Subgroups — #18072120

Order 4864.

794 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926568	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#64520	C₂	1	1	yes	maximal, characteristic
3	#14561476	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₉×#117	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	—	2	1	no	—
6	C₂²×#64520	—	2	1	no	—
7	#11926568	—	2	2	no	—
8	#11926568	C₄	1	4	yes	complemented
9	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
11	C₄×#64520	—	2	2	no	—
12	C₄×#64520	C₄	1	4	yes	complemented
13	C₂×#10597260	—	2	1	no	—
14	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₉×#117	C₄	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₄×C₂²⋊C₄×C₁₉	—	2	1	no	—
20	C₄×C₂²⋊C₄×C₁₉	—	2	1	no	—
21	C₁₉×#376	—	2	1	no	—
22	C₄³×C₁₉	—	2	1	no	—
23	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
26	C₁₉×#117	—	2	8	no	—
27	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊C₄×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	—	2	2	no	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	D₄	1	2	yes	—
38	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
39	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
40	C₂×#64520	—	2	8	no	—
41	C₂×#64520	—	2	4	no	—
42	C₂×#64520	—	2	4	no	—
43	C₂×#64520	—	4	4	no	—
44	C₂²×C₁₉⋊C₈	—	2	2	no	—
45	C₂²×C₁₉⋊C₈	—	2	2	no	—
46	C₁₉×#95	—	2	2	no	—
47	C₁₉×#95	—	2	2	no	—
48	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
49	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
50	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
51	#10597260	—	2	4	no	—
52	C₄×C₁₉⋊C₈	—	2	2	no	—
53	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
54	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	—	2	4	no	—
56	C₂²×C₄×C₁₉	—	2	4	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	2	2	no	—
59	C₂²×C₄×C₁₉	—	2	4	no	—
60	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉	—	2	4	no	—
73	C₄⋊C₄×C₁₉	M₁₆	1	8	yes	complemented
74	C₂²⋊C₄×C₁₉	—	2	8	no	—
75	C₄²×C₁₉	—	2	8	no	—
76	C₄²×C₁₉	—	4	4	no	—
77	C₄²×C₁₉	—	4	2	no	—
78	C₄²×C₁₉	—	4	2	no	—
79	C₄²×C₁₉	C₄²	1	2	yes	—
80	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
82	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
83	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
84	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
85	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
89	#64520	—	2	16	no	—
90	#64520	—	4	8	no	—
91	C₂×C₁₉⋊C₈	—	2	8	no	—
92	C₂×C₁₉⋊C₈	—	2	4	no	—
93	C₂×C₁₉⋊C₈	—	2	4	no	—
94	C₂×C₁₉⋊C₈	—	4	4	no	—
95	#7903	—	19	1	no	maximal, Sylow
96	C₂³×C₁₉	—	2	2	no	—
97	C₂³×C₁₉	—	2	2	no	—
98	C₂³×C₁₉	—	2	2	no	—
99	C₂³×C₁₉	#95	1	2	yes	—
100	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉	—	2	2	no	—
102	C₂×C₄×C₁₉	—	2	2	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	4	no	—
106	C₂×C₄×C₁₉	—	2	4	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	2	no	—
109	C₂×C₄×C₁₉	—	2	4	no	—
110	C₂×C₄×C₁₉	—	2	4	no	—
111	C₂×C₄×C₁₉	—	2	8	no	—
112	C₂×C₄×C₁₉	—	4	4	no	—
113	C₂×C₄×C₁₉	—	4	4	no	—
114	C₂×C₄×C₁₉	#95	1	2	yes	—
115	C₂×C₄×C₁₉	#95	1	2	yes	—
116	C₂×C₄×C₁₉	#95	1	2	yes	—
117	C₂×C₄×C₁₉	#104	1	4	yes	—
118	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
125	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₁₉	#137	1	2	yes	—
127	C₁₉⋊C₈	—	2	8	no	—
128	C₁₉⋊C₈	—	4	4	no	—
129	C₂×#423	—	19	1	no	—
130	C₂×C₄×M₁₆	—	19	1	no	—
131	C₄×#117	D₁₉	1	1	yes	characteristic
132	#1279	—	19	4	no	—
133	C₂²×C₁₉	—	2	2	no	—
134	C₂²×C₁₉	—	2	2	no	—
135	C₂²×C₁₉	—	2	2	no	—
136	C₂²×C₁₉	—	4	4	no	—
137	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
138	C₂²×C₁₉	#407	1	2	yes	—
139	C₂²×C₁₉	#417	1	2	yes	—
140	C₂²×C₁₉	C₂×#104	1	2	yes	—
141	C₄×C₁₉	—	2	2	no	—
142	C₄×C₁₉	—	2	1	no	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	2	no	—
145	C₄×C₁₉	—	4	8	no	—
146	C₄×C₁₉	#339	1	4	yes	—
147	C₄×C₁₉	C₄×M₁₆	1	2	yes	—
148	C₄×C₁₉	#418	1	1	yes	characteristic
149	C₄×C₁₉	#419	1	1	yes	characteristic
150	C₄×C₁₉	#423	1	2	yes	—
151	C₄×C₁₉	#428	1	2	yes	—
152	C₂²×M₁₆	—	38	1	no	—
153	C₂²×M₁₆	—	38	1	no	—
154	C₂×#117	Dic₁₉	1	2	yes	—
155	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
156	#423	—	19	4	no	—
157	#423	—	38	2	no	—
158	C₂×C₄⋊C₈	—	19	2	no	—
159	C₂×C₄⋊C₈	—	38	1	no	—
160	C₄×M₁₆	—	19	4	no	—
161	C₄×M₁₆	—	38	2	no	—
162	C₂×#97	—	38	1	no	—
163	C₂×C₄×C₈	—	19	2	no	—
164	C₄×C₄⋊C₄	Dic₁₉	1	2	yes	—
165	C₄×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
166	C₄×C₂²⋊C₄	—	2	1	no	—
167	C₄×C₂²⋊C₄	—	2	1	no	—
168	#376	—	2	1	no	—
169	C₄³	—	2	1	no	—
170	C₂×C₁₉	—	4	2	no	—
171	C₂×C₁₉	#1489	1	4	yes	—
172	C₂×C₁₉	#1728	1	1	yes	characteristic
173	C₂×C₁₉	C₂×#339	1	1	yes	characteristic
174	C₂×C₁₉	#1736	1	1	yes	characteristic
175	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
176	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
177	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
178	C₂×M₁₆	—	38	8	no	—
179	C₂×M₁₆	—	38	4	no	—
180	C₂×M₁₆	—	38	4	no	—
181	C₂×M₁₆	—	76	4	no	—
182	C₂²×C₈	—	38	2	no	—
183	C₂²×C₈	—	38	2	no	—
184	#117	—	2	8	no	—
185	C₂×C₄⋊C₄	C₂×Dic₁₉	1	4	yes	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	Dic₃₈	1	2	yes	—
191	C₂×C₄²	C₄×D₁₉	1	2	yes	—
192	C₂×C₄²	C₄×D₁₉	1	2	yes	—
193	C₂×C₄²	D₇₆	1	2	yes	—
194	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
195	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
196	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
198	C₄⋊C₈	—	19	4	no	—
199	C₄⋊C₈	—	38	4	no	—
200	C₄⋊C₈	—	38	2	no	—
201	#97	—	38	4	no	—
202	C₄×C₈	—	19	4	no	—
203	C₄×C₈	—	38	2	no	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	C₁₉	#7903	1	1	yes	characteristic, complemented, Sylow
207	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
214	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
215	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
216	C₂²×C₄	C₄⋊Dic₁₉	1	2	yes	—
217	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
218	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
219	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
220	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
221	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
222	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
225	M₁₆	—	38	16	no	—
226	M₁₆	—	76	8	no	—
227	C₂×C₈	—	38	8	no	—
228	C₂×C₈	—	38	4	no	—
229	C₂×C₈	—	38	4	no	—
230	C₂×C₈	—	76	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	#64520	1	8	yes	complemented
233	C₂²⋊C₄	—	2	8	no	—
234	C₄²	—	2	8	no	—
235	C₄²	—	4	2	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	4	no	—
238	C₄²	C₄×Dic₁₉	1	2	yes	—
239	C₄²	Dic₁₉⋊C₄	1	2	yes	—
240	C₄²	Dic₁₉⋊C₄	1	2	yes	—
241	C₄²	C₄⋊Dic₁₉	1	2	yes	—
242	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
243	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
244	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
245	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
247	C₄²	C₂×D₇₆	1	1	yes	characteristic
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	C₁₉⋊#95	1	2	yes	—
252	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	C₁₉⋊#95	1	2	yes	—
267	C₂×C₄	C₁₉⋊#95	1	2	yes	—
268	C₂×C₄	C₁₉⋊#95	1	2	yes	—
269	C₂×C₄	#10597294	1	4	yes	—
270	C₂×C₄	C₂×#64520	1	4	yes	—
271	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
272	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
273	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
275	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
276	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
277	C₂×C₄	#10597395	1	2	yes	—
278	C₂×C₄	#10597397	1	2	yes	—
279	C₈	—	38	8	no	—
280	C₈	—	76	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	4	no	—
285	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
286	C₂²	#11926508	1	2	yes	—
287	C₂²	#11926545	1	2	yes	—
288	C₂²	C₂×#10597294	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	8	no	—
294	C₄	#11926303	1	4	yes	—
295	C₄	C₄×#64520	1	2	yes	—
296	C₄	#11926548	1	1	yes	characteristic
297	C₄	#11926552	1	1	yes	characteristic
298	C₄	#11926568	1	2	yes	—
299	C₄	#11926587	1	2	yes	—
300	C₂	—	4	2	no	—
301	C₂	#14562074	1	4	yes	—
302	C₂	#14563329	1	1	yes	characteristic
303	C₂	C₂×#11926303	1	1	yes	characteristic
304	C₂	#14563356	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄×C₁₉×#117	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄×#117	D₁₉
hypocenter	C₁₉	#7903
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
3. #14561476	C₂	semidirect
8. #11926568	C₄	semidirect
12. C₄×#64520	C₄	semidirect
73. C₄⋊C₄×C₁₉	M₁₆	semidirect
206. C₁₉	#7903	semidirect
232. C₄⋊C₄	#64520	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926568	2
C₂×C₄×#64520	2
#14561476	2
C₄×C₁₉×#117	2
#7903	19