Subgroups — #18075708

Order 4864.

454 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563655	C₂	1	2	yes	maximal
2	C₄×#10597275	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562394	C₂	1	1	yes	maximal, characteristic, complemented
4	#14561537	C₂	1	2	yes	maximal, complemented
5	C₈×#41×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597275	—	2	1	no	—
7	#11926568	C₂²	1	2	yes	—
8	C₂×#10597294	—	2	2	no	—
9	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
10	#11926307	—	2	1	no	—
11	#11926303	—	2	2	no	—
12	#11926297	—	2	1	no	—
13	#11926297	—	2	1	no	—
14	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
15	C₈×Q₈×C₁₉	C₄	1	1	yes	characteristic, complemented
16	C₈×Q₈×C₁₉	C₄	1	1	yes	characteristic, complemented
17	C₈×D₄×C₁₉	—	2	1	no	—
18	C₈×D₄×C₁₉	—	2	1	no	—
19	C₈×D₄×C₁₉	C₄	1	1	yes	characteristic, complemented
20	C₈×D₄×C₁₉	C₄	1	1	yes	characteristic, complemented
21	C₁₉×#431	—	2	1	no	—
22	C₁₉×#431	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₉	—	2	1	no	—
24	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
25	C₂×#41×C₁₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₉	—	2	1	no	—
27	C₂²×C₈×C₁₉	—	2	1	no	—
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₄×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
31	C₄×Q₈×C₁₉	Q₈	1	1	yes	characteristic
32	C₄×D₄×C₁₉	—	2	1	no	—
33	C₄×D₄×C₁₉	—	2	1	no	—
34	C₄×D₄×C₁₉	C₂×C₄	1	1	yes	characteristic
35	C₄×D₄×C₁₉	Q₈	1	1	yes	characteristic
36	C₁₉×#117	—	2	1	no	—
37	C₁₉×#117	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₁₉	—	2	1	no	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×C₄⋊Dic₁₉	—	2	2	no	—
41	C₂×C₄×Dic₁₉	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₁₉×C₄⋊C₈	—	2	1	no	—
44	C₁₉×C₄⋊C₈	—	2	1	no	—
45	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₁₉×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₁₉×C₂²⋊C₈	—	2	1	no	—
48	C₁₉×C₂²⋊C₈	—	2	1	no	—
49	C₁₉×C₂²⋊C₈	—	2	1	no	—
50	C₄×C₈×C₁₉	—	2	1	no	—
51	C₄×C₈×C₁₉	—	2	1	no	—
52	C₄×C₈×C₁₉	—	2	1	no	—
53	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
56	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
57	#10597294	—	4	4	no	—
58	C₁₉⋊#95	—	2	2	no	—
59	#10597275	—	2	4	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
62	#41×C₁₉	—	2	1	no	—
63	#41×C₁₉	—	2	1	no	—
64	#41×C₁₉	D₈	1	2	yes	—
65	#41×C₁₉	Q₁₆	1	2	yes	—
66	C₂×Q₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂×D₄×C₁₉	—	2	1	no	—
68	C₂×D₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₉	—	2	1	no	—
70	C₂²×C₄×C₁₉	—	2	1	no	—
71	C₂²×C₄×C₁₉	—	2	1	no	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
75	C₂²×Dic₁₉	—	4	2	no	—
76	C₂×C₈×C₁₉	—	2	2	no	—
77	C₂×C₈×C₁₉	—	2	1	no	—
78	C₂×C₈×C₁₉	—	2	1	no	—
79	C₂×C₈×C₁₉	—	4	1	no	—
80	C₂×C₈×C₁₉	—	4	2	no	—
81	C₂×C₈×C₁₉	—	4	2	no	—
82	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₄⋊C₄×C₁₉	—	2	1	no	—
87	C₄⋊C₄×C₁₉	—	2	1	no	—
88	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
89	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
90	C₂²⋊C₄×C₁₉	—	2	1	no	—
91	C₂²⋊C₄×C₁₉	—	2	1	no	—
92	C₂²⋊C₄×C₁₉	—	2	1	no	—
93	C₄²×C₁₉	—	2	1	no	—
94	C₄²×C₁₉	—	2	1	no	—
95	C₄²×C₁₉	—	2	1	no	—
96	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
99	C₄²×C₁₉	#41	1	1	yes	characteristic
100	C₄⋊Dic₁₉	—	2	4	no	—
101	C₄⋊Dic₁₉	—	4	2	no	—
102	C₄×Dic₁₉	—	4	2	no	—
103	C₄×Dic₁₉	—	4	2	no	—
104	#64520	—	4	2	no	—
105	#64520	—	4	2	no	—
106	C₂×C₁₉⋊C₈	—	4	2	no	—
107	#8667	—	19	1	no	maximal, Sylow
108	C₂³×C₁₉	—	2	1	no	—
109	C₂³×C₁₉	#121	1	1	yes	characteristic
110	Q₈×C₁₉	—	2	1	no	—
111	Q₈×C₁₉	#107	1	2	yes	complemented
112	D₄×C₁₉	—	2	2	no	—
113	D₄×C₁₉	—	2	2	no	—
114	D₄×C₁₉	—	2	1	no	—
115	D₄×C₁₉	#107	1	2	yes	complemented
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	2	2	no	—
120	C₂×C₄×C₁₉	—	2	2	no	—
121	C₂×C₄×C₁₉	—	2	1	no	—
122	C₂×C₄×C₁₉	—	2	1	no	—
123	C₂×C₄×C₁₉	—	4	1	no	—
124	C₂×C₄×C₁₉	—	4	2	no	—
125	C₂×C₄×C₁₉	—	4	1	no	—
126	C₂×C₄×C₁₉	—	4	1	no	—
127	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
130	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
131	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
133	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
135	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
136	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
137	C₂×Dic₁₉	—	4	4	no	—
138	C₂×Dic₁₉	—	8	2	no	—
139	C₂×Dic₁₉	—	8	2	no	—
140	C₈×C₁₉	—	2	1	no	—
141	C₈×C₁₉	—	2	1	no	—
142	C₈×C₁₉	—	2	1	no	—
143	C₈×C₁₉	—	4	1	no	—
144	C₈×C₁₉	—	4	1	no	—
145	C₈×C₁₉	#104	1	2	yes	complemented
146	C₁₉⋊C₈	—	8	2	no	—
147	C₈×#41	D₁₉	1	1	yes	characteristic
148	#1806	—	19	2	no	—
149	C₄×#107	—	19	1	no	—
150	#1567	—	19	1	no	—
151	#1303	—	19	2	no	—
152	C₂²×C₁₉	—	2	1	no	—
153	C₂²×C₁₉	—	2	1	no	—
154	C₂²×C₁₉	—	2	2	no	—
155	C₂²×C₁₉	—	4	1	no	—
156	C₂²×C₁₉	—	4	1	no	—
157	C₂²×C₁₉	#380	1	1	yes	characteristic
158	C₂²×C₁₉	#466	1	1	yes	characteristic
159	C₂²×C₁₉	#467	1	1	yes	characteristic
160	C₄×C₁₉	—	2	1	no	—
161	C₄×C₁₉	—	2	2	no	—
162	C₄×C₁₉	—	2	1	no	—
163	C₄×C₁₉	—	2	1	no	—
164	C₄×C₁₉	—	4	1	no	—
165	C₄×C₁₉	—	4	1	no	—
166	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
167	C₄×C₁₉	#421	1	1	yes	characteristic
168	C₄×C₁₉	C₂×#107	1	1	yes	characteristic
169	C₄×C₁₉	#427	1	1	yes	characteristic
170	C₄×C₁₉	#466	1	1	yes	characteristic
171	C₄×C₁₉	#467	1	1	yes	characteristic
172	Dic₁₉	—	8	4	no	—
173	C₄×#41	C₂×D₁₉	1	1	yes	characteristic
174	C₈×Q₈	Dic₁₉	1	1	yes	characteristic, complemented
175	C₈×Q₈	Dic₁₉	1	1	yes	characteristic, complemented
176	C₈×D₄	—	2	1	no	—
177	C₈×D₄	—	2	1	no	—
178	C₈×D₄	Dic₁₉	1	1	yes	characteristic, complemented
179	C₈×D₄	Dic₁₉	1	1	yes	characteristic, complemented
180	#431	—	2	1	no	—
181	#431	C₂×D₁₉	1	1	yes	characteristic
182	C₂×#107	—	38	1	no	—
183	#423	—	19	2	no	—
184	C₂×#104	—	38	2	no	—
185	C₂×C₄×C₈	—	2	1	no	—
186	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
187	C₄×C₄⋊C₄	—	19	2	no	—
188	#340	—	38	1	no	—
189	#339	—	38	2	no	—
190	#335	—	38	1	no	—
191	#335	—	38	1	no	—
192	C₂×C₁₉	—	2	1	no	—
193	C₂×C₁₉	—	2	1	no	—
194	C₂×C₁₉	—	4	2	no	—
195	C₂×C₁₉	#1806	1	1	yes	characteristic
196	C₂×C₁₉	#1808	1	1	yes	characteristic
197	C₂×C₁₉	#1814	1	1	yes	characteristic
198	C₂×#41	D₇₆	1	1	yes	characteristic
199	C₂×M₁₆	—	38	2	no	—
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
203	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
204	C₄×Q₈	Dic₃₈	1	1	yes	characteristic
205	C₄×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
206	C₄×D₄	—	2	1	no	—
207	C₄×D₄	—	2	1	no	—
208	C₄×D₄	Dic₃₈	1	1	yes	characteristic
209	C₄×D₄	C₂×Dic₁₉	1	1	yes	characteristic
210	#117	—	2	1	no	—
211	#117	D₇₆	1	1	yes	characteristic
212	C₂×C₄⋊C₄	—	38	2	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	38	2	no	—
215	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
216	#107	—	38	4	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	38	2	no	—
220	C₄⋊C₈	—	38	2	no	—
221	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
222	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
223	#104	—	76	4	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
231	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
232	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
233	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
234	#95	—	38	2	no	—
235	C₁₉	#8667	1	1	yes	characteristic, complemented, Sylow
236	#41	—	2	1	no	—
237	#41	—	2	1	no	—
238	#41	D₁₅₂	1	2	yes	—
239	#41	Dic₇₆	1	2	yes	—
240	C₂×Q₈	C₄⋊Dic₁₉	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	C₄⋊Dic₁₉	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	76	2	no	—
247	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
248	C₂²×C₄	#64541	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
250	M₁₆	—	76	2	no	—
251	M₁₆	—	76	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	—	76	2	no	—
259	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
260	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
261	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
262	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	38	4	no	—
266	C₄⋊C₄	—	76	2	no	—
267	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
268	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	76	2	no	—
276	C₄²	—	76	2	no	—
277	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
278	C₄²	#64541	1	1	yes	characteristic
279	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	#10597376	1	1	yes	characteristic
283	Q₈	—	2	1	no	—
284	Q₈	#10597275	1	2	yes	complemented
285	D₄	—	2	2	no	—
286	D₄	—	2	2	no	—
287	D₄	—	2	1	no	—
288	D₄	#10597275	1	2	yes	complemented
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	76	4	no	—
301	C₂×C₄	—	152	2	no	—
302	C₂×C₄	—	152	2	no	—
303	C₂×C₄	C₂×D₁₅₂	1	1	yes	characteristic
304	C₂×C₄	#10597350	1	1	yes	characteristic
305	C₂×C₄	#10597350	1	1	yes	characteristic
306	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
307	C₂×C₄	#10597370	1	1	yes	characteristic
308	C₂×C₄	#10597370	1	1	yes	characteristic
309	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
310	C₂×C₄	#10597372	1	1	yes	characteristic
311	C₂×C₄	#10597376	1	1	yes	characteristic
312	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	—	152	2	no	—
319	C₈	#10597294	1	2	yes	complemented
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	#11926404	1	1	yes	characteristic
326	C₂²	#11926780	1	1	yes	characteristic
327	C₂²	#11926785	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	152	4	no	—
335	C₄	C₂×#10597294	1	1	yes	characteristic
336	C₄	#11926561	1	1	yes	characteristic
337	C₄	C₂×#10597275	1	1	yes	characteristic
338	C₄	#11926582	1	1	yes	characteristic
339	C₄	#11926780	1	1	yes	characteristic
340	C₄	#11926785	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	4	2	no	—
344	C₂	#14563655	1	1	yes	characteristic
345	C₂	#14563664	1	1	yes	characteristic
346	C₂	#14563689	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₈×#41×C₁₉	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂×#41×C₁₉	D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₈×#41	D₁₉
hypocenter	C₁₉	#8667
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×#41

Complements

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

N	H ≅ G/N	type
2. C₄×#10597275	C₂	semidirect
3. #14562394	C₂	semidirect
4. #14561537	C₂	semidirect
15. C₈×Q₈×C₁₉	C₄	semidirect
16. C₈×Q₈×C₁₉	C₄	semidirect
19. C₈×D₄×C₁₉	C₄	semidirect
20. C₈×D₄×C₁₉	C₄	semidirect
111. Q₈×C₁₉	#107	semidirect
115. D₄×C₁₉	#107	semidirect
145. C₈×C₁₉	#104	semidirect
174. C₈×Q₈	Dic₁₉	semidirect
175. C₈×Q₈	Dic₁₉	semidirect
178. C₈×D₄	Dic₁₉	semidirect
179. C₈×D₄	Dic₁₉	semidirect
235. C₁₉	#8667	semidirect
284. Q₈	#10597275	semidirect
288. D₄	#10597275	semidirect
319. C₈	#10597294	semidirect

Maximal subgroups

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

subgroup	index
#14563655	2
C₄×#10597275	2
#14562394	2
#14561537	2
C₈×#41×C₁₉	2
#8667	19