Subgroups — #18075678

Order 4864.

378 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563773	C₂	1	1	yes	maximal, characteristic
2	#14562383	C₂	1	1	yes	maximal, characteristic
3	#14562064	C₂	1	2	yes	maximal
4	#14561532	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	#11926568	C₂²	1	2	yes	—
8	#11926309	—	2	1	no	—
9	#11926309	—	2	1	no	—
10	#11926297	—	2	1	no	—
11	#11926295	—	2	1	no	—
12	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
13	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₁₉	—	2	1	no	—
15	C₈×Q₈×C₁₉	—	2	1	no	—
16	C₈×Q₈×C₁₉	C₄	1	2	yes	—
17	C₈×Q₈×C₁₉	C₄	1	2	yes	—
18	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
19	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₉	—	2	1	no	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₂²×Q₈×C₁₉	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	—	2	1	no	—
24	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₄×Q₈×C₁₉	—	2	1	no	—
27	C₄×Q₈×C₁₉	—	2	1	no	—
28	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₄×Q₈×C₁₉	D₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
31	C₂×C₄⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂×#64520	—	2	2	no	—
35	C₁₉×C₄⋊C₈	—	2	2	no	—
36	C₁₉×C₄⋊C₈	—	2	2	no	—
37	C₁₉×C₄⋊C₈	—	2	1	no	—
38	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₉	—	2	2	no	—
40	C₄×C₈×C₁₉	—	2	2	no	—
41	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
45	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
49	C₄×C₁₉⋊C₈	—	2	2	no	—
50	C₂×Q₈×C₁₉	—	2	4	no	—
51	C₂×Q₈×C₁₉	—	2	1	no	—
52	C₂×Q₈×C₁₉	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₁₉	—	2	1	no	—
54	C₂²×C₄×C₁₉	—	2	1	no	—
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
58	C₂×C₈×C₁₉	—	2	2	no	—
59	C₂×C₈×C₁₉	—	2	2	no	—
60	C₂×C₈×C₁₉	—	2	2	no	—
61	C₂×C₈×C₁₉	—	4	2	no	—
62	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₄⋊C₄×C₁₉	—	2	2	no	—
67	C₄⋊C₄×C₁₉	—	2	2	no	—
68	C₄⋊C₄×C₁₉	—	2	1	no	—
69	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
70	C₄²×C₁₉	—	2	2	no	—
71	C₄²×C₁₉	—	2	2	no	—
72	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₉	#41	1	1	yes	characteristic
76	#64520	—	4	2	no	—
77	#64520	—	4	2	no	—
78	C₂×C₁₉⋊C₈	—	4	2	no	—
79	C₂×C₁₉⋊C₈	—	4	2	no	—
80	#8661	—	19	1	no	maximal, Sylow
81	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
82	Q₈×C₁₉	—	2	4	no	—
83	Q₈×C₁₉	—	2	2	no	—
84	Q₈×C₁₉	#108	1	4	yes	—
85	C₂×C₄×C₁₉	—	2	4	no	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	2	2	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	4	1	no	—
90	C₂×C₄×C₁₉	—	4	2	no	—
91	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
94	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
95	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₉	#135	1	2	yes	—
98	C₂×C₄×C₁₉	#137	1	2	yes	—
99	C₈×C₁₉	—	2	1	no	—
100	C₈×C₁₉	—	2	1	no	—
101	C₈×C₁₉	—	2	1	no	—
102	C₈×C₁₉	—	2	1	no	—
103	C₈×C₁₉	—	4	2	no	—
104	C₁₉⋊C₈	—	8	2	no	—
105	C₁₉⋊C₈	—	8	2	no	—
106	C₂×C₈×Q₈	D₁₉	1	1	yes	characteristic
107	#1837	—	19	1	no	—
108	#1564	—	19	1	no	—
109	#1487	—	19	2	no	—
110	#1301	—	19	2	no	—
111	C₂²×C₁₉	—	2	1	no	—
112	C₂²×C₁₉	—	2	1	no	—
113	C₂²×C₁₉	#380	1	1	yes	characteristic
114	C₂²×C₁₉	#479	1	2	yes	—
115	C₄×C₁₉	—	2	4	no	—
116	C₄×C₁₉	—	2	1	no	—
117	C₄×C₁₉	—	2	1	no	—
118	C₄×C₁₉	—	2	1	no	—
119	C₄×C₁₉	—	4	2	no	—
120	C₄×C₁₉	#416	1	1	yes	characteristic
121	C₄×C₁₉	#419	1	1	yes	characteristic
122	C₄×C₁₉	C₂×#108	1	2	yes	—
123	C₄×C₁₉	#464	1	2	yes	—
124	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
125	C₈×Q₈	—	2	1	no	—
126	C₈×Q₈	—	2	1	no	—
127	C₈×Q₈	Dic₁₉	1	2	yes	—
128	C₈×Q₈	Dic₁₉	1	2	yes	—
129	#431	—	19	2	no	—
130	#423	—	19	2	no	—
131	C₂×C₄⋊C₈	—	2	1	no	—
132	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
133	C₂×C₄×C₈	—	2	1	no	—
134	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
135	#341	—	38	1	no	—
136	#341	—	38	1	no	—
137	#335	—	38	1	no	—
138	#334	—	38	1	no	—
139	Q₈⋊C₈	—	38	4	no	—
140	C₂×C₁₉	—	2	2	no	—
141	C₂×C₁₉	#1805	1	1	yes	characteristic
142	C₂×C₁₉	#1810	1	1	yes	characteristic
143	C₂×C₁₉	#1814	1	1	yes	characteristic
144	C₂²×Q₈	Dic₃₈	1	1	yes	characteristic
145	C₂×M₁₆	—	38	2	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
148	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
149	C₄×Q₈	—	2	1	no	—
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	D₇₆	1	2	yes	—
152	C₄×Q₈	C₂×Dic₁₉	1	2	yes	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	Dic₃₈	1	1	yes	characteristic
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	38	2	no	—
161	C₄⋊C₈	—	38	2	no	—
162	C₄⋊C₈	—	38	2	no	—
163	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
164	C₂²⋊C₈	—	38	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	38	2	no	—
168	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
169	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
170	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
171	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
172	C₁₉	#8661	1	1	yes	characteristic, complemented, Sylow
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	C₄⋊Dic₁₉	1	2	yes	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
179	C₂²×C₄	#64541	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
181	M₁₆	—	76	2	no	—
182	M₁₆	—	76	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	76	2	no	—
188	C₂×C₈	—	76	2	no	—
189	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	C₂×D₇₆	1	1	yes	characteristic
200	C₄²	#64541	1	1	yes	characteristic
201	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
202	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
203	C₂³	#10597370	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	2	2	no	—
206	Q₈	#10597276	1	4	yes	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	#10597370	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
215	C₂×C₄	#10597372	1	1	yes	characteristic
216	C₂×C₄	#10597376	1	1	yes	characteristic
217	C₂×C₄	#10597376	1	1	yes	characteristic
218	C₂×C₄	#10597388	1	2	yes	—
219	C₂×C₄	#10597396	1	2	yes	—
220	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	4	2	no	—
226	C₈	—	152	2	no	—
227	C₈	—	152	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	#11926404	1	1	yes	characteristic
231	C₂²	#11926851	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	2	no	—
237	C₄	#11926540	1	1	yes	characteristic
238	C₄	#11926553	1	1	yes	characteristic
239	C₄	C₂×#10597276	1	2	yes	—
240	C₄	#11926766	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#14563650	1	1	yes	characteristic
243	C₂	#14563673	1	1	yes	characteristic
244	C₂	#14563689	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₈×Q₈×C₁₉	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂³×C₁₉	C₂²⋊Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₈×Q₈	D₁₉
hypocenter	C₁₉	#8661
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
172. C₁₉	#8661	semidirect

Maximal subgroups

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

subgroup	index
#14563773	2
#14562383	2
#14562064	2
#14561532	2
C₂×C₈×Q₈×C₁₉	2
#8661	19