Subgroups — #18081432

Order 4864.

482 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564572	C₂	1	2	yes	maximal
2	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₁₉⋊C₈)	C₂	1	2	yes	maximal
4	C₂×#11926266	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Q₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
9	#11926299	—	2	4	no	—
10	#11926297	—	2	2	no	—
11	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
12	#11926266	—	2	2	no	—
13	C₂²×Q₁₆×C₁₉	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₁₉	C₂²	1	2	yes	—
15	C₄×Q₁₆×C₁₉	—	2	2	no	—
16	C₄×Q₁₆×C₁₉	—	2	2	no	—
17	C₂×C₁₉×#107	C₄	1	1	yes	characteristic
18	C₂×C₁₉×#103	C₄	1	2	yes	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
21	C₂×Q₁₆×C₁₉	—	2	4	no	—
22	C₂×Q₁₆×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₉	—	2	2	no	—
26	C₄×Q₈×C₁₉	—	2	2	no	—
27	C₄×Q₈×C₁₉	—	2	2	no	—
28	C₄×Q₈×C₁₉	—	2	2	no	—
29	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
30	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	—	2	2	no	—
32	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
33	C₂²×C₁₉⋊C₈	—	2	1	no	—
34	C₂²×C₁₉⋊C₈	—	2	2	no	—
35	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
36	C₁₉×#107	—	2	2	no	—
37	C₁₉×#103	—	2	4	no	—
38	C₄×C₈×C₁₉	D₄	1	2	yes	—
39	C₄×C₈×C₁₉	D₄	1	2	yes	—
40	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
41	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
42	C₄×C₁₉⋊C₈	—	2	1	no	—
43	C₄×C₁₉⋊C₈	—	2	1	no	—
44	C₄×C₁₉⋊C₈	—	2	1	no	—
45	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
46	C₂×Q₈×C₁₉	—	2	2	no	—
47	C₂×Q₈×C₁₉	—	2	2	no	—
48	C₂×Q₈×C₁₉	—	2	2	no	—
49	C₂×Q₈×C₁₉	—	2	2	no	—
50	C₂×Q₈×C₁₉	—	2	2	no	—
51	C₂×Q₈×C₁₉	M₁₆	1	4	yes	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
57	Q₁₆×C₁₉	—	2	8	no	—
58	C₂×C₈×C₁₉	—	2	2	no	—
59	C₂×C₈×C₁₉	—	2	2	no	—
60	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
61	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₄⋊C₄×C₁₉	—	2	2	no	—
63	C₄⋊C₄×C₁₉	—	2	2	no	—
64	C₄⋊C₄×C₁₉	—	4	2	no	—
65	C₄⋊C₄×C₁₉	—	4	2	no	—
66	C₄²×C₁₉	—	4	2	no	—
67	C₄²×C₁₉	—	4	2	no	—
68	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₉	#41	1	1	yes	characteristic
72	C₂×C₁₉⋊C₈	—	2	2	no	—
73	C₂×C₁₉⋊C₈	—	2	2	no	—
74	C₂×C₁₉⋊C₈	—	2	2	no	—
75	C₂×C₁₉⋊C₈	—	4	2	no	—
76	C₂×C₁₉⋊C₈	—	4	2	no	—
77	C₂×C₁₉⋊C₈	—	4	4	no	—
78	C₂×C₁₉⋊C₈	Q₁₆	1	4	yes	complemented
79	#9662	—	19	1	no	maximal, Sylow
80	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
81	Q₈×C₁₉	—	2	4	no	—
82	Q₈×C₁₉	—	2	4	no	—
83	Q₈×C₁₉	—	4	2	no	—
84	Q₈×C₁₉	—	4	2	no	—
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	—	2	4	no	—
87	C₂×C₄×C₁₉	—	4	2	no	—
88	C₂×C₄×C₁₉	—	4	2	no	—
89	C₂×C₄×C₁₉	—	4	2	no	—
90	C₂×C₄×C₁₉	—	4	4	no	—
91	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
95	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
96	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
98	C₂×C₄×C₁₉	#131	1	2	yes	—
99	C₂×C₄×C₁₉	C₂×Q₁₆	1	2	yes	—
100	C₂×C₄×C₁₉	#135	1	2	yes	—
101	C₈×C₁₉	—	2	2	no	—
102	C₈×C₁₉	—	2	2	no	—
103	C₁₉⋊C₈	—	2	4	no	—
104	C₁₉⋊C₈	—	4	2	no	—
105	C₁₉⋊C₈	—	8	2	no	—
106	C₂×C₄×Q₁₆	D₁₉	1	1	yes	characteristic
107	#1984	—	19	2	no	—
108	C₂×#335	—	19	1	no	—
109	C₂×Q₈⋊C₈	—	19	2	no	—
110	C₂×#322	—	19	1	no	—
111	C₂²×C₁₉	#390	1	1	yes	characteristic
112	C₂²×C₁₉	#435	1	2	yes	—
113	C₂²×C₁₉	C₄×Q₁₆	1	2	yes	—
114	C₂²×C₁₉	#444	1	2	yes	—
115	C₄×C₁₉	—	2	1	no	—
116	C₄×C₁₉	—	2	1	no	—
117	C₄×C₁₉	—	4	4	no	—
118	C₄×C₁₉	—	8	2	no	—
119	C₄×C₁₉	#407	1	1	yes	characteristic
120	C₄×C₁₉	#408	1	1	yes	characteristic
121	C₄×C₁₉	#435	1	2	yes	—
122	C₄×C₁₉	#467	1	2	yes	—
123	C₄×C₁₉	#494	1	1	yes	characteristic
124	C₄×C₁₉	#495	1	1	yes	characteristic
125	C₂²×Q₁₆	Dic₁₉	1	1	yes	characteristic
126	C₂×C₄×Q₈	C₂×D₁₉	1	2	yes	—
127	C₄×Q₁₆	—	2	2	no	—
128	C₄×Q₁₆	—	2	2	no	—
129	C₂×#107	Dic₁₉	1	1	yes	characteristic
130	C₂×C₄⋊C₈	—	19	2	no	—
131	C₂×#103	Dic₁₉	1	2	yes	—
132	C₂×C₄×C₈	—	19	1	no	—
133	C₂×C₄×C₈	—	19	1	no	—
134	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
135	#336	—	38	4	no	—
136	#335	—	38	2	no	—
137	Q₈⋊C₈	—	38	4	no	—
138	#322	—	38	2	no	—
139	C₂×C₁₉	#1584	1	4	yes	—
140	C₂×C₁₉	#1955	1	1	yes	characteristic
141	C₂×C₁₉	#1969	1	1	yes	characteristic
142	C₂×C₁₉	#1975	1	1	yes	characteristic
143	C₂²×Q₈	C₂×Dic₁₉	1	2	yes	—
144	C₂×Q₁₆	—	2	4	no	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂²×C₈	—	19	2	no	—
147	C₂²×C₈	—	38	1	no	—
148	C₂²×C₈	—	38	2	no	—
149	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
150	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
151	C₄×Q₈	—	2	2	no	—
152	C₄×Q₈	—	2	2	no	—
153	C₄×Q₈	—	2	2	no	—
154	C₄×Q₈	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
159	#107	—	2	2	no	—
160	C₄⋊C₈	—	38	2	no	—
161	C₄⋊C₈	—	38	2	no	—
162	#103	—	2	4	no	—
163	C₄×C₈	—	19	2	no	—
164	C₄×C₈	—	38	1	no	—
165	C₄×C₈	—	38	1	no	—
166	C₄×C₈	—	38	1	no	—
167	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
168	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
169	C₁₉	#9662	1	1	yes	characteristic, complemented, Sylow
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	2	2	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	#64520	1	4	yes	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
179	C₂²×C₄	#64543	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
181	Q₁₆	—	2	8	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	19	4	no	—
185	C₂×C₈	—	38	2	no	—
186	C₂×C₈	—	38	2	no	—
187	C₂×C₈	—	38	2	no	—
188	C₂×C₈	—	76	2	no	—
189	C₂×C₈	—	76	2	no	—
190	C₂×C₈	—	76	4	no	—
191	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
192	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	—	4	2	no	—
199	C₄²	D₄×D₁₉	1	1	yes	characteristic
200	C₄²	#64543	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
203	C₂³	D₄×Dic₁₉	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	2	4	no	—
206	Q₈	—	4	2	no	—
207	Q₈	—	4	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	Q₁₆×D₁₉	1	2	yes	—
215	C₂×C₄	#10597365	1	2	yes	—
216	C₂×C₄	C₂×#64520	1	2	yes	—
217	C₂×C₄	D₄×Dic₁₉	1	1	yes	characteristic
218	C₂×C₄	#10597382	1	1	yes	characteristic
219	C₂×C₄	#10597384	1	1	yes	characteristic
220	C₂×C₄	#10597384	1	1	yes	characteristic
221	C₂×C₄	#10597386	1	1	yes	characteristic
222	C₂×C₄	#10597394	1	2	yes	—
223	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
224	C₈	—	2	2	no	—
225	C₈	—	2	2	no	—
226	C₈	—	38	4	no	—
227	C₈	—	76	2	no	—
228	C₈	—	152	2	no	—
229	C₂²	#11926447	1	1	yes	characteristic
230	C₂²	#11926616	1	2	yes	—
231	C₂²	Q₁₆×Dic₁₉	1	2	yes	—
232	C₂²	#11926666	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	4	4	no	—
236	C₄	—	8	2	no	—
237	C₄	#11926508	1	1	yes	characteristic
238	C₄	#11926511	1	1	yes	characteristic
239	C₄	#11926616	1	2	yes	—
240	C₄	#11926782	1	2	yes	—
241	C₄	#11926922	1	1	yes	characteristic
242	C₄	#11926929	1	1	yes	characteristic
243	C₂	#14562486	1	4	yes	—
244	C₂	#14564434	1	1	yes	characteristic
245	C₂	#14564497	1	1	yes	characteristic
246	C₂	#14564529	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₄×Q₁₆×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₄×Q₁₆	D₁₉
hypocenter	C₁₉	#9662
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
78. C₂×C₁₉⋊C₈	Q₁₆	semidirect
169. C₁₉	#9662	semidirect

Maximal subgroups

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

subgroup	index
#14564572	2
C₂×#11926297	2
C₂×Q₈⋊(C₁₉⋊C₈)	2
C₂×#11926266	2
C₂×C₄×Q₁₆×C₁₉	2
#9662	19