Subgroups — #18075608

Order 4864.

438 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563763	C₂	1	1	yes	maximal, characteristic
2	#14563640	C₂	1	2	yes	maximal
3	C₂×#11926295	C₂	1	1	yes	maximal, characteristic
4	#14561519	C₂	1	2	yes	maximal
5	C₂×C₈×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#10597292	—	2	2	no	—
8	#11926414	C₂²	1	2	yes	—
9	#11926307	—	2	1	no	—
10	#11926307	—	2	1	no	—
11	#11926306	—	2	2	no	—
12	#11926295	—	2	2	no	—
13	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₁₉	—	2	2	no	—
15	C₈×Q₈×C₁₉	C₄	1	4	yes	complemented
16	C₂×C₁₉×C₄⋊C₈	—	2	1	no	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₉	—	2	1	no	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	—	2	1	no	—
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
24	C₄×Q₈×C₁₉	—	2	1	no	—
25	C₄×Q₈×C₁₉	—	2	1	no	—
26	C₄×Q₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₄×Q₈×C₁₉	Q₈	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
32	C₂×C₄⋊Dic₁₉	—	2	2	no	—
33	C₂×C₄⋊Dic₁₉	—	2	2	no	—
34	C₂×C₄⋊Dic₁₉	—	2	2	no	—
35	C₂²×C₁₉⋊C₈	—	2	2	no	—
36	C₁₉×C₄⋊C₈	—	2	4	no	—
37	C₁₉×C₄⋊C₈	—	2	1	no	—
38	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₉	—	2	4	no	—
40	C₄×C₈×C₁₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₉	D₄	1	2	yes	—
42	#10597292	—	4	4	no	—
43	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
44	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
45	C₂×Q₈×C₁₉	—	2	1	no	—
46	C₂×Q₈×C₁₉	—	2	2	no	—
47	C₂×Q₈×C₁₉	C₄⋊C₄	1	2	yes	—
48	C₂×Q₈×C₁₉	SD₁₆	1	4	yes	—
49	C₂²×C₄×C₁₉	—	2	1	no	—
50	C₂²×C₄×C₁₉	—	2	1	no	—
51	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
54	C₂²×Dic₁₉	—	4	2	no	—
55	C₂×C₈×C₁₉	—	2	2	no	—
56	C₂×C₈×C₁₉	—	2	2	no	—
57	C₂×C₈×C₁₉	—	2	2	no	—
58	C₂×C₈×C₁₉	—	4	2	no	—
59	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
61	C₄⋊C₄×C₁₉	—	2	2	no	—
62	C₄⋊C₄×C₁₉	—	2	1	no	—
63	C₄⋊C₄×C₁₉	—	2	2	no	—
64	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
65	C₄²×C₁₉	—	2	2	no	—
66	C₄²×C₁₉	—	2	2	no	—
67	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₁₉	#41	1	1	yes	characteristic
71	C₄⋊Dic₁₉	—	4	2	no	—
72	C₄⋊Dic₁₉	—	4	2	no	—
73	C₄⋊Dic₁₉	—	4	2	no	—
74	C₄⋊Dic₁₉	—	4	2	no	—
75	C₄⋊Dic₁₉	—	8	2	no	—
76	C₂×C₁₉⋊C₈	—	4	2	no	—
77	C₂×C₁₉⋊C₈	—	4	2	no	—
78	C₂×C₁₉⋊C₈	—	4	2	no	—
79	#8647	—	19	1	no	maximal, Sylow
80	C₂³×C₁₉	#121	1	1	yes	characteristic
81	Q₈×C₁₉	—	2	2	no	—
82	Q₈×C₁₉	—	2	4	no	—
83	Q₈×C₁₉	#106	1	4	yes	—
84	C₂×C₄×C₁₉	—	2	2	no	—
85	C₂×C₄×C₁₉	—	2	4	no	—
86	C₂×C₄×C₁₉	—	2	1	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	4	1	no	—
90	C₂×C₄×C₁₉	—	4	1	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	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
98	C₂×C₄×C₁₉	#135	1	2	yes	—
99	C₂×C₄×C₁₉	#135	1	2	yes	—
100	C₂×C₄×C₁₉	#137	1	2	yes	—
101	C₂×Dic₁₉	—	8	2	no	—
102	C₂×Dic₁₉	—	8	2	no	—
103	C₂×Dic₁₉	—	8	2	no	—
104	C₈×C₁₉	—	2	2	no	—
105	C₈×C₁₉	—	2	2	no	—
106	C₈×C₁₉	—	4	2	no	—
107	C₁₉⋊C₈	—	8	2	no	—
108	C₂×C₈×Q₈	D₁₉	1	1	yes	characteristic
109	#1834	—	19	1	no	—
110	#1803	—	19	2	no	—
111	C₂×#334	—	19	1	no	—
112	#1296	—	19	2	no	—
113	C₂²×C₁₉	#380	1	1	yes	characteristic
114	C₂²×C₁₉	#464	1	2	yes	—
115	C₂²×C₁₉	#465	1	2	yes	—
116	C₂²×C₁₉	#472	1	1	yes	characteristic
117	C₂²×C₁₉	#473	1	1	yes	characteristic
118	C₄×C₁₉	—	2	4	no	—
119	C₄×C₁₉	—	2	1	no	—
120	C₄×C₁₉	—	2	1	no	—
121	C₄×C₁₉	—	4	2	no	—
122	C₄×C₁₉	#416	1	1	yes	characteristic
123	C₄×C₁₉	#419	1	1	yes	characteristic
124	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
125	C₄×C₁₉	#427	1	1	yes	characteristic
126	C₄×C₁₉	#465	1	2	yes	—
127	C₄×C₁₉	#479	1	2	yes	—
128	Dic₁₉	—	16	2	no	—
129	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
130	C₈×Q₈	—	2	2	no	—
131	C₈×Q₈	Dic₁₉	1	4	yes	complemented
132	C₂×C₄⋊C₈	—	2	1	no	—
133	C₂×C₄⋊C₈	—	19	2	no	—
134	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
135	C₂×#103	—	38	2	no	—
136	C₂×C₄×C₈	—	2	1	no	—
137	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
138	#384	—	19	2	no	—
139	#340	—	38	1	no	—
140	#340	—	38	1	no	—
141	#340	—	38	2	no	—
142	#334	—	38	2	no	—
143	C₂×C₁₉	#1678	1	2	yes	—
144	C₂×C₁₉	#1680	1	2	yes	—
145	C₂×C₁₉	#1805	1	1	yes	characteristic
146	C₂×C₁₉	#1807	1	1	yes	characteristic
147	C₂×C₁₉	#1816	1	1	yes	characteristic
148	C₂²×Q₈	D₇₆	1	1	yes	characteristic
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	—	38	2	no	—
151	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
153	C₄×Q₈	—	2	1	no	—
154	C₄×Q₈	—	2	1	no	—
155	C₄×Q₈	Dic₃₈	1	2	yes	—
156	C₄×Q₈	C₂×Dic₁₉	1	2	yes	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	C₂×C₄⋊C₄	—	38	2	no	—
159	C₂×C₄⋊C₄	—	38	2	no	—
160	C₂×C₄⋊C₄	—	38	2	no	—
161	C₂×C₄⋊C₄	D₇₆	1	1	yes	characteristic
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	—	2	1	no	—
166	C₄⋊C₈	—	38	2	no	—
167	C₄⋊C₈	—	38	2	no	—
168	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
169	#103	—	76	4	no	—
170	C₄×C₈	—	2	4	no	—
171	C₄×C₈	C₂×Dic₁₉	1	2	yes	—
172	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
173	C₁₉	#8647	1	1	yes	characteristic, complemented, Sylow
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	#64516	1	4	yes	—
177	C₂×Q₈	C₄⋊Dic₁₉	1	2	yes	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	76	2	no	—
181	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
182	C₂²×C₄	#64541	1	1	yes	characteristic
183	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	76	2	no	—
189	C₂×C₈	—	76	2	no	—
190	C₂×C₈	—	76	2	no	—
191	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
192	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	76	2	no	—
197	C₄⋊C₄	—	76	2	no	—
198	C₄⋊C₄	—	76	2	no	—
199	C₄⋊C₄	—	76	2	no	—
200	C₄⋊C₄	—	152	2	no	—
201	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
205	C₄²	#64541	1	1	yes	characteristic
206	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
207	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
208	C₂³	#10597376	1	1	yes	characteristic
209	Q₈	—	2	2	no	—
210	Q₈	—	2	4	no	—
211	Q₈	#10597274	1	4	yes	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	152	2	no	—
220	C₂×C₄	—	152	2	no	—
221	C₂×C₄	—	152	2	no	—
222	C₂×C₄	C₂×#64516	1	2	yes	—
223	C₂×C₄	#10597350	1	2	yes	—
224	C₂×C₄	#10597370	1	1	yes	characteristic
225	C₂×C₄	#10597370	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
227	C₂×C₄	#10597372	1	1	yes	characteristic
228	C₂×C₄	#10597376	1	1	yes	characteristic
229	C₂×C₄	#10597388	1	2	yes	—
230	C₂×C₄	#10597396	1	2	yes	—
231	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	—	4	2	no	—
235	C₈	—	152	2	no	—
236	C₂²	#11926404	1	1	yes	characteristic
237	C₂²	#11926766	1	2	yes	—
238	C₂²	#11926773	1	2	yes	—
239	C₂²	#11926814	1	1	yes	characteristic
240	C₂²	#11926819	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	4	2	no	—
245	C₄	—	304	2	no	—
246	C₄	#11926540	1	1	yes	characteristic
247	C₄	#11926553	1	1	yes	characteristic
248	C₄	C₂×#10597274	1	1	yes	characteristic
249	C₄	#11926582	1	1	yes	characteristic
250	C₄	#11926773	1	2	yes	—
251	C₄	#11926851	1	2	yes	—
252	C₂	#14563051	1	2	yes	—
253	C₂	#14563065	1	2	yes	—
254	C₂	#14563650	1	1	yes	characteristic
255	C₂	#14563660	1	1	yes	characteristic
256	C₂	#14563699	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₈×Q₈×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	#121
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₈×Q₈	D₁₉
hypocenter	C₁₉	#8647
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
15. C₈×Q₈×C₁₉	C₄	semidirect
131. C₈×Q₈	Dic₁₉	semidirect
173. C₁₉	#8647	semidirect

Maximal subgroups

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

subgroup	index
#14563763	2
#14563640	2
C₂×#11926295	2
#14561519	2
C₂×C₈×Q₈×C₁₉	2
#8647	19