Subgroups — #18079084

Order 4864.

560 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574550	C₂	1	1	yes	maximal, characteristic
2	#14564475	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562312	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11926287	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561532	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#2935	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597320	C₂²	1	1	yes	characteristic
9	#11926920	—	2	1	no	—
10	#11926892	—	2	1	no	—
11	#11926739	—	2	2	no	—
12	#11926677	C₄	1	4	yes	—
13	#11926597	C₂²	1	1	yes	characteristic
14	#11926568	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×#10597280	—	2	1	no	—
17	C₂×#10597280	—	2	1	no	—
18	#11926290	—	2	1	no	—
19	#11926288	—	2	1	no	—
20	#11926287	—	2	1	no	—
21	#11926287	—	2	1	no	—
22	C₂×C₁₉×#131	—	2	1	no	—
23	C₁₉×#532	C₂²	1	1	yes	characteristic
24	C₁₉×#436	—	2	1	no	—
25	C₁₉×#435	—	2	2	no	—
26	C₈×D₄×C₁₉	—	2	1	no	—
27	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₉×#408	—	2	1	no	—
29	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
30	C₂×#41×C₁₉	—	2	1	no	—
31	C₂²×D₇₆	—	2	1	no	—
32	C₂²×D₇₆	—	2	1	no	—
33	C₂²×D₇₆	C₂×C₄	1	2	yes	—
34	C₁₉×#131	—	4	2	no	—
35	C₁₉×#131	—	4	2	no	—
36	C₂×M₁₆×C₁₉	—	2	2	no	—
37	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
38	C₂×M₁₆×C₁₉	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₉	—	2	2	no	—
40	C₂²×C₈×C₁₉	—	2	1	no	—
41	C₁₉×C₄⋊Q₈	D₄	1	1	yes	characteristic
42	C₁₉×#127	D₄	1	1	yes	characteristic
43	C₁₉×#124	—	2	1	no	—
44	C₁₉×#121	—	2	1	no	—
45	C₁₉×#121	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₁₉	—	2	1	no	—
47	C₄×D₄×C₁₉	D₄	1	2	yes	—
48	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
49	C₂×#64525	—	2	4	no	—
50	C₂×#64520	—	2	1	no	—
51	C₂²×C₁₉⋊C₈	—	2	1	no	—
52	#10597320	—	2	1	no	—
53	#10597320	—	2	1	no	—
54	#10597320	—	2	1	no	—
55	#10597320	—	2	1	no	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
58	C₁₉×C₄⋊C₈	D₄	1	1	yes	characteristic
59	C₁₉×C₂²⋊C₈	—	2	2	no	—
60	C₁₉×C₂²⋊C₈	—	2	2	no	—
61	C₁₉×#97	—	2	1	no	—
62	C₄×C₈×C₁₉	—	2	1	no	—
63	C₂²⋊(C₁₉⋊C₈)	C₂×C₄	1	2	yes	—
64	#10597280	—	4	2	no	—
65	#10597280	—	4	2	no	—
66	#10597266	—	2	4	no	—
67	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
69	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
70	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
71	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
72	C₄×C₁₉⋊C₈	—	2	1	no	—
73	C₂³×D₁₉	—	2	2	no	—
74	#41×C₁₉	—	4	1	no	—
75	#41×C₁₉	—	4	1	no	—
76	#41×C₁₉	—	4	1	no	—
77	#41×C₁₉	—	4	1	no	—
78	C₂×Q₈×C₁₉	—	2	1	no	—
79	C₂×D₄×C₁₉	—	2	2	no	—
80	C₂×D₄×C₁₉	—	2	1	no	—
81	C₂×D₄×C₁₉	—	2	1	no	—
82	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
83	C₂²×C₄×C₁₉	—	2	2	no	—
84	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
87	C₂×D₇₆	—	2	2	no	—
88	C₂×D₇₆	—	2	2	no	—
89	C₂×D₇₆	—	4	1	no	—
90	C₂×D₇₆	—	4	1	no	—
91	C₂×D₇₆	—	4	1	no	—
92	C₂×D₇₆	—	4	1	no	—
93	C₂×D₇₆	—	4	1	no	—
94	C₂×D₇₆	—	4	1	no	—
95	C₂×D₇₆	—	4	1	no	—
96	C₂×D₇₆	—	4	1	no	—
97	C₂×D₇₆	—	8	2	no	—
98	C₂×D₇₆	—	8	1	no	—
99	C₂×D₇₆	—	8	1	no	—
100	M₁₆×C₁₉	—	2	1	no	—
101	M₁₆×C₁₉	—	2	1	no	—
102	M₁₆×C₁₉	—	2	1	no	—
103	M₁₆×C₁₉	—	4	4	no	—
104	M₁₆×C₁₉	D₈	1	2	yes	complemented
105	C₂×C₈×C₁₉	—	2	1	no	—
106	C₂×C₈×C₁₉	—	2	1	no	—
107	C₂×C₈×C₁₉	—	2	1	no	—
108	C₂×C₈×C₁₉	—	2	1	no	—
109	C₂×C₈×C₁₉	—	4	4	no	—
110	C₂×C₈×C₁₉	—	4	1	no	—
111	C₂×C₈×C₁₉	—	4	1	no	—
112	C₄⋊C₄×C₁₉	—	2	1	no	—
113	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
114	C₂²⋊C₄×C₁₉	—	2	2	no	—
115	C₂²⋊C₄×C₁₉	—	2	2	no	—
116	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₁₉	#41	1	1	yes	characteristic
120	#64525	—	8	4	no	—
121	#64520	—	4	1	no	—
122	#64520	—	4	1	no	—
123	C₂×C₁₉⋊C₈	—	2	2	no	—
124	C₂×C₁₉⋊C₈	—	4	1	no	—
125	C₂×C₁₉⋊C₈	—	4	1	no	—
126	C₂×C₁₉⋊C₈	—	8	1	no	—
127	#9273	—	19	1	no	maximal, Sylow
128	C₂³×C₁₉	—	2	2	no	—
129	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
130	C₂²×D₁₉	—	4	2	no	—
131	C₂²×D₁₉	—	4	2	no	—
132	C₂²×D₁₉	—	4	2	no	—
133	C₂²×D₁₉	—	8	2	no	—
134	C₂²×D₁₉	—	8	2	no	—
135	Q₈×C₁₉	—	4	1	no	—
136	Q₈×C₁₉	—	4	1	no	—
137	D₄×C₁₉	—	4	2	no	—
138	D₄×C₁₉	—	4	2	no	—
139	D₄×C₁₉	—	4	2	no	—
140	D₄×C₁₉	—	4	1	no	—
141	D₄×C₁₉	—	4	1	no	—
142	D₄×C₁₉	—	4	1	no	—
143	D₄×C₁₉	—	4	1	no	—
144	C₂×C₄×C₁₉	—	2	1	no	—
145	C₂×C₄×C₁₉	—	2	1	no	—
146	C₂×C₄×C₁₉	—	2	2	no	—
147	C₂×C₄×C₁₉	—	4	1	no	—
148	C₂×C₄×C₁₉	—	4	2	no	—
149	C₂×C₄×C₁₉	—	4	2	no	—
150	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
153	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
154	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
155	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
156	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
157	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
158	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
159	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
160	D₇₆	—	4	4	no	—
161	D₇₆	—	8	2	no	—
162	D₇₆	—	8	2	no	—
163	D₇₆	—	8	1	no	—
164	D₇₆	—	8	1	no	—
165	D₇₆	—	8	1	no	—
166	D₇₆	—	8	1	no	—
167	D₇₆	—	8	1	no	—
168	D₇₆	—	8	1	no	—
169	C₈×C₁₉	—	2	2	no	—
170	C₈×C₁₉	—	4	1	no	—
171	C₈×C₁₉	—	4	1	no	—
172	C₈×C₁₉	—	4	1	no	—
173	C₁₉⋊C₈	—	4	2	no	—
174	C₁₉⋊C₈	—	8	1	no	—
175	C₁₉⋊C₈	—	8	1	no	—
176	#3205	—	19	1	no	—
177	#2935	D₁₉	1	1	yes	characteristic, complemented
178	#1965	—	19	1	no	—
179	#1547	—	19	1	no	—
180	C₂×#331	—	19	1	no	—
181	#1301	—	19	1	no	—
182	#1283	—	19	1	no	—
183	C₂²×C₁₉	—	2	1	no	—
184	C₂²×C₁₉	—	2	1	no	—
185	C₂²×C₁₉	—	4	2	no	—
186	C₂²×C₁₉	—	4	2	no	—
187	C₂²×C₁₉	—	4	2	no	—
188	C₂²×C₁₉	#386	1	1	yes	characteristic
189	C₂²×C₁₉	C₄×D₈	1	1	yes	characteristic
190	C₂²×C₁₉	#442	1	1	yes	characteristic
191	C₂×D₁₉	—	8	2	no	—
192	C₂×D₁₉	—	8	2	no	—
193	C₂×D₁₉	—	8	2	no	—
194	C₂×D₁₉	—	16	2	no	—
195	C₂×D₁₉	—	16	2	no	—
196	C₄×C₁₉	—	2	1	no	—
197	C₄×C₁₉	—	2	1	no	—
198	C₄×C₁₉	—	2	1	no	—
199	C₄×C₁₉	—	4	2	no	—
200	C₄×C₁₉	#447	1	1	yes	characteristic
201	C₄×C₁₉	#449	1	1	yes	characteristic
202	C₄×C₁₉	#459	1	1	yes	characteristic
203	C₄×C₁₉	#463	1	1	yes	characteristic
204	C₄×C₁₉	#471	1	1	yes	characteristic
205	C₄×C₁₉	#472	1	1	yes	characteristic
206	C₂×#131	—	2	1	no	—
207	#532	C₂×D₁₉	1	1	yes	characteristic
208	C₂×#127	—	19	1	no	—
209	#493	—	38	1	no	—
210	#486	—	38	1	no	—
211	#459	—	38	2	no	—
212	#447	—	19	4	no	—
213	#436	—	2	1	no	—
214	#435	—	2	2	no	—
215	C₈×D₄	—	2	1	no	—
216	#431	—	19	1	no	—
217	#423	—	19	1	no	—
218	C₂×C₄⋊C₈	—	19	1	no	—
219	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
220	C₂×#100	—	38	1	no	—
221	C₂×#100	—	38	1	no	—
222	#408	—	2	1	no	—
223	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
224	#332	—	38	1	no	—
225	#331	—	38	1	no	—
226	#331	—	38	1	no	—
227	#331	—	38	1	no	—
228	C₂×C₁₉	—	2	1	no	—
229	C₂×C₁₉	—	2	1	no	—
230	C₂×C₁₉	—	8	2	no	—
231	C₂×C₁₉	#1879	1	1	yes	characteristic
232	C₂×C₁₉	#1907	1	1	yes	characteristic
233	C₂×C₁₉	#1914	1	1	yes	characteristic
234	D₁₉	—	16	2	no	—
235	C₂×#41	—	2	1	no	—
236	C₂²×D₄	—	19	2	no	—
237	C₂²×D₄	—	38	1	no	—
238	C₂²×D₄	—	38	1	no	—
239	C₂×D₈	—	38	4	no	—
240	#131	—	4	2	no	—
241	#131	—	4	2	no	—
242	C₂×M₁₆	—	2	2	no	—
243	C₂×M₁₆	—	38	1	no	—
244	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
245	C₂×M₁₆	C₂²⋊D₁₉	1	1	yes	characteristic
246	C₂²×C₈	—	2	2	no	—
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	38	1	no	—
249	C₄⋊Q₈	D₇₆	1	1	yes	characteristic
250	#127	—	38	1	no	—
251	#127	—	38	1	no	—
252	#127	—	38	1	no	—
253	#127	—	38	1	no	—
254	#127	D₇₆	1	1	yes	characteristic
255	#124	—	2	1	no	—
256	#121	—	2	1	no	—
257	#121	C₄×D₁₉	1	2	yes	—
258	C₄×D₄	—	2	1	no	—
259	C₄×D₄	C₂²⋊D₁₉	1	2	yes	—
260	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
261	C₄⋊C₈	—	2	1	no	—
262	C₄⋊C₈	—	38	1	no	—
263	C₄⋊C₈	—	38	1	no	—
264	C₄⋊C₈	—	38	1	no	—
265	C₄⋊C₈	—	38	1	no	—
266	C₄⋊C₈	—	38	1	no	—
267	C₄⋊C₈	C₂²⋊D₁₉	1	1	yes	characteristic
268	C₄⋊C₈	C₂²⋊D₁₉	1	1	yes	characteristic
269	#102	—	38	4	no	—
270	#100	—	76	2	no	—
271	#100	—	76	2	no	—
272	C₂²⋊C₈	—	2	2	no	—
273	C₂²⋊C₈	—	2	2	no	—
274	C₂²⋊C₈	—	19	2	no	—
275	#97	—	2	1	no	—
276	C₄×C₈	—	2	1	no	—
277	C₄×C₈	—	38	1	no	—
278	C₁₉	#9273	1	1	yes	characteristic, complemented, Sylow
279	C₂⁴	—	38	2	no	—
280	#41	—	4	1	no	—
281	#41	—	4	1	no	—
282	#41	—	4	1	no	—
283	#41	—	4	1	no	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	2	1	no	—
288	C₂×D₄	—	38	2	no	—
289	C₂×D₄	—	38	2	no	—
290	C₂×D₄	—	76	1	no	—
291	C₂×D₄	—	76	1	no	—
292	C₂×D₄	—	76	1	no	—
293	C₂×D₄	—	76	1	no	—
294	C₂×D₄	—	76	1	no	—
295	C₂×D₄	—	76	1	no	—
296	C₂×D₄	—	76	1	no	—
297	C₂×D₄	—	76	1	no	—
298	C₂×D₄	—	152	1	no	—
299	C₂×D₄	—	152	1	no	—
300	C₂×D₄	—	152	2	no	—
301	C₂×D₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
304	C₂²×C₄	#64541	1	1	yes	characteristic
305	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
306	D₈	—	152	4	no	—
307	M₁₆	—	2	1	no	—
308	M₁₆	—	2	1	no	—
309	M₁₆	—	2	1	no	—
310	M₁₆	—	4	4	no	—
311	M₁₆	—	76	1	no	—
312	M₁₆	—	76	1	no	—
313	M₁₆	#64525	1	2	yes	complemented
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	4	4	no	—
319	C₂×C₈	—	4	1	no	—
320	C₂×C₈	—	4	1	no	—
321	C₂×C₈	—	38	2	no	—
322	C₂×C₈	—	76	1	no	—
323	C₂×C₈	—	76	1	no	—
324	C₂×C₈	—	152	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₄²	C₂×D₇₆	1	1	yes	characteristic
330	C₄²	#64541	1	1	yes	characteristic
331	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
332	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
333	C₂³	—	2	2	no	—
334	C₂³	—	76	2	no	—
335	C₂³	—	76	2	no	—
336	C₂³	—	76	2	no	—
337	C₂³	—	152	2	no	—
338	C₂³	—	152	2	no	—
339	C₂³	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
340	Q₈	—	4	1	no	—
341	Q₈	—	4	1	no	—
342	D₄	—	4	2	no	—
343	D₄	—	4	2	no	—
344	D₄	—	4	2	no	—
345	D₄	—	4	1	no	—
346	D₄	—	4	1	no	—
347	D₄	—	4	1	no	—
348	D₄	—	4	1	no	—
349	D₄	—	76	4	no	—
350	D₄	—	152	2	no	—
351	D₄	—	152	2	no	—
352	D₄	—	152	1	no	—
353	D₄	—	152	1	no	—
354	D₄	—	152	1	no	—
355	D₄	—	152	1	no	—
356	D₄	—	152	1	no	—
357	D₄	—	152	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
365	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
366	C₂×C₄	#10597375	1	1	yes	characteristic
367	C₂×C₄	#10597376	1	1	yes	characteristic
368	C₂×C₄	#10597376	1	1	yes	characteristic
369	C₂×C₄	C₂×#64525	1	1	yes	characteristic
370	C₂×C₄	#10597378	1	1	yes	characteristic
371	C₂×C₄	#10597395	1	1	yes	characteristic
372	C₂×C₄	#10597396	1	1	yes	characteristic
373	C₂×C₄	#10597399	1	1	yes	characteristic
374	C₈	—	2	2	no	—
375	C₈	—	4	1	no	—
376	C₈	—	4	1	no	—
377	C₈	—	4	1	no	—
378	C₈	—	76	2	no	—
379	C₈	—	152	1	no	—
380	C₈	—	152	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	2	1	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	304	2	no	—
387	C₂²	—	304	2	no	—
388	C₂²	—	152	2	no	—
389	C₂²	—	152	2	no	—
390	C₂²	—	152	2	no	—
391	C₂²	#11926423	1	1	yes	characteristic
392	C₂²	C₄×#64525	1	1	yes	characteristic
393	C₂²	#11926654	1	1	yes	characteristic
394	C₄	—	2	1	no	—
395	C₄	—	2	1	no	—
396	C₄	—	2	1	no	—
397	C₄	—	4	2	no	—
398	C₄	#11926678	1	1	yes	characteristic
399	C₄	#11926689	1	1	yes	characteristic
400	C₄	#11926739	1	1	yes	characteristic
401	C₄	#11926759	1	1	yes	characteristic
402	C₄	#11926807	1	1	yes	characteristic
403	C₄	#11926814	1	1	yes	characteristic
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	8	2	no	—
407	C₂	—	304	2	no	—
408	C₂	#14563983	1	1	yes	characteristic
409	C₂	#14564143	1	1	yes	characteristic
410	C₂	#14564182	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926423
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2935	C₂
socle	C₂²×C₁₉	#386
prime radical	#14574550	C₂
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2935	D₁₉
hypocenter	C₁₉	#9273
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2935

Complements

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

N	H ≅ G/N	type
2. #14564475	C₂	semidirect
3. #14562312	C₂	semidirect
4. C₂×#11926287	C₂	semidirect
5. #14561532	C₂	semidirect
6. #14561486	C₂	semidirect
7. C₁₉×#2935	C₂	semidirect
104. M₁₆×C₁₉	D₈	semidirect
177. #2935	D₁₉	semidirect
278. C₁₉	#9273	semidirect
313. M₁₆	#64525	semidirect

Maximal subgroups

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

subgroup	index
#14574550	2
#14564475	2
#14562312	2
C₂×#11926287	2
#14561532	2
#14561486	2
C₁₉×#2935	2
#9273	19