Subgroups — #18082055

Order 4864.

522 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926785	C₂	1	1	yes	maximal, characteristic
2	#14564620	C₂	1	2	yes	maximal
3	#14563664	C₂	1	1	yes	maximal, characteristic
4	#14562208	C₂	1	2	yes	maximal
5	C₈×C₂²⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₇₆	C₄	1	1	yes	characteristic, complemented
7	C₂×#10597370	C₂²	1	2	yes	—
8	#11926785	—	2	1	no	—
9	#11926785	—	2	1	no	—
10	#11926785	—	2	1	no	—
11	#11926785	—	2	1	no	—
12	C₂×#10597275	—	2	1	no	—
13	C₂×#10597275	C₄	1	1	yes	characteristic, complemented
14	C₂×#10597273	C₄	1	2	yes	complemented
15	#11926404	C₂²	1	2	yes	—
16	#11926310	—	2	4	no	—
17	#11926307	—	2	1	no	—
18	#11926282	—	2	4	no	—
19	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×C₂²⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₉	—	2	1	no	—
22	C₄×C₂²⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#336	—	2	1	no	—
24	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃₈	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	4	2	no	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
35	C₂×C₄²×C₁₉	—	2	1	no	—
36	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
37	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
38	C₂×C₄⋊Dic₁₉	—	2	2	no	—
39	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
40	#10597370	—	2	2	no	—
41	#10597370	—	2	2	no	—
42	#10597370	—	2	2	no	—
43	#10597370	—	2	2	no	—
44	C₂×Dic₁₉⋊C₄	—	2	2	no	—
45	C₂×Dic₇₆	—	2	1	no	—
46	C₂×Dic₇₆	—	2	1	no	—
47	C₂×Dic₇₆	—	4	2	no	—
48	C₁₉×C₂²⋊C₈	—	2	1	no	—
49	C₁₉×C₂²⋊C₈	D₄	1	2	yes	—
50	C₄×C₈×C₁₉	—	4	2	no	—
51	C₁₉×#95	—	2	1	no	—
52	C₁₉⋊#95	—	2	2	no	—
53	#10597275	—	2	1	no	—
54	#10597275	—	2	1	no	—
55	#10597275	—	4	2	no	—
56	#10597273	—	2	2	no	—
57	#10597273	—	2	2	no	—
58	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	—	4	1	no	—
61	C₂²×C₄×C₁₉	—	4	1	no	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
68	C₂²×Dic₁₉	—	2	2	no	—
69	C₂²×Dic₁₉	—	2	2	no	—
70	C₂²×Dic₁₉	—	4	2	no	—
71	C₂×Dic₃₈	—	2	2	no	—
72	C₂×Dic₃₈	—	2	2	no	—
73	C₂×Dic₃₈	—	4	2	no	—
74	C₂×Dic₃₈	—	4	2	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	2	2	no	—
78	C₂×C₈×C₁₉	—	2	2	no	—
79	C₂×C₈×C₁₉	—	4	1	no	—
80	C₂×C₈×C₁₉	—	4	1	no	—
81	C₂×C₈×C₁₉	—	4	1	no	—
82	C₂×C₈×C₁₉	—	4	1	no	—
83	C₂×C₈×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₁₉	—	2	2	no	—
89	C₂²⋊C₄×C₁₉	Q₁₆	1	4	yes	complemented
90	C₄²×C₁₉	—	4	1	no	—
91	C₄²×C₁₉	—	4	1	no	—
92	C₂²⋊Dic₁₉	—	4	2	no	—
93	C₂²⋊Dic₁₉	—	4	2	no	—
94	C₂²⋊Dic₁₉	—	4	4	no	—
95	C₄⋊Dic₁₉	—	2	2	no	—
96	C₄⋊Dic₁₉	—	2	2	no	—
97	C₄⋊Dic₁₉	—	4	2	no	—
98	C₄⋊Dic₁₉	—	4	2	no	—
99	Dic₁₉⋊C₄	—	4	2	no	—
100	Dic₁₉⋊C₄	—	4	2	no	—
101	Dic₇₆	—	4	2	no	—
102	Dic₇₆	—	4	2	no	—
103	#9782	—	19	1	no	maximal, Sylow
104	C₂³×C₁₉	—	4	1	no	—
105	C₂³×C₁₉	—	4	1	no	—
106	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
107	C₂³×C₁₉	#121	1	1	yes	characteristic
108	C₂³×C₁₉	#127	1	1	yes	characteristic
109	C₂³×C₁₉	C₂×Q₁₆	1	2	yes	—
110	C₂³×C₁₉	#135	1	2	yes	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	4	no	—
118	C₂×C₄×C₁₉	—	4	1	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	4	1	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
127	C₂×Dic₁₉	—	4	2	no	—
128	C₂×Dic₁₉	—	4	2	no	—
129	C₂×Dic₁₉	—	8	2	no	—
130	C₂×Dic₁₉	—	8	2	no	—
131	C₂×Dic₁₉	—	8	2	no	—
132	C₂×Dic₁₉	—	8	2	no	—
133	C₂×Dic₁₉	—	8	2	no	—
134	Dic₃₈	—	4	2	no	—
135	Dic₃₈	—	4	2	no	—
136	Dic₃₈	—	8	2	no	—
137	C₈×C₁₉	—	2	1	no	—
138	C₈×C₁₉	—	2	1	no	—
139	C₈×C₁₉	—	2	1	no	—
140	C₈×C₁₉	—	4	2	no	—
141	C₈×C₁₉	#99	1	2	yes	—
142	C₂×#467	—	19	1	no	—
143	#1996	—	19	2	no	—
144	#1808	—	19	1	no	—
145	C₈×C₂²⋊C₄	D₁₉	1	1	yes	characteristic
146	#1520	—	19	2	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	2	1	no	—
149	C₂²×C₁₉	—	2	1	no	—
150	C₂²×C₁₉	—	2	1	no	—
151	C₂²×C₁₉	—	2	1	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	4	2	no	—
154	C₂²×C₁₉	—	4	2	no	—
155	C₂²×C₁₉	#390	1	1	yes	characteristic
156	C₂²×C₁₉	C₄×Q₁₆	1	2	yes	—
157	C₂²×C₁₉	#467	1	2	yes	—
158	C₂²×C₁₉	#494	1	1	yes	characteristic
159	C₂²×C₁₉	#495	1	1	yes	characteristic
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	1	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	4	4	no	—
164	C₄×C₁₉	C₂×#99	1	1	yes	characteristic
165	C₄×C₁₉	#410	1	1	yes	characteristic
166	Dic₁₉	—	8	2	no	—
167	Dic₁₉	—	8	2	no	—
168	Dic₁₉	—	16	2	no	—
169	C₂²×Q₁₆	—	19	1	no	—
170	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
171	C₂×C₂²⋊Q₈	—	19	2	no	—
172	#467	—	38	1	no	—
173	#467	—	38	1	no	—
174	#467	—	38	1	no	—
175	#467	—	38	1	no	—
176	C₂×#107	—	19	1	no	—
177	C₂×#107	—	38	1	no	—
178	C₂×#103	—	19	2	no	—
179	C₂×C₂²⋊C₈	C₂×D₁₉	1	1	yes	characteristic
180	C₂×C₄×C₈	—	2	1	no	—
181	#380	—	19	2	no	—
182	C₄×C₂²⋊C₄	C₂×D₁₉	1	1	yes	characteristic
183	#342	—	38	4	no	—
184	#340	—	38	1	no	—
185	#336	—	2	1	no	—
186	#328	—	38	4	no	—
187	C₂×C₁₉	—	2	1	no	—
188	C₂×C₁₉	—	2	2	no	—
189	C₂×C₁₉	—	2	2	no	—
190	C₂×C₁₉	—	2	1	no	—
191	C₂×C₁₉	#1964	1	1	yes	characteristic
192	C₂×C₁₉	#1969	1	1	yes	characteristic
193	C₂×C₁₉	#1976	1	1	yes	characteristic
194	C₂²×Q₈	—	19	2	no	—
195	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
196	C₂×Q₁₆	—	38	1	no	—
197	C₂×Q₁₆	—	38	1	no	—
198	C₂×Q₁₆	—	76	2	no	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	—	4	2	no	—
201	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
202	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
203	C₂²×C₈	D₇₆	1	1	yes	characteristic
204	C₂²×C₈	D₇₆	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
206	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
207	C₂²⋊Q₈	—	38	2	no	—
208	C₂²⋊Q₈	—	38	2	no	—
209	C₂²⋊Q₈	—	38	2	no	—
210	C₂²⋊Q₈	—	38	2	no	—
211	C₂×C₄⋊C₄	—	19	2	no	—
212	C₂×C₄⋊C₄	—	38	2	no	—
213	C₂×C₄⋊C₄	—	38	2	no	—
214	C₂×C₂²⋊C₄	—	38	2	no	—
215	C₂×C₂²⋊C₄	—	38	2	no	—
216	C₂×C₂²⋊C₄	D₇₆	1	2	yes	—
217	C₂×C₄²	—	2	1	no	—
218	#107	—	38	1	no	—
219	#107	—	38	1	no	—
220	#107	—	76	2	no	—
221	#103	—	38	2	no	—
222	#103	—	38	2	no	—
223	C₂²⋊C₈	—	2	1	no	—
224	C₂²⋊C₈	D₇₆	1	2	yes	—
225	C₄×C₈	—	4	2	no	—
226	#95	—	2	1	no	—
227	#95	—	38	2	no	—
228	C₁₉	#9782	1	1	yes	characteristic, complemented, Sylow
229	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
230	C₂×Q₈	—	38	2	no	—
231	C₂×Q₈	—	38	2	no	—
232	C₂×Q₈	—	76	2	no	—
233	C₂×Q₈	—	76	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	38	2	no	—
238	C₂²×C₄	—	38	2	no	—
239	C₂²×C₄	—	76	2	no	—
240	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
243	C₂²×C₄	#64541	1	1	yes	characteristic
244	C₂²×C₄	#64541	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
246	Q₁₆	—	76	2	no	—
247	Q₁₆	—	76	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
258	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
259	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
260	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
261	C₄⋊C₄	—	38	2	no	—
262	C₄⋊C₄	—	38	2	no	—
263	C₄⋊C₄	—	76	2	no	—
264	C₄⋊C₄	—	76	2	no	—
265	C₄⋊C₄	—	76	2	no	—
266	C₄⋊C₄	—	76	2	no	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	76	2	no	—
269	C₂²⋊C₄	—	76	2	no	—
270	C₂²⋊C₄	—	76	4	no	—
271	C₂²⋊C₄	Dic₇₆	1	4	yes	complemented
272	C₄²	—	4	1	no	—
273	C₄²	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	C₄×D₇₆	1	1	yes	characteristic
277	C₂³	#10597320	1	1	yes	characteristic
278	C₂³	#10597350	1	2	yes	—
279	C₂³	C₂×Dic₇₆	1	2	yes	—
280	C₂³	#10597376	1	1	yes	characteristic
281	Q₈	—	76	2	no	—
282	Q₈	—	76	2	no	—
283	Q₈	—	152	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	76	2	no	—
297	C₂×C₄	—	76	2	no	—
298	C₂×C₄	—	152	2	no	—
299	C₂×C₄	—	152	2	no	—
300	C₂×C₄	—	152	2	no	—
301	C₂×C₄	—	152	2	no	—
302	C₂×C₄	—	152	2	no	—
303	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
304	C₂×C₄	#10597321	1	1	yes	characteristic
305	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
306	C₂×C₄	#10597376	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	4	2	no	—
311	C₈	#10597263	1	2	yes	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	#11926446	1	1	yes	characteristic
321	C₂²	C₄×Dic₇₆	1	2	yes	—
322	C₂²	#11926785	1	2	yes	—
323	C₂²	#11926924	1	1	yes	characteristic
324	C₂²	#11926928	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	4	no	—
329	C₄	—	304	2	no	—
330	C₄	—	152	2	no	—
331	C₄	—	152	2	no	—
332	C₄	C₂×#10597263	1	1	yes	characteristic
333	C₄	#11926517	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	2	1	no	—
336	C₂	—	2	2	no	—
337	C₂	—	2	1	no	—
338	C₂	#14564471	1	1	yes	characteristic
339	C₂	#14564496	1	1	yes	characteristic
340	C₂	#14564533	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926446
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₈×C₂²⋊C₄×C₁₉	C₂
socle	C₂²×C₁₉	#390
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₈×C₂²⋊C₄	D₁₉
hypocenter	C₁₉	#9782
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₈×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
6. C₂²×Dic₇₆	C₄	semidirect
13. C₂×#10597275	C₄	semidirect
14. C₂×#10597273	C₄	semidirect
89. C₂²⋊C₄×C₁₉	Q₁₆	semidirect
228. C₁₉	#9782	semidirect
271. C₂²⋊C₄	Dic₇₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926785	2
#14564620	2
#14563664	2
#14562208	2
C₈×C₂²⋊C₄×C₁₉	2
#9782	19