Subgroups — #18088306

Order 4864.

642 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926499	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926314	C₂	1	2	yes	maximal
3	C₂×#11926309	C₂	1	1	yes	maximal, characteristic
4	#14563345	C₂	1	2	yes	maximal
5	C₁₉×#1735	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂²×#64520	C₂²	1	1	yes	characteristic
8	C₂×#10597372	C₂²	1	1	yes	characteristic
9	#11926499	—	2	2	no	—
10	#11926499	C₄	1	4	yes	complemented
11	C₂×#10597272	—	2	1	no	—
12	C₂×C₈×Dic₁₉	—	2	1	no	—
13	#11926314	C₄	1	8	yes	complemented
14	#11926309	—	2	1	no	—
15	#11926309	—	2	1	no	—
16	#11926304	—	2	4	no	—
17	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×#117	C₂²	1	2	yes	—
19	C₁₉×#340	—	2	2	no	—
20	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	D₄	1	2	yes	—
27	C₁₉×#117	—	2	2	no	—
28	C₁₉×#117	—	2	2	no	—
29	C₁₉×#117	—	2	2	no	—
30	C₁₉×#117	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
32	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
33	C₂×C₄²×C₁₉	—	2	2	no	—
34	C₂×C₂²⋊Dic₁₉	—	2	1	no	—
35	#10597372	—	2	1	no	—
36	#10597372	—	2	1	no	—
37	#10597372	C₂×C₄	1	2	yes	—
38	#10597372	D₄	1	2	yes	—
39	C₂×C₄⋊Dic₁₉	Q₈	1	2	yes	—
40	C₂×C₄×Dic₁₉	—	2	1	no	—
41	C₂×#64520	—	2	1	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	—	2	2	no	—
48	C₂×#64520	C₂×C₄	1	4	yes	—
49	C₂×#64520	C₂×C₄	1	4	yes	—
50	C₂×#64520	C₂×C₄	1	2	yes	—
51	C₂²×C₁₉⋊C₈	—	2	1	no	—
52	C₂²×C₁₉⋊C₈	—	2	2	no	—
53	#10597272	—	2	4	no	—
54	C₈×Dic₁₉	—	2	4	no	—
55	C₂⁴×C₁₉	#41	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	—	4	2	no	—
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	#41	1	2	yes	—
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	C₂²×C₄×C₁₉	#41	1	2	yes	—
66	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
67	C₂²×Dic₁₉	—	2	2	no	—
68	C₂×C₈×C₁₉	—	2	2	no	—
69	C₂×C₈×C₁₉	—	2	2	no	—
70	C₂×C₈×C₁₉	—	2	4	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
75	C₄⋊C₄×C₁₉	—	2	4	no	—
76	C₄⋊C₄×C₁₉	—	4	2	no	—
77	C₂²⋊C₄×C₁₉	—	2	4	no	—
78	C₂²⋊C₄×C₁₉	—	4	2	no	—
79	C₄²×C₁₉	—	2	4	no	—
80	C₄²×C₁₉	—	4	2	no	—
81	C₂²⋊Dic₁₉	—	2	2	no	—
82	C₂²⋊Dic₁₉	—	2	2	no	—
83	C₄⋊Dic₁₉	—	2	2	no	—
84	C₄⋊Dic₁₉	C₄⋊C₄	1	4	yes	—
85	C₄×Dic₁₉	—	2	2	no	—
86	C₄×Dic₁₉	—	2	2	no	—
87	#64520	—	2	2	no	—
88	#64520	—	2	4	no	—
89	#64520	—	2	4	no	—
90	#64520	—	2	4	no	—
91	#64520	—	4	2	no	—
92	#64520	—	4	2	no	—
93	#64520	C₄²	1	4	yes	—
94	C₂×C₁₉⋊C₈	—	2	2	no	—
95	C₂×C₁₉⋊C₈	—	2	4	no	—
96	C₂×C₁₉⋊C₈	—	2	4	no	—
97	C₂×C₁₉⋊C₈	—	4	2	no	—
98	C₂×C₁₉⋊C₈	—	4	2	no	—
99	#10828	—	19	1	no	maximal, Sylow
100	C₂³×C₁₉	—	4	2	no	—
101	C₂³×C₁₉	#117	1	2	yes	—
102	C₂³×C₁₉	#117	1	1	yes	characteristic
103	C₂³×C₁₉	#121	1	1	yes	characteristic
104	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂³×C₁₉	#126	1	2	yes	—
106	C₂×C₄×C₁₉	—	4	4	no	—
107	C₂×C₄×C₁₉	—	8	4	no	—
108	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#117	1	2	yes	—
115	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
121	C₂×C₄×C₁₉	#123	1	2	yes	—
122	C₂×C₄×C₁₉	#123	1	2	yes	—
123	C₂×C₄×C₁₉	#124	1	2	yes	—
124	C₂×C₄×C₁₉	#125	1	2	yes	—
125	C₂×C₄×C₁₉	#126	1	2	yes	—
126	C₂×Dic₁₉	—	2	4	no	—
127	C₂×Dic₁₉	—	4	4	no	—
128	C₈×C₁₉	—	2	2	no	—
129	C₈×C₁₉	—	2	2	no	—
130	C₁₉⋊C₈	—	2	4	no	—
131	C₁₉⋊C₈	—	4	4	no	—
132	C₂×#405	—	19	1	no	—
133	C₂×#344	—	19	2	no	—
134	C₂×#341	—	19	1	no	—
135	#1735	D₁₉	1	1	yes	characteristic
136	#1733	—	19	2	no	—
137	C₂²×C₁₉	—	2	2	no	—
138	C₂²×C₁₉	—	2	2	no	—
139	C₂²×C₁₉	—	2	2	no	—
140	C₂²×C₁₉	—	4	2	no	—
141	C₂²×C₁₉	—	4	2	no	—
142	C₂²×C₁₉	#376	1	1	yes	characteristic
143	C₂²×C₁₉	#380	1	1	yes	characteristic
144	C₂²×C₁₉	#383	1	1	yes	characteristic
145	C₂²×C₁₉	#387	1	2	yes	—
146	C₂²×C₁₉	#388	1	2	yes	—
147	C₄×C₁₉	—	8	4	no	—
148	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₁₉	#381	1	1	yes	characteristic
151	C₄×C₁₉	#382	1	1	yes	characteristic
152	C₄×C₁₉	#387	1	2	yes	—
153	C₄×C₁₉	#388	1	2	yes	—
154	Dic₁₉	—	4	4	no	—
155	C₂²×M₁₆	—	19	1	no	—
156	C₂²×M₁₆	—	19	2	no	—
157	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
158	C₂×#117	—	19	1	no	—
159	C₂×#117	C₂×D₁₉	1	2	yes	—
160	#405	—	19	4	no	—
161	#405	—	38	2	no	—
162	C₂×#97	—	38	1	no	—
163	C₂×C₄×C₈	—	38	1	no	—
164	#344	—	19	8	no	—
165	#341	—	38	1	no	—
166	#341	—	38	1	no	—
167	#340	—	2	2	no	—
168	#339	—	38	4	no	—
169	C₂×C₁₉	—	2	2	no	—
170	C₂×C₁₉	—	2	1	no	—
171	C₂×C₁₉	—	2	1	no	—
172	C₂×C₁₉	—	2	2	no	—
173	C₂×C₁₉	#1434	1	1	yes	characteristic
174	C₂×C₁₉	#1814	1	2	yes	—
175	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
176	C₂×M₁₆	—	19	4	no	—
177	C₂×M₁₆	—	19	2	no	—
178	C₂×M₁₆	—	19	4	no	—
179	C₂×M₁₆	—	38	2	no	—
180	C₂×M₁₆	—	38	1	no	—
181	C₂×M₁₆	—	38	2	no	—
182	C₂×M₁₆	—	38	2	no	—
183	C₂×M₁₆	—	38	2	no	—
184	C₂×M₁₆	—	38	2	no	—
185	C₂×M₁₆	—	38	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	38	1	no	—
189	C₂²×C₈	—	38	2	no	—
190	C₂²×C₈	C₄×D₁₉	1	2	yes	—
191	C₂²×C₈	D₇₆	1	1	yes	characteristic
192	C₂²×C₈	D₇₆	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
194	#117	—	2	2	no	—
195	#117	—	2	2	no	—
196	#117	—	2	2	no	—
197	#117	—	19	2	no	—
198	#117	—	19	2	no	—
199	#117	—	38	1	no	—
200	#117	—	38	1	no	—
201	#117	C₄×D₁₉	1	4	yes	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	19	2	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₂²⋊C₄	—	38	1	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	38	1	no	—
208	#97	—	38	4	no	—
209	C₄×C₈	—	38	4	no	—
210	C₁₉	#10828	1	1	yes	characteristic, complemented, Sylow
211	C₂⁴	#64543	1	1	yes	characteristic
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	—	38	2	no	—
214	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
216	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
217	C₂²×C₄	#64541	1	2	yes	—
218	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
219	C₂²×C₄	#64543	1	2	yes	—
220	C₂²×C₄	#64543	1	2	yes	—
221	C₂²×C₄	Q₈×D₁₉	1	1	yes	characteristic
222	C₂²×C₄	#64545	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
224	M₁₆	—	19	4	no	—
225	M₁₆	—	38	2	no	—
226	M₁₆	—	38	4	no	—
227	M₁₆	—	38	4	no	—
228	M₁₆	—	38	4	no	—
229	M₁₆	—	76	2	no	—
230	M₁₆	—	76	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	38	4	no	—
237	C₂×C₈	—	38	2	no	—
238	C₂×C₈	—	38	4	no	—
239	C₂×C₈	—	76	2	no	—
240	C₂×C₈	—	76	2	no	—
241	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
242	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	19	4	no	—
246	C₄⋊C₄	—	38	2	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	38	2	no	—
250	C₂²⋊C₄	—	38	2	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	4	2	no	—
253	C₄²	—	38	2	no	—
254	C₄²	—	38	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	#10597323	1	2	yes	—
257	C₂³	#10597325	1	2	yes	—
258	C₂³	#10597338	1	1	yes	characteristic
259	C₂³	#10597343	1	1	yes	characteristic
260	C₂³	#10597384	1	1	yes	characteristic
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	8	4	no	—
263	C₂×C₄	—	38	4	no	—
264	C₂×C₄	—	76	4	no	—
265	C₂×C₄	C₄²×D₁₉	1	1	yes	characteristic
266	C₂×C₄	#10597318	1	1	yes	characteristic
267	C₂×C₄	C₄×D₇₆	1	2	yes	—
268	C₂×C₄	#10597323	1	1	yes	characteristic
269	C₂×C₄	#10597323	1	1	yes	characteristic
270	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	2	yes	—
271	C₂×C₄	#10597331	1	2	yes	—
272	C₂×C₄	#10597332	1	2	yes	—
273	C₂×C₄	#10597333	1	2	yes	—
274	C₂×C₄	#10597335	1	2	yes	—
275	C₂×C₄	C₄⋊C₄×D₁₉	1	1	yes	characteristic
276	C₂×C₄	#10597338	1	2	yes	—
277	C₂×C₄	#10597341	1	1	yes	characteristic
278	C₂×C₄	#10597344	1	2	yes	—
279	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
281	C₂×C₄	#10597381	1	2	yes	—
282	C₂×C₄	#10597392	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	2	2	no	—
285	C₈	—	38	4	no	—
286	C₈	—	76	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	#11926389	1	1	yes	characteristic
293	C₂²	#11926401	1	1	yes	characteristic
294	C₂²	#11926411	1	1	yes	characteristic
295	C₂²	#11926429	1	2	yes	—
296	C₂²	#11926432	1	2	yes	—
297	C₄	—	8	4	no	—
298	C₄	—	76	4	no	—
299	C₄	C₄×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
300	C₄	#11926395	1	1	yes	characteristic
301	C₄	#11926405	1	1	yes	characteristic
302	C₄	#11926408	1	1	yes	characteristic
303	C₄	#11926425	1	2	yes	—
304	C₄	#11926436	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	—	2	2	no	—
309	C₂	#14561859	1	1	yes	characteristic
310	C₂	#14563686	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597341
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1735	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1735	D₁₉
hypocenter	C₁₉	#10828
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1735

Complements

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

N	H ≅ G/N	type
10. #11926499	C₄	semidirect
13. #11926314	C₄	semidirect
210. C₁₉	#10828	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926499	2
C₂×#11926314	2
C₂×#11926309	2
#14563345	2
C₁₉×#1735	2
#10828	19