Subgroups — #18085158

Order 4864.

562 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926947	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11926835	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926831	C₂	1	1	yes	maximal, characteristic
4	#14564428	C₂	1	1	yes	maximal, characteristic
5	C₂×#11926292	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561469	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#333	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10597315	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
10	#11926947	—	2	2	no	—
11	#11926947	—	2	2	no	—
12	#11926835	—	2	2	no	—
13	#11926835	—	2	2	no	—
14	#11926831	—	2	2	no	—
15	#11926831	—	2	2	no	—
16	C₂×#10597264	—	2	2	no	—
17	C₂×#10597274	—	2	1	no	—
18	C₂×#10597265	—	2	1	no	—
19	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
20	C₂×#10597278	—	2	1	no	—
21	C₂×#10597266	—	2	1	no	—
22	C₂×(C₂×D₁₉)⋊C₈	—	2	2	no	—
23	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
24	#11926292	—	2	2	no	—
25	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
27	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₁₉×#333	—	2	2	no	—
29	C₂²×Q₈×C₁₉	—	2	1	no	—
30	C₂²×D₇₆	D₄	1	1	yes	characteristic
31	C₂²×C₄×D₁₉	—	2	1	no	—
32	C₂²×Dic₃₈	—	2	1	no	—
33	C₂²×C₈×C₁₉	—	2	1	no	—
34	C₂²×C₈×C₁₉	—	2	1	no	—
35	C₁₉×C₄⋊Q₈	—	2	1	no	—
36	C₁₉×C₄⋊Q₈	—	2	1	no	—
37	C₁₉×C₄⋊Q₈	—	2	1	no	—
38	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
41	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
42	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
43	C₂×C₄⋊Dic₁₉	—	2	1	no	—
44	C₂×C₄⋊Dic₁₉	—	2	1	no	—
45	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
46	C₂²×C₁₉⋊C₈	—	2	1	no	—
47	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
48	C₄×D₇₆	—	2	1	no	—
49	C₄×D₇₆	—	2	1	no	—
50	C₄×D₇₆	—	2	1	no	—
51	C₄×D₇₆	—	2	1	no	—
52	#10597315	—	2	1	no	—
53	#10597315	—	2	1	no	—
54	#10597315	—	2	1	no	—
55	#10597315	D₄	1	2	yes	—
56	C₁₉×C₄⋊C₈	—	2	1	no	—
57	C₁₉×C₄⋊C₈	—	2	1	no	—
58	C₁₉×C₄⋊C₈	—	2	1	no	—
59	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
60	#10597278	—	2	4	no	—
61	(C₂×D₁₉)⋊C₈	—	4	4	no	—
62	#10597274	—	2	4	no	—
63	#10597266	—	2	4	no	—
64	#10597265	—	2	4	no	—
65	#10597264	—	2	8	no	—
66	C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
67	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
68	C₄×C₁₉⋊C₈	—	2	1	no	—
69	C₄×C₁₉⋊C₈	—	2	1	no	—
70	C₂³×D₁₉	—	2	1	no	—
71	C₂×Q₈×C₁₉	—	2	2	no	—
72	C₂×Q₈×C₁₉	—	4	1	no	—
73	C₂×Q₈×C₁₉	—	8	2	no	—
74	C₂²×C₄×C₁₉	—	4	1	no	—
75	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
78	C₂²×Dic₁₉	—	2	1	no	—
79	C₂²×Dic₁₉	—	4	1	no	—
80	C₂×D₇₆	—	2	1	no	—
81	C₂×D₇₆	—	2	1	no	—
82	C₂×D₇₆	—	2	1	no	—
83	C₂×D₇₆	—	2	1	no	—
84	C₂×D₇₆	—	2	1	no	—
85	C₂×D₇₆	Q₁₆	1	2	yes	—
86	C₂×C₄×D₁₉	—	4	1	no	—
87	C₂×C₄×D₁₉	—	4	1	no	—
88	C₂×C₄×D₁₉	—	4	2	no	—
89	C₂×C₄×D₁₉	—	4	2	no	—
90	C₂×Dic₃₈	—	2	2	no	—
91	C₂×Dic₃₈	—	4	1	no	—
92	C₂×Dic₃₈	—	8	2	no	—
93	C₂×C₈×C₁₉	—	2	2	no	—
94	C₂×C₈×C₁₉	—	4	1	no	—
95	C₂×C₈×C₁₉	—	4	1	no	—
96	C₂×C₈×C₁₉	—	4	1	no	—
97	C₂×C₈×C₁₉	—	4	2	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	4	1	no	—
101	C₄⋊C₄×C₁₉	—	4	1	no	—
102	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
106	(C₂×D₁₉)⋊C₄	—	4	1	no	—
107	(C₂×D₁₉)⋊C₄	—	4	1	no	—
108	C₄⋊Dic₁₉	—	2	2	no	—
109	C₄⋊Dic₁₉	—	2	1	no	—
110	C₄⋊Dic₁₉	—	2	1	no	—
111	C₄⋊Dic₁₉	—	2	2	no	—
112	C₄⋊Dic₁₉	—	4	1	no	—
113	C₄⋊Dic₁₉	—	4	1	no	—
114	C₂×C₁₉⋊C₈	—	2	2	no	—
115	C₂×C₁₉⋊C₈	—	2	2	no	—
116	C₂×C₁₉⋊C₈	—	2	2	no	—
117	C₂×C₁₉⋊C₈	—	4	1	no	—
118	C₂×C₁₉⋊C₈	—	4	1	no	—
119	C₂×C₁₉⋊C₈	SD₁₆	1	4	yes	—
120	#10278	—	19	1	no	maximal, Sylow
121	C₂³×C₁₉	#121	1	1	yes	characteristic
122	C₂²×D₁₉	—	2	2	no	—
123	C₂²×D₁₉	—	4	1	no	—
124	C₂²×D₁₉	—	4	2	no	—
125	C₂²×D₁₉	—	4	1	no	—
126	C₂²×D₁₉	—	4	2	no	—
127	Q₈×C₁₉	—	8	2	no	—
128	Q₈×C₁₉	—	8	2	no	—
129	C₂×C₄×C₁₉	—	2	2	no	—
130	C₂×C₄×C₁₉	—	4	2	no	—
131	C₂×C₄×C₁₉	—	8	2	no	—
132	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
133	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
134	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
137	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
138	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
139	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#137	1	2	yes	—
143	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
144	C₂×Dic₁₉	—	4	2	no	—
145	C₂×Dic₁₉	—	4	1	no	—
146	C₂×Dic₁₉	—	4	2	no	—
147	C₂×Dic₁₉	—	8	2	no	—
148	D₇₆	—	2	2	no	—
149	D₇₆	—	2	2	no	—
150	D₇₆	—	4	1	no	—
151	D₇₆	—	4	1	no	—
152	C₄×D₁₉	—	8	2	no	—
153	C₄×D₁₉	—	8	1	no	—
154	C₄×D₁₉	—	8	1	no	—
155	Dic₃₈	—	8	2	no	—
156	Dic₃₈	—	8	2	no	—
157	C₈×C₁₉	—	4	2	no	—
158	C₈×C₁₉	—	8	1	no	—
159	C₁₉⋊C₈	—	2	4	no	—
160	C₁₉⋊C₈	—	4	2	no	—
161	C₂×#500	—	19	1	no	—
162	C₂×#476	—	19	1	no	—
163	C₂×#476	—	19	1	no	—
164	#1954	—	19	1	no	—
165	C₂×#333	—	19	1	no	—
166	C₂×#333	D₁₉	1	1	yes	characteristic, complemented
167	#1277	—	19	1	no	—
168	C₂²×C₁₉	#392	1	1	yes	characteristic
169	C₂²×C₁₉	#461	1	2	yes	—
170	C₂²×C₁₉	#467	1	1	yes	characteristic
171	C₂²×C₁₉	#470	1	1	yes	characteristic
172	C₂²×C₁₉	#472	1	2	yes	—
173	C₂×D₁₉	—	4	4	no	—
174	C₂×D₁₉	—	4	2	no	—
175	C₂×D₁₉	—	8	2	no	—
176	C₂×D₁₉	—	8	1	no	—
177	C₂×D₁₉	—	8	1	no	—
178	C₄×C₁₉	—	2	1	no	—
179	C₄×C₁₉	—	2	1	no	—
180	C₄×C₁₉	—	8	2	no	—
181	C₄×C₁₉	#448	1	2	yes	—
182	C₄×C₁₉	#451	1	1	yes	characteristic
183	C₄×C₁₉	#451	1	1	yes	characteristic
184	C₄×C₁₉	#452	1	1	yes	characteristic
185	C₄×C₁₉	#452	1	1	yes	characteristic
186	C₄×C₁₉	#492	1	1	yes	characteristic
187	C₄×C₁₉	#497	1	1	yes	characteristic
188	Dic₁₉	—	8	1	no	—
189	Dic₁₉	—	8	2	no	—
190	C₂×C₄⋊Q₈	—	19	1	no	—
191	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
192	C₂×C₄×D₄	—	19	1	no	—
193	#500	—	38	2	no	—
194	#500	—	38	2	no	—
195	#476	—	38	2	no	—
196	#476	—	38	2	no	—
197	#476	—	38	2	no	—
198	#476	—	38	2	no	—
199	C₂×#107	—	38	2	no	—
200	C₂×#106	—	38	1	no	—
201	C₂×#106	—	38	1	no	—
202	C₂×C₄⋊C₈	—	19	1	no	—
203	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
204	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
205	C₂×#102	—	38	1	no	—
206	C₂×#102	—	38	1	no	—
207	C₂×C₂²⋊C₈	—	38	2	no	—
208	C₂×C₄×C₈	—	19	1	no	—
209	#333	—	2	2	no	—
210	#333	—	38	2	no	—
211	C₂×C₁₉	#1675	1	2	yes	—
212	C₂×C₁₉	#1695	1	2	yes	—
213	C₂×C₁₉	#2001	1	1	yes	characteristic
214	C₂×C₁₉	#2013	1	1	yes	characteristic
215	C₂×C₁₉	#2018	1	1	yes	characteristic
216	D₁₉	—	8	2	no	—
217	C₂²×Q₈	—	2	1	no	—
218	C₂²×Q₈	—	38	1	no	—
219	C₂²×D₄	—	19	1	no	—
220	C₂³×C₄	—	38	1	no	—
221	C₂²×C₈	—	2	1	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	19	2	no	—
224	C₂²×C₈	—	38	1	no	—
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	2	1	no	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	19	2	no	—
229	C₄⋊Q₈	—	38	1	no	—
230	C₄⋊Q₈	—	38	1	no	—
231	C₄⋊Q₈	—	38	1	no	—
232	C₄⋊Q₈	D₇₆	1	2	yes	—
233	C₄×D₄	—	38	1	no	—
234	C₄×D₄	—	38	1	no	—
235	C₄×D₄	—	38	1	no	—
236	C₄×D₄	—	38	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	19	1	no	—
240	C₂×C₄⋊C₄	—	38	1	no	—
241	C₂×C₄⋊C₄	—	38	1	no	—
242	C₂×C₂²⋊C₄	—	38	1	no	—
243	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
244	#107	—	38	8	no	—
245	#106	—	38	4	no	—
246	#106	—	38	4	no	—
247	C₄⋊C₈	—	2	1	no	—
248	C₄⋊C₈	—	2	1	no	—
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	19	2	no	—
251	C₄⋊C₈	—	38	1	no	—
252	C₄⋊C₈	C₂²⋊D₁₉	1	2	yes	—
253	#102	—	38	4	no	—
254	#102	—	38	4	no	—
255	C₂²⋊C₈	—	76	4	no	—
256	C₄×C₈	—	38	1	no	—
257	C₄×C₈	—	38	1	no	—
258	C₁₉	#10278	1	1	yes	characteristic, complemented, Sylow
259	C₂⁴	—	38	1	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	4	1	no	—
262	C₂×Q₈	—	8	2	no	—
263	C₂×Q₈	—	38	2	no	—
264	C₂×Q₈	—	76	1	no	—
265	C₂×Q₈	—	152	2	no	—
266	C₂×D₄	—	19	2	no	—
267	C₂×D₄	—	38	1	no	—
268	C₂×D₄	—	38	1	no	—
269	C₂×D₄	—	38	1	no	—
270	C₂×D₄	—	38	1	no	—
271	C₂×D₄	—	38	1	no	—
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	38	1	no	—
274	C₂²×C₄	—	76	1	no	—
275	C₂²×C₄	—	76	1	no	—
276	C₂²×C₄	—	76	1	no	—
277	C₂²×C₄	—	76	2	no	—
278	C₂²×C₄	—	76	2	no	—
279	C₂²×C₄	#64541	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
281	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	—	19	4	no	—
288	C₂×C₈	—	38	2	no	—
289	C₂×C₈	—	38	2	no	—
290	C₂×C₈	—	38	2	no	—
291	C₂×C₈	—	76	1	no	—
292	C₂×C₈	—	76	1	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	4	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	38	2	no	—
298	C₄⋊C₄	—	38	1	no	—
299	C₄⋊C₄	—	38	1	no	—
300	C₄⋊C₄	—	38	2	no	—
301	C₄⋊C₄	—	76	1	no	—
302	C₄⋊C₄	—	76	1	no	—
303	C₂²⋊C₄	—	76	1	no	—
304	C₂²⋊C₄	—	76	1	no	—
305	C₄²	C₂×D₇₆	1	1	yes	characteristic
306	C₄²	D₄×D₁₉	1	1	yes	characteristic
307	C₄²	D₄×D₁₉	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
309	C₂³	—	38	2	no	—
310	C₂³	—	76	1	no	—
311	C₂³	—	76	2	no	—
312	C₂³	—	76	1	no	—
313	C₂³	—	76	2	no	—
314	C₂³	#10597330	1	1	yes	characteristic
315	Q₈	—	8	2	no	—
316	Q₈	—	8	2	no	—
317	Q₈	—	152	2	no	—
318	Q₈	—	152	2	no	—
319	D₄	—	38	2	no	—
320	D₄	—	38	2	no	—
321	D₄	—	76	1	no	—
322	D₄	—	76	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	2	no	—
326	C₂×C₄	—	76	1	no	—
327	C₂×C₄	—	76	2	no	—
328	C₂×C₄	—	76	2	no	—
329	C₂×C₄	—	152	2	no	—
330	C₂×C₄	—	152	1	no	—
331	C₂×C₄	—	152	1	no	—
332	C₂×C₄	—	152	2	no	—
333	C₂×C₄	#10597328	1	1	yes	characteristic
334	C₂×C₄	#10597328	1	1	yes	characteristic
335	C₂×C₄	#10597330	1	1	yes	characteristic
336	C₂×C₄	SD₁₆×D₁₉	1	2	yes	—
337	C₂×C₄	#10597361	1	1	yes	characteristic
338	C₂×C₄	#10597362	1	1	yes	characteristic
339	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
340	C₂×C₄	#10597364	1	2	yes	—
341	C₂×C₄	#10597364	1	1	yes	characteristic
342	C₂×C₄	#10597376	1	1	yes	characteristic
343	C₂×C₄	#10597383	1	1	yes	characteristic
344	C₂×C₄	#10597386	1	1	yes	characteristic
345	C₈	—	4	2	no	—
346	C₈	—	8	1	no	—
347	C₈	—	38	4	no	—
348	C₈	—	76	2	no	—
349	C₂²	—	76	2	no	—
350	C₂²	—	76	4	no	—
351	C₂²	—	152	2	no	—
352	C₂²	—	152	1	no	—
353	C₂²	—	152	1	no	—
354	C₂²	#11926453	1	1	yes	characteristic
355	C₂²	#11926748	1	2	yes	—
356	C₂²	#11926784	1	1	yes	characteristic
357	C₂²	#11926801	1	1	yes	characteristic
358	C₂²	#11926813	1	2	yes	—
359	C₄	—	2	1	no	—
360	C₄	—	2	1	no	—
361	C₄	—	8	2	no	—
362	C₄	—	152	1	no	—
363	C₄	—	152	2	no	—
364	C₄	#11926684	1	2	yes	—
365	C₄	#11926698	1	1	yes	characteristic
366	C₄	#11926701	1	1	yes	characteristic
367	C₄	#11926703	1	1	yes	characteristic
368	C₄	#11926709	1	1	yes	characteristic
369	C₄	#11926915	1	1	yes	characteristic
370	C₄	#11926937	1	1	yes	characteristic
371	C₂	—	152	2	no	—
372	C₂	#14563033	1	2	yes	—
373	C₂	#14563157	1	2	yes	—
374	C₂	#14564644	1	1	yes	characteristic
375	C₂	#14564719	1	1	yes	characteristic
376	C₂	#14564743	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597330
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#333	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂²×D₇₆	D₄
hypercenter	C₂×#333	D₁₉
hypocenter	C₁₉	#10278
LCS	G ▷ C₂×C₄²×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
1. C₂×#11926947	C₂	semidirect
5. C₂×#11926292	C₂	semidirect
6. #14561469	C₂	semidirect
7. C₂×C₁₉×#333	C₂	semidirect
166. C₂×#333	D₁₉	semidirect
258. C₁₉	#10278	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926947	2
C₂×#11926835	2
C₂×#11926831	2
#14564428	2
C₂×#11926292	2
#14561469	2
C₂×C₁₉×#333	2
#10278	19