Subgroups — #18088262

Order 4864.

694 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926309	C₂	1	1	yes	maximal, characteristic
2	#14563352	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926304	C₂	1	2	yes	maximal
4	#14563331	C₂	1	2	yes	maximal
5	C₂²×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂×#10597372	C₂²	1	2	yes	—
8	#11926309	C₄	1	4	yes	complemented
9	#11926307	C₄	1	4	yes	complemented
10	#11926304	—	2	4	no	—
11	#11926298	—	2	4	no	—
12	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₁₉×#97	—	2	4	no	—
16	C₂×C₁₉×#97	C₄	1	4	yes	complemented
17	C₂³×C₄×C₁₉	—	2	1	no	—
18	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₁₉	—	2	2	no	—
20	C₂²×C₈×C₁₉	—	2	2	no	—
21	C₂²×C₈×C₁₉	—	2	2	no	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₉	D₄	1	2	yes	—
28	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₉	D₄	1	4	yes	—
33	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
34	#10597372	—	2	2	no	—
35	#10597372	—	2	2	no	—
36	#10597372	—	2	2	no	—
37	#10597372	—	2	2	no	—
38	C₂×C₄⋊Dic₁₉	C₂×C₄	1	4	yes	—
39	C₂×C₄×Dic₁₉	—	2	2	no	—
40	C₂×#64520	—	2	2	no	—
41	C₂×#64520	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	C₂×C₄	1	4	yes	—
46	C₂²×C₁₉⋊C₈	—	2	2	no	—
47	C₁₉×#97	—	2	8	no	—
48	C₂⁴×C₁₉	#41	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	2	no	—
50	C₂²×C₄×C₁₉	—	2	4	no	—
51	C₂²×C₄×C₁₉	—	4	4	no	—
52	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	#41	1	2	yes	—
59	C₂²×C₄×C₁₉	#41	1	2	yes	—
60	C₂²×C₄×C₁₉	#41	1	2	yes	—
61	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
62	C₂²×Dic₁₉	—	2	4	no	—
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	2	no	—
65	C₂×C₈×C₁₉	—	2	2	no	—
66	C₂×C₈×C₁₉	—	2	2	no	—
67	C₂×C₈×C₁₉	—	2	4	no	—
68	C₂×C₈×C₁₉	—	2	4	no	—
69	C₂×C₈×C₁₉	—	2	4	no	—
70	C₂×C₈×C₁₉	—	2	4	no	—
71	C₂×C₈×C₁₉	C₄²	1	4	yes	—
72	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
73	C₄²×C₁₉	—	2	2	no	—
74	C₄²×C₁₉	—	2	4	no	—
75	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
76	C₂²⋊Dic₁₉	—	4	2	no	—
77	C₂²⋊Dic₁₉	—	4	2	no	—
78	C₄⋊Dic₁₉	—	2	4	no	—
79	C₄⋊Dic₁₉	—	2	4	no	—
80	C₄×Dic₁₉	—	4	2	no	—
81	C₄×Dic₁₉	—	4	2	no	—
82	#64520	—	2	4	no	—
83	#64520	—	2	4	no	—
84	#64520	—	4	2	no	—
85	#64520	—	4	2	no	—
86	C₂×C₁₉⋊C₈	—	2	4	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	#10819	—	19	1	no	maximal, Sylow
90	C₂³×C₁₉	—	2	4	no	—
91	C₂³×C₁₉	#117	1	1	yes	characteristic
92	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
94	C₂³×C₁₉	#123	1	2	yes	—
95	C₂³×C₁₉	#126	1	2	yes	—
96	C₂×C₄×C₁₉	—	2	4	no	—
97	C₂×C₄×C₁₉	—	4	8	no	—
98	C₂×C₄×C₁₉	—	4	4	no	—
99	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#117	1	2	yes	—
103	C₂×C₄×C₁₉	#117	1	2	yes	—
104	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#121	1	2	yes	—
112	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₁₉	#123	1	2	yes	—
114	C₂×C₄×C₁₉	#124	1	2	yes	—
115	C₂×C₄×C₁₉	#125	1	2	yes	—
116	C₂×C₄×C₁₉	#126	1	2	yes	—
117	C₂×Dic₁₉	—	4	4	no	—
118	C₂×Dic₁₉	—	4	8	no	—
119	C₈×C₁₉	—	2	4	no	—
120	C₈×C₁₉	—	2	4	no	—
121	C₁₉⋊C₈	—	4	4	no	—
122	C₂²×#97	D₁₉	1	1	yes	characteristic
123	C₂×#341	—	19	1	no	—
124	#1735	—	19	1	no	—
125	C₂×#339	—	19	2	no	—
126	#1729	—	19	2	no	—
127	C₂²×C₁₉	—	2	4	no	—
128	C₂²×C₁₉	—	2	4	no	—
129	C₂²×C₁₉	—	2	2	no	—
130	C₂²×C₁₉	—	2	2	no	—
131	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₁₉	#381	1	1	yes	characteristic
133	C₂²×C₁₉	#383	1	1	yes	characteristic
134	C₂²×C₁₉	#387	1	2	yes	—
135	C₂²×C₁₉	#388	1	2	yes	—
136	C₂²×C₁₉	#444	1	4	yes	—
137	C₄×C₁₉	—	4	4	no	—
138	C₄×C₁₉	#376	1	1	yes	characteristic
139	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
140	C₄×C₁₉	#380	1	1	yes	characteristic
141	C₄×C₁₉	#382	1	1	yes	characteristic
142	C₄×C₁₉	#387	1	2	yes	—
143	C₄×C₁₉	#388	1	2	yes	—
144	Dic₁₉	—	8	4	no	—
145	C₂²×M₁₆	—	19	2	no	—
146	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
147	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
148	C₂×#117	—	19	2	no	—
149	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
150	C₂×#97	—	2	4	no	—
151	C₂×#97	Dic₁₉	1	4	yes	complemented
152	#341	—	19	4	no	—
153	#340	—	19	4	no	—
154	#339	—	38	4	no	—
155	#336	—	38	4	no	—
156	C₂×C₁₉	—	2	1	no	—
157	C₂×C₁₉	—	2	1	no	—
158	C₂×C₁₉	—	2	2	no	—
159	C₂×C₁₉	#1434	1	1	yes	characteristic
160	C₂×C₁₉	#1773	1	2	yes	—
161	C₂×C₁₉	#1938	1	4	yes	—
162	C₂³×C₄	—	2	1	no	—
163	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
164	C₂×M₁₆	—	19	4	no	—
165	C₂×M₁₆	—	38	2	no	—
166	C₂×M₁₆	—	38	2	no	—
167	C₂×M₁₆	—	38	2	no	—
168	C₂×M₁₆	—	38	2	no	—
169	C₂×M₁₆	—	38	2	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	38	2	no	—
175	C₂²×C₈	Dic₃₈	1	2	yes	—
176	C₂²×C₈	C₄×D₁₉	1	2	yes	—
177	C₂²×C₈	C₄×D₁₉	1	2	yes	—
178	C₂²×C₈	D₇₆	1	2	yes	—
179	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
180	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
181	#117	—	38	2	no	—
182	#117	—	38	2	no	—
183	#117	—	38	2	no	—
184	#117	—	38	2	no	—
185	C₂×C₄⋊C₄	—	19	4	no	—
186	C₂×C₂²⋊C₄	—	38	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	1	no	—
189	C₂×C₄²	—	38	2	no	—
190	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₁₉	1	4	yes	—
192	#97	—	2	8	no	—
193	C₁₉	#10819	1	1	yes	characteristic, complemented, Sylow
194	C₂⁴	#64541	1	1	yes	characteristic
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	38	4	no	—
199	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
203	C₂²×C₄	#64541	1	2	yes	—
204	C₂²×C₄	#64541	1	1	yes	characteristic
205	C₂²×C₄	#64541	1	2	yes	—
206	C₂²×C₄	#64541	1	2	yes	—
207	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
208	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
209	M₁₆	—	38	4	no	—
210	M₁₆	—	38	4	no	—
211	M₁₆	—	76	2	no	—
212	M₁₆	—	76	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	38	4	no	—
222	C₂×C₈	—	76	2	no	—
223	C₂×C₈	—	76	2	no	—
224	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
225	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
226	C₄⋊C₄	—	38	4	no	—
227	C₄⋊C₄	—	38	4	no	—
228	C₂²⋊C₄	—	76	2	no	—
229	C₂²⋊C₄	—	76	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	2	4	no	—
232	C₄²	—	76	2	no	—
233	C₄²	—	76	2	no	—
234	C₄²	C₂²⋊Dic₁₉	1	4	yes	—
235	C₂³	—	2	4	no	—
236	C₂³	C₄×Dic₃₈	1	1	yes	characteristic
237	C₂³	C₄×D₇₆	1	1	yes	characteristic
238	C₂³	#10597322	1	2	yes	—
239	C₂³	#10597372	1	1	yes	characteristic
240	C₂³	#10597375	1	2	yes	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	—	76	8	no	—
245	C₂×C₄	—	76	4	no	—
246	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
247	C₂×C₄	#10597316	1	2	yes	—
248	C₂×C₄	#10597318	1	2	yes	—
249	C₂×C₄	#10597318	1	2	yes	—
250	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
251	C₂×C₄	#10597321	1	2	yes	—
252	C₂×C₄	#10597322	1	2	yes	—
253	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
254	C₂×C₄	#10597370	1	2	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
256	C₂×C₄	#10597372	1	1	yes	characteristic
257	C₂×C₄	#10597372	1	1	yes	characteristic
258	C₂×C₄	#10597372	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
260	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
261	C₂×C₄	#10597375	1	2	yes	—
262	C₂×C₄	#10597376	1	2	yes	—
263	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
264	C₈	—	2	4	no	—
265	C₈	—	2	4	no	—
266	C₈	—	76	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
272	C₂²	#11926407	1	1	yes	characteristic
273	C₂²	#11926412	1	1	yes	characteristic
274	C₂²	#11926428	1	2	yes	—
275	C₂²	#11926435	1	2	yes	—
276	C₂²	#11926667	1	4	yes	—
277	C₄	—	4	4	no	—
278	C₄	—	152	4	no	—
279	C₄	#11926390	1	1	yes	characteristic
280	C₄	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
281	C₄	#11926404	1	1	yes	characteristic
282	C₄	#11926410	1	1	yes	characteristic
283	C₄	#11926428	1	2	yes	—
284	C₄	#11926435	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	2	2	no	—
288	C₂	#14561862	1	1	yes	characteristic
289	C₂	#14563519	1	2	yes	—
290	C₂	#14564346	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂²×C₁₉×#97	C₂
socle	C₂³×C₁₉	C₄×D₄
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂²×#97	D₁₉
hypocenter	C₁₉	#10819
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
8. #11926309	C₄	semidirect
9. #11926307	C₄	semidirect
16. C₂×C₁₉×#97	C₄	semidirect
151. C₂×#97	Dic₁₉	semidirect
193. C₁₉	#10819	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926309	2
#14563352	2
C₂×#11926304	2
#14563331	2
C₂²×C₁₉×#97	2
#10819	19