Subgroups — #18070881

Order 4864.

432 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563778	C₂	1	1	yes	maximal, characteristic, complemented
2	#14563381	C₂	1	1	yes	maximal, characteristic, complemented
3	#14561461	C₂	1	2	yes	maximal, complemented
4	#14561456	C₂	1	2	yes	maximal
5	C₁₉×#2861	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
7	#11926299	—	2	1	no	—
8	#11926299	—	2	1	no	—
9	#11926299	—	2	1	no	—
10	#11926299	C₂²	1	2	yes	—
11	#11926299	C₂²	1	2	yes	—
12	C₂×C₁₉×#125	C₄	1	1	yes	characteristic
13	C₂×C₁₉×#124	C₄	1	1	yes	characteristic
14	C₁₉×#391	C₄	1	1	yes	characteristic
15	C₁₉×#391	C₂²	1	1	yes	characteristic
16	C₁₉×#390	C₄	1	1	yes	characteristic
17	C₁₉×#390	C₂²	1	1	yes	characteristic
18	C₁₉×#388	—	2	1	no	—
19	C₁₉×#388	—	2	1	no	—
20	C₁₉×#388	—	2	1	no	—
21	C₁₉×#388	—	2	1	no	—
22	C₄³×C₁₉	C₂²	1	1	yes	characteristic
23	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₁₉×#125	—	4	2	no	—
26	C₁₉×#124	—	4	2	no	—
27	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
32	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
33	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
34	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
35	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
44	C₂²×C₁₉⋊C₈	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₁₉×#95	—	2	1	no	—
47	C₁₉×#95	—	2	1	no	—
48	C₁₉×#95	—	2	1	no	—
49	C₁₉×#95	—	2	1	no	—
50	C₄×C₁₉⋊C₈	—	2	4	no	—
51	C₂⁴×C₁₉	—	2	1	no	—
52	C₂×Q₈×C₁₉	—	2	2	no	—
53	C₂×Q₈×C₁₉	—	4	1	no	—
54	C₂×Q₈×C₁₉	—	4	1	no	—
55	C₂×D₄×C₁₉	—	2	2	no	—
56	C₂×D₄×C₁₉	—	4	1	no	—
57	C₂×D₄×C₁₉	—	4	1	no	—
58	C₂²×C₄×C₁₉	—	2	1	no	—
59	C₂²×C₄×C₁₉	—	2	1	no	—
60	C₂²×C₄×C₁₉	—	2	1	no	—
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₁₉	—	4	1	no	—
69	C₄⋊C₄×C₁₉	—	4	1	no	—
70	C₄⋊C₄×C₁₉	—	4	2	no	—
71	C₄⋊C₄×C₁₉	—	4	2	no	—
72	C₂²⋊C₄×C₁₉	—	4	2	no	—
73	C₂²⋊C₄×C₁₉	—	4	2	no	—
74	C₂²⋊C₄×C₁₉	—	4	2	no	—
75	C₂²⋊C₄×C₁₉	—	4	2	no	—
76	C₂²⋊C₄×C₁₉	—	4	2	no	—
77	C₂²⋊C₄×C₁₉	—	4	2	no	—
78	C₄²×C₁₉	—	2	2	no	—
79	C₄²×C₁₉	—	2	2	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	2	no	—
82	C₄²×C₁₉	—	2	2	no	—
83	C₄²×C₁₉	—	4	1	no	—
84	C₄²×C₁₉	—	4	1	no	—
85	C₂×C₁₉⋊C₈	—	2	4	no	—
86	C₂×C₁₉⋊C₈	—	4	2	no	—
87	C₂×C₁₉⋊C₈	—	4	2	no	—
88	C₂×C₁₉⋊C₈	—	4	2	no	—
89	C₂×C₁₉⋊C₈	—	4	2	no	—
90	C₂×C₁₉⋊C₈	—	4	2	no	—
91	#7656	—	19	1	no	maximal, Sylow
92	C₂³×C₁₉	—	4	1	no	—
93	C₂³×C₁₉	—	4	1	no	—
94	C₂³×C₁₉	—	4	1	no	—
95	C₂³×C₁₉	—	4	1	no	—
96	C₂³×C₁₉	—	4	1	no	—
97	C₂³×C₁₉	—	4	1	no	—
98	C₂³×C₁₉	—	4	1	no	—
99	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	Q₈×C₁₉	—	8	2	no	—
101	D₄×C₁₉	—	8	2	no	—
102	C₂×C₄×C₁₉	—	2	1	no	—
103	C₂×C₄×C₁₉	—	2	1	no	—
104	C₂×C₄×C₁₉	—	2	1	no	—
105	C₂×C₄×C₁₉	—	2	1	no	—
106	C₂×C₄×C₁₉	—	2	1	no	—
107	C₂×C₄×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
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	1	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	1	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	4	1	no	—
125	C₂×C₄×C₁₉	—	4	1	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	#100	1	2	yes	—
128	C₂×C₄×C₁₉	#104	1	2	yes	—
129	C₂×C₄×C₁₉	#131	1	2	yes	—
130	C₂×C₄×C₁₉	#135	1	2	yes	—
131	C₂×C₄×C₁₉	#135	1	2	yes	—
132	C₁₉⋊C₈	—	4	4	no	—
133	C₁₉⋊C₈	—	8	2	no	—
134	#2861	D₁₉	1	1	yes	characteristic
135	#1839	—	19	1	no	—
136	#1744	—	19	1	no	—
137	#1272	—	19	2	no	—
138	#1270	—	19	2	no	—
139	C₂²×C₁₉	—	8	1	no	—
140	C₂²×C₁₉	—	8	1	no	—
141	C₂²×C₁₉	—	8	1	no	—
142	C₂²×C₁₉	—	8	1	no	—
143	C₂²×C₁₉	—	8	1	no	—
144	C₂²×C₁₉	—	8	1	no	—
145	C₂²×C₁₉	—	8	1	no	—
146	C₂²×C₁₉	#408	1	1	yes	characteristic
147	C₂²×C₁₉	#410	1	1	yes	characteristic
148	C₂²×C₁₉	C₂×#100	1	1	yes	characteristic
149	C₂²×C₁₉	#416	1	1	yes	characteristic
150	C₂²×C₁₉	#416	1	1	yes	characteristic
151	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
152	C₂²×C₁₉	#421	1	1	yes	characteristic
153	C₄×C₁₉	—	2	2	no	—
154	C₄×C₁₉	—	2	2	no	—
155	C₄×C₁₉	—	2	2	no	—
156	C₄×C₁₉	—	2	2	no	—
157	C₄×C₁₉	—	2	2	no	—
158	C₄×C₁₉	—	4	1	no	—
159	C₄×C₁₉	—	4	1	no	—
160	C₄×C₁₉	—	8	1	no	—
161	C₄×C₁₉	—	8	1	no	—
162	C₄×C₁₉	—	8	1	no	—
163	C₂×#125	Dic₁₉	1	1	yes	characteristic
164	C₂×#124	Dic₁₉	1	1	yes	characteristic
165	C₂×C₄×C₈	—	38	1	no	—
166	#391	Dic₁₉	1	1	yes	characteristic
167	#391	C₂×D₁₉	1	1	yes	characteristic
168	#390	Dic₁₉	1	1	yes	characteristic
169	#390	C₂×D₁₉	1	1	yes	characteristic
170	#388	—	2	1	no	—
171	#388	—	2	1	no	—
172	#388	—	2	1	no	—
173	#388	—	2	1	no	—
174	C₄³	C₂×D₁₉	1	1	yes	characteristic
175	#336	—	19	2	no	—
176	#336	—	19	2	no	—
177	#336	—	38	1	no	—
178	#336	—	38	1	no	—
179	#336	—	38	1	no	—
180	C₂×C₁₉	—	16	1	no	—
181	C₂×C₁₉	#1469	1	1	yes	characteristic
182	C₂×C₁₉	#1482	1	1	yes	characteristic
183	C₂×C₁₉	#1494	1	1	yes	characteristic
184	C₂×C₁₉	#1500	1	1	yes	characteristic
185	C₂×C₁₉	#1521	1	1	yes	characteristic
186	C₂×C₁₉	#1530	1	1	yes	characteristic
187	C₂×C₁₉	#1543	1	1	yes	characteristic
188	C₂²×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
189	C₂²×D₄	C₂×Dic₁₉	1	1	yes	characteristic
190	C₂²×C₈	—	38	2	no	—
191	C₂²×C₈	—	38	2	no	—
192	#125	—	4	2	no	—
193	#124	—	4	2	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
197	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	2	1	no	—
200	C₂×C₂²⋊C₄	—	2	1	no	—
201	C₂×C₂²⋊C₄	—	2	1	no	—
202	C₂×C₂²⋊C₄	—	2	1	no	—
203	C₂×C₂²⋊C₄	—	2	1	no	—
204	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
205	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
207	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
208	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
209	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
210	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
211	C₄×C₈	—	38	4	no	—
212	#95	—	2	1	no	—
213	#95	—	2	1	no	—
214	#95	—	2	1	no	—
215	#95	—	2	1	no	—
216	C₁₉	#7656	1	1	yes	characteristic, complemented, Sylow
217	C₂⁴	—	2	1	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	4	1	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂×D₄	—	2	2	no	—
222	C₂×D₄	—	4	1	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
228	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
229	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
230	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
231	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
234	C₂×C₈	—	38	4	no	—
235	C₂×C₈	—	76	2	no	—
236	C₂×C₈	—	76	2	no	—
237	C₂×C₈	—	76	2	no	—
238	C₂×C₈	—	76	2	no	—
239	C₂×C₈	—	76	2	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₂²⋊C₄	—	4	2	no	—
245	C₂²⋊C₄	—	4	2	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₂²⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	4	2	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	4	1	no	—
256	C₄²	—	4	1	no	—
257	C₂³	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	4	1	no	—
261	C₂³	—	4	1	no	—
262	C₂³	—	4	1	no	—
263	C₂³	—	4	1	no	—
264	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
265	Q₈	—	8	2	no	—
266	D₄	—	8	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	#10597290	1	2	yes	—
293	C₂×C₄	#10597294	1	2	yes	—
294	C₂×C₄	#10597394	1	2	yes	—
295	C₂×C₄	#10597396	1	2	yes	—
296	C₂×C₄	#10597396	1	2	yes	—
297	C₈	—	76	4	no	—
298	C₈	—	152	2	no	—
299	C₂²	—	8	1	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	1	no	—
302	C₂²	—	8	1	no	—
303	C₂²	—	8	1	no	—
304	C₂²	—	8	1	no	—
305	C₂²	—	8	1	no	—
306	C₂²	#11926511	1	1	yes	characteristic
307	C₂²	#11926518	1	1	yes	characteristic
308	C₂²	C₂×#10597290	1	1	yes	characteristic
309	C₂²	#11926540	1	1	yes	characteristic
310	C₂²	#11926540	1	1	yes	characteristic
311	C₂²	C₂×#10597294	1	1	yes	characteristic
312	C₂²	#11926561	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	1	no	—
323	C₂	—	16	1	no	—
324	C₂	#14561987	1	1	yes	characteristic
325	C₂	#14562043	1	1	yes	characteristic
326	C₂	#14562095	1	1	yes	characteristic
327	C₂	#14562121	1	1	yes	characteristic
328	C₂	#14562212	1	1	yes	characteristic
329	C₂	#14562246	1	1	yes	characteristic
330	C₂	#14562297	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2861	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₉	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2861	D₁₉
hypocenter	C₁₉	#7656
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #2861

Complements

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

N	H ≅ G/N	type
1. #14563778	C₂	semidirect
2. #14563381	C₂	semidirect
3. #14561461	C₂	semidirect
216. C₁₉	#7656	semidirect

Maximal subgroups

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

subgroup	index
#14563778	2
#14563381	2
#14561461	2
#14561456	2
C₁₉×#2861	2
#7656	19