Subgroups — #18074267

Order 4864.

966 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563631	C₂	1	1	yes	maximal, characteristic, complemented
2	#14562307	C₂	1	2	yes	maximal, complemented
3	#14562060	C₂	1	2	yes	maximal
4	#14561959	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	#11926568	C₂²	1	2	yes	—
8	C₂×#10597293	—	2	1	no	—
9	#11926313	—	2	1	no	—
10	#11926290	—	2	4	no	—
11	D₄⋊(C₁₉⋊C₈)	—	2	4	no	—
12	#11926270	—	2	2	no	—
13	C₂³×Q₈×C₁₉	—	2	1	no	—
14	D₄×Q₈×C₁₉	—	2	4	no	—
15	D₄×Q₈×C₁₉	—	2	4	no	—
16	C₂×C₁₉×C₄⋊Q₈	C₄	1	2	yes	—
17	C₂×C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₉×C₂²⋊Q₈	—	2	1	no	—
19	C₂×C₁₉×C₂²⋊Q₈	—	2	2	no	—
20	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₁₉	C₄	1	2	yes	—
22	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₂²×Q₈×C₁₉	—	2	8	no	—
24	C₂²×Q₈×C₁₉	—	2	2	no	—
25	C₂²×Q₈×C₁₉	—	2	1	no	—
26	C₂²×Q₈×C₁₉	—	4	4	no	—
27	C₂²×Q₈×C₁₉	—	4	2	no	—
28	C₂²×Q₈×C₁₉	—	4	2	no	—
29	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
30	C₂²×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₁₉	—	2	2	no	—
32	C₂³×C₄×C₁₉	—	2	1	no	—
33	C₁₉×C₄⋊Q₈	—	2	4	no	—
34	C₁₉×C₄⋊Q₈	—	2	4	no	—
35	C₁₉×C₄⋊Q₈	—	2	2	no	—
36	C₁₉×C₄⋊Q₈	D₄	1	4	yes	—
37	C₁₉×C₂²⋊Q₈	—	2	8	no	—
38	C₁₉×C₂²⋊Q₈	—	2	16	no	—
39	C₄×Q₈×C₁₉	—	2	2	no	—
40	C₄×Q₈×C₁₉	—	2	2	no	—
41	C₄×D₄×C₁₉	—	2	4	no	—
42	C₄×D₄×C₁₉	—	2	4	no	—
43	C₄×D₄×C₁₉	—	2	2	no	—
44	C₄×D₄×C₁₉	D₄	1	4	yes	—
45	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
46	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
48	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₁₉	D₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
53	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
55	C₂×#64520	—	2	2	no	—
56	#10597293	—	4	2	no	—
57	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
58	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
59	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
61	C₄×C₁₉⋊C₈	—	2	2	no	—
62	C₂⁴×C₁₉	—	2	1	no	—
63	C₂×Q₈×C₁₉	—	2	8	no	—
64	C₂×Q₈×C₁₉	—	2	2	no	—
65	C₂×Q₈×C₁₉	—	2	4	no	—
66	C₂×Q₈×C₁₉	—	2	2	no	—
67	C₂×Q₈×C₁₉	—	4	4	no	—
68	C₂×Q₈×C₁₉	—	4	16	no	—
69	C₂×Q₈×C₁₉	—	4	16	no	—
70	C₂×Q₈×C₁₉	—	4	4	no	—
71	C₂×Q₈×C₁₉	—	4	1	no	—
72	C₂×Q₈×C₁₉	—	4	1	no	—
73	C₂×Q₈×C₁₉	—	4	2	no	—
74	C₂×Q₈×C₁₉	D₈	1	2	yes	—
75	C₂×Q₈×C₁₉	SD₁₆	1	2	yes	—
76	C₂×D₄×C₁₉	—	2	4	no	—
77	C₂×D₄×C₁₉	—	2	1	no	—
78	C₂×D₄×C₁₉	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₁₉	—	2	8	no	—
80	C₂²×C₄×C₁₉	—	2	2	no	—
81	C₂²×C₄×C₁₉	—	2	1	no	—
82	C₂²×C₄×C₁₉	—	2	4	no	—
83	C₂²×C₄×C₁₉	—	4	4	no	—
84	C₂²×C₄×C₁₉	—	4	4	no	—
85	C₂²×C₄×C₁₉	—	4	1	no	—
86	C₂²×C₄×C₁₉	—	4	1	no	—
87	C₂²×C₄×C₁₉	—	4	2	no	—
88	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
89	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
90	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
93	C₄⋊C₄×C₁₉	—	2	8	no	—
94	C₄⋊C₄×C₁₉	—	2	1	no	—
95	C₄⋊C₄×C₁₉	—	2	2	no	—
96	C₄⋊C₄×C₁₉	—	2	4	no	—
97	C₄⋊C₄×C₁₉	—	2	4	no	—
98	C₄⋊C₄×C₁₉	—	2	2	no	—
99	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
100	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
101	C₂²⋊C₄×C₁₉	—	2	8	no	—
102	C₂²⋊C₄×C₁₉	—	2	4	no	—
103	C₄²×C₁₉	—	2	2	no	—
104	C₄²×C₁₉	—	2	2	no	—
105	C₄²×C₁₉	C₂×D₄	1	2	yes	—
106	C₄²×C₁₉	C₂×D₄	1	2	yes	—
107	#64520	—	4	2	no	—
108	#64520	—	4	2	no	—
109	C₂×C₁₉⋊C₈	—	4	2	no	—
110	C₂×C₁₉⋊C₈	—	4	2	no	—
111	#8388	—	19	1	no	maximal, Sylow
112	C₂³×C₁₉	—	2	4	no	—
113	C₂³×C₁₉	—	2	2	no	—
114	C₂³×C₁₉	—	4	2	no	—
115	C₂³×C₁₉	—	4	1	no	—
116	C₂³×C₁₉	—	4	1	no	—
117	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	Q₈×C₁₉	—	2	4	no	—
119	Q₈×C₁₉	—	2	4	no	—
120	Q₈×C₁₉	—	4	16	no	—
121	Q₈×C₁₉	—	4	4	no	—
122	Q₈×C₁₉	—	4	4	no	—
123	D₄×C₁₉	—	2	4	no	—
124	D₄×C₁₉	—	2	2	no	—
125	D₄×C₁₉	#101	1	4	yes	complemented
126	C₂×C₄×C₁₉	—	2	2	no	—
127	C₂×C₄×C₁₉	—	2	2	no	—
128	C₂×C₄×C₁₉	—	2	2	no	—
129	C₂×C₄×C₁₉	—	2	2	no	—
130	C₂×C₄×C₁₉	—	2	4	no	—
131	C₂×C₄×C₁₉	—	2	2	no	—
132	C₂×C₄×C₁₉	—	4	1	no	—
133	C₂×C₄×C₁₉	—	4	16	no	—
134	C₂×C₄×C₁₉	—	4	2	no	—
135	C₂×C₄×C₁₉	—	4	1	no	—
136	C₂×C₄×C₁₉	—	4	2	no	—
137	C₂×C₄×C₁₉	—	4	1	no	—
138	C₂×C₄×C₁₉	—	4	4	no	—
139	C₂×C₄×C₁₉	—	4	4	no	—
140	C₂×C₄×C₁₉	—	4	2	no	—
141	C₂×C₄×C₁₉	#102	1	4	yes	—
142	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₁₉	#120	1	2	yes	—
145	C₂×C₄×C₁₉	#120	1	2	yes	—
146	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
149	C₁₉⋊C₈	—	8	2	no	—
150	C₁₉⋊C₈	—	8	2	no	—
151	C₂×D₄×Q₈	D₁₉	1	1	yes	characteristic
152	#1801	—	19	1	no	—
153	#1546	—	19	2	no	—
154	#1486	—	19	2	no	—
155	#1461	—	19	1	no	—
156	C₂²×C₁₉	—	2	1	no	—
157	C₂²×C₁₉	—	2	1	no	—
158	C₂²×C₁₉	—	2	4	no	—
159	C₂²×C₁₉	—	4	4	no	—
160	C₂²×C₁₉	—	4	4	no	—
161	C₂²×C₁₉	—	4	2	no	—
162	C₂²×C₁₉	—	4	2	no	—
163	C₂²×C₁₉	#379	1	1	yes	characteristic
164	C₂²×C₁₉	C₂×#102	1	1	yes	characteristic
165	C₂²×C₁₉	#418	1	1	yes	characteristic
166	C₄×C₁₉	—	2	2	no	—
167	C₄×C₁₉	—	2	1	no	—
168	C₄×C₁₉	—	2	4	no	—
169	C₄×C₁₉	—	4	2	no	—
170	C₄×C₁₉	—	4	4	no	—
171	C₄×C₁₉	—	4	2	no	—
172	C₄×C₁₉	C₂×#101	1	2	yes	—
173	C₄×C₁₉	#447	1	2	yes	—
174	C₄×C₁₉	#450	1	2	yes	—
175	C₂³×Q₈	—	2	1	no	—
176	D₄×Q₈	—	2	4	no	—
177	D₄×Q₈	—	2	4	no	—
178	C₂×C₄⋊Q₈	Dic₁₉	1	2	yes	—
179	C₂×C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
180	C₂×C₂²⋊Q₈	—	2	1	no	—
181	C₂×C₂²⋊Q₈	—	2	2	no	—
182	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
183	C₂×C₄×D₄	Dic₁₉	1	2	yes	—
184	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
185	#431	—	19	2	no	—
186	#423	—	19	2	no	—
187	C₂×#101	—	38	1	no	—
188	#343	—	38	1	no	—
189	#332	—	38	4	no	—
190	D₄⋊C₈	—	38	4	no	—
191	#324	—	38	2	no	—
192	C₂×C₁₉	—	2	1	no	—
193	C₂×C₁₉	—	2	1	no	—
194	C₂×C₁₉	—	4	4	no	—
195	C₂×C₁₉	#1784	1	1	yes	characteristic
196	C₂×C₁₉	#1786	1	1	yes	characteristic
197	C₂×C₁₉	#1796	1	1	yes	characteristic
198	C₂²×Q₈	—	2	8	no	—
199	C₂²×Q₈	—	2	2	no	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×Q₈	—	4	4	no	—
202	C₂²×Q₈	—	4	2	no	—
203	C₂²×Q₈	—	4	2	no	—
204	C₂²×Q₈	C₂²⋊D₁₉	1	1	yes	characteristic
205	C₂²×D₄	C₂×Dic₁₉	1	1	yes	characteristic
206	C₂³×C₄	—	2	2	no	—
207	C₂³×C₄	—	2	1	no	—
208	C₂×M₁₆	—	38	2	no	—
209	C₄⋊Q₈	—	2	4	no	—
210	C₄⋊Q₈	—	2	4	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	C₂²⋊D₁₉	1	4	yes	—
213	C₂²⋊Q₈	—	2	8	no	—
214	C₂²⋊Q₈	—	2	16	no	—
215	C₄×Q₈	—	2	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×D₄	—	2	4	no	—
218	C₄×D₄	—	2	4	no	—
219	C₄×D₄	—	2	2	no	—
220	C₄×D₄	C₂²⋊D₁₉	1	4	yes	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
224	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
225	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	2	yes	—
226	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	1	no	—
229	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
230	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
231	C₄⋊C₈	—	38	2	no	—
232	C₄⋊C₈	—	38	2	no	—
233	C₄⋊C₈	—	38	2	no	—
234	#101	—	76	2	no	—
235	C₂²⋊C₈	—	38	2	no	—
236	C₄×C₈	—	38	2	no	—
237	C₁₉	#8388	1	1	yes	characteristic, complemented, Sylow
238	C₂⁴	—	2	1	no	—
239	C₂×Q₈	—	2	8	no	—
240	C₂×Q₈	—	2	2	no	—
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂×Q₈	—	4	16	no	—
245	C₂×Q₈	—	4	16	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	4	no	—
250	C₂×Q₈	#64525	1	2	yes	—
251	C₂×Q₈	#64526	1	2	yes	—
252	C₂×D₄	—	2	4	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	C₂²⋊Dic₁₉	1	2	yes	—
255	C₂²×C₄	—	2	8	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	1	no	—
258	C₂²×C₄	—	2	4	no	—
259	C₂²×C₄	—	4	4	no	—
260	C₂²×C₄	—	4	1	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	4	no	—
264	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
265	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
266	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
269	M₁₆	—	76	2	no	—
270	M₁₆	—	76	2	no	—
271	C₂×C₈	—	76	2	no	—
272	C₂×C₈	—	76	2	no	—
273	C₄⋊C₄	—	2	8	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	4	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	C₂²⋊Dic₁₉	1	4	yes	—
280	C₄⋊C₄	C₂²⋊Dic₁₉	1	2	yes	—
281	C₂²⋊C₄	—	2	8	no	—
282	C₂²⋊C₄	—	2	4	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
286	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
287	C₂³	—	2	4	no	—
288	C₂³	—	2	2	no	—
289	C₂³	—	4	2	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	4	1	no	—
292	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
293	Q₈	—	2	4	no	—
294	Q₈	—	2	4	no	—
295	Q₈	—	4	16	no	—
296	Q₈	—	4	4	no	—
297	Q₈	—	4	4	no	—
298	D₄	—	2	4	no	—
299	D₄	—	2	2	no	—
300	D₄	#10597293	1	4	yes	complemented
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	16	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	#10597289	1	4	yes	—
317	C₂×C₄	C₂×#64525	1	1	yes	characteristic
318	C₂×C₄	#10597378	1	2	yes	—
319	C₂×C₄	C₂×#64526	1	1	yes	characteristic
320	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
321	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
322	C₂×C₄	#10597399	1	2	yes	—
323	C₂×C₄	#10597399	1	2	yes	—
324	C₈	—	152	2	no	—
325	C₈	—	152	2	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	4	no	—
329	C₂²	—	4	4	no	—
330	C₂²	—	4	4	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	#11926400	1	1	yes	characteristic
334	C₂²	C₂×#10597289	1	1	yes	characteristic
335	C₂²	#11926549	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	4	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	4	no	—
341	C₄	—	4	2	no	—
342	C₄	C₂×#10597293	1	2	yes	—
343	C₄	#11926678	1	2	yes	—
344	C₄	#11926695	1	2	yes	—
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	4	4	no	—
348	C₂	#14563559	1	1	yes	characteristic
349	C₂	#14563566	1	1	yes	characteristic
350	C₂	#14563610	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#11926400
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×D₄×Q₈×C₁₉	C₂
socle	C₂²×C₁₉	#379
prime radical	C₂²×D₄×C₁₉	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×D₄×Q₈	D₁₉
hypocenter	C₁₉	#8388
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #14563631	C₂	semidirect
2. #14562307	C₂	semidirect
4. #14561959	C₂	semidirect
125. D₄×C₁₉	#101	semidirect
237. C₁₉	#8388	semidirect
300. D₄	#10597293	semidirect

Maximal subgroups

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

subgroup	index
#14563631	2
#14562307	2
#14562060	2
#14561959	2
C₂×D₄×Q₈×C₁₉	2
#8388	19