Subgroups — #18081598

Order 4864.

622 conjugacy classes of subgroups.

The subgroup lattice has 372 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	C₄⋊C₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11926290	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×#332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597320	C₂²	1	1	yes	characteristic
9	C₂×#10597393	—	2	2	no	—
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×C₈×Dic₁₉	—	2	1	no	—
14	#11926445	—	2	1	no	—
15	#11926445	—	2	1	no	—
16	#11926445	C₄	1	4	yes	—
17	C₄²×Dic₁₉	C₂²	1	1	yes	characteristic
18	#11926298	—	2	1	no	—
19	#11926290	—	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×#332	—	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	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	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
39	#10597393	—	4	4	no	—
40	#10597393	—	4	4	no	—
41	C₂×(C₂×D₁₉)⋊C₄	—	2	4	no	—
42	C₂×(C₂×D₁₉)⋊C₄	—	2	4	no	—
43	C₂×C₄×Dic₁₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
45	C₂×C₄×Dic₁₉	D₄	1	1	yes	characteristic
46	C₂×C₄×Dic₁₉	D₄	1	2	yes	—
47	C₂²×C₁₉⋊C₈	—	2	1	no	—
48	C₂²×C₁₉⋊C₈	—	2	1	no	—
49	#10597320	—	2	1	no	—
50	#10597320	—	2	1	no	—
51	#10597320	—	2	1	no	—
52	#10597320	—	2	1	no	—
53	C₁₉×C₄⋊C₈	—	2	1	no	—
54	C₁₉×C₄⋊C₈	—	2	1	no	—
55	C₁₉×C₄⋊C₈	—	2	1	no	—
56	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
57	Dic₁₉⋊C₈	—	4	4	no	—
58	C₈×Dic₁₉	—	4	2	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
62	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
63	C₂³×D₁₉	—	2	2	no	—
64	C₂×Q₈×C₁₉	—	4	1	no	—
65	C₂×Q₈×C₁₉	—	4	1	no	—
66	C₂×Q₈×C₁₉	—	4	2	no	—
67	C₂×Q₈×C₁₉	—	4	2	no	—
68	C₂²×C₄×C₁₉	—	2	2	no	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
73	C₂²×Dic₁₉	C₂²⋊C₄	1	2	yes	—
74	C₂×D₇₆	—	2	2	no	—
75	C₂×D₇₆	—	2	2	no	—
76	C₂×D₇₆	—	4	4	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	2	no	—
82	C₂×D₇₆	—	4	2	no	—
83	C₂×D₇₆	—	4	1	no	—
84	C₂×D₇₆	—	4	1	no	—
85	C₂×D₇₆	—	4	1	no	—
86	C₂×D₇₆	—	4	1	no	—
87	C₂×C₈×C₁₉	—	2	2	no	—
88	C₂×C₈×C₁₉	—	4	1	no	—
89	C₂×C₈×C₁₉	—	4	1	no	—
90	C₂×C₈×C₁₉	—	4	1	no	—
91	C₂×C₈×C₁₉	—	4	2	no	—
92	C₄⋊C₄×C₁₉	—	2	2	no	—
93	C₄⋊C₄×C₁₉	—	2	2	no	—
94	C₄⋊C₄×C₁₉	—	4	1	no	—
95	C₄⋊C₄×C₁₉	—	4	1	no	—
96	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₉	#41	1	1	yes	characteristic
99	C₄²×C₁₉	#41	1	1	yes	characteristic
100	(C₂×D₁₉)⋊C₄	—	4	8	no	—
101	(C₂×D₁₉)⋊C₄	—	4	8	no	—
102	C₄×Dic₁₉	—	2	1	no	—
103	C₄×Dic₁₉	—	2	1	no	—
104	C₄×Dic₁₉	—	2	1	no	—
105	C₄×Dic₁₉	—	2	1	no	—
106	C₄×Dic₁₉	—	4	2	no	—
107	C₄×Dic₁₉	D₈	1	4	yes	—
108	C₄×Dic₁₉	SD₁₆	1	4	yes	—
109	C₂×C₁₉⋊C₈	—	2	2	no	—
110	C₂×C₁₉⋊C₈	—	4	1	no	—
111	C₂×C₁₉⋊C₈	—	4	1	no	—
112	C₂×C₁₉⋊C₈	—	4	1	no	—
113	C₂×C₁₉⋊C₈	—	4	2	no	—
114	#9698	—	19	1	no	maximal, Sylow
115	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂²×D₁₉	—	4	2	no	—
117	C₂²×D₁₉	—	4	4	no	—
118	C₂²×D₁₉	—	4	4	no	—
119	C₂²×D₁₉	—	4	2	no	—
120	C₂²×D₁₉	—	4	2	no	—
121	Q₈×C₁₉	—	8	2	no	—
122	Q₈×C₁₉	—	8	1	no	—
123	Q₈×C₁₉	—	8	1	no	—
124	C₂×C₄×C₁₉	—	2	2	no	—
125	C₂×C₄×C₁₉	—	4	2	no	—
126	C₂×C₄×C₁₉	—	4	4	no	—
127	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
132	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
135	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
137	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
139	C₂×Dic₁₉	—	2	2	no	—
140	C₂×Dic₁₉	—	2	2	no	—
141	C₂×Dic₁₉	—	2	2	no	—
142	C₂×Dic₁₉	—	2	2	no	—
143	C₂×Dic₁₉	#102	1	4	yes	—
144	C₂×Dic₁₉	#102	1	4	yes	—
145	D₇₆	—	8	2	no	—
146	D₇₆	—	8	2	no	—
147	D₇₆	—	8	1	no	—
148	D₇₆	—	8	1	no	—
149	D₇₆	—	8	1	no	—
150	D₇₆	—	8	1	no	—
151	D₇₆	—	8	2	no	—
152	D₇₆	—	8	1	no	—
153	D₇₆	—	8	1	no	—
154	C₈×C₁₉	—	4	2	no	—
155	C₈×C₁₉	—	8	1	no	—
156	C₁₉⋊C₈	—	4	2	no	—
157	C₁₉⋊C₈	—	8	1	no	—
158	#2858	—	19	1	no	—
159	#1842	—	19	1	no	—
160	C₄×C₄⋊C₈	—	19	1	no	—
161	C₂×#332	—	19	1	no	—
162	C₂×#332	D₁₉	1	1	yes	characteristic, complemented
163	C₂×#331	—	19	1	no	—
164	C₂×#331	—	19	1	no	—
165	C₂²×C₁₉	#390	1	1	yes	characteristic
166	C₂²×C₁₉	#413	1	2	yes	—
167	C₂²×C₁₉	C₂×#102	1	2	yes	—
168	C₂²×C₁₉	#416	1	1	yes	characteristic
169	C₂²×C₁₉	#417	1	1	yes	characteristic
170	C₂×D₁₉	—	8	4	no	—
171	C₂×D₁₉	—	8	4	no	—
172	C₂×D₁₉	—	8	2	no	—
173	C₂×D₁₉	—	8	2	no	—
174	C₂×D₁₉	—	8	2	no	—
175	C₄×C₁₉	—	2	1	no	—
176	C₄×C₁₉	—	2	1	no	—
177	C₄×C₁₉	—	8	2	no	—
178	C₄×C₁₉	#463	1	1	yes	characteristic
179	C₄×C₁₉	#463	1	1	yes	characteristic
180	C₄×C₁₉	#464	1	1	yes	characteristic
181	C₄×C₁₉	#464	1	1	yes	characteristic
182	C₄×C₁₉	#486	1	2	yes	—
183	C₄×C₁₉	#492	1	1	yes	characteristic
184	C₄×C₁₉	#493	1	1	yes	characteristic
185	Dic₁₉	—	2	4	no	—
186	Dic₁₉	—	2	4	no	—
187	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
188	C₂×#127	—	19	1	no	—
189	C₂×#124	—	38	2	no	—
190	C₂×C₄⋊C₈	—	19	1	no	—
191	C₂×C₄⋊C₈	—	19	1	no	—
192	C₂×C₄⋊C₈	—	38	2	no	—
193	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
194	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
195	C₂×C₄×C₈	—	38	1	no	—
196	#390	—	19	4	no	—
197	#390	—	38	1	no	—
198	#390	—	38	1	no	—
199	C₄³	—	19	1	no	—
200	#336	—	38	1	no	—
201	#332	—	2	2	no	—
202	#332	—	38	2	no	—
203	#331	—	38	2	no	—
204	#331	—	38	2	no	—
205	C₂×C₁₉	#1540	1	2	yes	—
206	C₂×C₁₉	#1547	1	2	yes	—
207	C₂×C₁₉	#1957	1	1	yes	characteristic
208	C₂×C₁₉	#1967	1	1	yes	characteristic
209	C₂×C₁₉	#1983	1	1	yes	characteristic
210	D₁₉	—	16	2	no	—
211	C₂²×Q₈	—	2	1	no	—
212	C₂²×D₄	—	19	2	no	—
213	C₂²×D₄	—	38	1	no	—
214	C₂²×D₄	—	38	1	no	—
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	2	1	no	—
217	C₂²×C₈	—	38	1	no	—
218	C₂²×C₈	—	38	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	2	1	no	—
223	#127	—	38	1	no	—
224	#127	—	38	1	no	—
225	#127	—	38	1	no	—
226	#127	—	38	1	no	—
227	#124	—	76	4	no	—
228	#124	—	76	4	no	—
229	C₂×C₄⋊C₄	—	2	1	no	—
230	C₂×C₄⋊C₄	C₄×D₁₉	1	2	yes	—
231	C₂×C₂²⋊C₄	—	38	4	no	—
232	C₂×C₂²⋊C₄	—	38	4	no	—
233	C₂×C₄²	—	19	2	no	—
234	C₂×C₄²	—	19	1	no	—
235	C₂×C₄²	—	19	1	no	—
236	C₂×C₄²	—	19	2	no	—
237	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
238	C₄⋊C₈	—	2	1	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	19	2	no	—
242	C₄⋊C₈	—	38	1	no	—
243	C₄⋊C₈	—	38	1	no	—
244	C₄⋊C₈	—	38	1	no	—
245	C₄⋊C₈	—	76	4	no	—
246	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
247	C₄×C₈	—	76	2	no	—
248	C₁₉	#9698	1	1	yes	characteristic, complemented, Sylow
249	C₂⁴	—	38	2	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×D₄	—	38	2	no	—
255	C₂×D₄	—	38	2	no	—
256	C₂×D₄	—	76	4	no	—
257	C₂×D₄	—	76	1	no	—
258	C₂×D₄	—	76	1	no	—
259	C₂×D₄	—	76	1	no	—
260	C₂×D₄	—	76	1	no	—
261	C₂×D₄	—	76	2	no	—
262	C₂×D₄	—	76	2	no	—
263	C₂×D₄	—	76	1	no	—
264	C₂×D₄	—	76	1	no	—
265	C₂×D₄	—	76	1	no	—
266	C₂×D₄	—	76	1	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	19	2	no	—
269	C₂²×C₄	—	19	2	no	—
270	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
271	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	2	no	—
278	C₂×C₈	—	38	2	no	—
279	C₂×C₈	—	76	1	no	—
280	C₂×C₈	—	76	1	no	—
281	C₂×C₈	—	76	1	no	—
282	C₂×C₈	—	76	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	76	8	no	—
288	C₂²⋊C₄	—	76	8	no	—
289	C₄²	—	19	4	no	—
290	C₄²	—	19	4	no	—
291	C₄²	—	38	1	no	—
292	C₄²	—	38	1	no	—
293	C₄²	—	38	1	no	—
294	C₄²	—	38	1	no	—
295	C₄²	—	76	2	no	—
296	C₄²	#64541	1	1	yes	characteristic
297	C₄²	D₄×D₁₉	1	1	yes	characteristic
298	C₄²	#64543	1	1	yes	characteristic
299	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
300	C₂³	—	76	4	no	—
301	C₂³	—	76	2	no	—
302	C₂³	—	76	2	no	—
303	C₂³	—	76	2	no	—
304	C₂³	—	76	4	no	—
305	C₂³	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
306	Q₈	—	8	2	no	—
307	Q₈	—	8	1	no	—
308	Q₈	—	8	1	no	—
309	D₄	—	152	2	no	—
310	D₄	—	152	2	no	—
311	D₄	—	152	2	no	—
312	D₄	—	152	1	no	—
313	D₄	—	152	1	no	—
314	D₄	—	152	1	no	—
315	D₄	—	152	1	no	—
316	D₄	—	152	1	no	—
317	D₄	—	152	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	19	4	no	—
322	C₂×C₄	—	19	4	no	—
323	C₂×C₄	—	38	2	no	—
324	C₂×C₄	—	38	2	no	—
325	C₂×C₄	—	38	2	no	—
326	C₂×C₄	—	38	2	no	—
327	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
328	C₂×C₄	#10597330	1	1	yes	characteristic
329	C₂×C₄	#10597331	1	1	yes	characteristic
330	C₂×C₄	D₈×D₁₉	1	2	yes	—
331	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
332	C₂×C₄	#10597360	1	1	yes	characteristic
333	C₂×C₄	#10597362	1	1	yes	characteristic
334	C₂×C₄	#10597364	1	1	yes	characteristic
335	C₂×C₄	#10597365	1	1	yes	characteristic
336	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
337	C₂×C₄	#10597385	1	1	yes	characteristic
338	C₂×C₄	#10597386	1	1	yes	characteristic
339	C₈	—	4	2	no	—
340	C₈	—	8	1	no	—
341	C₈	—	76	2	no	—
342	C₈	—	152	1	no	—
343	C₂²	—	152	4	no	—
344	C₂²	—	152	2	no	—
345	C₂²	—	152	2	no	—
346	C₂²	—	152	2	no	—
347	C₂²	—	152	4	no	—
348	C₂²	#11926444	1	1	yes	characteristic
349	C₂²	#11926527	1	2	yes	—
350	C₂²	#102×D₁₉	1	2	yes	—
351	C₂²	#11926539	1	1	yes	characteristic
352	C₂²	#11926544	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	8	2	no	—
356	C₄	—	38	4	no	—
357	C₄	—	38	4	no	—
358	C₄	#11926761	1	1	yes	characteristic
359	C₄	#11926762	1	1	yes	characteristic
360	C₄	#11926768	1	1	yes	characteristic
361	C₄	#11926769	1	1	yes	characteristic
362	C₄	#11926889	1	2	yes	—
363	C₄	#11926915	1	1	yes	characteristic
364	C₄	#11926919	1	1	yes	characteristic
365	C₂	—	304	2	no	—
366	C₂	#14562283	1	2	yes	—
367	C₂	#14562311	1	2	yes	—
368	C₂	#14564439	1	1	yes	characteristic
369	C₂	#14564483	1	1	yes	characteristic
370	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×#332	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#332	D₁₉
hypocenter	C₁₉	#9698
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#332

Complements

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

N	H ≅ G/N	type
2. #14563786	C₂	semidirect
3. C₄⋊C₈×Dic₁₉	C₂	semidirect
4. C₂×#11926290	C₂	semidirect
7. C₂×C₁₉×#332	C₂	semidirect
162. C₂×#332	D₁₉	semidirect
248. C₁₉	#9698	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
C₄⋊C₈×Dic₁₉	2
C₂×#11926290	2
C₂×#11926287	2
C₂×C₁₉×#332	2
#9698	19