
153	C₄×C₈	—	19	2	no	—
154	C₄×C₈	—	19	2	no	—
155	C₄×C₈	—	19	2	no	—
156	C₄×C₈	—	19	2	no	—
157	C₄×C₈	—	38	2	no	—
158	C₄×C₈	—	38	2	no	—
159	C₄×C₈	—	38	2	no	—
160	C₁₉	#8807	1	1	yes	characteristic, complemented, Sylow
161	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
162	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
163	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
164	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
165	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	4	4	no	—
168	C₂×C₈	—	19	2	no	—
169	C₂×C₈	—	19	2	no	—
170	C₂×C₈	—	19	2	no	—
171	C₂×C₈	—	19	2	no	—
172	C₂×C₈	—	19	2	no	—
173	C₂×C₈	—	19	2	no	—
174	C₂×C₈	—	19	2	no	—
175	C₂×C₈	—	19	2	no	—
176	C₂×C₈	—	38	1	no	—
177	C₂×C₈	—	38	1	no	—
178	C₂×C₈	—	38	1	no	—
179	C₂×C₈	—	38	1	no	—
180	C₂×C₈	—	76	2	no	—
181	C₂×C₈	—	76	4	no	—
182	C₄²	—	2	4	no	—
183	C₄²	—	2	4	no	—
184	C₄²	—	2	1	no	—
185	C₄²	—	2	1	no	—
186	C₄²	C₈×D₁₉	1	2	yes	—
187	C₄²	#64515	1	2	yes	—
188	C₄²	D₄×D₁₉	1	1	yes	characteristic
189	C₄²	D₄×D₁₉	1	1	yes	characteristic
190	C₄²	Q₈×D₁₉	1	1	yes	characteristic
191	C₄²	Q₈×D₁₉	1	1	yes	characteristic
192	C₂³	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	Dic₁₉⋊C₈	1	4	yes	—
199	C₂×C₄	#10597279	1	4	yes	—
200	C₂×C₄	#10597334	1	1	yes	characteristic
201	C₂×C₄	#10597334	1	1	yes	characteristic
202	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
203	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
204	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
205	C₂×C₄	C₂×#64515	1	1	yes	characteristic
206	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
207	C₂×C₄	M₁₆×D₁₉	1	1	yes	characteristic
208	C₂×C₄	#10597386	1	1	yes	characteristic
209	C₂×C₄	#10597390	1	1	yes	characteristic
210	C₂×C₄	#10597396	1	4	yes	—
211	C₈	—	8	2	no	—
212	C₈	—	19	4	no	—
213	C₈	—	19	4	no	—
214	C₈	—	38	2	no	—
215	C₈	—	38	2	no	—
216	C₈	—	152	2	no	—
217	C₂²	#11926413	1	1	yes	characteristic
218	C₂²	C₂×Dic₁₉⋊C₈	1	1	yes	characteristic
219	C₂²	#11926567	1	1	yes	characteristic
220	C₂²	#11926580	1	2	yes	—
221	C₂²	C₂×#10597279	1	2	yes	—
222	C₄	—	2	4	no	—
223	C₄	—	2	4	no	—
224	C₄	—	2	1	no	—
225	C₄	—	2	1	no	—
226	C₄	C₄⋊C₈×D₁₉	1	1	yes	characteristic
227	C₄	C₄⋊C₈×D₁₉	1	1	yes	characteristic
228	C₄	#11926566	1	1	yes	characteristic
229	C₄	#11926566	1	1	yes	characteristic
230	C₄	#11926926	1	2	yes	—
231	C₄	#11926943	1	2	yes	—
232	C₂	#14562381	1	2	yes	—
233	C₂	#14562396	1	2	yes	—
234	C₂	#14563786	1	1	yes	characteristic
235	C₂	#14563797	1	1	yes	characteristic
236	C₂	#14563806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1831	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1831	D₁₉
hypocenter	C₁₉	#8807
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1831

Complements

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

N	H ≅ G/N	type
37. C₄×C₁₉⋊C₈	C₈	semidirect
160. C₁₉	#8807	semidirect

Maximal subgroups

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

subgroup	index
#14563752	2
C₄²×C₁₉⋊C₈	2
C₂×#11926296	2
C₂×#11926294	2
C₁₉×#1831	2
#8807	19