Subgroups — #18079224

Order 4864.

558 conjugacy classes of subgroups.

The subgroup lattice has 392 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	#14564459	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562304	C₂	1	1	yes	maximal, characteristic, complemented
4	#14562195	C₂	1	1	yes	maximal, characteristic, complemented
5	#14561495	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561466	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₉×#2932	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597320	C₂²	1	1	yes	characteristic
9	#11926920	C₄	1	2	yes	complemented
10	#11926892	C₄	1	2	yes	complemented
11	#11926739	—	2	2	no	—
12	#11926677	—	2	2	no	—
13	#11926597	C₂²	1	1	yes	characteristic
14	#11926597	C₂²	1	1	yes	characteristic
15	C₂×#10597278	—	2	1	no	—
16	C₂×#10597278	—	2	1	no	—
17	#11926410	C₂²	1	1	yes	characteristic
18	#11926287	—	2	2	no	—
19	#11926278	—	2	2	no	—
20	C₂×C₁₉×#131	—	2	1	no	—
21	C₁₉×#532	C₂²	1	1	yes	characteristic
22	C₁₉×#435	—	2	2	no	—
23	C₁₉×#435	—	2	2	no	—
24	C₁₉×#423	C₂²	1	1	yes	characteristic
25	C₁₉×#408	—	2	1	no	—
26	C₂×C₁₉×#97	C₂²	1	1	yes	characteristic
27	C₂×#41×C₁₉	—	2	1	no	—
28	C₂²×D₇₆	—	2	1	no	—
29	C₂²×D₇₆	—	2	1	no	—
30	C₂²×D₇₆	—	2	1	no	—
31	C₁₉×#131	—	4	2	no	—
32	C₁₉×#131	—	4	2	no	—
33	C₂×M₁₆×C₁₉	—	2	2	no	—
34	C₂×M₁₆×C₁₉	—	2	1	no	—
35	C₂²×C₈×C₁₉	—	2	2	no	—
36	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
38	C₁₉×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
39	C₁₉×#127	C₂×C₄	1	1	yes	characteristic
40	C₁₉×#124	—	2	1	no	—
41	C₁₉×#121	—	2	1	no	—
42	C₁₉×#121	D₄	1	2	yes	—
43	C₄×D₄×C₁₉	—	2	1	no	—
44	C₄×D₄×C₁₉	D₄	1	2	yes	—
45	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
46	C₂×#64525	—	2	4	no	—
47	C₂×C₄⋊Dic₁₉	—	2	1	no	—
48	#10597320	—	2	1	no	—
49	#10597320	—	2	1	no	—
50	#10597320	—	2	1	no	—
51	#10597320	C₂×C₄	1	2	yes	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
54	C₁₉×C₂²⋊C₈	—	2	4	no	—
55	C₁₉×#97	—	2	1	no	—
56	C₁₉×#97	—	2	1	no	—
57	C₁₉⋊#95	—	2	1	no	—
58	C₁₉⋊#95	—	2	1	no	—
59	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
61	#10597278	—	4	2	no	—
62	#10597278	—	4	2	no	—
63	#10597266	—	2	4	no	—
64	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
65	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
66	C₄×C₁₉⋊C₈	—	2	1	no	—
67	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
68	C₂³×D₁₉	—	4	1	no	—
69	#41×C₁₉	—	4	1	no	—
70	#41×C₁₉	—	4	1	no	—
71	#41×C₁₉	—	4	1	no	—
72	#41×C₁₉	—	4	1	no	—
73	C₂×Q₈×C₁₉	—	2	1	no	—
74	C₂×D₄×C₁₉	—	2	2	no	—
75	C₂×D₄×C₁₉	—	2	1	no	—
76	C₂×D₄×C₁₉	—	2	1	no	—
77	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₁₉	—	2	2	no	—
79	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
82	C₂²×Dic₁₉	—	4	1	no	—
83	C₂×D₇₆	—	2	2	no	—
84	C₂×D₇₆	—	2	2	no	—
85	C₂×D₇₆	—	2	2	no	—
86	C₂×D₇₆	—	4	1	no	—
87	C₂×D₇₆	—	4	1	no	—
88	C₂×D₇₆	—	4	1	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	—	8	2	no	—
94	C₂×D₇₆	—	8	2	no	—
95	M₁₆×C₁₉	—	4	4	no	—
96	M₁₆×C₁₉	—	4	1	no	—
97	M₁₆×C₁₉	—	4	1	no	—
98	C₂×C₈×C₁₉	—	2	1	no	—
99	C₂×C₈×C₁₉	—	2	1	no	—
100	C₂×C₈×C₁₉	—	2	2	no	—
101	C₂×C₈×C₁₉	—	2	1	no	—
102	C₂×C₈×C₁₉	—	2	1	no	—
103	C₂×C₈×C₁₉	—	2	1	no	—
104	C₂×C₈×C₁₉	—	4	4	no	—
105	C₂×C₈×C₁₉	D₈	1	2	yes	complemented
106	C₄⋊C₄×C₁₉	—	2	1	no	—
107	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
108	C₂²⋊C₄×C₁₉	—	2	2	no	—
109	C₂²⋊C₄×C₁₉	—	2	2	no	—
110	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₁₉	#41	1	1	yes	characteristic
114	#64525	—	8	4	no	—
115	C₄⋊Dic₁₉	—	4	1	no	—
116	C₄⋊Dic₁₉	—	4	1	no	—
117	C₂×C₁₉⋊C₈	—	2	2	no	—
118	C₂×C₁₉⋊C₈	—	4	1	no	—
119	#9293	—	19	1	no	maximal, Sylow
120	C₂³×C₁₉	—	2	2	no	—
121	C₂³×C₁₉	#121	1	1	yes	characteristic
122	C₂²×D₁₉	—	4	2	no	—
123	C₂²×D₁₉	—	8	2	no	—
124	C₂²×D₁₉	—	8	2	no	—
125	C₂²×D₁₉	—	8	1	no	—
126	C₂²×D₁₉	—	8	1	no	—
127	Q₈×C₁₉	—	4	1	no	—
128	Q₈×C₁₉	—	4	1	no	—
129	D₄×C₁₉	—	4	2	no	—
130	D₄×C₁₉	—	4	2	no	—
131	D₄×C₁₉	—	4	2	no	—
132	D₄×C₁₉	—	4	1	no	—
133	D₄×C₁₉	—	4	1	no	—
134	D₄×C₁₉	—	8	1	no	—
135	C₂×C₄×C₁₉	—	2	2	no	—
136	C₂×C₄×C₁₉	—	4	1	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	2	no	—
139	C₂×C₄×C₁₉	—	4	2	no	—
140	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#123	1	1	yes	characteristic
146	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
148	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×Dic₁₉	—	8	1	no	—
151	C₂×Dic₁₉	—	8	1	no	—
152	C₂×Dic₁₉	—	8	1	no	—
153	D₇₆	—	4	4	no	—
154	D₇₆	—	4	2	no	—
155	D₇₆	—	4	2	no	—
156	D₇₆	—	8	2	no	—
157	D₇₆	—	8	2	no	—
158	D₇₆	—	8	2	no	—
159	D₇₆	—	8	1	no	—
160	D₇₆	—	8	1	no	—
161	C₈×C₁₉	—	2	2	no	—
162	C₈×C₁₉	—	4	2	no	—
163	C₈×C₁₉	—	4	1	no	—
164	C₁₉⋊C₈	—	4	2	no	—
165	C₁₉⋊C₈	—	8	1	no	—
166	#3205	—	19	1	no	—
167	#2932	D₁₉	1	1	yes	characteristic, complemented
168	#1961	—	19	1	no	—
169	#1545	—	19	1	no	—
170	#1517	—	19	1	no	—
171	#1286	—	19	1	no	—
172	#1275	—	19	1	no	—
173	C₂²×C₁₉	—	2	1	no	—
174	C₂²×C₁₉	—	2	1	no	—
175	C₂²×C₁₉	—	4	2	no	—
176	C₂²×C₁₉	—	4	2	no	—
177	C₂²×C₁₉	—	4	2	no	—
178	C₂²×C₁₉	#386	1	1	yes	characteristic
179	C₂²×C₁₉	#459	1	1	yes	characteristic
180	C₂²×C₁₉	#463	1	1	yes	characteristic
181	C₂×D₁₉	—	8	2	no	—
182	C₂×D₁₉	—	8	2	no	—
183	C₂×D₁₉	—	8	2	no	—
184	C₂×D₁₉	—	16	2	no	—
185	C₂×D₁₉	—	16	2	no	—
186	C₄×C₁₉	—	2	1	no	—
187	C₄×C₁₉	—	2	1	no	—
188	C₄×C₁₉	—	2	1	no	—
189	C₄×C₁₉	—	2	1	no	—
190	C₄×C₁₉	—	4	2	no	—
191	C₄×C₁₉	C₄×D₈	1	1	yes	characteristic
192	C₄×C₁₉	#442	1	1	yes	characteristic
193	C₄×C₁₉	#447	1	1	yes	characteristic
194	C₄×C₁₉	#449	1	1	yes	characteristic
195	Dic₁₉	—	16	1	no	—
196	C₂×#131	—	2	1	no	—
197	#532	C₂×D₁₉	1	1	yes	characteristic
198	C₂×#127	—	19	1	no	—
199	#493	—	19	2	no	—
200	#486	—	19	2	no	—
201	#459	—	38	2	no	—
202	#447	—	38	2	no	—
203	#435	—	2	2	no	—
204	#435	—	2	2	no	—
205	#431	—	19	1	no	—
206	#431	—	19	1	no	—
207	#423	C₂×D₁₉	1	1	yes	characteristic
208	C₂×#102	—	38	1	no	—
209	C₂×#102	—	38	1	no	—
210	#408	—	2	1	no	—
211	C₂×#97	C₂×D₁₉	1	1	yes	characteristic
212	#382	—	19	1	no	—
213	#331	—	38	2	no	—
214	#327	—	38	2	no	—
215	C₂×C₁₉	—	2	1	no	—
216	C₂×C₁₉	—	2	1	no	—
217	C₂×C₁₉	—	8	2	no	—
218	C₂×C₁₉	#1879	1	1	yes	characteristic
219	C₂×C₁₉	#1903	1	1	yes	characteristic
220	C₂×C₁₉	#1910	1	1	yes	characteristic
221	D₁₉	—	16	2	no	—
222	C₂×#41	—	2	1	no	—
223	C₂²×D₄	—	38	1	no	—
224	C₂²×D₄	—	38	1	no	—
225	C₂²×D₄	—	38	1	no	—
226	C₂×D₈	—	38	4	no	—
227	#131	—	4	2	no	—
228	#131	—	4	2	no	—
229	C₂×M₁₆	—	2	2	no	—
230	C₂×M₁₆	—	2	1	no	—
231	C₂²×C₈	—	2	2	no	—
232	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
233	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
234	C₄⋊Q₈	C₄×D₁₉	1	1	yes	characteristic
235	#127	—	19	2	no	—
236	#127	—	38	1	no	—
237	#127	—	38	1	no	—
238	#127	—	38	1	no	—
239	#127	C₄×D₁₉	1	1	yes	characteristic
240	#124	—	2	1	no	—
241	#121	—	2	1	no	—
242	#121	D₇₆	1	2	yes	—
243	C₄×D₄	—	2	1	no	—
244	C₄×D₄	C₂²⋊D₁₉	1	2	yes	—
245	C₂×C₄⋊C₄	—	38	1	no	—
246	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	38	1	no	—
249	C₄⋊C₈	—	38	1	no	—
250	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
251	#102	—	38	4	no	—
252	#102	—	76	2	no	—
253	#102	—	76	2	no	—
254	C₂²⋊C₈	—	2	4	no	—
255	C₂²⋊C₈	—	38	1	no	—
256	C₂²⋊C₈	—	38	1	no	—
257	#97	—	2	1	no	—
258	#97	—	2	1	no	—
259	C₄×C₈	—	19	2	no	—
260	C₄×C₈	—	38	1	no	—
261	#95	—	38	1	no	—
262	#95	—	38	1	no	—
263	C₁₉	#9293	1	1	yes	characteristic, complemented, Sylow
264	C₂⁴	—	76	1	no	—
265	#41	—	4	1	no	—
266	#41	—	4	1	no	—
267	#41	—	4	1	no	—
268	#41	—	4	1	no	—
269	C₂×Q₈	—	2	1	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	1	no	—
272	C₂×D₄	—	2	1	no	—
273	C₂×D₄	—	38	2	no	—
274	C₂×D₄	—	38	2	no	—
275	C₂×D₄	—	38	2	no	—
276	C₂×D₄	—	76	1	no	—
277	C₂×D₄	—	76	1	no	—
278	C₂×D₄	—	76	1	no	—
279	C₂×D₄	—	76	1	no	—
280	C₂×D₄	—	76	1	no	—
281	C₂×D₄	—	76	1	no	—
282	C₂×D₄	—	76	1	no	—
283	C₂×D₄	—	152	2	no	—
284	C₂×D₄	—	152	2	no	—
285	C₂×D₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	76	1	no	—
288	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
289	C₂²×C₄	#64541	1	1	yes	characteristic
290	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
291	D₈	—	152	4	no	—
292	M₁₆	—	4	4	no	—
293	M₁₆	—	4	1	no	—
294	M₁₆	—	4	1	no	—
295	C₂×C₈	—	2	1	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	2	1	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	4	4	no	—
302	C₂×C₈	—	38	2	no	—
303	C₂×C₈	—	76	1	no	—
304	C₂×C₈	#64525	1	2	yes	complemented
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	76	1	no	—
307	C₄⋊C₄	—	76	1	no	—
308	C₄⋊C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	2	2	no	—
311	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
312	C₄²	#64541	1	1	yes	characteristic
313	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
314	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
315	C₂³	—	2	2	no	—
316	C₂³	—	76	2	no	—
317	C₂³	—	152	2	no	—
318	C₂³	—	152	2	no	—
319	C₂³	—	152	1	no	—
320	C₂³	—	152	1	no	—
321	C₂³	#10597376	1	1	yes	characteristic
322	Q₈	—	4	1	no	—
323	Q₈	—	4	1	no	—
324	D₄	—	4	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	1	no	—
328	D₄	—	4	1	no	—
329	D₄	—	8	1	no	—
330	D₄	—	76	4	no	—
331	D₄	—	76	2	no	—
332	D₄	—	76	2	no	—
333	D₄	—	152	2	no	—
334	D₄	—	152	2	no	—
335	D₄	—	152	2	no	—
336	D₄	—	152	1	no	—
337	D₄	—	152	1	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	152	1	no	—
344	C₂×C₄	—	152	1	no	—
345	C₂×C₄	—	152	1	no	—
346	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
347	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
348	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
349	C₂×C₄	#10597375	1	1	yes	characteristic
350	C₂×C₄	#10597376	1	1	yes	characteristic
351	C₂×C₄	C₂×#64525	1	1	yes	characteristic
352	C₂×C₄	#10597378	1	1	yes	characteristic
353	C₂×C₄	#10597395	1	1	yes	characteristic
354	C₂×C₄	#10597396	1	1	yes	characteristic
355	C₂×C₄	#10597399	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	4	2	no	—
358	C₈	—	4	1	no	—
359	C₈	—	76	2	no	—
360	C₈	—	152	1	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	4	2	no	—
364	C₂²	—	4	2	no	—
365	C₂²	—	4	2	no	—
366	C₂²	—	304	2	no	—
367	C₂²	—	304	2	no	—
368	C₂²	—	152	2	no	—
369	C₂²	—	152	2	no	—
370	C₂²	—	152	2	no	—
371	C₂²	#11926423	1	1	yes	characteristic
372	C₂²	#11926739	1	1	yes	characteristic
373	C₂²	#11926759	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	2	1	no	—
378	C₄	—	4	2	no	—
379	C₄	—	304	1	no	—
380	C₄	C₄×#64525	1	1	yes	characteristic
381	C₄	#11926654	1	1	yes	characteristic
382	C₄	#11926678	1	1	yes	characteristic
383	C₄	#11926689	1	1	yes	characteristic
384	C₂	—	2	1	no	—
385	C₂	—	2	1	no	—
386	C₂	—	8	2	no	—
387	C₂	—	304	2	no	—
388	C₂	#14563983	1	1	yes	characteristic
389	C₂	#14564115	1	1	yes	characteristic
390	C₂	#14564162	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926423
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2932	C₂
socle	C₂²×C₁₉	#386
prime radical	#14574550	C₂
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2932	D₁₉
hypocenter	C₁₉	#9293
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2932

Complements

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

N	H ≅ G/N	type
2. #14564459	C₂	semidirect
3. #14562304	C₂	semidirect
4. #14562195	C₂	semidirect
5. #14561495	C₂	semidirect
6. #14561466	C₂	semidirect
7. C₁₉×#2932	C₂	semidirect
9. #11926920	C₄	semidirect
10. #11926892	C₄	semidirect
105. C₂×C₈×C₁₉	D₈	semidirect
167. #2932	D₁₉	semidirect
263. C₁₉	#9293	semidirect
304. C₂×C₈	#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
#14564459	2
#14562304	2
#14562195	2
#14561495	2
#14561466	2
C₁₉×#2932	2
#9293	19