Subgroups — #18081856

Order 4864.

526 conjugacy classes of subgroups.

The subgroup lattice has 358 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	#14564561	C₂	1	1	yes	maximal, characteristic
3	#14564559	C₂	1	1	yes	maximal, characteristic, complemented
4	#14563505	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11926294	C₂	1	1	yes	maximal, characteristic, complemented
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	#11926892	—	2	2	no	—
10	#11926892	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#10597292	—	2	2	no	—
13	C₂×#10597278	—	2	2	no	—
14	C₂×#10597266	—	2	1	no	—
15	C₂×#10597266	—	2	1	no	—
16	C₂×#10597272	—	2	1	no	—
17	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
19	#11926298	—	2	1	no	—
20	#11926294	—	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	—	2	1	no	—
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	C₂×C₄	1	2	yes	—
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	C₂×C₄⋊Dic₁₉	D₄	1	2	yes	—
40	C₂×C₄×Dic₁₉	—	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×C₄⋊C₈	—	2	1	no	—
51	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
52	#10597292	—	4	4	no	—
53	C₁₉⋊#95	—	2	1	no	—
54	#10597278	—	4	4	no	—
55	#10597272	—	4	2	no	—
56	#10597266	—	2	4	no	—
57	#10597266	—	2	4	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
60	C₄×C₁₉⋊C₈	C₂×C₄	1	2	yes	—
61	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
62	C₂³×D₁₉	—	4	1	no	—
63	C₂×Q₈×C₁₉	—	2	2	no	—
64	C₂×Q₈×C₁₉	—	4	1	no	—
65	C₂×Q₈×C₁₉	—	4	2	no	—
66	C₂×Q₈×C₁₉	—	4	2	no	—
67	C₂²×C₄×C₁₉	—	4	1	no	—
68	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
71	C₂²×Dic₁₉	—	2	2	no	—
72	C₂×D₇₆	—	2	2	no	—
73	C₂×D₇₆	—	2	2	no	—
74	C₂×D₇₆	—	2	2	no	—
75	C₂×D₇₆	—	4	1	no	—
76	C₂×D₇₆	—	4	1	no	—
77	C₂×D₇₆	—	4	1	no	—
78	C₂×D₇₆	—	4	2	no	—
79	C₂×D₇₆	—	4	2	no	—
80	C₂×D₇₆	—	4	1	no	—
81	C₂×D₇₆	—	4	1	no	—
82	C₂×D₇₆	—	4	1	no	—
83	C₂×D₇₆	—	4	1	no	—
84	C₂×D₇₆	—	8	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	—	2	2	no	—
91	C₄⋊C₄×C₁₉	—	2	2	no	—
92	C₄⋊C₄×C₁₉	—	4	1	no	—
93	C₄⋊C₄×C₁₉	—	4	1	no	—
94	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₉	#41	1	1	yes	characteristic
98	C₄⋊Dic₁₉	—	2	2	no	—
99	C₄⋊Dic₁₉	—	2	2	no	—
100	C₄×Dic₁₉	—	4	1	no	—
101	C₄×Dic₁₉	—	4	1	no	—
102	C₂×C₁₉⋊C₈	—	2	1	no	—
103	C₂×C₁₉⋊C₈	—	2	1	no	—
104	C₂×C₁₉⋊C₈	—	2	1	no	—
105	C₂×C₁₉⋊C₈	—	2	1	no	—
106	C₂×C₁₉⋊C₈	—	4	1	no	—
107	C₂×C₁₉⋊C₈	—	4	2	no	—
108	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
109	C₂×C₁₉⋊C₈	C₂²⋊C₄	1	2	yes	—
110	#9746	—	19	1	no	maximal, Sylow
111	C₂³×C₁₉	#121	1	1	yes	characteristic
112	C₂²×D₁₉	—	4	2	no	—
113	C₂²×D₁₉	—	8	1	no	—
114	C₂²×D₁₉	—	8	1	no	—
115	C₂²×D₁₉	—	8	2	no	—
116	C₂²×D₁₉	—	8	2	no	—
117	Q₈×C₁₉	—	4	4	no	—
118	Q₈×C₁₉	—	8	1	no	—
119	Q₈×C₁₉	—	8	1	no	—
120	C₂×C₄×C₁₉	—	2	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	AGL(1,ℤ/8)	1	2	yes	—
134	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
135	C₂×Dic₁₉	—	4	2	no	—
136	C₂×Dic₁₉	—	4	2	no	—
137	C₂×Dic₁₉	—	4	2	no	—
138	D₇₆	—	4	2	no	—
139	D₇₆	—	4	2	no	—
140	D₇₆	—	4	4	no	—
141	D₇₆	—	8	2	no	—
142	D₇₆	—	8	2	no	—
143	D₇₆	—	8	1	no	—
144	D₇₆	—	8	1	no	—
145	D₇₆	—	8	1	no	—
146	D₇₆	—	8	1	no	—
147	C₈×C₁₉	—	4	2	no	—
148	C₈×C₁₉	—	8	1	no	—
149	C₁₉⋊C₈	—	2	2	no	—
150	C₁₉⋊C₈	—	2	2	no	—
151	C₁₉⋊C₈	—	8	1	no	—
152	C₂×#486	—	19	1	no	—
153	#1982	—	19	1	no	—
154	#1981	—	19	1	no	—
155	#1770	—	19	1	no	—
156	C₂×#334	—	19	1	no	—
157	C₂×#332	D₁₉	1	1	yes	characteristic, complemented
158	C₂×#331	—	19	1	no	—
159	C₂²×C₁₉	#390	1	1	yes	characteristic
160	C₂²×C₁₉	#463	1	1	yes	characteristic
161	C₂²×C₁₉	#464	1	1	yes	characteristic
162	C₂²×C₁₉	#469	1	2	yes	—
163	C₂²×C₁₉	#471	1	2	yes	—
164	C₂×D₁₉	—	8	2	no	—
165	C₂×D₁₉	—	8	2	no	—
166	C₂×D₁₉	—	8	2	no	—
167	C₂×D₁₉	—	16	2	no	—
168	C₂×D₁₉	—	16	2	no	—
169	C₄×C₁₉	—	2	1	no	—
170	C₄×C₁₉	—	2	1	no	—
171	C₄×C₁₉	—	8	2	no	—
172	C₄×C₁₉	#416	1	1	yes	characteristic
173	C₄×C₁₉	#417	1	1	yes	characteristic
174	C₄×C₁₉	#442	1	2	yes	—
175	C₄×C₁₉	#463	1	1	yes	characteristic
176	C₄×C₁₉	#464	1	1	yes	characteristic
177	C₄×C₁₉	#496	1	1	yes	characteristic
178	C₄×C₁₉	#496	1	1	yes	characteristic
179	Dic₁₉	—	8	2	no	—
180	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
181	C₂×#127	—	19	1	no	—
182	#486	—	19	4	no	—
183	#486	—	38	2	no	—
184	C₂×C₄⋊C₈	—	19	1	no	—
185	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
186	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
187	C₂×#103	—	38	2	no	—
188	C₂×#102	—	38	1	no	—
189	C₂×#102	—	38	1	no	—
190	C₂×#102	—	38	2	no	—
191	C₂×#97	—	38	1	no	—
192	C₂×C₄×C₈	—	19	1	no	—
193	C₄×C₄⋊C₄	—	19	1	no	—
194	#336	—	38	1	no	—
195	#334	—	38	2	no	—
196	#332	—	2	2	no	—
197	#331	—	38	2	no	—
198	C₂×C₁₉	#1701	1	2	yes	—
199	C₂×C₁₉	#1702	1	2	yes	—
200	C₂×C₁₉	#1972	1	1	yes	characteristic
201	C₂×C₁₉	#1977	1	1	yes	characteristic
202	C₂×C₁₉	#1983	1	1	yes	characteristic
203	D₁₉	—	16	2	no	—
204	C₂²×Q₈	—	2	1	no	—
205	C₂²×D₄	—	38	1	no	—
206	C₂²×D₄	—	38	1	no	—
207	C₂²×D₄	—	38	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	19	1	no	—
211	C₂²×C₈	—	19	1	no	—
212	C₂²×C₈	—	38	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	2	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	C₄×D₁₉	1	2	yes	—
217	#127	—	19	2	no	—
218	#127	—	38	1	no	—
219	#127	—	38	1	no	—
220	#127	—	38	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	19	2	no	—
224	C₂×C₄²	—	38	1	no	—
225	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	38	1	no	—
230	C₄⋊C₈	—	38	1	no	—
231	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
232	#103	—	76	4	no	—
233	#102	—	38	4	no	—
234	#102	—	38	4	no	—
235	#102	—	76	4	no	—
236	#97	—	76	2	no	—
237	C₄×C₈	—	19	2	no	—
238	C₄×C₈	—	19	2	no	—
239	#95	—	38	1	no	—
240	C₁₉	#9746	1	1	yes	characteristic, complemented, Sylow
241	C₂⁴	—	76	1	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×D₄	—	38	2	no	—
247	C₂×D₄	—	38	2	no	—
248	C₂×D₄	—	38	2	no	—
249	C₂×D₄	—	76	1	no	—
250	C₂×D₄	—	76	1	no	—
251	C₂×D₄	—	76	1	no	—
252	C₂×D₄	—	76	2	no	—
253	C₂×D₄	—	76	1	no	—
254	C₂×D₄	—	76	2	no	—
255	C₂×D₄	—	76	1	no	—
256	C₂×D₄	—	76	1	no	—
257	C₂×D₄	—	76	1	no	—
258	C₂×D₄	—	152	2	no	—
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	—	38	2	no	—
261	C₂²×C₄	#64541	1	1	yes	characteristic
262	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
263	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	2	no	—
269	C₂×C₈	—	19	2	no	—
270	C₂×C₈	—	19	2	no	—
271	C₂×C₈	—	38	1	no	—
272	C₂×C₈	—	38	1	no	—
273	C₂×C₈	—	38	1	no	—
274	C₂×C₈	—	38	1	no	—
275	C₂×C₈	—	76	1	no	—
276	C₂×C₈	—	76	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	38	2	no	—
282	C₄⋊C₄	—	38	2	no	—
283	C₄²	—	76	1	no	—
284	C₄²	—	76	1	no	—
285	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
286	C₄²	D₄×D₁₉	1	1	yes	characteristic
287	C₄²	#64543	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
289	C₂³	—	76	2	no	—
290	C₂³	—	152	2	no	—
291	C₂³	—	152	2	no	—
292	C₂³	—	152	1	no	—
293	C₂³	—	152	1	no	—
294	C₂³	#10597330	1	1	yes	characteristic
295	Q₈	—	4	4	no	—
296	Q₈	—	8	1	no	—
297	Q₈	—	8	1	no	—
298	D₄	—	76	2	no	—
299	D₄	—	76	2	no	—
300	D₄	—	76	4	no	—
301	D₄	—	152	2	no	—
302	D₄	—	152	2	no	—
303	D₄	—	152	1	no	—
304	D₄	—	152	1	no	—
305	D₄	—	152	1	no	—
306	D₄	—	152	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	—	76	2	no	—
311	C₂×C₄	—	76	2	no	—
312	C₂×C₄	—	76	2	no	—
313	C₂×C₄	C₂²⋊C₄×D₁₉	1	1	yes	characteristic
314	C₂×C₄	(C₂²⋊D₁₉)⋊C₄	1	1	yes	characteristic
315	C₂×C₄	#10597331	1	1	yes	characteristic
316	C₂×C₄	#10597357	1	2	yes	—
317	C₂×C₄	#10597360	1	1	yes	characteristic
318	C₂×C₄	#10597360	1	2	yes	—
319	C₂×C₄	#10597362	1	1	yes	characteristic
320	C₂×C₄	#10597364	1	1	yes	characteristic
321	C₂×C₄	#10597365	1	1	yes	characteristic
322	C₂×C₄	C₄×C₂²⋊D₁₉	1	1	yes	characteristic
323	C₂×C₄	#10597385	1	1	yes	characteristic
324	C₂×C₄	#10597386	1	1	yes	characteristic
325	C₈	—	4	2	no	—
326	C₈	—	8	1	no	—
327	C₈	—	38	2	no	—
328	C₈	—	38	2	no	—
329	C₈	—	152	1	no	—
330	C₂²	—	304	2	no	—
331	C₂²	—	304	2	no	—
332	C₂²	—	152	2	no	—
333	C₂²	—	152	2	no	—
334	C₂²	—	152	2	no	—
335	C₂²	#11926444	1	1	yes	characteristic
336	C₂²	#11926762	1	1	yes	characteristic
337	C₂²	#11926769	1	1	yes	characteristic
338	C₂²	#11926796	1	2	yes	—
339	C₂²	#11926810	1	2	yes	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	8	2	no	—
343	C₄	—	152	2	no	—
344	C₄	#11926539	1	1	yes	characteristic
345	C₄	#11926544	1	1	yes	characteristic
346	C₄	#11926653	1	2	yes	—
347	C₄	#11926761	1	1	yes	characteristic
348	C₄	#11926768	1	1	yes	characteristic
349	C₄	#11926930	1	1	yes	characteristic
350	C₄	#11926932	1	1	yes	characteristic
351	C₂	—	304	2	no	—
352	C₂	#14563199	1	2	yes	—
353	C₂	#14563206	1	2	yes	—
354	C₂	#14564508	1	1	yes	characteristic
355	C₂	#14564539	1	1	yes	characteristic
356	C₂	#14564564	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597330
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#332	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂×#10597320	C₂²
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#332	D₁₉
hypocenter	C₁₉	#9746
LCS	G ▷ C₄²×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
3. #14564559	C₂	semidirect
4. #14563505	C₂	semidirect
5. C₂×#11926294	C₂	semidirect
7. C₂×C₁₉×#332	C₂	semidirect
10. #11926892	C₄	semidirect
157. C₂×#332	D₁₉	semidirect
240. C₁₉	#9746	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
#14564561	2
#14564559	2
#14563505	2
C₂×#11926294	2
C₂×#11926287	2
C₂×C₁₉×#332	2
#9746	19