Subgroups — #18081513

Order 4864.

720 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	C₄×#10597320	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11926271	C₂	1	4	yes	maximal, complemented
4	C₁₉×#1767	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#10597320	C₄	1	1	yes	characteristic
6	C₂×C₄×D₇₆	—	2	1	no	—
7	C₂×C₄×D₇₆	—	2	1	no	—
8	C₂×C₄×D₇₆	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
10	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
11	#11926446	—	2	1	no	—
12	#11926446	—	2	1	no	—
13	#11926446	C₄	1	2	yes	—
14	#11926414	C₄	1	1	yes	characteristic
15	#11926299	—	2	1	no	—
16	#11926271	—	2	8	no	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₁₉×#97	—	2	1	no	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₁₉×#336	—	2	1	no	—
21	C₂²×D₇₆	—	2	1	no	—
22	C₂²×D₇₆	—	2	1	no	—
23	C₂²×D₇₆	C₂×C₄	1	2	yes	—
24	C₂²×C₄×D₁₉	—	2	2	no	—
25	C₂²×C₈×C₁₉	—	2	2	no	—
26	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
33	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
34	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
35	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
36	C₂×C₄⋊Dic₁₉	—	2	1	no	—
37	C₂×C₄⋊Dic₁₉	—	2	1	no	—
38	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	#10597320	—	2	4	no	—
41	C₄×D₇₆	—	2	4	no	—
42	C₄×D₇₆	—	2	4	no	—
43	C₄×D₇₆	—	4	4	no	—
44	C₄×D₇₆	—	4	2	no	—
45	C₄×D₇₆	—	4	2	no	—
46	C₁₉×C₄⋊C₈	—	2	2	no	—
47	C₁₉×C₄⋊C₈	D₄	1	4	yes	complemented
48	C₁₉×#97	—	4	2	no	—
49	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
50	C₄×C₁₉⋊C₈	—	2	4	no	—
51	C₂³×D₁₉	—	2	2	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	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₁₉	—	2	2	no	—
60	C₂×D₇₆	—	2	4	no	—
61	C₂×D₇₆	—	2	4	no	—
62	C₂×D₇₆	—	2	4	no	—
63	C₂×D₇₆	—	2	4	no	—
64	C₂×D₇₆	—	2	2	no	—
65	C₂×D₇₆	—	4	2	no	—
66	C₂×D₇₆	—	4	2	no	—
67	C₂×D₇₆	—	4	2	no	—
68	C₂×D₇₆	—	4	2	no	—
69	C₂×D₇₆	M₁₆	1	4	yes	—
70	C₂×C₄×D₁₉	—	4	4	no	—
71	C₂×C₄×D₁₉	—	4	2	no	—
72	C₂×C₄×D₁₉	—	4	2	no	—
73	C₂×C₄×D₁₉	—	4	4	no	—
74	C₂×C₈×C₁₉	—	2	4	no	—
75	C₂×C₈×C₁₉	—	4	2	no	—
76	C₂×C₈×C₁₉	—	4	2	no	—
77	C₄²×C₁₉	—	2	2	no	—
78	C₄²×C₁₉	—	2	2	no	—
79	C₄²×C₁₉	—	2	2	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	4	1	no	—
82	C₄²×C₁₉	D₈	1	2	yes	—
83	C₄²×C₁₉	SD₁₆	1	2	yes	—
84	C₄²×C₁₉	C₂×D₄	1	2	yes	—
85	C₄²×C₁₉	#41	1	2	yes	—
86	(C₂×D₁₉)⋊C₄	—	4	4	no	—
87	(C₂×D₁₉)⋊C₄	—	4	4	no	—
88	(C₂×D₁₉)⋊C₄	—	4	2	no	—
89	(C₂×D₁₉)⋊C₄	—	4	2	no	—
90	C₄⋊Dic₁₉	—	2	4	no	—
91	C₄⋊Dic₁₉	—	4	2	no	—
92	C₄⋊Dic₁₉	—	4	2	no	—
93	C₂×C₁₉⋊C₈	—	2	4	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	#9678	—	19	1	no	maximal, Sylow
97	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×D₁₉	—	2	4	no	—
99	C₂²×D₁₉	—	4	2	no	—
100	C₂²×D₁₉	—	4	2	no	—
101	C₂²×D₁₉	—	4	2	no	—
102	C₂²×D₁₉	—	4	2	no	—
103	C₂²×D₁₉	—	4	2	no	—
104	C₂²×D₁₉	—	4	2	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	#102	1	2	yes	—
117	C₂×C₄×C₁₉	#102	1	2	yes	—
118	C₂×C₄×C₁₉	#104	1	2	yes	—
119	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₁₉	#121	1	2	yes	—
121	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
124	C₂×C₄×C₁₉	#131	1	2	yes	—
125	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
128	C₂×Dic₁₉	—	4	2	no	—
129	C₂×Dic₁₉	—	4	2	no	—
130	C₂×Dic₁₉	—	4	2	no	—
131	D₇₆	—	2	8	no	—
132	D₇₆	—	4	8	no	—
133	D₇₆	—	4	8	no	—
134	D₇₆	—	4	4	no	—
135	C₄×D₁₉	—	8	4	no	—
136	C₄×D₁₉	—	8	4	no	—
137	C₈×C₁₉	—	4	4	no	—
138	C₁₉⋊C₈	—	4	4	no	—
139	C₄×#127	—	19	1	no	—
140	#1767	D₁₉	1	1	yes	characteristic, complemented
141	C₄×C₄⋊C₈	—	19	1	no	—
142	C₂×D₄⋊C₈	—	19	4	no	—
143	C₂²×C₁₉	#390	1	1	yes	characteristic
144	C₂²×C₁₉	#407	1	1	yes	characteristic
145	C₂²×C₁₉	#408	1	1	yes	characteristic
146	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
147	C₂²×C₁₉	#418	1	1	yes	characteristic
148	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
149	C₂²×C₁₉	#421	1	1	yes	characteristic
150	C₂×D₁₉	—	4	4	no	—
151	C₂×D₁₉	—	4	4	no	—
152	C₂×D₁₉	—	4	4	no	—
153	C₂×D₁₉	—	8	2	no	—
154	C₂×D₁₉	—	8	2	no	—
155	C₂×D₁₉	—	8	2	no	—
156	C₂×D₁₉	—	8	2	no	—
157	C₄×C₁₉	—	2	1	no	—
158	C₄×C₁₉	—	2	1	no	—
159	C₄×C₁₉	—	2	2	no	—
160	C₄×C₁₉	—	2	2	no	—
161	C₄×C₁₉	—	2	2	no	—
162	C₄×C₁₉	—	4	1	no	—
163	C₄×C₁₉	#435	1	4	yes	—
164	C₄×C₁₉	#459	1	2	yes	—
165	C₄×C₁₉	#460	1	2	yes	—
166	Dic₁₉	—	8	2	no	—
167	C₂×#127	—	19	1	no	—
168	C₂×C₄×D₄	—	19	2	no	—
169	C₂×C₄×D₄	—	38	1	no	—
170	C₂×C₄×D₄	—	38	1	no	—
171	C₂×C₄⋊C₈	—	38	1	no	—
172	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
173	C₂×#97	—	2	1	no	—
174	C₂×C₄×C₈	—	19	2	no	—
175	#390	—	19	2	no	—
176	#390	—	38	1	no	—
177	#390	—	38	1	no	—
178	#384	—	19	1	no	—
179	C₄³	C₂×D₁₉	1	1	yes	characteristic
180	#336	—	2	1	no	—
181	#336	—	38	1	no	—
182	D₄⋊C₈	—	38	8	no	—
183	C₂×C₁₉	#1486	1	1	yes	characteristic
184	C₂×C₁₉	#1490	1	1	yes	characteristic
185	C₂×C₁₉	#1492	1	1	yes	characteristic
186	C₂×C₁₉	#1496	1	1	yes	characteristic
187	C₂×C₁₉	#1955	1	1	yes	characteristic
188	C₂×C₁₉	#1980	1	1	yes	characteristic
189	C₂×C₁₉	#1982	1	1	yes	characteristic
190	D₁₉	—	8	4	no	—
191	C₂²×D₄	—	19	2	no	—
192	C₂²×D₄	—	38	1	no	—
193	C₂²×D₄	—	38	1	no	—
194	C₂³×C₄	—	38	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	38	2	no	—
197	#127	—	38	4	no	—
198	C₄×D₄	—	38	4	no	—
199	C₄×D₄	—	38	4	no	—
200	C₄×D₄	—	76	4	no	—
201	C₄×D₄	—	76	2	no	—
202	C₄×D₄	—	76	2	no	—
203	C₂×C₄⋊C₄	—	19	2	no	—
204	C₂×C₄⋊C₄	—	38	1	no	—
205	C₂×C₄⋊C₄	—	38	1	no	—
206	C₂×C₂²⋊C₄	—	38	2	no	—
207	C₂×C₂²⋊C₄	—	38	2	no	—
208	C₂×C₂²⋊C₄	—	38	2	no	—
209	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
211	C₂×C₄²	D₇₆	1	1	yes	characteristic
212	C₂×C₄²	D₇₆	1	1	yes	characteristic
213	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
214	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	—	38	4	no	—
218	C₄⋊C₈	D₇₆	1	4	yes	complemented
219	#97	—	4	2	no	—
220	C₄×C₈	—	38	4	no	—
221	C₁₉	#9678	1	1	yes	characteristic, complemented, Sylow
222	C₂⁴	—	38	2	no	—
223	C₂×D₄	—	19	4	no	—
224	C₂×D₄	—	38	4	no	—
225	C₂×D₄	—	38	4	no	—
226	C₂×D₄	—	38	4	no	—
227	C₂×D₄	—	38	2	no	—
228	C₂×D₄	—	38	4	no	—
229	C₂×D₄	—	76	2	no	—
230	C₂×D₄	—	76	2	no	—
231	C₂×D₄	—	76	2	no	—
232	C₂×D₄	—	76	2	no	—
233	C₂²×C₄	—	38	2	no	—
234	C₂²×C₄	—	76	4	no	—
235	C₂²×C₄	—	76	2	no	—
236	C₂²×C₄	—	76	2	no	—
237	C₂²×C₄	—	76	4	no	—
238	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
239	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
240	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
241	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	38	4	no	—
249	C₂×C₈	—	76	2	no	—
250	C₂×C₈	—	76	2	no	—
251	C₄⋊C₄	—	38	4	no	—
252	C₄⋊C₄	—	76	2	no	—
253	C₄⋊C₄	—	76	2	no	—
254	C₂²⋊C₄	—	76	4	no	—
255	C₂²⋊C₄	—	76	4	no	—
256	C₂²⋊C₄	—	76	2	no	—
257	C₂²⋊C₄	—	76	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	4	1	no	—
263	C₄²	#64525	1	2	yes	—
264	C₄²	#64527	1	2	yes	—
265	C₄²	C₂×D₇₆	1	2	yes	—
266	C₄²	#64541	1	2	yes	—
267	C₂³	—	38	4	no	—
268	C₂³	—	76	2	no	—
269	C₂³	—	76	2	no	—
270	C₂³	—	76	2	no	—
271	C₂³	—	76	2	no	—
272	C₂³	—	76	2	no	—
273	C₂³	—	76	2	no	—
274	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
275	D₄	—	38	8	no	—
276	D₄	—	76	8	no	—
277	D₄	—	76	8	no	—
278	D₄	—	76	4	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	—	2	1	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	—	76	2	no	—
291	C₂×C₄	—	76	2	no	—
292	C₂×C₄	—	76	2	no	—
293	C₂×C₄	—	152	4	no	—
294	C₂×C₄	—	152	4	no	—
295	C₂×C₄	#10597266	1	2	yes	—
296	C₂×C₄	#10597266	1	2	yes	—
297	C₂×C₄	#10597282	1	2	yes	—
298	C₂×C₄	C₄×D₇₆	1	2	yes	—
299	C₂×C₄	#10597320	1	1	yes	characteristic
300	C₂×C₄	#10597321	1	1	yes	characteristic
301	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
302	C₂×C₄	#10597353	1	2	yes	—
303	C₂×C₄	#10597376	1	2	yes	—
304	C₂×C₄	C₂×#64525	1	1	yes	characteristic
305	C₂×C₄	C₂×#64527	1	1	yes	characteristic
306	C₂×C₄	#10597395	1	2	yes	—
307	C₈	—	4	4	no	—
308	C₈	—	76	4	no	—
309	C₂²	—	76	4	no	—
310	C₂²	—	76	4	no	—
311	C₂²	—	76	4	no	—
312	C₂²	—	152	2	no	—
313	C₂²	—	152	2	no	—
314	C₂²	—	152	2	no	—
315	C₂²	—	152	2	no	—
316	C₂²	#11926446	1	1	yes	characteristic
317	C₂²	#11926507	1	1	yes	characteristic
318	C₂²	#11926512	1	1	yes	characteristic
319	C₂²	C₂×#10597266	1	1	yes	characteristic
320	C₂²	#11926548	1	1	yes	characteristic
321	C₂²	C₂×#10597282	1	1	yes	characteristic
322	C₂²	#11926560	1	1	yes	characteristic
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	152	2	no	—
330	C₄	#11926617	1	4	yes	—
331	C₄	#11926739	1	2	yes	—
332	C₄	#11926744	1	2	yes	—
333	C₂	—	152	4	no	—
334	C₂	#14562059	1	1	yes	characteristic
335	C₂	#14562077	1	1	yes	characteristic
336	C₂	#14562085	1	1	yes	characteristic
337	C₂	#14562103	1	1	yes	characteristic
338	C₂	#14564433	1	1	yes	characteristic
339	C₂	#14564554	1	1	yes	characteristic
340	C₂	#14564562	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1767	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂×#10597320	C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1767	D₁₉
hypocenter	C₁₉	#9678
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1767

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊(C₁₉⋊C₈)	C₂	semidirect
3. C₂×#11926271	C₂	semidirect
4. C₁₉×#1767	C₂	semidirect
47. C₁₉×C₄⋊C₈	D₄	semidirect
140. #1767	D₁₉	semidirect
218. C₄⋊C₈	D₇₆	semidirect
221. C₁₉	#9678	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597320	2
C₄×C₄⋊(C₁₉⋊C₈)	2
C₂×#11926271	2
C₁₉×#1767	2
#9678	19