Subgroups — #18088435

Order 4864.

982 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10597274	C₂	1	2	yes	maximal
2	C₂×#11926307	C₂	1	2	yes	maximal
3	#14561865	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₁₉	C₂²	1	4	yes	—
6	C₂×#10597274	—	2	4	no	—
7	C₂×#10597274	—	2	4	no	—
8	C₂×#10597274	C₄	1	8	yes	complemented
9	C₂×C₁₉⋊#95	—	2	4	no	—
10	#11926307	—	2	4	no	—
11	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₉	—	2	4	no	—
15	C₂×C₄×C₈×C₁₉	C₄	1	4	yes	complemented
16	C₂³×C₄×C₁₉	D₄	1	2	yes	—
17	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
18	C₂³×Dic₁₉	—	2	4	no	—
19	C₂²×C₈×C₁₉	—	2	2	no	—
20	C₂²×C₈×C₁₉	—	2	2	no	—
21	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₉	D₄	1	2	yes	—
25	C₂²×C₈×C₁₉	D₄	1	2	yes	—
26	C₂²×C₈×C₁₉	D₄	1	4	yes	—
27	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
28	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
33	C₂×C₄⋊Dic₁₉	—	2	4	no	—
34	C₂×C₄⋊Dic₁₉	—	2	4	no	—
35	C₂×C₄⋊Dic₁₉	—	2	4	no	—
36	C₂×C₄⋊Dic₁₉	—	2	4	no	—
37	C₂×C₄⋊Dic₁₉	—	2	4	no	—
38	C₂×C₄⋊Dic₁₉	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₁₉	—	2	8	no	—
40	C₁₉⋊#95	—	4	8	no	—
41	#10597274	—	2	16	no	—
42	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	—	2	4	no	—
44	C₂²×C₄×C₁₉	—	2	4	no	—
45	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₁₉	SD₁₆	1	8	yes	—
47	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	#41	1	2	yes	—
56	C₂²×C₄×C₁₉	#41	1	2	yes	—
57	C₂²×Dic₁₉	—	2	8	no	—
58	C₂²×Dic₁₉	—	4	4	no	—
59	C₂²×Dic₁₉	—	4	4	no	—
60	C₂²×Dic₁₉	—	4	8	no	—
61	C₂²×Dic₁₉	—	4	8	no	—
62	C₂×C₈×C₁₉	—	2	2	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	4	no	—
65	C₂×C₈×C₁₉	—	2	4	no	—
66	C₂×C₈×C₁₉	C₄²	1	4	yes	—
67	C₂×C₈×C₁₉	C₂²⋊C₄	1	8	yes	—
68	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
69	C₂×C₈×C₁₉	C₄⋊C₄	1	8	yes	—
70	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
71	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
72	C₄²×C₁₉	—	2	2	no	—
73	C₄²×C₁₉	—	2	2	no	—
74	C₄²×C₁₉	—	2	4	no	—
75	C₄⋊Dic₁₉	—	2	8	no	—
76	C₄⋊Dic₁₉	—	2	8	no	—
77	C₄⋊Dic₁₉	—	4	4	no	—
78	C₄⋊Dic₁₉	—	4	4	no	—
79	#10861	—	19	1	no	maximal, Sylow
80	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₁₉	C₄×D₄	1	2	yes	—
82	C₂³×C₁₉	#121	1	2	yes	—
83	C₂³×C₁₉	#127	1	1	yes	characteristic
84	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
85	C₂³×C₁₉	C₂×SD₁₆	1	4	yes	—
86	C₂³×C₁₉	#135	1	4	yes	—
87	C₂×C₄×C₁₉	—	2	4	no	—
88	C₂×C₄×C₁₉	—	2	4	no	—
89	C₂×C₄×C₁₉	—	2	16	no	—
90	C₂×C₄×C₁₉	#106	1	8	yes	complemented
91	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
98	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
102	C₂×C₄×C₁₉	#121	1	2	yes	—
103	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₁₉	#124	1	2	yes	—
106	C₂×C₄×C₁₉	#125	1	2	yes	—
107	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
109	C₂×Dic₁₉	—	4	8	no	—
110	C₂×Dic₁₉	—	4	16	no	—
111	C₂×Dic₁₉	—	8	8	no	—
112	C₂×Dic₁₉	—	8	4	no	—
113	C₂×Dic₁₉	—	8	4	no	—
114	C₈×C₁₉	—	2	4	no	—
115	C₈×C₁₉	#95	1	8	yes	complemented
116	C₂²×#106	—	19	2	no	—
117	C₂²×C₄×C₈	D₁₉	1	1	yes	characteristic
118	C₂×#340	—	19	2	no	—
119	C₄⋊#95	—	19	2	no	—
120	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₁₉	#380	1	1	yes	characteristic
122	C₂²×C₁₉	#384	1	1	yes	characteristic
123	C₂²×C₁₉	#389	1	2	yes	—
124	C₂²×C₁₉	#390	1	2	yes	—
125	C₂²×C₁₉	C₂×#106	1	2	yes	—
126	C₂²×C₁₉	#427	1	2	yes	—
127	C₂²×C₁₉	C₄×SD₁₆	1	8	yes	—
128	C₂²×C₁₉	#465	1	8	yes	—
129	C₂²×C₁₉	#492	1	2	yes	—
130	C₂²×C₁₉	#495	1	2	yes	—
131	C₂²×C₁₉	#498	1	2	yes	—
132	C₂²×C₁₉	#499	1	2	yes	—
133	C₄×C₁₉	—	2	8	no	—
134	C₄×C₁₉	C₂×#95	1	1	yes	characteristic
135	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₁₉	#380	1	1	yes	characteristic
137	C₄×C₁₉	#382	1	1	yes	characteristic
138	C₄×C₁₉	#389	1	2	yes	—
139	C₄×C₁₉	#391	1	2	yes	—
140	Dic₁₉	—	8	8	no	—
141	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
142	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
143	C₂²×C₄⋊C₄	—	19	4	no	—
144	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
145	C₂×#106	—	19	8	no	—
146	C₂×#106	—	38	4	no	—
147	C₂×#106	—	38	4	no	—
148	C₂×C₄×C₈	—	2	4	no	—
149	C₂×C₄×C₈	Dic₁₉	1	4	yes	complemented
150	C₂×#95	—	38	4	no	—
151	#340	—	38	4	no	—
152	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
153	C₂×C₁₉	C₄×#106	1	2	yes	—
154	C₂×C₁₉	#1807	1	2	yes	—
155	C₂×C₁₉	#1844	1	1	yes	characteristic
156	C₂×C₁₉	#1846	1	1	yes	characteristic
157	C₂×C₁₉	#1941	1	4	yes	—
158	C₂×C₁₉	#1968	1	4	yes	—
159	C₂³×C₄	—	38	4	no	—
160	C₂³×C₄	D₇₆	1	2	yes	—
161	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	Dic₃₈	1	2	yes	—
165	C₂²×C₈	Dic₃₈	1	2	yes	—
166	C₂²×C₈	C₄×D₁₉	1	4	yes	—
167	C₂²×C₈	D₇₆	1	2	yes	—
168	C₂²×C₈	D₇₆	1	2	yes	—
169	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
170	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
171	C₂²×C₈	C₂²⋊D₁₉	1	4	yes	—
172	C₂×C₄⋊C₄	—	19	8	no	—
173	C₂×C₄⋊C₄	—	38	4	no	—
174	C₂×C₄⋊C₄	—	38	4	no	—
175	C₂×C₄⋊C₄	—	38	4	no	—
176	C₂×C₄⋊C₄	—	38	4	no	—
177	C₂×C₄⋊C₄	—	38	4	no	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	Dic₃₈	1	2	yes	—
181	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
182	#106	—	38	16	no	—
183	C₄×C₈	—	2	8	no	—
184	#95	—	76	8	no	—
185	C₁₉	#10861	1	1	yes	characteristic, complemented, Sylow
186	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	38	8	no	—
190	C₂²×C₄	—	76	4	no	—
191	C₂²×C₄	—	76	4	no	—
192	C₂²×C₄	—	76	8	no	—
193	C₂²×C₄	—	76	8	no	—
194	C₂²×C₄	#64516	1	8	yes	—
195	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
196	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
197	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
198	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
200	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
202	C₂²×C₄	#64541	1	2	yes	—
203	C₂²×C₄	#64541	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
211	C₂×C₈	Dic₁₉⋊C₄	1	8	yes	—
212	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
213	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
214	C₂×C₈	(C₂×D₁₉)⋊C₄	1	8	yes	—
215	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
216	C₄⋊C₄	—	38	8	no	—
217	C₄⋊C₄	—	38	8	no	—
218	C₄⋊C₄	—	76	4	no	—
219	C₄⋊C₄	—	76	4	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	2	no	—
223	C₂³	#10597315	1	1	yes	characteristic
224	C₂³	C₄×D₇₆	1	2	yes	—
225	C₂³	#10597320	1	1	yes	characteristic
226	C₂³	C₂×#64516	1	4	yes	—
227	C₂³	#10597350	1	4	yes	—
228	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
229	C₂³	#10597376	1	2	yes	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	16	no	—
233	C₂×C₄	—	76	16	no	—
234	C₂×C₄	—	76	8	no	—
235	C₂×C₄	—	152	4	no	—
236	C₂×C₄	—	152	4	no	—
237	C₂×C₄	—	152	8	no	—
238	C₂×C₄	#10597274	1	8	yes	complemented
239	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
240	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
241	C₂×C₄	#10597315	1	1	yes	characteristic
242	C₂×C₄	#10597315	1	1	yes	characteristic
243	C₂×C₄	#10597316	1	2	yes	—
244	C₂×C₄	C₄×D₇₆	1	2	yes	—
245	C₂×C₄	#10597321	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
247	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
248	C₂×C₄	#10597370	1	2	yes	—
249	C₂×C₄	#10597370	1	2	yes	—
250	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
251	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
252	C₂×C₄	#10597372	1	1	yes	characteristic
253	C₂×C₄	#10597372	1	1	yes	characteristic
254	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
255	C₂×C₄	#10597376	1	2	yes	—
256	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
257	C₈	—	2	4	no	—
258	C₈	C₁₉⋊#95	1	8	yes	complemented
259	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
260	C₂²	#11926404	1	1	yes	characteristic
261	C₂²	#11926414	1	1	yes	characteristic
262	C₂²	#11926442	1	2	yes	—
263	C₂²	#11926446	1	2	yes	—
264	C₂²	C₂×#10597274	1	2	yes	—
265	C₂²	#11926582	1	2	yes	—
266	C₂²	C₄×#64516	1	8	yes	—
267	C₂²	#11926773	1	8	yes	—
268	C₂²	#11926917	1	2	yes	—
269	C₂²	#11926928	1	2	yes	—
270	C₂²	#11926942	1	2	yes	—
271	C₂²	#11926946	1	2	yes	—
272	C₄	—	2	8	no	—
273	C₄	—	152	8	no	—
274	C₄	C₂×C₁₉⋊#95	1	1	yes	characteristic
275	C₄	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
276	C₄	#11926404	1	1	yes	characteristic
277	C₄	#11926410	1	1	yes	characteristic
278	C₄	#11926442	1	2	yes	—
279	C₄	#11926450	1	2	yes	—
280	C₂	#14561865	1	1	yes	characteristic
281	C₂	C₄×#10597274	1	2	yes	—
282	C₂	#14563660	1	2	yes	—
283	C₂	#14563792	1	1	yes	characteristic
284	C₂	#14563799	1	1	yes	characteristic
285	C₂	#14564363	1	4	yes	—
286	C₂	#14564489	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂²×C₄×C₈×C₁₉	C₂
socle, prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂²×C₄×C₈	D₁₉
hypocenter	C₁₉	#10861
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₂×#10597274	C₄	semidirect
15. C₂×C₄×C₈×C₁₉	C₄	semidirect
90. C₂×C₄×C₁₉	#106	semidirect
115. C₈×C₁₉	#95	semidirect
149. C₂×C₄×C₈	Dic₁₉	semidirect
185. C₁₉	#10861	semidirect
238. C₂×C₄	#10597274	semidirect
258. C₈	C₁₉⋊#95	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10597274	2
C₂×#11926307	2
#14561865	2
C₂²×C₄×C₈×C₁₉	2
#10861	19