Subgroups — #18075703

Order 4864.

454 conjugacy classes of subgroups.

The subgroup lattice has 330 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×#10597274	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562391	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×#10597274	—	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	#11926295	—	2	1	no	—
13	#11926295	—	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	#10597274	—	2	4	no	—
60	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
61	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
62	#41×C₁₉	—	2	2	no	—
63	#41×C₁₉	SD₁₆	1	4	yes	—
64	C₂×Q₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂×D₄×C₁₉	—	2	1	no	—
66	C₂×D₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₉	—	2	1	no	—
68	C₂²×C₄×C₁₉	—	2	1	no	—
69	C₂²×C₄×C₁₉	—	2	1	no	—
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
73	C₂²×Dic₁₉	—	4	2	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	—	2	1	no	—
76	C₂×C₈×C₁₉	—	2	1	no	—
77	C₂×C₈×C₁₉	—	4	1	no	—
78	C₂×C₈×C₁₉	—	4	2	no	—
79	C₂×C₈×C₁₉	—	4	2	no	—
80	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₄⋊C₄×C₁₉	—	2	1	no	—
85	C₄⋊C₄×C₁₉	—	2	1	no	—
86	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
87	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
88	C₂²⋊C₄×C₁₉	—	2	1	no	—
89	C₂²⋊C₄×C₁₉	—	2	1	no	—
90	C₂²⋊C₄×C₁₉	—	2	1	no	—
91	C₄²×C₁₉	—	2	1	no	—
92	C₄²×C₁₉	—	2	1	no	—
93	C₄²×C₁₉	—	2	1	no	—
94	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
97	C₄²×C₁₉	#41	1	1	yes	characteristic
98	C₄⋊Dic₁₉	—	2	4	no	—
99	C₄⋊Dic₁₉	—	4	2	no	—
100	C₄×Dic₁₉	—	4	2	no	—
101	C₄×Dic₁₉	—	4	2	no	—
102	#64520	—	4	2	no	—
103	#64520	—	4	2	no	—
104	C₂×C₁₉⋊C₈	—	4	2	no	—
105	#8666	—	19	1	no	maximal, Sylow
106	C₂³×C₁₉	—	2	1	no	—
107	C₂³×C₁₉	#121	1	1	yes	characteristic
108	Q₈×C₁₉	—	2	1	no	—
109	Q₈×C₁₉	#106	1	2	yes	complemented
110	D₄×C₁₉	—	2	4	no	—
111	D₄×C₁₉	—	2	1	no	—
112	D₄×C₁₉	#106	1	2	yes	complemented
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	4	no	—
117	C₂×C₄×C₁₉	—	2	1	no	—
118	C₂×C₄×C₁₉	—	2	1	no	—
119	C₂×C₄×C₁₉	—	4	1	no	—
120	C₂×C₄×C₁₉	—	4	2	no	—
121	C₂×C₄×C₁₉	—	4	1	no	—
122	C₂×C₄×C₁₉	—	4	1	no	—
123	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
126	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
127	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
130	C₂×C₄×C₁₉	#135	1	2	yes	—
131	C₂×Dic₁₉	—	4	4	no	—
132	C₂×Dic₁₉	—	8	2	no	—
133	C₂×Dic₁₉	—	8	2	no	—
134	C₈×C₁₉	—	2	1	no	—
135	C₈×C₁₉	—	2	1	no	—
136	C₈×C₁₉	—	2	1	no	—
137	C₈×C₁₉	—	4	1	no	—
138	C₈×C₁₉	—	4	1	no	—
139	C₈×C₁₉	#104	1	2	yes	complemented
140	C₁₉⋊C₈	—	8	2	no	—
141	C₈×#41	D₁₉	1	1	yes	characteristic
142	#1806	—	19	2	no	—
143	C₄×#106	—	19	1	no	—
144	#1566	—	19	1	no	—
145	#1303	—	19	2	no	—
146	C₂²×C₁₉	—	2	1	no	—
147	C₂²×C₁₉	—	2	1	no	—
148	C₂²×C₁₉	—	2	2	no	—
149	C₂²×C₁₉	—	4	1	no	—
150	C₂²×C₁₉	—	4	1	no	—
151	C₂²×C₁₉	#380	1	1	yes	characteristic
152	C₂²×C₁₉	#465	1	2	yes	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	2	no	—
155	C₄×C₁₉	—	2	1	no	—
156	C₄×C₁₉	—	2	1	no	—
157	C₄×C₁₉	—	4	1	no	—
158	C₄×C₁₉	—	4	1	no	—
159	C₄×C₁₉	C₂×#104	1	1	yes	characteristic
160	C₄×C₁₉	#421	1	1	yes	characteristic
161	C₄×C₁₉	C₂×#106	1	1	yes	characteristic
162	C₄×C₁₉	#427	1	1	yes	characteristic
163	C₄×C₁₉	#465	1	2	yes	—
164	Dic₁₉	—	8	4	no	—
165	C₄×#41	C₂×D₁₉	1	1	yes	characteristic
166	C₈×Q₈	Dic₁₉	1	1	yes	characteristic, complemented
167	C₈×Q₈	Dic₁₉	1	1	yes	characteristic, complemented
168	C₈×D₄	—	2	1	no	—
169	C₈×D₄	—	2	1	no	—
170	C₈×D₄	Dic₁₉	1	1	yes	characteristic, complemented
171	C₈×D₄	Dic₁₉	1	1	yes	characteristic, complemented
172	#431	—	2	1	no	—
173	#431	C₂×D₁₉	1	1	yes	characteristic
174	C₂×#106	—	38	1	no	—
175	#423	—	19	2	no	—
176	C₂×#104	—	38	2	no	—
177	C₂×C₄×C₈	—	2	1	no	—
178	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
179	C₄×C₄⋊C₄	—	19	2	no	—
180	#340	—	38	1	no	—
181	#339	—	38	2	no	—
182	#334	—	38	1	no	—
183	#334	—	38	1	no	—
184	C₂×C₁₉	—	2	2	no	—
185	C₂×C₁₉	—	4	2	no	—
186	C₂×C₁₉	#1806	1	1	yes	characteristic
187	C₂×C₁₉	#1807	1	1	yes	characteristic
188	C₂×C₁₉	#1814	1	1	yes	characteristic
189	C₂×#41	D₇₆	1	1	yes	characteristic
190	C₂×M₁₆	—	38	2	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
194	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
195	C₄×Q₈	Dic₃₈	1	1	yes	characteristic
196	C₄×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
197	C₄×D₄	—	2	1	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	Dic₃₈	1	1	yes	characteristic
200	C₄×D₄	C₂×Dic₁₉	1	1	yes	characteristic
201	#117	—	2	1	no	—
202	#117	D₇₆	1	1	yes	characteristic
203	C₂×C₄⋊C₄	—	38	2	no	—
204	C₂×C₄²	—	2	1	no	—
205	C₂×C₄²	—	38	2	no	—
206	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
207	#106	—	38	4	no	—
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	38	2	no	—
211	C₄⋊C₈	—	38	2	no	—
212	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
213	C₄⋊C₈	C₂×Dic₁₉	1	1	yes	characteristic
214	#104	—	76	4	no	—
215	C₂²⋊C₈	—	2	1	no	—
216	C₂²⋊C₈	—	2	1	no	—
217	C₂²⋊C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
222	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
223	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
225	#95	—	38	2	no	—
226	C₁₉	#8666	1	1	yes	characteristic, complemented, Sylow
227	#41	—	2	2	no	—
228	#41	#64516	1	4	yes	—
229	C₂×Q₈	C₄⋊Dic₁₉	1	1	yes	characteristic
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₄⋊Dic₁₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	76	2	no	—
236	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
237	C₂²×C₄	#64541	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
239	M₁₆	—	76	2	no	—
240	M₁₆	—	76	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	2	no	—
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	76	2	no	—
248	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
250	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
251	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	38	4	no	—
255	C₄⋊C₄	—	76	2	no	—
256	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
257	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	76	2	no	—
265	C₄²	—	76	2	no	—
266	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
267	C₄²	#64541	1	1	yes	characteristic
268	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	#10597376	1	1	yes	characteristic
272	Q₈	—	2	1	no	—
273	Q₈	#10597274	1	2	yes	complemented
274	D₄	—	2	4	no	—
275	D₄	—	2	1	no	—
276	D₄	#10597274	1	2	yes	complemented
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	76	4	no	—
288	C₂×C₄	—	152	2	no	—
289	C₂×C₄	—	152	2	no	—
290	C₂×C₄	C₂×#64516	1	2	yes	—
291	C₂×C₄	#10597350	1	2	yes	—
292	C₂×C₄	#10597370	1	1	yes	characteristic
293	C₂×C₄	#10597370	1	1	yes	characteristic
294	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
295	C₂×C₄	#10597372	1	1	yes	characteristic
296	C₂×C₄	#10597376	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	1	no	—
303	C₈	—	152	2	no	—
304	C₈	#10597294	1	2	yes	complemented
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	#11926404	1	1	yes	characteristic
311	C₂²	#11926773	1	2	yes	—
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	152	4	no	—
319	C₄	C₂×#10597294	1	1	yes	characteristic
320	C₄	#11926561	1	1	yes	characteristic
321	C₄	C₂×#10597274	1	1	yes	characteristic
322	C₄	#11926582	1	1	yes	characteristic
323	C₄	#11926773	1	2	yes	—
324	C₂	—	2	2	no	—
325	C₂	—	4	2	no	—
326	C₂	#14563655	1	1	yes	characteristic
327	C₂	#14563660	1	1	yes	characteristic
328	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	#8666
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×#10597274	C₂	semidirect
3. #14562391	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
109. Q₈×C₁₉	#106	semidirect
112. D₄×C₁₉	#106	semidirect
139. C₈×C₁₉	#104	semidirect
166. C₈×Q₈	Dic₁₉	semidirect
167. C₈×Q₈	Dic₁₉	semidirect
170. C₈×D₄	Dic₁₉	semidirect
171. C₈×D₄	Dic₁₉	semidirect
226. C₁₉	#8666	semidirect
273. Q₈	#10597274	semidirect
276. D₄	#10597274	semidirect
304. 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×#10597274	2
#14562391	2
#14561537	2
C₈×#41×C₁₉	2
#8666	19