Subgroups — #18081902

Order 4864.

502 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574063	C₂	1	1	yes	maximal, characteristic
2	#14562383	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562129	C₂	1	2	yes	maximal, complemented
4	#14562033	C₂	1	2	yes	maximal, complemented
5	C₁₉×#1770	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10597350	—	2	1	no	—
7	C₂×#10597350	—	2	1	no	—
8	#11927094	C₂²	1	2	yes	—
9	#11926928	—	2	1	no	—
10	#11926928	—	2	1	no	—
11	#11926924	C₄	1	1	yes	characteristic, complemented
12	#11926921	C₄	1	1	yes	characteristic, complemented
13	#11926917	C₄	1	2	yes	complemented
14	#11926597	C₂²	1	2	yes	—
15	#11926297	—	2	1	no	—
16	#11926295	—	2	1	no	—
17	#11926277	—	2	4	no	—
18	#11926271	—	2	4	no	—
19	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×#97	—	2	1	no	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
23	C₁₉×#336	—	2	1	no	—
24	C₂×#64541	—	2	2	no	—
25	C₂²×C₈×C₁₉	—	2	1	no	—
26	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
29	C₂×C₄²×C₁₉	—	2	1	no	—
30	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
31	#10597376	—	2	2	no	—
32	#10597376	—	2	2	no	—
33	C₂×Dic₇₆	—	2	1	no	—
34	C₂×Dic₇₆	—	2	1	no	—
35	#10597350	—	4	2	no	—
36	#10597350	—	4	2	no	—
37	#10597350	—	4	2	no	—
38	#10597350	—	4	2	no	—
39	C₂×D₁₅₂	—	2	1	no	—
40	C₂×D₁₅₂	—	2	1	no	—
41	C₂×#64516	—	2	2	no	—
42	C₂×#64516	—	2	2	no	—
43	#10597321	—	2	2	no	—
44	#10597320	C₂×C₄	1	2	yes	—
45	C₄×D₇₆	—	2	2	no	—
46	C₄×D₇₆	—	2	2	no	—
47	#10597315	C₂×C₄	1	2	yes	—
48	C₁₉×C₄⋊C₈	—	2	1	no	—
49	C₁₉×C₄⋊C₈	D₄	1	2	yes	—
50	C₁₉×#97	—	4	2	no	—
51	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
55	C₁₉×#95	D₄	1	2	yes	—
56	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
57	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
58	C₄×C₁₉⋊C₈	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	2	no	—
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	C₂×C₂²⋊D₁₉	—	4	2	no	—
64	#64541	—	4	2	no	—
65	#64541	—	4	2	no	—
66	#64541	—	4	2	no	—
67	#64541	—	4	2	no	—
68	C₂×D₇₆	—	2	2	no	—
69	C₂×D₇₆	—	2	2	no	—
70	C₂×D₇₆	—	2	2	no	—
71	C₂×C₄×D₁₉	—	4	2	no	—
72	C₂×Dic₃₈	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	1	no	—
74	C₂×C₈×C₁₉	—	2	1	no	—
75	C₂×C₈×C₁₉	—	2	2	no	—
76	C₂×C₈×C₁₉	—	4	1	no	—
77	C₂×C₈×C₁₉	—	4	1	no	—
78	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₉	D₈	1	2	yes	—
83	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
84	C₄⋊C₄×C₁₉	—	4	2	no	—
85	C₄²×C₁₉	—	4	1	no	—
86	C₄²×C₁₉	—	4	1	no	—
87	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₉	#41	1	1	yes	characteristic
91	(C₂×D₁₉)⋊C₄	—	4	2	no	—
92	(C₂×D₁₉)⋊C₄	—	4	2	no	—
93	C₄⋊Dic₁₉	—	2	2	no	—
94	C₄⋊Dic₁₉	—	2	2	no	—
95	C₂×C₁₉⋊C₈	—	4	2	no	—
96	Dic₇₆	—	4	2	no	—
97	Dic₇₆	—	4	2	no	—
98	D₁₅₂	—	4	2	no	—
99	D₁₅₂	—	4	2	no	—
100	#64516	—	4	4	no	—
101	#64516	—	4	4	no	—
102	#9753	—	19	1	no	maximal, Sylow
103	C₂³×C₁₉	#121	1	1	yes	characteristic
104	C₂²×D₁₉	—	4	2	no	—
105	C₂×C₄×C₁₉	—	2	2	no	—
106	C₂×C₄×C₁₉	—	4	1	no	—
107	C₂×C₄×C₁₉	—	4	2	no	—
108	C₂×C₄×C₁₉	—	4	2	no	—
109	C₂×C₄×C₁₉	—	4	2	no	—
110	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
115	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂²⋊D₁₉	—	8	2	no	—
121	C₂²⋊D₁₉	—	8	2	no	—
122	C₂×Dic₁₉	—	4	2	no	—
123	D₇₆	—	4	2	no	—
124	D₇₆	—	4	2	no	—
125	D₇₆	—	4	4	no	—
126	D₇₆	—	8	2	no	—
127	C₄×D₁₉	—	8	2	no	—
128	C₄×D₁₉	—	8	2	no	—
129	Dic₃₈	—	4	2	no	—
130	Dic₃₈	—	4	2	no	—
131	C₈×C₁₉	—	2	1	no	—
132	C₈×C₁₉	—	2	1	no	—
133	C₈×C₁₉	—	4	2	no	—
134	C₈×C₁₉	#102	1	2	yes	—
135	C₈×C₁₉	#102	1	2	yes	—
136	C₁₉⋊C₈	—	8	2	no	—
137	#3147	—	19	1	no	—
138	#1770	D₁₉	1	1	yes	characteristic, complemented
139	#1564	—	19	1	no	—
140	#1502	—	19	2	no	—
141	#1480	—	19	2	no	—
142	C₂²×C₁₉	—	2	1	no	—
143	C₂²×C₁₉	—	2	1	no	—
144	C₂²×C₁₉	#390	1	1	yes	characteristic
145	C₂²×C₁₉	#459	1	1	yes	characteristic
146	C₂²×C₁₉	#460	1	1	yes	characteristic
147	C₂×D₁₉	—	8	2	no	—
148	C₂×D₁₉	—	8	2	no	—
149	C₂×D₁₉	—	8	2	no	—
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	2	1	no	—
153	C₄×C₁₉	—	8	2	no	—
154	C₄×C₁₉	C₂×#102	1	1	yes	characteristic
155	C₄×C₁₉	#418	1	1	yes	characteristic
156	C₄×C₁₉	#444	1	2	yes	—
157	C₄×C₁₉	#459	1	1	yes	characteristic
158	C₄×C₁₉	#460	1	1	yes	characteristic
159	Dic₁₉	—	8	2	no	—
160	C₂×#135	—	38	1	no	—
161	C₂×#135	—	38	1	no	—
162	#532	—	19	2	no	—
163	#495	—	38	1	no	—
164	#495	—	38	1	no	—
165	#494	—	19	1	no	—
166	#493	—	19	1	no	—
167	#492	—	19	2	no	—
168	#431	—	19	2	no	—
169	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
170	C₂×#97	—	2	1	no	—
171	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
172	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
173	#336	—	2	1	no	—
174	#335	—	38	1	no	—
175	#334	—	38	1	no	—
176	#327	—	38	4	no	—
177	D₄⋊C₈	—	38	4	no	—
178	C₂×C₁₉	—	2	1	no	—
179	C₂×C₁₉	—	2	1	no	—
180	C₂×C₁₉	#1975	1	1	yes	characteristic
181	C₂×C₁₉	#1976	1	1	yes	characteristic
182	C₂×C₁₉	#1982	1	1	yes	characteristic
183	D₁₉	—	16	2	no	—
184	C₂×#41	—	38	2	no	—
185	#135	—	76	2	no	—
186	#135	—	76	2	no	—
187	#135	—	76	2	no	—
188	#135	—	76	2	no	—
189	C₂×Q₁₆	—	38	1	no	—
190	C₂×Q₁₆	—	38	1	no	—
191	C₂×SD₁₆	—	38	2	no	—
192	C₂×SD₁₆	—	38	2	no	—
193	C₂×D₈	—	38	1	no	—
194	C₂×D₈	—	38	1	no	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
198	C₄⋊Q₈	—	19	2	no	—
199	#127	—	19	2	no	—
200	#124	—	38	2	no	—
201	#121	—	38	2	no	—
202	#121	—	38	2	no	—
203	C₄×D₄	—	38	2	no	—
204	C₄×D₄	—	38	2	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	38	2	no	—
210	C₄⋊C₈	D₇₆	1	2	yes	—
211	C₂²⋊C₈	—	38	2	no	—
212	#97	—	4	2	no	—
213	C₄×C₈	—	38	2	no	—
214	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
215	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
216	C₄×C₈	D₇₆	1	1	yes	characteristic
217	C₄×C₈	D₇₆	1	1	yes	characteristic
218	#95	D₇₆	1	2	yes	—
219	C₁₉	#9753	1	1	yes	characteristic, complemented, Sylow
220	#41	—	76	2	no	—
221	#41	—	76	2	no	—
222	#41	—	76	2	no	—
223	#41	—	76	2	no	—
224	C₂×Q₈	—	38	2	no	—
225	C₂×D₄	—	38	2	no	—
226	C₂×D₄	—	38	2	no	—
227	C₂×D₄	—	38	2	no	—
228	C₂×D₄	—	76	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	76	2	no	—
231	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
232	C₂²×C₄	#64541	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
234	Q₁₆	—	76	2	no	—
235	Q₁₆	—	76	2	no	—
236	SD₁₆	—	76	4	no	—
237	SD₁₆	—	76	4	no	—
238	D₈	—	76	2	no	—
239	D₈	—	76	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	76	2	no	—
246	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
247	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
248	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
249	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
250	C₂×C₈	#64525	1	2	yes	—
251	C₂×C₈	#64527	1	2	yes	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	—	38	2	no	—
254	C₄⋊C₄	—	38	2	no	—
255	C₂²⋊C₄	—	76	2	no	—
256	C₂²⋊C₄	—	76	2	no	—
257	C₄²	—	4	1	no	—
258	C₄²	—	4	1	no	—
259	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
260	C₄²	C₂×D₇₆	1	1	yes	characteristic
261	C₄²	C₂×D₇₆	1	1	yes	characteristic
262	C₄²	#64541	1	1	yes	characteristic
263	C₂³	—	76	2	no	—
264	C₂³	#10597376	1	1	yes	characteristic
265	Q₈	—	76	2	no	—
266	Q₈	—	76	2	no	—
267	D₄	—	76	2	no	—
268	D₄	—	76	2	no	—
269	D₄	—	76	4	no	—
270	D₄	—	152	2	no	—
271	D₄	—	152	2	no	—
272	D₄	—	152	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	76	2	no	—
279	C₂×C₄	—	152	2	no	—
280	C₂×C₄	—	152	2	no	—
281	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
282	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
283	C₂×C₄	#10597320	1	1	yes	characteristic
284	C₂×C₄	#10597321	1	1	yes	characteristic
285	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	#10597376	1	1	yes	characteristic
287	C₂×C₄	C₂×#64525	1	1	yes	characteristic
288	C₂×C₄	C₂×#64527	1	1	yes	characteristic
289	C₂×C₄	#10597395	1	1	yes	characteristic
290	C₂×C₄	#10597395	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	4	2	no	—
294	C₈	—	152	2	no	—
295	C₈	#10597266	1	2	yes	—
296	C₈	#10597266	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	152	2	no	—
300	C₂²	—	152	2	no	—
301	C₂²	—	152	2	no	—
302	C₂²	#11926446	1	1	yes	characteristic
303	C₂²	#11926739	1	1	yes	characteristic
304	C₂²	#11926744	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	152	2	no	—
310	C₄	C₂×#10597266	1	1	yes	characteristic
311	C₄	#11926548	1	1	yes	characteristic
312	C₄	#11926667	1	2	yes	—
313	C₄	#11926739	1	1	yes	characteristic
314	C₄	#11926744	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	304	2	no	—
318	C₂	#14564528	1	1	yes	characteristic
319	C₂	#14564533	1	1	yes	characteristic
320	C₂	#14564562	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1770	C₂
socle	C₂²×C₁₉	#390
prime radical	#14574063	C₂
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1770	D₁₉
hypocenter	C₁₉	#9753
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1770

Complements

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

N	H ≅ G/N	type
2. #14562383	C₂	semidirect
3. #14562129	C₂	semidirect
4. #14562033	C₂	semidirect
5. C₁₉×#1770	C₂	semidirect
11. #11926924	C₄	semidirect
12. #11926921	C₄	semidirect
13. #11926917	C₄	semidirect
138. #1770	D₁₉	semidirect
219. C₁₉	#9753	semidirect

Maximal subgroups

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

subgroup	index
#14574063	2
#14562383	2
#14562129	2
#14562033	2
C₁₉×#1770	2
#9753	19