Subgroups — #18085138

Order 4864.

562 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926831	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926821	C₂	1	1	yes	maximal, characteristic
3	#14564429	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926293	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11926290	C₂	1	1	yes	maximal, characteristic, complemented
6	#14561470	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₉×#487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₇₆	C₂²	1	1	yes	characteristic
9	#11926831	—	2	2	no	—
10	#11926831	—	2	2	no	—
11	#11926821	—	2	2	no	—
12	#11926821	—	2	2	no	—
13	C₂×#10597264	—	2	1	no	—
14	C₂×#10597265	—	2	1	no	—
15	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×#10597266	—	2	1	no	—
19	C₂×#10597266	—	2	1	no	—
20	C₂×(C₂×D₁₉)⋊C₈	—	2	2	no	—
21	#11926293	—	2	2	no	—
22	#11926290	—	2	2	no	—
23	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
24	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₁₉×#487	—	2	2	no	—
26	C₁₉×#487	—	2	2	no	—
27	C₂×C₁₉×#103	—	2	2	no	—
28	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₁₉	—	2	1	no	—
30	C₂²×Q₈×C₁₉	—	2	1	no	—
31	C₂²×D₇₆	D₄	1	1	yes	characteristic
32	C₂²×C₄×D₁₉	—	2	1	no	—
33	C₂²×C₈×C₁₉	D₄	1	2	yes	—
34	C₁₉×C₄⋊Q₈	—	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	—	2	1	no	—
39	C₁₉×C₄⋊Q₈	—	2	1	no	—
40	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
41	C₁₉×C₄⋊Q₈	D₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
46	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
47	C₂×(C₂×D₁₉)⋊C₄	—	2	1	no	—
48	C₂×C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
49	C₂²×C₁₉⋊C₈	—	2	1	no	—
50	C₂²×C₁₉⋊C₈	—	2	1	no	—
51	C₂²×C₁₉⋊C₈	—	2	1	no	—
52	C₄×D₇₆	—	2	1	no	—
53	C₄×D₇₆	—	2	1	no	—
54	C₄×D₇₆	—	2	1	no	—
55	C₄×D₇₆	—	2	1	no	—
56	C₁₉×#103	—	2	8	no	—
57	C₄×C₈×C₁₉	—	2	1	no	—
58	C₄×C₈×C₁₉	—	2	1	no	—
59	(C₂×D₁₉)⋊C₈	—	4	4	no	—
60	#10597266	—	2	4	no	—
61	#10597266	—	2	4	no	—
62	#10597265	—	2	4	no	—
63	#10597264	—	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)	—	2	1	no	—
68	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
69	C₄⋊(C₁₉⋊C₈)	D₄	1	2	yes	—
70	C₂³×D₁₉	—	2	1	no	—
71	C₂×Q₈×C₁₉	—	2	2	no	—
72	C₂×Q₈×C₁₉	—	2	2	no	—
73	C₂×Q₈×C₁₉	—	4	1	no	—
74	C₂×Q₈×C₁₉	—	4	1	no	—
75	C₂×Q₈×C₁₉	—	4	2	no	—
76	C₂×Q₈×C₁₉	—	4	2	no	—
77	C₂×Q₈×C₁₉	—	8	2	no	—
78	C₂²×C₄×C₁₉	—	4	1	no	—
79	C₂²×C₄×C₁₉	—	4	1	no	—
80	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
83	C₂²×Dic₁₉	—	2	1	no	—
84	C₂×D₇₆	—	2	1	no	—
85	C₂×D₇₆	—	2	1	no	—
86	C₂×D₇₆	—	2	1	no	—
87	C₂×D₇₆	—	2	1	no	—
88	C₂×D₇₆	—	2	1	no	—
89	C₂×D₇₆	Q₁₆	1	2	yes	—
90	C₂×C₄×D₁₉	—	4	1	no	—
91	C₂×C₄×D₁₉	—	4	1	no	—
92	C₂×C₄×D₁₉	—	4	2	no	—
93	C₂×C₄×D₁₉	—	4	2	no	—
94	C₂×C₈×C₁₉	—	2	2	no	—
95	C₂×C₈×C₁₉	—	2	2	no	—
96	C₂×C₈×C₁₉	—	2	2	no	—
97	C₄⋊C₄×C₁₉	—	2	2	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	—	2	2	no	—
100	C₄⋊C₄×C₁₉	—	2	2	no	—
101	C₄⋊C₄×C₁₉	—	4	1	no	—
102	C₄⋊C₄×C₁₉	—	4	1	no	—
103	C₄⋊C₄×C₁₉	—	4	1	no	—
104	C₄⋊C₄×C₁₉	—	4	1	no	—
105	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
109	(C₂×D₁₉)⋊C₄	—	4	1	no	—
110	(C₂×D₁₉)⋊C₄	—	4	1	no	—
111	C₄⋊Dic₁₉	—	2	1	no	—
112	C₄⋊Dic₁₉	—	2	1	no	—
113	C₂×C₁₉⋊C₈	—	2	2	no	—
114	C₂×C₁₉⋊C₈	—	2	2	no	—
115	C₂×C₁₉⋊C₈	—	4	1	no	—
116	C₂×C₁₉⋊C₈	—	4	1	no	—
117	C₂×C₁₉⋊C₈	—	4	1	no	—
118	C₂×C₁₉⋊C₈	—	4	1	no	—
119	C₂×C₁₉⋊C₈	—	4	1	no	—
120	C₂×C₁₉⋊C₈	—	4	2	no	—
121	#10275	—	19	1	no	maximal, Sylow
122	C₂³×C₁₉	#121	1	1	yes	characteristic
123	C₂²×D₁₉	—	2	2	no	—
124	C₂²×D₁₉	—	4	1	no	—
125	C₂²×D₁₉	—	4	2	no	—
126	C₂²×D₁₉	—	4	1	no	—
127	C₂²×D₁₉	—	4	2	no	—
128	Q₈×C₁₉	—	4	4	no	—
129	Q₈×C₁₉	—	8	2	no	—
130	Q₈×C₁₉	—	8	2	no	—
131	Q₈×C₁₉	—	8	1	no	—
132	Q₈×C₁₉	—	8	1	no	—
133	C₂×C₄×C₁₉	—	2	2	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	4	2	no	—
136	C₂×C₄×C₁₉	—	8	2	no	—
137	C₂×C₄×C₁₉	—	8	2	no	—
138	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
139	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
140	C₂×C₄×C₁₉	#120	1	1	yes	characteristic
141	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
142	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
144	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
145	C₂×C₄×C₁₉	#135	1	2	yes	—
146	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
147	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
149	C₂×C₄×C₁₉	#137	1	1	yes	characteristic
150	C₂×Dic₁₉	—	4	1	no	—
151	C₂×Dic₁₉	—	4	2	no	—
152	D₇₆	—	2	2	no	—
153	D₇₆	—	2	2	no	—
154	D₇₆	—	4	1	no	—
155	D₇₆	—	4	1	no	—
156	C₄×D₁₉	—	8	2	no	—
157	C₄×D₁₉	—	8	1	no	—
158	C₄×D₁₉	—	8	1	no	—
159	C₈×C₁₉	—	4	2	no	—
160	C₁₉⋊C₈	—	4	2	no	—
161	C₁₉⋊C₈	—	4	2	no	—
162	C₁₉⋊C₈	—	8	1	no	—
163	C₂×#487	D₁₉	1	1	yes	characteristic, complemented
164	C₂×#476	—	19	1	no	—
165	C₂×#474	—	19	1	no	—
166	#1954	—	19	1	no	—
167	C₂×#333	—	19	1	no	—
168	C₂×#332	—	19	1	no	—
169	#1277	—	19	1	no	—
170	C₂²×C₁₉	#392	1	1	yes	characteristic
171	C₂²×C₁₉	#463	1	2	yes	—
172	C₂²×C₁₉	#467	1	1	yes	characteristic
173	C₂²×C₁₉	#470	1	1	yes	characteristic
174	C₂²×C₁₉	#471	1	2	yes	—
175	C₂×D₁₉	—	4	4	no	—
176	C₂×D₁₉	—	4	2	no	—
177	C₂×D₁₉	—	8	2	no	—
178	C₂×D₁₉	—	8	1	no	—
179	C₂×D₁₉	—	8	1	no	—
180	C₄×C₁₉	—	2	1	no	—
181	C₄×C₁₉	—	2	1	no	—
182	C₄×C₁₉	—	8	2	no	—
183	C₄×C₁₉	—	8	2	no	—
184	C₄×C₁₉	#449	1	2	yes	—
185	C₄×C₁₉	#451	1	1	yes	characteristic
186	C₄×C₁₉	#451	1	1	yes	characteristic
187	C₄×C₁₉	#452	1	1	yes	characteristic
188	C₄×C₁₉	#452	1	1	yes	characteristic
189	C₄×C₁₉	#495	1	1	yes	characteristic
190	C₄×C₁₉	#496	1	1	yes	characteristic
191	Dic₁₉	—	8	1	no	—
192	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
193	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
194	C₂×C₄×D₄	—	19	1	no	—
195	#487	—	2	2	no	—
196	#487	—	2	2	no	—
197	#476	—	38	2	no	—
198	#476	—	38	2	no	—
199	#474	—	38	2	no	—
200	#474	—	38	2	no	—
201	C₂×#107	—	38	1	no	—
202	C₂×#106	—	38	1	no	—
203	C₂×C₄⋊C₈	—	19	1	no	—
204	C₂×C₄⋊C₈	—	19	1	no	—
205	C₂×C₄⋊C₈	—	19	1	no	—
206	C₂×#103	—	2	2	no	—
207	C₂×#102	—	38	1	no	—
208	C₂×#102	—	38	1	no	—
209	C₂×C₂²⋊C₈	—	38	2	no	—
210	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
211	#333	—	38	2	no	—
212	#332	—	38	2	no	—
213	C₂×C₁₉	#1683	1	2	yes	—
214	C₂×C₁₉	#1703	1	2	yes	—
215	C₂×C₁₉	#2001	1	1	yes	characteristic
216	C₂×C₁₉	#2012	1	1	yes	characteristic
217	C₂×C₁₉	#2019	1	1	yes	characteristic
218	D₁₉	—	8	2	no	—
219	C₂²×Q₈	—	2	1	no	—
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×D₄	—	19	1	no	—
222	C₂³×C₄	—	38	1	no	—
223	C₂²×C₈	—	38	1	no	—
224	C₂²×C₈	—	38	1	no	—
225	C₂²×C₈	—	38	1	no	—
226	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
234	C₄⋊Q₈	C₂²⋊D₁₉	1	2	yes	—
235	C₄×D₄	—	38	1	no	—
236	C₄×D₄	—	38	1	no	—
237	C₄×D₄	—	38	1	no	—
238	C₄×D₄	—	38	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	1	no	—
242	C₂×C₄⋊C₄	—	2	1	no	—
243	C₂×C₄⋊C₄	—	19	1	no	—
244	C₂×C₂²⋊C₄	—	38	1	no	—
245	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
246	#107	—	38	4	no	—
247	#106	—	38	4	no	—
248	C₄⋊C₈	—	19	2	no	—
249	C₄⋊C₈	—	19	2	no	—
250	C₄⋊C₈	—	38	1	no	—
251	C₄⋊C₈	—	38	1	no	—
252	C₄⋊C₈	—	38	1	no	—
253	C₄⋊C₈	—	38	1	no	—
254	#103	—	2	8	no	—
255	#102	—	38	4	no	—
256	#102	—	38	4	no	—
257	C₂²⋊C₈	—	76	4	no	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₁₉	#10275	1	1	yes	characteristic, complemented, Sylow
261	C₂⁴	—	38	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	4	2	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	8	2	no	—
269	C₂×D₄	—	19	2	no	—
270	C₂×D₄	—	38	1	no	—
271	C₂×D₄	—	38	1	no	—
272	C₂×D₄	—	38	1	no	—
273	C₂×D₄	—	38	1	no	—
274	C₂×D₄	—	38	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	—	4	1	no	—
277	C₂²×C₄	—	38	1	no	—
278	C₂²×C₄	—	76	1	no	—
279	C₂²×C₄	—	76	2	no	—
280	C₂²×C₄	—	76	2	no	—
281	C₂²×C₄	—	76	1	no	—
282	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
283	C₂²×C₄	#64543	1	1	yes	characteristic
284	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	38	2	no	—
289	C₂×C₈	—	38	2	no	—
290	C₂×C₈	—	76	1	no	—
291	C₂×C₈	—	76	1	no	—
292	C₂×C₈	—	76	1	no	—
293	C₂×C₈	—	76	1	no	—
294	C₂×C₈	—	76	1	no	—
295	C₂×C₈	—	76	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	38	1	no	—
305	C₄⋊C₄	—	38	1	no	—
306	C₂²⋊C₄	—	76	1	no	—
307	C₂²⋊C₄	—	76	1	no	—
308	C₄²	D₄×D₁₉	1	1	yes	characteristic
309	C₄²	D₄×D₁₉	1	1	yes	characteristic
310	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
312	C₂³	—	38	2	no	—
313	C₂³	—	76	1	no	—
314	C₂³	—	76	2	no	—
315	C₂³	—	76	1	no	—
316	C₂³	—	76	2	no	—
317	C₂³	#10597384	1	1	yes	characteristic
318	Q₈	—	4	4	no	—
319	Q₈	—	8	2	no	—
320	Q₈	—	8	2	no	—
321	Q₈	—	8	1	no	—
322	Q₈	—	8	1	no	—
323	D₄	—	38	2	no	—
324	D₄	—	38	2	no	—
325	D₄	—	76	1	no	—
326	D₄	—	76	1	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	—	8	2	no	—
332	C₂×C₄	—	76	1	no	—
333	C₂×C₄	—	76	2	no	—
334	C₂×C₄	—	152	1	no	—
335	C₂×C₄	—	152	1	no	—
336	C₂×C₄	—	152	2	no	—
337	C₂×C₄	Q₁₆×D₁₉	1	1	yes	characteristic
338	C₂×C₄	#10597364	1	1	yes	characteristic
339	C₂×C₄	#10597364	1	1	yes	characteristic
340	C₂×C₄	#10597365	1	1	yes	characteristic
341	C₂×C₄	#10597383	1	1	yes	characteristic
342	C₂×C₄	#10597383	1	1	yes	characteristic
343	C₂×C₄	#10597385	1	1	yes	characteristic
344	C₂×C₄	#10597385	1	1	yes	characteristic
345	C₂×C₄	#10597386	1	1	yes	characteristic
346	C₂×C₄	#10597395	1	2	yes	—
347	C₂×C₄	#10597396	1	2	yes	—
348	C₂×C₄	#10597399	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	76	2	no	—
351	C₈	—	76	2	no	—
352	C₈	—	152	1	no	—
353	C₂²	—	76	4	no	—
354	C₂²	—	76	2	no	—
355	C₂²	—	152	1	no	—
356	C₂²	—	152	2	no	—
357	C₂²	—	152	1	no	—
358	C₂²	#11926452	1	1	yes	characteristic
359	C₂²	#11926756	1	2	yes	—
360	C₂²	#11926782	1	1	yes	characteristic
361	C₂²	#11926799	1	1	yes	characteristic
362	C₂²	#11926804	1	2	yes	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	8	2	no	—
366	C₄	—	8	2	no	—
367	C₄	—	152	1	no	—
368	C₄	#11926689	1	2	yes	—
369	C₄	#11926698	1	1	yes	characteristic
370	C₄	#11926698	1	1	yes	characteristic
371	C₄	#11926705	1	1	yes	characteristic
372	C₄	#11926705	1	1	yes	characteristic
373	C₄	#11926926	1	1	yes	characteristic
374	C₄	#11926934	1	1	yes	characteristic
375	C₂	—	152	2	no	—
376	C₂	#14563083	1	2	yes	—
377	C₂	#14563207	1	2	yes	—
378	C₂	#14564642	1	1	yes	characteristic
379	C₂	#14564712	1	1	yes	characteristic
380	C₂	#14564747	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#10597384
commutator, cosocle	C₂×C₄²×C₁₉	C₂³
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#487	C₂
socle	C₂³×C₁₉	#121
prime radical	C₂²×D₇₆	D₄
hypercenter	C₂×#487	D₁₉
hypocenter	C₁₉	#10275
LCS	G ▷ C₂×C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#487

Complements

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

N	H ≅ G/N	type
4. C₂×#11926293	C₂	semidirect
5. C₂×#11926290	C₂	semidirect
6. #14561470	C₂	semidirect
7. C₂×C₁₉×#487	C₂	semidirect
163. C₂×#487	D₁₉	semidirect
260. C₁₉	#10275	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926831	2
C₂×#11926821	2
#14564429	2
C₂×#11926293	2
C₂×#11926290	2
#14561470	2
C₂×C₁₉×#487	2
#10275	19