Subgroups — #18081869

Order 4864.

510 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926892	C₂	1	1	yes	maximal, characteristic
2	#14564562	C₂	1	1	yes	maximal, characteristic
3	#14563504	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11926296	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11926293	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11926287	C₂	1	1	yes	maximal, characteristic
7	C₁₉×#1987	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597320	C₂²	1	1	yes	characteristic
9	#11926892	—	2	2	no	—
10	#11926892	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#10597266	—	2	1	no	—
14	C₂×#10597266	—	2	1	no	—
15	C₂×#10597266	—	2	2	no	—
16	C₂×#10597260	—	2	1	no	—
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	#11926299	—	2	1	no	—
19	#11926296	—	2	2	no	—
20	#11926293	—	2	2	no	—
21	#11926287	—	2	2	no	—
22	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
25	C₁₉×#340	—	2	2	no	—
26	C₂²×Q₈×C₁₉	—	2	1	no	—
27	C₂²×D₇₆	—	2	1	no	—
28	C₂²×D₇₆	—	2	1	no	—
29	C₂²×D₇₆	—	2	1	no	—
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₁₉×C₄⋊Q₈	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	—	2	1	no	—
33	C₁₉×C₄⋊Q₈	—	2	1	no	—
34	C₁₉×C₄⋊Q₈	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
38	C₂×C₄²×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂²×C₁₉⋊C₈	—	2	1	no	—
41	C₂²×C₁₉⋊C₈	—	2	1	no	—
42	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
43	C₂²×C₁₉⋊C₈	D₄	1	1	yes	characteristic
44	#10597320	—	2	1	no	—
45	#10597320	—	2	1	no	—
46	#10597320	—	2	1	no	—
47	#10597320	C₂×C₄	1	2	yes	—
48	C₁₉×C₄⋊C₈	—	2	1	no	—
49	C₁₉×C₄⋊C₈	—	2	1	no	—
50	C₁₉×#95	—	2	1	no	—
51	#10597266	—	2	4	no	—
52	#10597266	—	2	4	no	—
53	#10597266	—	4	4	no	—
54	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
55	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
56	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
57	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
58	#10597260	—	4	2	no	—
59	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
60	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
61	C₂³×D₁₉	—	4	1	no	—
62	C₂×Q₈×C₁₉	—	2	2	no	—
63	C₂×Q₈×C₁₉	—	4	1	no	—
64	C₂×Q₈×C₁₉	—	8	2	no	—
65	C₂²×C₄×C₁₉	—	2	2	no	—
66	C₂²×C₄×C₁₉	—	4	1	no	—
67	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
70	C₂×D₇₆	—	2	2	no	—
71	C₂×D₇₆	—	2	2	no	—
72	C₂×D₇₆	—	2	2	no	—
73	C₂×D₇₆	—	4	1	no	—
74	C₂×D₇₆	—	4	1	no	—
75	C₂×D₇₆	—	4	1	no	—
76	C₂×D₇₆	—	4	2	no	—
77	C₂×D₇₆	—	4	2	no	—
78	C₂×D₇₆	—	4	1	no	—
79	C₂×D₇₆	—	4	1	no	—
80	C₂×D₇₆	—	4	1	no	—
81	C₂×D₇₆	—	4	1	no	—
82	C₂×D₇₆	—	8	2	no	—
83	C₂×C₈×C₁₉	—	4	1	no	—
84	C₂×C₈×C₁₉	—	4	2	no	—
85	C₄⋊C₄×C₁₉	—	2	2	no	—
86	C₄⋊C₄×C₁₉	—	2	2	no	—
87	C₄⋊C₄×C₁₉	—	2	2	no	—
88	C₄⋊C₄×C₁₉	—	2	2	no	—
89	C₄⋊C₄×C₁₉	—	4	1	no	—
90	C₄⋊C₄×C₁₉	—	4	1	no	—
91	C₄²×C₁₉	—	4	1	no	—
92	C₄²×C₁₉	—	4	1	no	—
93	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₉	#41	1	1	yes	characteristic
97	C₂×C₁₉⋊C₈	—	2	1	no	—
98	C₂×C₁₉⋊C₈	—	2	1	no	—
99	C₂×C₁₉⋊C₈	—	2	1	no	—
100	C₂×C₁₉⋊C₈	—	2	1	no	—
101	C₂×C₁₉⋊C₈	—	2	2	no	—
102	C₂×C₁₉⋊C₈	—	4	1	no	—
103	C₂×C₁₉⋊C₈	—	4	1	no	—
104	C₂×C₁₉⋊C₈	—	4	1	no	—
105	C₂×C₁₉⋊C₈	—	4	2	no	—
106	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
107	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
108	#9747	—	19	1	no	maximal, Sylow
109	C₂³×C₁₉	#121	1	1	yes	characteristic
110	C₂²×D₁₉	—	4	2	no	—
111	C₂²×D₁₉	—	8	1	no	—
112	C₂²×D₁₉	—	8	1	no	—
113	C₂²×D₁₉	—	8	2	no	—
114	C₂²×D₁₉	—	8	2	no	—
115	Q₈×C₁₉	—	8	2	no	—
116	Q₈×C₁₉	—	8	2	no	—
117	C₂×C₄×C₁₉	—	2	2	no	—
118	C₂×C₄×C₁₉	—	4	2	no	—
119	C₂×C₄×C₁₉	—	4	2	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	2	no	—
122	C₂×C₄×C₁₉	—	8	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	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
127	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
131	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
132	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#137	1	2	yes	—
134	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
135	D₇₆	—	4	2	no	—
136	D₇₆	—	4	2	no	—
137	D₇₆	—	4	4	no	—
138	D₇₆	—	8	2	no	—
139	D₇₆	—	8	2	no	—
140	D₇₆	—	8	1	no	—
141	D₇₆	—	8	1	no	—
142	D₇₆	—	8	1	no	—
143	D₇₆	—	8	1	no	—
144	C₈×C₁₉	—	8	1	no	—
145	C₁₉⋊C₈	—	2	2	no	—
146	C₁₉⋊C₈	—	2	2	no	—
147	C₁₉⋊C₈	—	4	2	no	—
148	C₁₉⋊C₈	—	8	1	no	—
149	C₂×#486	—	19	1	no	—
150	#1987	D₁₉	1	1	yes	characteristic, complemented
151	#1982	—	19	1	no	—
152	#1770	—	19	1	no	—
153	C₂×#335	—	19	1	no	—
154	C₂×#333	—	19	1	no	—
155	C₂×#331	—	19	1	no	—
156	C₂²×C₁₉	#390	1	1	yes	characteristic
157	C₂²×C₁₉	#463	1	1	yes	characteristic
158	C₂²×C₁₉	#464	1	1	yes	characteristic
159	C₂²×C₁₉	#468	1	2	yes	—
160	C₂²×C₁₉	#472	1	2	yes	—
161	C₂×D₁₉	—	8	2	no	—
162	C₂×D₁₉	—	8	2	no	—
163	C₂×D₁₉	—	8	2	no	—
164	C₂×D₁₉	—	16	2	no	—
165	C₂×D₁₉	—	16	2	no	—
166	C₄×C₁₉	—	2	1	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	8	2	no	—
169	C₄×C₁₉	—	8	2	no	—
170	C₄×C₁₉	#416	1	1	yes	characteristic
171	C₄×C₁₉	#417	1	1	yes	characteristic
172	C₄×C₁₉	#440	1	2	yes	—
173	C₄×C₁₉	#463	1	1	yes	characteristic
174	C₄×C₁₉	#464	1	1	yes	characteristic
175	C₄×C₁₉	#496	1	1	yes	characteristic
176	C₄×C₁₉	#497	1	1	yes	characteristic
177	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
178	C₂×#127	—	19	1	no	—
179	#486	—	19	4	no	—
180	#486	—	38	2	no	—
181	C₂×C₄⋊C₈	—	19	1	no	—
182	C₂×C₄⋊C₈	—	19	1	no	—
183	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
184	C₂×#102	—	38	1	no	—
185	C₂×#102	—	38	1	no	—
186	C₂×#102	—	38	2	no	—
187	C₂×#97	—	38	1	no	—
188	C₂×C₄×C₈	—	19	1	no	—
189	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
190	#340	—	2	2	no	—
191	#336	—	38	1	no	—
192	#335	—	38	2	no	—
193	#333	—	38	2	no	—
194	#331	—	38	2	no	—
195	C₂×C₁₉	#1697	1	2	yes	—
196	C₂×C₁₉	#1698	1	2	yes	—
197	C₂×C₁₉	#1973	1	1	yes	characteristic
198	C₂×C₁₉	#1978	1	1	yes	characteristic
199	C₂×C₁₉	#1983	1	1	yes	characteristic
200	D₁₉	—	16	2	no	—
201	C₂²×Q₈	—	2	1	no	—
202	C₂²×D₄	—	38	1	no	—
203	C₂²×D₄	—	38	1	no	—
204	C₂²×D₄	—	38	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	19	1	no	—
207	C₂²×C₈	—	19	1	no	—
208	C₂²×C₈	—	38	1	no	—
209	C₂²×C₈	—	38	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	C₄×D₁₉	1	2	yes	—
214	#127	—	19	2	no	—
215	#127	—	38	1	no	—
216	#127	—	38	1	no	—
217	#127	—	38	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	2	yes	—
221	C₂×C₄²	—	2	1	no	—
222	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	19	2	no	—
226	C₄⋊C₈	—	38	1	no	—
227	C₄⋊C₈	—	38	1	no	—
228	C₄⋊C₈	—	38	1	no	—
229	#102	—	38	4	no	—
230	#102	—	38	4	no	—
231	#102	—	76	4	no	—
232	#97	—	76	2	no	—
233	C₄×C₈	—	19	2	no	—
234	C₄×C₈	—	19	2	no	—
235	#95	—	2	1	no	—
236	C₁₉	#9747	1	1	yes	characteristic, complemented, Sylow
237	C₂⁴	—	76	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	8	2	no	—
241	C₂×D₄	—	38	2	no	—
242	C₂×D₄	—	38	2	no	—
243	C₂×D₄	—	38	2	no	—
244	C₂×D₄	—	76	1	no	—
245	C₂×D₄	—	76	2	no	—
246	C₂×D₄	—	76	2	no	—
247	C₂×D₄	—	76	1	no	—
248	C₂×D₄	—	76	1	no	—
249	C₂×D₄	—	76	1	no	—
250	C₂×D₄	—	76	1	no	—
251	C₂×D₄	—	76	1	no	—
252	C₂×D₄	—	76	1	no	—
253	C₂×D₄	—	152	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
257	C₂²×C₄	#64543	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	19	2	no	—
262	C₂×C₈	—	19	2	no	—
263	C₂×C₈	—	38	1	no	—
264	C₂×C₈	—	38	1	no	—
265	C₂×C₈	—	38	1	no	—
266	C₂×C₈	—	38	1	no	—
267	C₂×C₈	—	38	2	no	—
268	C₂×C₈	—	76	1	no	—
269	C₂×C₈	—	76	1	no	—
270	C₂×C₈	—	76	1	no	—
271	C₂×C₈	—	76	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
281	C₄²	#64541	1	1	yes	characteristic
282	C₄²	D₄×D₁₉	1	1	yes	characteristic
283	C₄²	D₄×D₁₉	1	1	yes	characteristic
284	C₂³	—	76	2	no	—
285	C₂³	—	152	2	no	—
286	C₂³	—	152	2	no	—
287	C₂³	—	152	1	no	—
288	C₂³	—	152	1	no	—
289	C₂³	#10597385	1	1	yes	characteristic
290	Q₈	—	8	2	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	76	2	no	—
293	D₄	—	76	2	no	—
294	D₄	—	76	4	no	—
295	D₄	—	152	2	no	—
296	D₄	—	152	2	no	—
297	D₄	—	152	1	no	—
298	D₄	—	152	1	no	—
299	D₄	—	152	1	no	—
300	D₄	—	152	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
308	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
309	C₂×C₄	#10597330	1	1	yes	characteristic
310	C₂×C₄	#10597331	1	1	yes	characteristic
311	C₂×C₄	#10597360	1	1	yes	characteristic
312	C₂×C₄	#10597362	1	1	yes	characteristic
313	C₂×C₄	#10597364	1	1	yes	characteristic
314	C₂×C₄	#10597365	1	1	yes	characteristic
315	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
316	C₂×C₄	#10597386	1	1	yes	characteristic
317	C₂×C₄	#10597388	1	2	yes	—
318	C₂×C₄	#10597395	1	2	yes	—
319	C₈	—	8	1	no	—
320	C₈	—	38	2	no	—
321	C₈	—	38	2	no	—
322	C₈	—	76	2	no	—
323	C₈	—	152	1	no	—
324	C₂²	—	304	2	no	—
325	C₂²	—	304	2	no	—
326	C₂²	—	152	2	no	—
327	C₂²	—	152	2	no	—
328	C₂²	—	152	2	no	—
329	C₂²	#11926444	1	1	yes	characteristic
330	C₂²	#11926761	1	1	yes	characteristic
331	C₂²	#11926768	1	1	yes	characteristic
332	C₂²	#11926792	1	2	yes	—
333	C₂²	#11926812	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	#11926539	1	1	yes	characteristic
339	C₄	#11926544	1	1	yes	characteristic
340	C₄	#11926642	1	2	yes	—
341	C₄	#11926762	1	1	yes	characteristic
342	C₄	#11926769	1	1	yes	characteristic
343	C₄	#11926934	1	1	yes	characteristic
344	C₄	#11926936	1	1	yes	characteristic
345	C₂	—	304	2	no	—
346	C₂	#14563170	1	2	yes	—
347	C₂	#14563177	1	2	yes	—
348	C₂	#14564519	1	1	yes	characteristic
349	C₂	#14564544	1	1	yes	characteristic
350	C₂	#14564564	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597385
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1987	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1987	D₁₉
hypocenter	C₁₉	#9747
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1987

Complements

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

N	H ≅ G/N	type
3. #14563504	C₂	semidirect
4. C₂×#11926296	C₂	semidirect
5. C₂×#11926293	C₂	semidirect
7. C₁₉×#1987	C₂	semidirect
10. #11926892	C₄	semidirect
150. #1987	D₁₉	semidirect
236. C₁₉	#9747	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926892	2
#14564562	2
#14563504	2
C₂×#11926296	2
C₂×#11926293	2
C₂×#11926287	2
C₁₉×#1987	2
#9747	19