Subgroups — #18069093

Order 4864.

442 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₂²⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#14563329	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₉×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂²×#64520	C₄	1	2	yes	—
5	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	1	no	—
6	C₂×C₄×C₁₉⋊C₈	—	2	1	no	—
7	#11926299	—	2	1	no	—
8	#11926299	—	2	1	no	—
9	#11926299	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
12	C₂×C₄×C₈×C₁₉	—	2	1	no	—
13	C₁₉×#336	C₄	1	2	yes	—
14	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	—	2	2	no	—
18	C₂×C₄²×C₁₉	—	4	2	no	—
19	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
23	C₂×#64520	—	2	2	no	—
24	C₂×#64520	—	4	2	no	—
25	C₂×#64520	—	4	2	no	—
26	C₂×#64520	C₈	1	4	yes	complemented
27	C₂²×C₁₉⋊C₈	—	2	2	no	—
28	C₂²×C₁₉⋊C₈	—	2	2	no	—
29	C₁₉×C₂²⋊C₈	—	2	4	no	—
30	C₄×C₈×C₁₉	—	4	2	no	—
31	C₂²⋊(C₁₉⋊C₈)	—	2	4	no	—
32	C₄×C₁₉⋊C₈	—	4	2	no	—
33	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	—	2	2	no	—
35	C₂²×C₄×C₁₉	—	2	1	no	—
36	C₂²×C₄×C₁₉	—	2	1	no	—
37	C₂²×C₄×C₁₉	—	2	2	no	—
38	C₂²×C₄×C₁₉	—	2	2	no	—
39	C₂²×C₄×C₁₉	—	2	2	no	—
40	C₂²×C₄×C₁₉	—	2	2	no	—
41	C₂²×C₄×C₁₉	—	4	1	no	—
42	C₂²×C₄×C₁₉	—	4	1	no	—
43	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
51	C₂×C₈×C₁₉	—	2	4	no	—
52	C₂×C₈×C₁₉	—	4	2	no	—
53	C₂×C₈×C₁₉	—	4	2	no	—
54	C₄²×C₁₉	—	2	2	no	—
55	C₄²×C₁₉	—	2	2	no	—
56	C₄²×C₁₉	—	4	1	no	—
57	C₄²×C₁₉	—	4	1	no	—
58	#64520	—	4	4	no	—
59	#64520	—	4	4	no	—
60	C₂×C₁₉⋊C₈	—	2	4	no	—
61	C₂×C₁₉⋊C₈	—	2	4	no	—
62	C₂×C₁₉⋊C₈	—	4	2	no	—
63	C₂×C₁₉⋊C₈	—	4	2	no	—
64	C₂×C₁₉⋊C₈	—	4	2	no	—
65	C₂×C₁₉⋊C₈	—	4	2	no	—
66	#7026	—	19	1	no	maximal, Sylow
67	C₂³×C₁₉	—	2	1	no	—
68	C₂³×C₁₉	—	2	1	no	—
69	C₂³×C₁₉	—	2	1	no	—
70	C₂³×C₁₉	—	2	1	no	—
71	C₂³×C₁₉	—	2	1	no	—
72	C₂³×C₁₉	—	2	1	no	—
73	C₂³×C₁₉	#95	1	1	yes	characteristic
74	C₂³×C₁₉	C₄⋊C₈	1	2	yes	—
75	C₂×C₄×C₁₉	—	2	1	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	—	2	1	no	—
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	2	1	no	—
80	C₂×C₄×C₁₉	—	2	1	no	—
81	C₂×C₄×C₁₉	—	2	1	no	—
82	C₂×C₄×C₁₉	—	2	1	no	—
83	C₂×C₄×C₁₉	—	2	1	no	—
84	C₂×C₄×C₁₉	—	2	1	no	—
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	2	2	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	4	4	no	—
90	C₂×C₄×C₁₉	—	4	4	no	—
91	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
92	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
93	C₂×C₄×C₁₉	C₂²⋊C₈	1	2	yes	—
94	C₂×C₄×C₁₉	#99	1	2	yes	—
95	C₂×C₄×C₁₉	#100	1	1	yes	characteristic
96	C₂×C₄×C₁₉	#101	1	1	yes	characteristic
97	C₂×C₄×C₁₉	#104	1	2	yes	—
98	C₂×C₄×C₁₉	#104	1	2	yes	—
99	C₂×C₄×C₁₉	#108	1	2	yes	—
100	C₂×C₄×C₁₉	#108	1	2	yes	—
101	C₈×C₁₉	—	4	4	no	—
102	C₁₉⋊C₈	—	4	4	no	—
103	C₁₉⋊C₈	—	4	4	no	—
104	C₄×C₂²⋊C₈	—	19	1	no	—
105	C₄×C₂²⋊C₈	D₁₉	1	1	yes	characteristic
106	#1728	—	19	1	no	—
107	C₂²×C₁₉	—	2	2	no	—
108	C₂²×C₁₉	—	2	2	no	—
109	C₂²×C₁₉	—	2	2	no	—
110	C₂²×C₁₉	—	4	1	no	—
111	C₂²×C₁₉	—	4	1	no	—
112	C₂²×C₁₉	—	4	1	no	—
113	C₂²×C₁₉	—	4	1	no	—
114	C₂²×C₁₉	#336	1	1	yes	characteristic
115	C₂²×C₁₉	#339	1	1	yes	characteristic
116	C₂²×C₁₉	#339	1	1	yes	characteristic
117	C₂²×C₁₉	#341	1	1	yes	characteristic
118	C₂²×C₁₉	#341	1	1	yes	characteristic
119	C₂²×C₁₉	#342	1	1	yes	characteristic
120	C₂²×C₁₉	#343	1	1	yes	characteristic
121	C₄×C₁₉	—	2	1	no	—
122	C₄×C₁₉	—	2	1	no	—
123	C₄×C₁₉	—	2	2	no	—
124	C₄×C₁₉	—	2	2	no	—
125	C₄×C₁₉	—	2	2	no	—
126	C₄×C₁₉	—	2	2	no	—
127	C₄×C₁₉	#323	1	2	yes	—
128	C₄×C₁₉	#324	1	2	yes	—
129	C₂²×M₁₆	—	19	2	no	—
130	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
131	C₂×C₂²⋊C₈	—	2	1	no	—
132	C₂×C₂²⋊C₈	—	38	1	no	—
133	C₂×C₄×C₈	—	2	1	no	—
134	C₂×C₄×C₈	—	38	1	no	—
135	#336	—	19	2	no	—
136	#336	—	38	1	no	—
137	#336	—	38	1	no	—
138	#336	Dic₁₉	1	2	yes	—
139	C₂×C₁₉	—	4	2	no	—
140	C₂×C₁₉	#1278	1	1	yes	characteristic
141	C₂×C₁₉	#1278	1	1	yes	characteristic
142	C₂×C₁₉	#1280	1	1	yes	characteristic
143	C₂×C₁₉	#1297	1	1	yes	characteristic
144	C₂×C₁₉	#1298	1	1	yes	characteristic
145	C₂×C₁₉	#1299	1	1	yes	characteristic
146	C₂×C₁₉	#1299	1	1	yes	characteristic
147	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
148	C₂³×C₄	C₄×D₁₉	1	1	yes	characteristic
149	C₂³×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
150	C₂×M₁₆	—	19	4	no	—
151	C₂×M₁₆	—	38	2	no	—
152	C₂×M₁₆	—	76	2	no	—
153	C₂×M₁₆	—	76	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	38	2	no	—
156	C₂²×C₈	—	38	2	no	—
157	C₂×C₄²	—	4	2	no	—
158	C₂×C₄²	C₄×D₁₉	1	1	yes	characteristic
159	C₂×C₄²	D₇₆	1	1	yes	characteristic
160	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
161	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
162	C₂²⋊C₈	—	2	4	no	—
163	C₂²⋊C₈	—	38	4	no	—
164	C₄×C₈	—	4	2	no	—
165	C₄×C₈	—	76	2	no	—
166	C₁₉	#7026	1	1	yes	characteristic, complemented, Sylow
167	C₂⁴	Dic₁₉⋊C₄	1	1	yes	characteristic
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	—	4	1	no	—
177	C₂²×C₄	C₈×D₁₉	1	2	yes	—
178	C₂²×C₄	#64515	1	2	yes	—
179	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
180	C₂²×C₄	Dic₁₉⋊C₄	1	1	yes	characteristic
181	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
182	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
183	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊Dic₁₉	1	1	yes	characteristic
185	M₁₆	—	76	4	no	—
186	M₁₆	—	76	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	38	4	no	—
191	C₂×C₈	—	38	4	no	—
192	C₂×C₈	—	76	2	no	—
193	C₂×C₈	—	76	2	no	—
194	C₂×C₈	—	76	2	no	—
195	C₂×C₈	—	76	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	4	1	no	—
199	C₄²	—	4	1	no	—
200	C₂³	—	2	1	no	—
201	C₂³	—	2	1	no	—
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	1	no	—
206	C₂³	Dic₁₉⋊C₈	1	2	yes	—
207	C₂³	C₁₉⋊#95	1	1	yes	characteristic
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	#10597262	1	2	yes	—
225	C₂×C₄	#10597263	1	2	yes	—
226	C₂×C₄	C₈×Dic₁₉	1	1	yes	characteristic
227	C₂×C₄	#10597272	1	1	yes	characteristic
228	C₂×C₄	#10597276	1	2	yes	—
229	C₂×C₄	(C₂×D₁₉)⋊C₈	1	2	yes	—
230	C₂×C₄	#10597279	1	2	yes	—
231	C₂×C₄	#10597280	1	1	yes	characteristic
232	C₂×C₄	#10597281	1	1	yes	characteristic
233	C₂×C₄	#10597294	1	2	yes	—
234	C₈	—	4	4	no	—
235	C₈	—	76	4	no	—
236	C₈	—	76	4	no	—
237	C₂²	—	2	2	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	#11926298	1	1	yes	characteristic
245	C₂²	#11926303	1	1	yes	characteristic
246	C₂²	#11926304	1	1	yes	characteristic
247	C₂²	#11926308	1	1	yes	characteristic
248	C₂²	#11926309	1	1	yes	characteristic
249	C₂²	#11926310	1	1	yes	characteristic
250	C₂²	#11926312	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	#11926267	1	2	yes	—
258	C₄	#11926269	1	2	yes	—
259	C₂	—	4	2	no	—
260	C₂	#14561471	1	1	yes	characteristic
261	C₂	#14561472	1	1	yes	characteristic
262	C₂	#14561477	1	1	yes	characteristic
263	C₂	#14561521	1	1	yes	characteristic
264	C₂	#14561523	1	1	yes	characteristic
265	C₂	#14561525	1	1	yes	characteristic
266	C₂	#14561526	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	(C₂×D₁₉)⋊C₄
commutator	C₂×C₄×C₁₉	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₁₉
Fitting	C₄×C₁₉×C₂²⋊C₈	C₂
socle	C₂³×C₁₉	#95
prime radical	C₂⁴×C₁₉	C₄⋊C₄
cosocle	C₂²×C₄²×C₁₉	C₂²
hypercenter	C₄×C₂²⋊C₈	D₁₉
hypocenter	C₁₉	#7026
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₄×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
26. C₂×#64520	C₈	semidirect
166. C₁₉	#7026	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₂²⋊(C₁₉⋊C₈)	2
#14563329	2
C₄×C₁₉×C₂²⋊C₈	2
#7026	19