Subgroups — #18090736

Order 4864.

654 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10597276	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	C₂×C₁₉×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂×#10597372	C₂²	1	2	yes	—
8	C₂×#10597276	—	2	4	no	—
9	C₂×#10597276	—	2	2	no	—
10	#11926307	C₄	1	4	yes	complemented
11	#11926304	—	2	4	no	—
12	#11926298	—	2	4	no	—
13	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#336	—	2	2	no	—
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	1	no	—
22	C₂²×C₈×C₁₉	—	2	1	no	—
23	C₂²×C₈×C₁₉	—	2	2	no	—
24	C₂²×C₈×C₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₉	D₄	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	2	no	—
31	C₂×C₄²×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	Q₈	1	2	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	D₄	1	4	yes	—
46	C₂²×C₁₉⋊C₈	—	2	2	no	—
47	#10597276	—	2	8	no	—
48	C₂⁴×C₁₉	#41	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	—	2	4	no	—
50	C₂²×C₄×C₁₉	—	2	2	no	—
51	C₂²×C₄×C₁₉	—	4	2	no	—
52	C₂²×C₄×C₁₉	—	4	2	no	—
53	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	#41	1	2	yes	—
61	C₂²×C₄×C₁₉	#41	1	2	yes	—
62	C₂²×C₄×C₁₉	#41	1	2	yes	—
63	C₂²×Dic₁₉	—	2	4	no	—
64	C₂×C₈×C₁₉	—	2	4	no	—
65	C₂×C₈×C₁₉	—	2	2	no	—
66	C₂×C₈×C₁₉	—	2	4	no	—
67	C₂×C₈×C₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	—	2	2	no	—
69	C₂×C₈×C₁₉	—	2	4	no	—
70	C₂×C₈×C₁₉	—	4	1	no	—
71	C₂×C₈×C₁₉	—	4	1	no	—
72	C₂×C₈×C₁₉	—	4	2	no	—
73	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
74	C₄²×C₁₉	—	2	4	no	—
75	C₄²×C₁₉	—	4	1	no	—
76	C₄²×C₁₉	—	4	1	no	—
77	C₂²⋊Dic₁₉	—	4	2	no	—
78	C₂²⋊Dic₁₉	—	4	2	no	—
79	C₄⋊Dic₁₉	—	2	4	no	—
80	C₄⋊Dic₁₉	—	2	4	no	—
81	C₄×Dic₁₉	—	4	2	no	—
82	C₄×Dic₁₉	—	4	2	no	—
83	#64520	—	2	4	no	—
84	#64520	—	2	4	no	—
85	#64520	—	4	2	no	—
86	#64520	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	2	4	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	#11258	—	19	1	no	maximal, Sylow
91	C₂³×C₁₉	—	2	2	no	—
92	C₂³×C₁₉	—	2	2	no	—
93	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
94	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂³×C₁₉	#124	1	2	yes	—
96	C₂³×C₁₉	#125	1	1	yes	characteristic
97	C₂³×C₁₉	#126	1	2	yes	—
98	C₂×C₄×C₁₉	—	2	4	no	—
99	C₂×C₄×C₁₉	—	4	8	no	—
100	C₂×C₄×C₁₉	—	4	2	no	—
101	C₂×C₄×C₁₉	—	4	2	no	—
102	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#117	1	2	yes	—
104	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
106	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#121	1	2	yes	—
109	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#123	1	2	yes	—
113	C₂×C₄×C₁₉	#123	1	2	yes	—
114	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
116	C₂×C₄×C₁₉	#125	1	2	yes	—
117	C₂×C₄×C₁₉	#125	1	2	yes	—
118	C₂×C₄×C₁₉	#126	1	2	yes	—
119	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×Dic₁₉	—	4	4	no	—
121	C₂×Dic₁₉	—	4	4	no	—
122	C₂×Dic₁₉	—	4	4	no	—
123	C₈×C₁₉	—	2	4	no	—
124	C₈×C₁₉	—	4	2	no	—
125	C₁₉⋊C₈	—	4	4	no	—
126	C₂²×#108	—	19	1	no	—
127	#1735	—	19	1	no	—
128	C₂×#339	—	19	2	no	—
129	#1729	—	19	2	no	—
130	C₂×#336	D₁₉	1	1	yes	characteristic
131	C₂²×C₁₉	—	2	4	no	—
132	C₂²×C₁₉	—	2	4	no	—
133	C₂²×C₁₉	—	2	1	no	—
134	C₂²×C₁₉	—	2	1	no	—
135	C₂²×C₁₉	—	2	1	no	—
136	C₂²×C₁₉	—	2	1	no	—
137	C₂²×C₁₉	#388	1	2	yes	—
138	C₂²×C₁₉	#389	1	1	yes	characteristic
139	C₂²×C₁₉	#393	1	1	yes	characteristic
140	C₂²×C₁₉	#400	1	1	yes	characteristic
141	C₂²×C₁₉	#400	1	2	yes	—
142	C₂²×C₁₉	#443	1	2	yes	—
143	C₂²×C₁₉	#444	1	2	yes	—
144	C₄×C₁₉	—	4	4	no	—
145	C₄×C₁₉	#382	1	1	yes	characteristic
146	C₄×C₁₉	#385	1	1	yes	characteristic
147	C₄×C₁₉	#387	1	2	yes	—
148	C₄×C₁₉	#394	1	1	yes	characteristic
149	C₄×C₁₉	#397	1	1	yes	characteristic
150	C₄×C₁₉	#398	1	2	yes	—
151	Dic₁₉	—	8	4	no	—
152	C₂²×M₁₆	—	19	2	no	—
153	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
154	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
155	C₂×#117	—	19	2	no	—
156	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
157	C₂×#108	—	38	4	no	—
158	C₂×#108	—	38	2	no	—
159	#340	—	19	4	no	—
160	#339	—	38	4	no	—
161	#336	—	2	2	no	—
162	#336	—	38	4	no	—
163	C₂×C₁₉	—	2	1	no	—
164	C₂×C₁₉	—	2	1	no	—
165	C₂×C₁₉	—	2	2	no	—
166	C₂×C₁₉	#1442	1	1	yes	characteristic
167	C₂×C₁₉	#1929	1	4	yes	—
168	C₂×C₁₉	#1945	1	1	yes	characteristic
169	C₂×C₁₉	#1946	1	1	yes	characteristic
170	C₂³×C₄	—	2	1	no	—
171	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
172	C₂×M₁₆	—	19	4	no	—
173	C₂×M₁₆	—	38	2	no	—
174	C₂×M₁₆	—	38	2	no	—
175	C₂×M₁₆	—	38	2	no	—
176	C₂×M₁₆	—	38	2	no	—
177	C₂×M₁₆	—	38	2	no	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	1	no	—
184	C₂²×C₈	—	38	2	no	—
185	C₂²×C₈	Dic₃₈	1	2	yes	—
186	C₂²×C₈	C₄×D₁₉	1	2	yes	—
187	C₂²×C₈	D₇₆	1	2	yes	—
188	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
189	#117	—	38	2	no	—
190	#117	—	38	2	no	—
191	#117	—	38	2	no	—
192	#117	—	38	2	no	—
193	C₂×C₄⋊C₄	—	19	4	no	—
194	C₂×C₂²⋊C₄	—	38	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	1	no	—
198	C₂×C₄²	—	38	2	no	—
199	C₂×C₄²	Dic₃₈	1	2	yes	—
200	#108	—	38	8	no	—
201	C₁₉	#11258	1	1	yes	characteristic, complemented, Sylow
202	C₂⁴	#64541	1	1	yes	characteristic
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	4	2	no	—
206	C₂²×C₄	—	4	2	no	—
207	C₂²×C₄	—	38	4	no	—
208	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
209	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
212	C₂²×C₄	#64541	1	1	yes	characteristic
213	C₂²×C₄	D₄×D₁₉	1	2	yes	—
214	C₂²×C₄	#64543	1	2	yes	—
215	C₂²×C₄	#64543	1	2	yes	—
216	C₂²×C₄	#64545	1	2	yes	—
217	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
218	M₁₆	—	38	4	no	—
219	M₁₆	—	38	4	no	—
220	M₁₆	—	76	2	no	—
221	M₁₆	—	76	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	38	4	no	—
232	C₂×C₈	—	76	2	no	—
233	C₂×C₈	—	76	2	no	—
234	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
235	C₄⋊C₄	—	38	4	no	—
236	C₄⋊C₄	—	38	4	no	—
237	C₂²⋊C₄	—	76	2	no	—
238	C₂²⋊C₄	—	76	2	no	—
239	C₄²	—	2	4	no	—
240	C₄²	—	4	1	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	76	2	no	—
243	C₄²	—	76	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	#10597316	1	1	yes	characteristic
247	C₂³	C₄×D₇₆	1	1	yes	characteristic
248	C₂³	#10597331	1	2	yes	—
249	C₂³	#10597344	1	2	yes	—
250	C₂³	#10597370	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	76	4	no	—
256	C₂×C₄	—	76	4	no	—
257	C₂×C₄	—	76	4	no	—
258	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
259	C₂×C₄	#10597315	1	1	yes	characteristic
260	C₂×C₄	#10597324	1	2	yes	—
261	C₂×C₄	#10597324	1	2	yes	—
262	C₂×C₄	#10597325	1	2	yes	—
263	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
264	C₂×C₄	#10597332	1	2	yes	—
265	C₂×C₄	#10597336	1	2	yes	—
266	C₂×C₄	#10597336	1	2	yes	—
267	C₂×C₄	#10597338	1	2	yes	—
268	C₂×C₄	#10597340	1	2	yes	—
269	C₂×C₄	#10597341	1	2	yes	—
270	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
271	C₂×C₄	#10597376	1	1	yes	characteristic
272	C₂×C₄	#10597381	1	1	yes	characteristic
273	C₂×C₄	#10597382	1	1	yes	characteristic
274	C₂×C₄	#10597383	1	1	yes	characteristic
275	C₂×C₄	#10597393	1	1	yes	characteristic
276	C₈	—	2	4	no	—
277	C₈	—	4	2	no	—
278	C₈	—	76	4	no	—
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	#11926436	1	2	yes	—
286	C₂²	#11926442	1	1	yes	characteristic
287	C₂²	#11926456	1	1	yes	characteristic
288	C₂²	#11926483	1	1	yes	characteristic
289	C₂²	#11926486	1	2	yes	—
290	C₂²	#11926662	1	2	yes	—
291	C₂²	#11926667	1	2	yes	—
292	C₄	—	4	4	no	—
293	C₄	—	152	4	no	—
294	C₄	#11926409	1	1	yes	characteristic
295	C₄	#11926416	1	1	yes	characteristic
296	C₄	#11926431	1	2	yes	—
297	C₄	#11926461	1	1	yes	characteristic
298	C₄	#11926469	1	1	yes	characteristic
299	C₄	#11926476	1	2	yes	—
300	C₂	—	2	1	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	#14561891	1	1	yes	characteristic
304	C₂	#14564284	1	4	yes	—
305	C₂	#14564383	1	1	yes	characteristic
306	C₂	#14564388	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#336	C₂
socle	C₂³×C₁₉	C₄×D₄
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#336	D₁₉
hypocenter	C₁₉	#11258
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂×#336

Complements

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

N	H ≅ G/N	type
10. #11926307	C₄	semidirect
201. C₁₉	#11258	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10597276	2
#14563352	2
C₂×#11926304	2
#14563331	2
C₂×C₁₉×#336	2
#11258	19