Subgroups — #18072011

Order 4864.

662 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926601	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926568	C₂	1	1	yes	maximal, characteristic
3	#14561470	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	—	2	1	no	—
6	#11926601	—	2	2	no	—
7	#11926601	C₄	1	4	yes	—
8	#11926568	—	2	2	no	—
9	#11926568	C₄	1	4	yes	—
10	C₂×C₄⋊(C₁₉⋊C₈)	—	2	1	no	—
11	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
14	C₂×#10597260	—	2	1	no	—
15	C₂²×M₁₆×C₁₉	C₄	1	2	yes	—
16	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₁₉	—	2	2	no	—
18	C₄×M₁₆×C₁₉	—	2	2	no	—
19	C₂×C₁₉×#97	C₄	1	2	yes	—
20	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
21	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
24	C₂×M₁₆×C₁₉	—	2	4	no	—
25	C₂×M₁₆×C₁₉	—	2	8	no	—
26	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₁₉	D₄	1	2	yes	—
32	C₂×C₄²×C₁₉	D₄	1	2	yes	—
33	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
35	C₂×#64520	—	2	4	no	—
36	C₂×#64520	—	4	4	no	—
37	C₂²×C₁₉⋊C₈	—	2	2	no	—
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₁₉×#97	—	2	4	no	—
40	C₄×C₈×C₁₉	—	2	2	no	—
41	C₄×C₈×C₁₉	—	2	2	no	—
42	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
43	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
44	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
45	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
46	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
47	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
48	#10597260	—	2	4	no	—
49	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	—	2	4	no	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	—	2	2	no	—
53	C₂²×C₄×C₁₉	—	4	2	no	—
54	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
66	M₁₆×C₁₉	—	2	16	no	—
67	C₂×C₈×C₁₉	—	2	4	no	—
68	C₂×C₈×C₁₉	—	2	4	no	—
69	C₂×C₈×C₁₉	M₁₆	1	8	yes	complemented
70	C₄²×C₁₉	C₄²	1	2	yes	—
71	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
73	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
75	C₄²×C₁₉	C₄⋊C₄	1	2	yes	—
76	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
80	#64520	—	4	8	no	—
81	C₂×C₁₉⋊C₈	—	2	4	no	—
82	C₂×C₁₉⋊C₈	—	2	4	no	—
83	C₂×C₁₉⋊C₈	—	4	2	no	—
84	C₂×C₁₉⋊C₈	—	4	4	no	—
85	C₂×C₁₉⋊C₈	—	4	2	no	—
86	#7876	—	19	1	no	maximal, Sylow
87	C₂³×C₁₉	—	2	2	no	—
88	C₂³×C₁₉	—	2	2	no	—
89	C₂³×C₁₉	—	2	2	no	—
90	C₂³×C₁₉	#95	1	2	yes	—
91	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	2	4	no	—
97	C₂×C₄×C₁₉	—	2	4	no	—
98	C₂×C₄×C₁₉	—	2	4	no	—
99	C₂×C₄×C₁₉	—	2	2	no	—
100	C₂×C₄×C₁₉	—	2	2	no	—
101	C₂×C₄×C₁₉	—	4	2	no	—
102	C₂×C₄×C₁₉	—	4	2	no	—
103	C₂×C₄×C₁₉	#95	1	2	yes	—
104	C₂×C₄×C₁₉	#95	1	2	yes	—
105	C₂×C₄×C₁₉	#95	1	2	yes	—
106	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
113	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
114	C₂×C₄×C₁₉	#137	1	2	yes	—
115	C₈×C₁₉	—	2	8	no	—
116	C₁₉⋊C₈	—	4	4	no	—
117	C₁₉⋊C₈	—	4	4	no	—
118	C₂×#432	—	19	1	no	—
119	C₂×#423	—	19	1	no	—
120	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
121	#1277	—	19	4	no	—
122	C₂²×C₁₉	—	2	2	no	—
123	C₂²×C₁₉	—	2	2	no	—
124	C₂²×C₁₉	—	2	2	no	—
125	C₂²×C₁₉	—	4	4	no	—
126	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
127	C₂²×C₁₉	#423	1	2	yes	—
128	C₂²×C₁₉	#428	1	2	yes	—
129	C₂²×C₁₉	#430	1	2	yes	—
130	C₄×C₁₉	—	2	2	no	—
131	C₄×C₁₉	—	2	2	no	—
132	C₄×C₁₉	—	2	2	no	—
133	C₄×C₁₉	—	2	1	no	—
134	C₄×C₁₉	—	2	1	no	—
135	C₄×C₁₉	C₄×M₁₆	1	2	yes	—
136	C₄×C₁₉	#407	1	2	yes	—
137	C₄×C₁₉	#417	1	2	yes	—
138	C₄×C₁₉	#418	1	1	yes	characteristic
139	C₄×C₁₉	#419	1	1	yes	characteristic
140	C₂²×M₁₆	—	38	1	no	—
141	C₂²×M₁₆	Dic₁₉	1	2	yes	—
142	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
143	#432	—	19	4	no	—
144	#432	—	38	2	no	—
145	#423	—	19	4	no	—
146	#423	—	38	2	no	—
147	C₂×C₄⋊C₈	—	19	2	no	—
148	C₂×C₄⋊C₈	—	19	2	no	—
149	C₂×C₄⋊C₈	—	38	1	no	—
150	C₂×C₂²⋊C₈	—	38	1	no	—
151	C₄×M₁₆	—	2	2	no	—
152	C₄×M₁₆	—	2	2	no	—
153	C₂×#97	—	38	1	no	—
154	C₂×#97	Dic₁₉	1	2	yes	—
155	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
156	C₂×C₁₉	—	4	2	no	—
157	C₂×C₁₉	#1569	1	4	yes	—
158	C₂×C₁₉	#1728	1	1	yes	characteristic
159	C₂×C₁₉	#1736	1	1	yes	characteristic
160	C₂×C₁₉	#1738	1	1	yes	characteristic
161	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
162	C₂³×C₄	D₇₆	1	1	yes	characteristic
163	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	—	2	8	no	—
166	C₂×M₁₆	—	38	4	no	—
167	C₂×M₁₆	—	76	4	no	—
168	C₂²×C₈	—	38	2	no	—
169	C₂²×C₈	—	38	2	no	—
170	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
171	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
172	C₂×C₄²	C₄×D₁₉	1	2	yes	—
173	C₂×C₄²	C₄×D₁₉	1	2	yes	—
174	C₂×C₄²	D₇₆	1	1	yes	characteristic
175	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
176	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
177	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
178	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
179	C₄⋊C₈	—	19	4	no	—
180	C₄⋊C₈	—	19	4	no	—
181	C₄⋊C₈	—	38	4	no	—
182	C₄⋊C₈	—	38	2	no	—
183	C₄⋊C₈	—	38	2	no	—
184	C₂²⋊C₈	—	38	4	no	—
185	#97	—	2	4	no	—
186	#97	—	38	4	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	2	2	no	—
189	C₁₉	#7876	1	1	yes	characteristic, complemented, Sylow
190	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
196	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
197	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
198	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
199	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
201	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
202	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
203	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
204	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
207	M₁₆	—	2	16	no	—
208	M₁₆	—	76	8	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	38	4	no	—
212	C₂×C₈	—	38	4	no	—
213	C₂×C₈	—	76	2	no	—
214	C₂×C₈	—	76	4	no	—
215	C₂×C₈	—	76	2	no	—
216	C₂×C₈	#64520	1	8	yes	complemented
217	C₄²	C₄×Dic₁₉	1	2	yes	—
218	C₄²	Dic₁₉⋊C₄	1	2	yes	—
219	C₄²	Dic₁₉⋊C₄	1	2	yes	—
220	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
221	C₄²	(C₂×D₁₉)⋊C₄	1	2	yes	—
222	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
223	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
224	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
225	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
226	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₁₉⋊#95	1	2	yes	—
231	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	C₁₉⋊#95	1	2	yes	—
244	C₂×C₄	C₁₉⋊#95	1	2	yes	—
245	C₂×C₄	C₁₉⋊#95	1	2	yes	—
246	C₂×C₄	#10597354	1	2	yes	—
247	C₂×C₄	#10597355	1	2	yes	—
248	C₂×C₄	C₂×#64520	1	4	yes	—
249	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
250	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
251	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
252	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
255	C₈	—	2	8	no	—
256	C₈	—	76	4	no	—
257	C₈	—	76	4	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	4	no	—
262	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
263	C₂²	#11926568	1	2	yes	—
264	C₂²	#11926586	1	2	yes	—
265	C₂²	#11926593	1	2	yes	—
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	2	1	no	—
271	C₄	C₄×#64520	1	2	yes	—
272	C₄	#11926508	1	2	yes	—
273	C₄	#11926546	1	2	yes	—
274	C₄	#11926550	1	1	yes	characteristic
275	C₄	#11926554	1	1	yes	characteristic
276	C₂	—	4	2	no	—
277	C₂	#14562402	1	4	yes	—
278	C₂	#14563329	1	1	yes	characteristic
279	C₂	#14563355	1	1	yes	characteristic
280	C₂	#14563362	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₄×M₁₆×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₉
hypocenter	C₁₉	#7876
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #14561470	C₂	semidirect
69. C₂×C₈×C₁₉	M₁₆	semidirect
189. C₁₉	#7876	semidirect
216. C₂×C₈	#64520	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926601	2
C₂×#11926568	2
#14561470	2
C₂×C₄×M₁₆×C₁₉	2
#7876	19