Subgroups — #18081400

Order 4864.

606 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₅₂	C₂	1	1	yes	maximal, characteristic
2	#14564428	C₂	1	2	yes	maximal, complemented
3	C₂×#11926271	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×C₁₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₉×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₅₂	C₄	1	1	yes	characteristic
7	C₂×C₄×D₇₆	C₂²	1	2	yes	—
8	C₄×D₁₅₂	—	2	2	no	—
9	C₄×D₁₅₂	—	2	2	no	—
10	C₂×#10597275	C₄	1	1	yes	characteristic
11	C₂×#10597278	C₄	1	2	yes	—
12	C₂×(C₂×D₁₉)⋊C₈	—	2	4	no	—
13	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
15	#11926271	—	2	4	no	—
16	C₈×C₁₉⋊C₈	—	2	2	no	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
19	C₁₉×#335	—	2	2	no	—
20	C₂²×D₇₆	C₂×C₄	1	2	yes	—
21	C₂²×C₄×D₁₉	—	2	2	no	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×(C₂×D₁₉)⋊C₄	—	2	2	no	—
27	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
28	C₂²×C₁₉⋊C₈	—	2	1	no	—
29	C₂²×C₁₉⋊C₈	D₄	1	2	yes	—
30	C₂×D₁₅₂	—	2	2	no	—
31	C₂×D₁₅₂	—	2	2	no	—
32	C₂×D₁₅₂	C₈	1	4	yes	complemented
33	C₄×D₇₆	—	2	2	no	—
34	C₄×D₇₆	—	2	2	no	—
35	C₄×D₇₆	—	2	2	no	—
36	C₄×D₇₆	—	2	2	no	—
37	C₁₉×C₄⋊C₈	—	2	2	no	—
38	C₁₉×C₄⋊C₈	—	2	2	no	—
39	C₄×C₈×C₁₉	D₄	1	2	yes	—
40	C₄×C₈×C₁₉	D₄	1	2	yes	—
41	#10597278	—	2	4	no	—
42	(C₂×D₁₉)⋊C₈	—	4	8	no	—
43	#10597275	—	2	2	no	—
44	C₄×C₁₉⋊C₈	—	2	1	no	—
45	C₄×C₁₉⋊C₈	—	2	1	no	—
46	C₄×C₁₉⋊C₈	—	2	1	no	—
47	C₄×C₁₉⋊C₈	D₄	1	2	yes	—
48	C₂³×D₁₉	—	2	2	no	—
49	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
52	C₂²×Dic₁₉	—	2	2	no	—
53	C₂×D₇₆	—	2	2	no	—
54	C₂×D₇₆	—	2	2	no	—
55	C₂×D₇₆	—	2	2	no	—
56	C₂×D₇₆	—	2	2	no	—
57	C₂×D₇₆	—	2	2	no	—
58	C₂×D₇₆	C₂×C₈	1	4	yes	—
59	C₂×C₄×D₁₉	—	4	2	no	—
60	C₂×C₄×D₁₉	—	4	2	no	—
61	C₂×C₄×D₁₉	—	4	4	no	—
62	C₂×C₄×D₁₉	—	4	4	no	—
63	C₂×C₈×C₁₉	—	2	1	no	—
64	C₂×C₈×C₁₉	—	2	1	no	—
65	C₂×C₈×C₁₉	—	4	2	no	—
66	C₂×C₈×C₁₉	—	4	4	no	—
67	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
68	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
71	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₁₉	#41	1	1	yes	characteristic
75	(C₂×D₁₉)⋊C₄	—	4	2	no	—
76	(C₂×D₁₉)⋊C₄	—	4	2	no	—
77	C₄⋊Dic₁₉	—	2	2	no	—
78	C₄⋊Dic₁₉	—	2	2	no	—
79	C₂×C₁₉⋊C₈	—	2	4	no	—
80	C₂×C₁₉⋊C₈	—	2	2	no	—
81	C₂×C₁₉⋊C₈	—	2	2	no	—
82	C₂×C₁₉⋊C₈	—	2	2	no	—
83	C₂×C₁₉⋊C₈	D₈	1	4	yes	complemented
84	D₁₅₂	—	2	8	no	—
85	#9657	—	19	1	no	maximal, Sylow
86	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
87	C₂²×D₁₉	—	2	4	no	—
88	C₂²×D₁₉	—	4	2	no	—
89	C₂²×D₁₉	—	4	2	no	—
90	C₂²×D₁₉	—	4	4	no	—
91	C₂²×D₁₉	—	4	4	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
96	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
97	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
98	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₁₉	#135	1	2	yes	—
105	C₂×Dic₁₉	—	4	2	no	—
106	C₂×Dic₁₉	—	4	4	no	—
107	D₇₆	—	2	4	no	—
108	D₇₆	—	2	4	no	—
109	D₇₆	—	4	2	no	—
110	D₇₆	—	4	2	no	—
111	C₄×D₁₉	—	8	4	no	—
112	C₄×D₁₉	—	8	2	no	—
113	C₄×D₁₉	—	8	2	no	—
114	C₈×C₁₉	—	2	2	no	—
115	C₈×C₁₉	—	8	2	no	—
116	C₈×C₁₉	C₂²⋊C₈	1	4	yes	complemented
117	C₁₉⋊C₈	—	2	4	no	—
118	C₁₉⋊C₈	—	2	4	no	—
119	C₂×C₄×D₈	—	19	1	no	—
120	#1954	—	19	2	no	—
121	C₂×#335	D₁₉	1	1	yes	characteristic, complemented
122	C₂×D₄⋊C₈	—	19	2	no	—
123	C₂×C₈²	—	19	1	no	—
124	C₂²×C₁₉	#390	1	1	yes	characteristic
125	C₂²×C₁₉	C₈×D₄	1	1	yes	characteristic
126	C₂²×C₁₉	#436	1	1	yes	characteristic
127	C₂²×C₁₉	C₄×D₈	1	2	yes	—
128	C₂²×C₁₉	#443	1	2	yes	—
129	C₂×D₁₉	—	4	8	no	—
130	C₂×D₁₉	—	4	4	no	—
131	C₂×D₁₉	—	8	4	no	—
132	C₂×D₁₉	—	8	2	no	—
133	C₂×D₁₉	—	8	2	no	—
134	C₄×C₁₉	—	2	1	no	—
135	C₄×C₁₉	—	2	1	no	—
136	C₄×C₁₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
137	C₄×C₁₉	#408	1	1	yes	characteristic
138	C₄×C₁₉	C₈×D₄	1	1	yes	characteristic
139	C₄×C₁₉	#436	1	1	yes	characteristic
140	C₄×C₁₉	#466	1	2	yes	—
141	C₄×C₁₉	#493	1	1	yes	characteristic
142	C₄×C₁₉	#495	1	1	yes	characteristic
143	Dic₁₉	—	8	2	no	—
144	C₂²×D₈	—	19	1	no	—
145	C₂×C₄×D₄	—	19	2	no	—
146	C₄×D₈	—	38	2	no	—
147	C₄×D₈	—	38	2	no	—
148	C₂×#107	—	19	1	no	—
149	C₂×C₄⋊C₈	C₂×D₁₉	1	2	yes	—
150	C₂×#102	—	19	2	no	—
151	C₂×C₂²⋊C₈	—	38	4	no	—
152	C₂×C₄×C₈	—	19	1	no	—
153	C₂×C₄×C₈	—	19	1	no	—
154	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
155	#335	—	2	2	no	—
156	D₄⋊C₈	—	38	4	no	—
157	C₈²	—	38	2	no	—
158	C₂×C₁₉	C₈×D₈	1	2	yes	—
159	C₂×C₁₉	#1589	1	2	yes	—
160	C₂×C₁₉	#1954	1	1	yes	characteristic
161	C₂×C₁₉	#1970	1	1	yes	characteristic
162	C₂×C₁₉	#1974	1	1	yes	characteristic
163	D₁₉	—	8	4	no	—
164	C₂²×D₄	—	19	2	no	—
165	C₂³×C₄	—	38	2	no	—
166	C₂×D₈	—	19	4	no	—
167	C₂×D₈	—	38	2	no	—
168	C₂×D₈	—	38	2	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	19	2	no	—
171	C₂²×C₈	—	38	1	no	—
172	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
173	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
174	C₄×D₄	—	38	2	no	—
175	C₄×D₄	—	38	2	no	—
176	C₄×D₄	—	38	2	no	—
177	C₄×D₄	—	38	2	no	—
178	C₂×C₄⋊C₄	—	19	2	no	—
179	C₂×C₂²⋊C₄	—	38	2	no	—
180	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
181	#107	—	38	2	no	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	2	2	no	—
184	#102	—	38	4	no	—
185	C₂²⋊C₈	—	76	8	no	—
186	C₄×C₈	—	19	2	no	—
187	C₄×C₈	—	38	1	no	—
188	C₄×C₈	—	38	1	no	—
189	C₄×C₈	—	38	1	no	—
190	C₄×C₈	D₇₆	1	2	yes	—
191	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
192	C₁₉	#9657	1	1	yes	characteristic, complemented, Sylow
193	C₂⁴	—	38	2	no	—
194	C₂×D₄	—	19	4	no	—
195	C₂×D₄	—	38	2	no	—
196	C₂×D₄	—	38	2	no	—
197	C₂×D₄	—	38	2	no	—
198	C₂×D₄	—	38	2	no	—
199	C₂×D₄	—	38	2	no	—
200	C₂²×C₄	—	38	2	no	—
201	C₂²×C₄	—	76	2	no	—
202	C₂²×C₄	—	76	2	no	—
203	C₂²×C₄	—	76	4	no	—
204	C₂²×C₄	—	76	4	no	—
205	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₁₉	1	1	yes	characteristic
207	C₂²×C₄	#64545	1	1	yes	characteristic
208	D₈	—	38	8	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	4	no	—
213	C₂×C₈	—	19	4	no	—
214	C₂×C₈	—	38	4	no	—
215	C₂×C₈	—	38	2	no	—
216	C₂×C₈	—	38	2	no	—
217	C₂×C₈	—	38	2	no	—
218	C₂×C₈	C₈×D₁₉	1	2	yes	—
219	C₂×C₈	#64515	1	2	yes	—
220	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
221	C₂×C₈	(C₂×D₁₉)⋊C₄	1	2	yes	—
222	C₄⋊C₄	—	38	2	no	—
223	C₄⋊C₄	—	38	2	no	—
224	C₂²⋊C₄	—	76	2	no	—
225	C₂²⋊C₄	—	76	2	no	—
226	C₄²	C₂×D₇₆	1	1	yes	characteristic
227	C₄²	D₄×D₁₉	1	1	yes	characteristic
228	C₄²	#64545	1	1	yes	characteristic
229	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
230	C₂³	—	38	4	no	—
231	C₂³	—	76	2	no	—
232	C₂³	—	76	4	no	—
233	C₂³	—	76	4	no	—
234	C₂³	—	76	2	no	—
235	C₂³	#10597339	1	1	yes	characteristic
236	D₄	—	38	4	no	—
237	D₄	—	38	4	no	—
238	D₄	—	76	2	no	—
239	D₄	—	76	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	76	4	no	—
242	C₂×C₄	—	76	2	no	—
243	C₂×C₄	—	152	2	no	—
244	C₂×C₄	—	152	2	no	—
245	C₂×C₄	—	152	4	no	—
246	C₂×C₄	#10597339	1	1	yes	characteristic
247	C₂×C₄	#10597341	1	1	yes	characteristic
248	C₂×C₄	#10597341	1	1	yes	characteristic
249	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
250	C₂×C₄	C₂×#64515	1	1	yes	characteristic
251	C₂×C₄	#10597353	1	1	yes	characteristic
252	C₂×C₄	#10597353	1	1	yes	characteristic
253	C₂×C₄	D₈×D₁₉	1	2	yes	—
254	C₂×C₄	#10597365	1	2	yes	—
255	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
256	C₂×C₄	#10597386	1	1	yes	characteristic
257	C₂×C₄	#10597393	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	8	2	no	—
260	C₈	—	38	4	no	—
261	C₈	—	38	4	no	—
262	C₈	(C₂×D₁₉)⋊C₈	1	4	yes	complemented
263	C₂²	—	76	4	no	—
264	C₂²	—	76	8	no	—
265	C₂²	—	152	2	no	—
266	C₂²	—	152	2	no	—
267	C₂²	—	152	4	no	—
268	C₂²	#11926445	1	1	yes	characteristic
269	C₂²	#11926609	1	1	yes	characteristic
270	C₂²	#11926621	1	1	yes	characteristic
271	C₂²	#11926626	1	2	yes	—
272	C₂²	#11926660	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	152	2	no	—
276	C₄	C₂×(C₂×D₁₉)⋊C₈	1	1	yes	characteristic
277	C₄	#11926512	1	1	yes	characteristic
278	C₄	#11926609	1	1	yes	characteristic
279	C₄	#11926621	1	1	yes	characteristic
280	C₄	#11926781	1	2	yes	—
281	C₄	#11926919	1	1	yes	characteristic
282	C₄	#11926929	1	1	yes	characteristic
283	C₂	—	152	4	no	—
284	C₂	#14562429	1	2	yes	—
285	C₂	#14562519	1	2	yes	—
286	C₂	#14564428	1	1	yes	characteristic
287	C₂	#14564499	1	1	yes	characteristic
288	C₂	#14564521	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₉
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#335	C₂
socle	C₂³×C₁₉	C₄×D₄
prime radical	C₂²×D₁₅₂	C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#335	D₁₉
hypocenter	C₁₉	#9657
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
2. #14564428	C₂	semidirect
3. C₂×#11926271	C₂	semidirect
4. C₂×C₈×C₁₉⋊C₈	C₂	semidirect
5. C₂×C₁₉×#335	C₂	semidirect
32. C₂×D₁₅₂	C₈	semidirect
83. C₂×C₁₉⋊C₈	D₈	semidirect
116. C₈×C₁₉	C₂²⋊C₈	semidirect
121. C₂×#335	D₁₉	semidirect
192. C₁₉	#9657	semidirect
262. C₈	(C₂×D₁₉)⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₅₂	2
#14564428	2
C₂×#11926271	2
C₂×C₈×C₁₉⋊C₈	2
C₂×C₁₉×#335	2
#9657	19