Subgroups — #18074197

Order 4864.

712 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#10597293	C₂	1	1	yes	maximal, characteristic
2	#14562237	C₂	1	2	yes	maximal
3	#14562082	C₂	1	2	yes	maximal
4	#14561979	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₉×#557	C₂	1	1	yes	maximal, characteristic
6	#11926601	C₂²	1	2	yes	—
7	C₂×#10597293	—	2	1	no	—
8	C₄×#64520	C₂²	1	2	yes	—
9	#11926313	—	2	1	no	—
10	#11926286	—	2	4	no	—
11	Q₈⋊(C₁₉⋊C₈)	—	2	4	no	—
12	#11926270	—	2	1	no	—
13	#11926270	—	2	1	no	—
14	C₁₉×#557	—	2	4	no	—
15	C₁₉×#557	—	2	4	no	—
16	C₂×C₁₉×C₄⋊Q₈	—	2	1	no	—
17	C₂×C₁₉×C₄⋊Q₈	C₄	1	1	yes	characteristic
18	C₂×C₁₉×#125	—	2	1	no	—
19	C₂×C₁₉×#125	—	2	1	no	—
20	C₂×C₁₉×#125	C₄	1	1	yes	characteristic
21	C₂×C₁₉×#125	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₁₉	—	2	1	no	—
23	C₂×C₄×Q₈×C₁₉	C₄	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₁₉	C₄	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
27	C₂²×Q₈×C₁₉	—	2	1	no	—
28	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×Q₈×C₁₉	D₄	1	1	yes	characteristic
30	C₁₉×C₄⋊Q₈	—	2	8	no	—
31	C₁₉×C₄⋊Q₈	—	2	2	no	—
32	C₁₉×C₄⋊Q₈	—	2	2	no	—
33	C₁₉×#125	—	2	8	no	—
34	C₁₉×#125	—	2	8	no	—
35	C₁₉×#125	—	2	2	no	—
36	C₁₉×#125	—	2	2	no	—
37	C₁₉×#125	—	2	2	no	—
38	C₁₉×#125	D₄	1	4	yes	—
39	C₄×Q₈×C₁₉	—	2	8	no	—
40	C₄×Q₈×C₁₉	—	2	2	no	—
41	C₄×Q₈×C₁₉	—	2	2	no	—
42	C₄×Q₈×C₁₉	—	2	2	no	—
43	C₄×Q₈×C₁₉	—	2	2	no	—
44	C₄×Q₈×C₁₉	—	2	2	no	—
45	C₄×Q₈×C₁₉	—	2	2	no	—
46	C₄×Q₈×C₁₉	—	2	2	no	—
47	C₄×Q₈×C₁₉	D₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
49	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
50	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
51	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₁₉	—	2	1	no	—
63	C₂×C₄²×C₁₉	—	2	1	no	—
64	C₂×C₄²×C₁₉	—	2	1	no	—
65	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
68	C₂×#64520	—	2	2	no	—
69	#10597293	—	2	4	no	—
70	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
71	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
72	#10597260	—	2	2	no	—
73	#10597260	—	2	2	no	—
74	C₄×C₁₉⋊C₈	—	2	2	no	—
75	C₂×Q₈×C₁₉	—	2	4	no	—
76	C₂×Q₈×C₁₉	—	2	4	no	—
77	C₂×Q₈×C₁₉	—	2	2	no	—
78	C₂×Q₈×C₁₉	—	2	1	no	—
79	C₂×Q₈×C₁₉	—	2	2	no	—
80	C₂×Q₈×C₁₉	—	4	1	no	—
81	C₂×Q₈×C₁₉	—	4	2	no	—
82	C₂×Q₈×C₁₉	—	4	2	no	—
83	C₂×Q₈×C₁₉	C₂²⋊C₄	1	2	yes	—
84	C₂²×C₄×C₁₉	—	2	1	no	—
85	C₂²×C₄×C₁₉	—	2	1	no	—
86	C₂²×C₄×C₁₉	—	2	1	no	—
87	C₂²×C₄×C₁₉	—	2	1	no	—
88	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
95	C₄⋊C₄×C₁₉	—	2	4	no	—
96	C₄⋊C₄×C₁₉	—	2	4	no	—
97	C₄⋊C₄×C₁₉	—	2	4	no	—
98	C₄⋊C₄×C₁₉	—	2	4	no	—
99	C₄⋊C₄×C₁₉	—	2	4	no	—
100	C₄⋊C₄×C₁₉	—	2	1	no	—
101	C₄⋊C₄×C₁₉	—	2	1	no	—
102	C₄⋊C₄×C₁₉	—	2	1	no	—
103	C₄⋊C₄×C₁₉	—	2	4	no	—
104	C₄⋊C₄×C₁₉	—	2	4	no	—
105	C₄⋊C₄×C₁₉	—	2	4	no	—
106	C₄⋊C₄×C₁₉	—	2	2	no	—
107	C₄⋊C₄×C₁₉	—	2	2	no	—
108	C₄⋊C₄×C₁₉	—	2	2	no	—
109	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
110	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
111	C₄⋊C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
112	C₄²×C₁₉	—	2	4	no	—
113	C₄²×C₁₉	—	2	4	no	—
114	C₄²×C₁₉	—	2	1	no	—
115	C₄²×C₁₉	—	2	1	no	—
116	C₄²×C₁₉	—	2	1	no	—
117	C₄²×C₁₉	—	2	1	no	—
118	C₄²×C₁₉	—	2	4	no	—
119	C₄²×C₁₉	C₂×D₄	1	2	yes	—
120	C₄²×C₁₉	C₂×D₄	1	2	yes	—
121	#64520	—	2	4	no	—
122	#64520	—	4	2	no	—
123	C₂×C₁₉⋊C₈	—	4	2	no	—
124	C₂×C₁₉⋊C₈	—	4	2	no	—
125	#8374	—	19	1	no	maximal, Sylow
126	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	Q₈×C₁₉	—	2	4	no	—
128	Q₈×C₁₉	—	2	2	no	—
129	Q₈×C₁₉	—	4	4	no	—
130	Q₈×C₁₉	—	4	8	no	—
131	Q₈×C₁₉	#101	1	4	yes	complemented
132	C₂×C₄×C₁₉	—	2	4	no	—
133	C₂×C₄×C₁₉	—	2	2	no	—
134	C₂×C₄×C₁₉	—	2	1	no	—
135	C₂×C₄×C₁₉	—	2	1	no	—
136	C₂×C₄×C₁₉	—	2	1	no	—
137	C₂×C₄×C₁₉	—	2	1	no	—
138	C₂×C₄×C₁₉	—	2	2	no	—
139	C₂×C₄×C₁₉	—	2	1	no	—
140	C₂×C₄×C₁₉	—	2	1	no	—
141	C₂×C₄×C₁₉	—	2	2	no	—
142	C₂×C₄×C₁₉	—	2	1	no	—
143	C₂×C₄×C₁₉	—	2	2	no	—
144	C₂×C₄×C₁₉	—	2	2	no	—
145	C₂×C₄×C₁₉	—	2	1	no	—
146	C₂×C₄×C₁₉	—	2	1	no	—
147	C₂×C₄×C₁₉	—	4	1	no	—
148	C₂×C₄×C₁₉	—	4	2	no	—
149	C₂×C₄×C₁₉	—	4	1	no	—
150	C₂×C₄×C₁₉	—	4	1	no	—
151	C₂×C₄×C₁₉	—	4	1	no	—
152	C₂×C₄×C₁₉	—	4	1	no	—
153	C₂×C₄×C₁₉	—	4	1	no	—
154	C₂×C₄×C₁₉	—	4	1	no	—
155	C₂×C₄×C₁₉	#104	1	2	yes	—
156	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₁₉	#120	1	2	yes	—
159	C₂×C₄×C₁₉	#120	1	2	yes	—
160	C₁₉⋊C₈	—	4	4	no	—
161	C₁₉⋊C₈	—	8	2	no	—
162	C₂×#557	D₁₉	1	1	yes	characteristic
163	C₄×#101	—	19	1	no	—
164	#1528	—	19	2	no	—
165	#1491	—	19	2	no	—
166	#1467	—	19	1	no	—
167	C₂²×C₁₉	—	2	1	no	—
168	C₂²×C₁₉	—	2	1	no	—
169	C₂²×C₁₉	#379	1	1	yes	characteristic
170	C₂²×C₁₉	C₂×#104	1	1	yes	characteristic
171	C₂²×C₁₉	#421	1	1	yes	characteristic
172	C₄×C₁₉	—	2	2	no	—
173	C₄×C₁₉	—	2	4	no	—
174	C₄×C₁₉	—	2	1	no	—
175	C₄×C₁₉	—	2	2	no	—
176	C₄×C₁₉	—	2	2	no	—
177	C₄×C₁₉	—	4	1	no	—
178	C₄×C₁₉	—	4	1	no	—
179	C₄×C₁₉	—	4	2	no	—
180	C₄×C₁₉	—	4	1	no	—
181	C₄×C₁₉	—	4	2	no	—
182	C₄×C₁₉	—	4	2	no	—
183	C₄×C₁₉	C₂×#101	1	2	yes	—
184	#557	—	2	4	no	—
185	#557	—	2	4	no	—
186	C₂×C₄⋊Q₈	—	2	1	no	—
187	C₂×C₄⋊Q₈	Dic₁₉	1	1	yes	characteristic
188	C₂×#125	—	2	1	no	—
189	C₂×#125	—	2	1	no	—
190	C₂×#125	Dic₁₉	1	1	yes	characteristic
191	C₂×#125	C₂×D₁₉	1	1	yes	characteristic
192	C₂×C₄×Q₈	—	2	1	no	—
193	C₂×C₄×Q₈	Dic₁₉	1	1	yes	characteristic
194	C₂×C₄×Q₈	Dic₁₉	1	1	yes	characteristic
195	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
196	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
197	#432	—	19	2	no	—
198	C₂×#101	—	38	1	no	—
199	C₄×M₁₆	—	19	2	no	—
200	#343	—	38	1	no	—
201	#330	—	38	4	no	—
202	Q₈⋊C₈	—	38	4	no	—
203	#324	—	38	1	no	—
204	#324	—	38	1	no	—
205	C₂×C₁₉	—	2	1	no	—
206	C₂×C₁₉	—	2	1	no	—
207	C₂×C₁₉	#1784	1	1	yes	characteristic
208	C₂×C₁₉	#1789	1	1	yes	characteristic
209	C₂×C₁₉	#1796	1	1	yes	characteristic
210	C₂²×Q₈	—	2	1	no	—
211	C₂²×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
212	C₂²×Q₈	C₂²⋊D₁₉	1	1	yes	characteristic
213	C₂×M₁₆	—	38	2	no	—
214	C₄⋊Q₈	—	2	8	no	—
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	—	2	2	no	—
217	#125	—	2	8	no	—
218	#125	—	2	8	no	—
219	#125	—	2	2	no	—
220	#125	—	2	2	no	—
221	#125	—	2	2	no	—
222	#125	C₂²⋊D₁₉	1	4	yes	—
223	C₄×Q₈	—	2	8	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	—	2	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	—	2	2	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×Q₈	C₂²⋊D₁₉	1	4	yes	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
241	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
242	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
243	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
244	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
245	C₂×C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
246	C₂×C₄²	—	2	1	no	—
247	C₂×C₄²	—	2	1	no	—
248	C₂×C₄²	—	2	1	no	—
249	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
250	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
251	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
252	C₄⋊C₈	—	38	2	no	—
253	#101	—	38	4	no	—
254	C₂²⋊C₈	—	38	2	no	—
255	#97	—	38	2	no	—
256	#97	—	38	2	no	—
257	C₄×C₈	—	38	2	no	—
258	C₁₉	#8374	1	1	yes	characteristic, complemented, Sylow
259	C₂×Q₈	—	2	4	no	—
260	C₂×Q₈	—	2	4	no	—
261	C₂×Q₈	—	2	1	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	2	no	—
266	C₂×Q₈	—	4	2	no	—
267	C₂×Q₈	C₂²⋊Dic₁₉	1	2	yes	—
268	C₂²×C₄	—	2	1	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
273	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
274	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
275	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
276	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
277	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
279	M₁₆	—	38	4	no	—
280	M₁₆	—	76	2	no	—
281	C₂×C₈	—	76	2	no	—
282	C₂×C₈	—	76	2	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	—	2	4	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	4	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	C₂²⋊Dic₁₉	1	2	yes	—
298	C₄⋊C₄	C₂²⋊Dic₁₉	1	2	yes	—
299	C₄⋊C₄	C₂²⋊Dic₁₉	1	2	yes	—
300	C₄²	—	2	4	no	—
301	C₄²	—	2	4	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	1	no	—
304	C₄²	—	2	1	no	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	4	no	—
307	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
308	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
309	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
310	Q₈	—	2	4	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	4	8	no	—
314	Q₈	#10597293	1	4	yes	complemented
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	#10597294	1	2	yes	—
339	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
340	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
341	C₂×C₄	#10597399	1	2	yes	—
342	C₂×C₄	#10597399	1	2	yes	—
343	C₈	—	76	4	no	—
344	C₈	—	152	2	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	#11926400	1	1	yes	characteristic
348	C₂²	C₂×#10597294	1	1	yes	characteristic
349	C₂²	#11926561	1	1	yes	characteristic
350	C₄	—	2	2	no	—
351	C₄	—	2	4	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	2	no	—
354	C₄	—	2	2	no	—
355	C₄	—	4	2	no	—
356	C₄	—	4	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	2	no	—
360	C₄	—	4	2	no	—
361	C₄	C₂×#10597293	1	2	yes	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	#14563559	1	1	yes	characteristic
365	C₂	#14563579	1	1	yes	characteristic
366	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×C₁₉×#557	C₂
socle	C₂²×C₁₉	#379
prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#557	D₁₉
hypocenter	C₁₉	#8374
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#557

Complements

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

N	H ≅ G/N	type
131. Q₈×C₁₉	#101	semidirect
258. C₁₉	#8374	semidirect
314. Q₈	#10597293	semidirect

Maximal subgroups

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

subgroup	index
C₄×#10597293	2
#14562237	2
#14562082	2
#14561979	2
C₂×C₁₉×#557	2
#8374	19