Subgroups — #18081639

Order 4864.

610 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14571993	C₂	1	1	yes	maximal, characteristic
2	#14563786	C₂	1	1	yes	maximal, characteristic, complemented
3	#14563490	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11926293	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11926287	C₂	1	1	yes	maximal, characteristic
6	C₂×#11926287	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₉×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597320	C₂²	1	1	yes	characteristic
9	C₂×#10597393	C₄	1	4	yes	—
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₁₉⋊C₈	—	2	2	no	—
13	C₂×#10597272	—	2	1	no	—
14	#11926445	—	2	1	no	—
15	#11926445	—	2	2	no	—
16	#11926445	—	2	1	no	—
17	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
18	#11926298	—	2	1	no	—
19	#11926293	—	2	2	no	—
20	#11926287	—	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	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
25	C₁₉×#333	—	2	2	no	—
26	C₂²×Q₈×C₁₉	C₂×C₄	1	2	yes	—
27	C₂²×D₇₆	—	2	1	no	—
28	C₂²×D₇₆	—	2	1	no	—
29	C₂²×D₇₆	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₁₉	—	2	1	no	—
31	C₂²×C₈×C₁₉	—	2	1	no	—
32	C₁₉×C₄⋊Q₈	—	2	1	no	—
33	C₁₉×C₄⋊Q₈	—	2	1	no	—
34	C₁₉×C₄⋊Q₈	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
38	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
39	#10597393	—	4	8	no	—
40	C₂×(C₂×D₁₉)⋊C₄	—	2	4	no	—
41	C₂×(C₂×D₁₉)⋊C₄	—	2	4	no	—
42	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
43	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
45	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
46	C₂²×C₁₉⋊C₈	—	2	1	no	—
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	#10597320	—	2	1	no	—
49	#10597320	—	2	1	no	—
50	#10597320	—	2	1	no	—
51	#10597320	—	2	1	no	—
52	C₁₉×C₄⋊C₈	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
56	#10597272	—	4	2	no	—
57	Dic₁₉⋊C₈	—	4	4	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
61	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
62	C₂³×D₁₉	—	2	2	no	—
63	C₂×Q₈×C₁₉	—	2	2	no	—
64	C₂×Q₈×C₁₉	—	2	2	no	—
65	C₂×Q₈×C₁₉	—	4	4	no	—
66	C₂²×C₄×C₁₉	—	2	2	no	—
67	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
71	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
72	C₂×D₇₆	—	2	2	no	—
73	C₂×D₇₆	—	2	2	no	—
74	C₂×D₇₆	—	4	4	no	—
75	C₂×D₇₆	—	4	1	no	—
76	C₂×D₇₆	—	4	1	no	—
77	C₂×D₇₆	—	4	1	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	—	4	2	no	—
83	C₂×D₇₆	—	4	1	no	—
84	C₂×D₇₆	—	4	2	no	—
85	C₂×C₈×C₁₉	—	2	2	no	—
86	C₂×C₈×C₁₉	—	4	1	no	—
87	C₂×C₈×C₁₉	—	4	1	no	—
88	C₂×C₈×C₁₉	—	4	1	no	—
89	C₂×C₈×C₁₉	—	4	2	no	—
90	C₄⋊C₄×C₁₉	—	4	1	no	—
91	C₄⋊C₄×C₁₉	—	4	1	no	—
92	C₄⋊C₄×C₁₉	—	4	1	no	—
93	C₄⋊C₄×C₁₉	—	4	1	no	—
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	C₄²×C₁₉	#41	1	1	yes	characteristic
98	(C₂×D₁₉)⋊C₄	—	4	8	no	—
99	(C₂×D₁₉)⋊C₄	—	4	8	no	—
100	C₄×Dic₁₉	—	2	1	no	—
101	C₄×Dic₁₉	—	2	1	no	—
102	C₄×Dic₁₉	—	2	2	no	—
103	C₄×Dic₁₉	—	2	1	no	—
104	C₄×Dic₁₉	—	2	2	no	—
105	C₄×Dic₁₉	—	2	1	no	—
106	C₄×Dic₁₉	—	2	4	no	—
107	C₂×C₁₉⋊C₈	—	2	2	no	—
108	C₂×C₁₉⋊C₈	—	4	1	no	—
109	C₂×C₁₉⋊C₈	—	4	1	no	—
110	C₂×C₁₉⋊C₈	—	4	1	no	—
111	C₂×C₁₉⋊C₈	—	4	2	no	—
112	#9707	—	19	1	no	maximal, Sylow
113	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂²×D₁₉	—	4	2	no	—
115	C₂²×D₁₉	—	4	4	no	—
116	C₂²×D₁₉	—	4	2	no	—
117	C₂²×D₁₉	—	4	4	no	—
118	C₂²×D₁₉	—	4	2	no	—
119	Q₈×C₁₉	—	8	2	no	—
120	Q₈×C₁₉	—	8	2	no	—
121	C₂×C₄×C₁₉	—	2	2	no	—
122	C₂×C₄×C₁₉	—	4	2	no	—
123	C₂×C₄×C₁₉	—	4	4	no	—
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	#121	1	1	yes	characteristic
128	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
132	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#137	1	2	yes	—
136	C₂×Dic₁₉	—	2	2	no	—
137	C₂×Dic₁₉	—	2	2	no	—
138	C₂×Dic₁₉	—	2	2	no	—
139	C₂×Dic₁₉	—	2	2	no	—
140	C₂×Dic₁₉	—	2	2	no	—
141	C₂×Dic₁₉	#100	1	4	yes	—
142	D₇₆	—	8	2	no	—
143	D₇₆	—	8	2	no	—
144	D₇₆	—	8	1	no	—
145	D₇₆	—	8	1	no	—
146	D₇₆	—	8	1	no	—
147	D₇₆	—	8	1	no	—
148	D₇₆	—	8	2	no	—
149	D₇₆	—	8	1	no	—
150	D₇₆	—	8	1	no	—
151	C₈×C₁₉	—	4	2	no	—
152	C₈×C₁₉	—	8	1	no	—
153	C₁₉⋊C₈	—	4	2	no	—
154	C₁₉⋊C₈	—	8	1	no	—
155	#2858	—	19	1	no	—
156	#1842	—	19	1	no	—
157	#1767	—	19	1	no	—
158	C₂×#333	—	19	1	no	—
159	C₂×#333	D₁₉	1	1	yes	characteristic, complemented
160	C₂×#331	—	19	1	no	—
161	C₂×#331	—	19	1	no	—
162	C₂²×C₁₉	#390	1	1	yes	characteristic
163	C₂²×C₁₉	C₂×#100	1	2	yes	—
164	C₂²×C₁₉	#416	1	1	yes	characteristic
165	C₂²×C₁₉	#417	1	2	yes	—
166	C₂²×C₁₉	#417	1	1	yes	characteristic
167	C₂×D₁₉	—	8	4	no	—
168	C₂×D₁₉	—	8	4	no	—
169	C₂×D₁₉	—	8	2	no	—
170	C₂×D₁₉	—	8	2	no	—
171	C₂×D₁₉	—	8	2	no	—
172	C₄×C₁₉	—	2	1	no	—
173	C₄×C₁₉	—	2	1	no	—
174	C₄×C₁₉	—	8	2	no	—
175	C₄×C₁₉	#463	1	1	yes	characteristic
176	C₄×C₁₉	#463	1	1	yes	characteristic
177	C₄×C₁₉	#464	1	1	yes	characteristic
178	C₄×C₁₉	#464	1	1	yes	characteristic
179	C₄×C₁₉	#489	1	2	yes	—
180	C₄×C₁₉	#496	1	1	yes	characteristic
181	C₄×C₁₉	#497	1	1	yes	characteristic
182	Dic₁₉	—	2	4	no	—
183	Dic₁₉	—	4	2	no	—
184	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
185	C₂×#127	—	19	1	no	—
186	C₂×#124	—	19	4	no	—
187	C₂×C₄⋊C₈	—	19	1	no	—
188	C₂×C₄⋊C₈	—	19	1	no	—
189	C₂×C₄⋊C₈	—	38	2	no	—
190	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
191	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
192	C₂×#97	—	38	1	no	—
193	#390	—	38	1	no	—
194	#390	—	38	1	no	—
195	#390	—	38	2	no	—
196	C₄³	—	19	1	no	—
197	#336	—	38	1	no	—
198	#333	—	2	2	no	—
199	#333	—	38	2	no	—
200	#331	—	38	2	no	—
201	#331	—	38	2	no	—
202	C₂×C₁₉	#1551	1	2	yes	—
203	C₂×C₁₉	#1560	1	2	yes	—
204	C₂×C₁₉	#1959	1	1	yes	characteristic
205	C₂×C₁₉	#1965	1	1	yes	characteristic
206	C₂×C₁₉	#1983	1	1	yes	characteristic
207	D₁₉	—	16	2	no	—
208	C₂²×Q₈	C₄×D₁₉	1	2	yes	—
209	C₂²×D₄	—	19	2	no	—
210	C₂²×D₄	—	38	1	no	—
211	C₂²×D₄	—	38	1	no	—
212	C₂²×C₈	—	2	1	no	—
213	C₂²×C₈	—	2	1	no	—
214	C₂²×C₈	—	38	1	no	—
215	C₂²×C₈	—	38	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	#127	—	38	1	no	—
221	#127	—	38	1	no	—
222	#127	—	38	1	no	—
223	#127	—	38	1	no	—
224	#124	—	76	8	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₂²⋊C₄	—	38	4	no	—
228	C₂×C₂²⋊C₄	—	38	4	no	—
229	C₂×C₄²	—	19	2	no	—
230	C₂×C₄²	—	19	1	no	—
231	C₂×C₄²	—	19	1	no	—
232	C₂×C₄²	—	19	2	no	—
233	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
234	C₄⋊C₈	—	2	1	no	—
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	19	2	no	—
238	C₄⋊C₈	—	38	1	no	—
239	C₄⋊C₈	—	38	1	no	—
240	C₄⋊C₈	—	38	1	no	—
241	C₄⋊C₈	—	76	4	no	—
242	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
243	#97	—	76	2	no	—
244	C₁₉	#9707	1	1	yes	characteristic, complemented, Sylow
245	C₂⁴	—	38	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	4	no	—
249	C₂×D₄	—	38	2	no	—
250	C₂×D₄	—	38	2	no	—
251	C₂×D₄	—	76	4	no	—
252	C₂×D₄	—	76	1	no	—
253	C₂×D₄	—	76	1	no	—
254	C₂×D₄	—	76	1	no	—
255	C₂×D₄	—	76	1	no	—
256	C₂×D₄	—	76	2	no	—
257	C₂×D₄	—	76	1	no	—
258	C₂×D₄	—	76	1	no	—
259	C₂×D₄	—	76	2	no	—
260	C₂×D₄	—	76	1	no	—
261	C₂×D₄	—	76	1	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	19	2	no	—
264	C₂²×C₄	—	19	2	no	—
265	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
266	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
267	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	4	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	38	2	no	—
274	C₂×C₈	—	76	1	no	—
275	C₂×C₈	—	76	1	no	—
276	C₂×C₈	—	76	1	no	—
277	C₂×C₈	—	76	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	76	8	no	—
283	C₂²⋊C₄	—	76	8	no	—
284	C₄²	—	38	4	no	—
285	C₄²	—	38	1	no	—
286	C₄²	—	38	2	no	—
287	C₄²	—	38	1	no	—
288	C₄²	—	38	2	no	—
289	C₄²	—	38	1	no	—
290	C₄²	—	38	1	no	—
291	C₄²	#64541	1	1	yes	characteristic
292	C₄²	D₄×D₁₉	1	1	yes	characteristic
293	C₄²	#64543	1	1	yes	characteristic
294	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
295	C₂³	—	76	2	no	—
296	C₂³	—	76	4	no	—
297	C₂³	—	76	2	no	—
298	C₂³	—	76	2	no	—
299	C₂³	—	76	4	no	—
300	C₂³	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
301	Q₈	—	8	2	no	—
302	Q₈	—	8	2	no	—
303	D₄	—	152	2	no	—
304	D₄	—	152	2	no	—
305	D₄	—	152	2	no	—
306	D₄	—	152	1	no	—
307	D₄	—	152	1	no	—
308	D₄	—	152	1	no	—
309	D₄	—	152	1	no	—
310	D₄	—	152	1	no	—
311	D₄	—	152	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	19	4	no	—
316	C₂×C₄	—	38	2	no	—
317	C₂×C₄	—	38	2	no	—
318	C₂×C₄	—	38	2	no	—
319	C₂×C₄	—	38	2	no	—
320	C₂×C₄	—	38	2	no	—
321	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
322	C₂×C₄	#10597330	1	1	yes	characteristic
323	C₂×C₄	#10597331	1	1	yes	characteristic
324	C₂×C₄	#10597360	1	2	yes	—
325	C₂×C₄	#10597360	1	1	yes	characteristic
326	C₂×C₄	#10597362	1	1	yes	characteristic
327	C₂×C₄	#10597364	1	2	yes	—
328	C₂×C₄	#10597364	1	1	yes	characteristic
329	C₂×C₄	#10597365	1	1	yes	characteristic
330	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
331	C₂×C₄	#10597385	1	1	yes	characteristic
332	C₂×C₄	#10597386	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	8	1	no	—
335	C₈	—	76	2	no	—
336	C₈	—	152	1	no	—
337	C₂²	—	152	4	no	—
338	C₂²	—	152	4	no	—
339	C₂²	—	152	2	no	—
340	C₂²	—	152	2	no	—
341	C₂²	—	152	2	no	—
342	C₂²	#11926444	1	1	yes	characteristic
343	C₂²	#100×D₁₉	1	2	yes	—
344	C₂²	#11926539	1	1	yes	characteristic
345	C₂²	#11926544	1	2	yes	—
346	C₂²	#11926544	1	1	yes	characteristic
347	C₄	—	2	1	no	—
348	C₄	—	2	1	no	—
349	C₄	—	8	2	no	—
350	C₄	—	38	4	no	—
351	C₄	—	76	2	no	—
352	C₄	#11926761	1	1	yes	characteristic
353	C₄	#11926762	1	1	yes	characteristic
354	C₄	#11926768	1	1	yes	characteristic
355	C₄	#11926769	1	1	yes	characteristic
356	C₄	#11926905	1	2	yes	—
357	C₄	#11926930	1	1	yes	characteristic
358	C₄	#11926937	1	1	yes	characteristic
359	C₂	—	304	2	no	—
360	C₂	#14562329	1	2	yes	—
361	C₂	#14562369	1	2	yes	—
362	C₂	#14564448	1	1	yes	characteristic
363	C₂	#14564473	1	1	yes	characteristic
364	C₂	#14564564	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₁₉
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#333	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#333	D₁₉
hypocenter	C₁₉	#9707
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
2. #14563786	C₂	semidirect
3. #14563490	C₂	semidirect
4. C₂×#11926293	C₂	semidirect
7. C₂×C₁₉×#333	C₂	semidirect
159. C₂×#333	D₁₉	semidirect
244. C₁₉	#9707	semidirect

Maximal subgroups

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

subgroup	index
#14571993	2
#14563786	2
#14563490	2
C₂×#11926293	2
C₂×#11926287	2
C₂×C₁₉×#333	2
#9707	19