Subgroups — #18069090

Order 4864.

622 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926299	C₂	1	2	yes	maximal
2	C₁₉×C₄⋊#95	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₁₉⋊C₈	—	2	2	no	—
4	#11926299	—	2	2	no	—
5	#11926299	C₄	1	4	yes	complemented
6	C₂²×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
7	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₁₉×#95	—	2	2	no	—
9	C₂³×C₄×C₁₉	—	2	2	no	—
10	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
11	C₂³×C₄×C₁₉	D₄	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₉	—	4	2	no	—
14	C₂×C₄⋊C₄×C₁₉	—	4	2	no	—
15	C₂×C₄⋊C₄×C₁₉	C₈	1	4	yes	complemented
16	C₂×C₄²×C₁₉	—	4	2	no	—
17	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
19	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
20	C₂²×C₁₉⋊C₈	—	2	8	no	—
21	C₂²×C₁₉⋊C₈	—	2	4	no	—
22	C₂²×C₁₉⋊C₈	—	4	4	no	—
23	C₂²×C₁₉⋊C₈	—	4	2	no	—
24	C₂²×C₁₉⋊C₈	—	4	2	no	—
25	C₁₉×#95	—	4	2	no	—
26	C₁₉×#95	—	4	2	no	—
27	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉	—	2	2	no	—
29	C₂²×C₄×C₁₉	—	2	4	no	—
30	C₂²×C₄×C₁₉	—	2	1	no	—
31	C₂²×C₄×C₁₉	—	2	1	no	—
32	C₂²×C₄×C₁₉	—	2	4	no	—
33	C₂²×C₄×C₁₉	—	2	4	no	—
34	C₂²×C₄×C₁₉	—	4	2	no	—
35	C₂²×C₄×C₁₉	—	4	2	no	—
36	C₂²×C₄×C₁₉	—	4	4	no	—
37	C₂²×C₄×C₁₉	—	4	4	no	—
38	C₂²×C₄×C₁₉	C₄²	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₉	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂×C₈	1	2	yes	—
43	C₂²×C₄×C₁₉	M₁₆	1	2	yes	—
44	C₂²×C₄×C₁₉	D₈	1	2	yes	—
45	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
46	C₂²×C₄×C₁₉	SD₁₆	1	2	yes	—
47	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
48	C₄⋊C₄×C₁₉	—	4	4	no	—
49	C₄⋊C₄×C₁₉	—	4	4	no	—
50	C₄²×C₁₉	—	4	2	no	—
51	C₄²×C₁₉	—	4	1	no	—
52	C₄²×C₁₉	—	4	1	no	—
53	C₂×C₁₉⋊C₈	—	2	8	no	—
54	C₂×C₁₉⋊C₈	—	4	8	no	—
55	C₂×C₁₉⋊C₈	—	4	8	no	—
56	C₂×C₁₉⋊C₈	—	4	4	no	—
57	C₂×C₁₉⋊C₈	—	4	4	no	—
58	#7025	—	19	1	no	maximal, Sylow
59	C₂³×C₁₉	#95	1	1	yes	characteristic
60	C₂³×C₁₉	C₂²⋊C₈	1	2	yes	—
61	C₂³×C₁₉	#99	1	2	yes	—
62	C₂³×C₁₉	#100	1	1	yes	characteristic
63	C₂³×C₁₉	#101	1	1	yes	characteristic
64	C₂³×C₁₉	#102	1	2	yes	—
65	C₂³×C₁₉	#103	1	2	yes	—
66	C₂³×C₁₉	#104	1	4	yes	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	2	no	—
71	C₂×C₄×C₁₉	—	2	4	no	—
72	C₂×C₄×C₁₉	—	2	4	no	—
73	C₂×C₄×C₁₉	—	2	4	no	—
74	C₂×C₄×C₁₉	—	2	4	no	—
75	C₂×C₄×C₁₉	—	2	2	no	—
76	C₂×C₄×C₁₉	—	2	2	no	—
77	C₂×C₄×C₁₉	—	2	2	no	—
78	C₂×C₄×C₁₉	—	2	1	no	—
79	C₂×C₄×C₁₉	—	2	2	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	—	4	4	no	—
84	C₂×C₄×C₁₉	—	4	8	no	—
85	C₂×C₄×C₁₉	—	8	4	no	—
86	C₂×C₄×C₁₉	—	8	2	no	—
87	C₂×C₄×C₁₉	—	8	2	no	—
88	C₂×C₄×C₁₉	C₄×C₈	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#97	1	1	yes	characteristic
90	C₂×C₄×C₁₉	#102	1	2	yes	—
91	C₂×C₄×C₁₉	#103	1	2	yes	—
92	C₂×C₄×C₁₉	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₁₉	#106	1	2	yes	—
94	C₂×C₄×C₁₉	#107	1	2	yes	—
95	C₁₉⋊C₈	—	4	8	no	—
96	C₂×#336	—	19	2	no	—
97	C₄⋊#95	D₁₉	1	1	yes	characteristic
98	C₂²×C₁₉	#323	1	2	yes	—
99	C₂²×C₁₉	#324	1	2	yes	—
100	C₂²×C₁₉	D₄⋊C₈	1	4	yes	—
101	C₂²×C₁₉	Q₈⋊C₈	1	4	yes	—
102	C₂²×C₁₉	#327	1	4	yes	—
103	C₂²×C₁₉	#328	1	4	yes	—
104	C₂²×C₁₉	#329	1	2	yes	—
105	C₂²×C₁₉	#330	1	2	yes	—
106	C₂²×C₁₉	#331	1	1	yes	characteristic
107	C₂²×C₁₉	#332	1	2	yes	—
108	C₂²×C₁₉	#333	1	1	yes	characteristic
109	C₂²×C₁₉	#336	1	1	yes	characteristic
110	C₂²×C₁₉	#339	1	2	yes	—
111	C₂²×C₁₉	#340	1	2	yes	—
112	C₂²×C₁₉	#342	1	1	yes	characteristic
113	C₂²×C₁₉	#343	1	1	yes	characteristic
114	C₄×C₁₉	—	2	1	no	—
115	C₄×C₁₉	—	2	1	no	—
116	C₄×C₁₉	—	2	2	no	—
117	C₄×C₁₉	—	2	2	no	—
118	C₄×C₁₉	—	2	2	no	—
119	C₄×C₁₉	—	2	2	no	—
120	C₄×C₁₉	—	2	2	no	—
121	C₄×C₁₉	—	8	4	no	—
122	C₂³×C₈	—	38	2	no	—
123	C₂²×C₄⋊C₄	Dic₁₉	1	2	yes	—
124	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
125	C₂×#95	—	2	2	no	—
126	#336	—	19	4	no	—
127	#336	—	38	2	no	—
128	C₂×C₁₉	#1270	1	2	yes	—
129	C₂×C₁₉	#1271	1	4	yes	—
130	C₂×C₁₉	#1272	1	2	yes	—
131	C₂×C₁₉	#1279	1	2	yes	—
132	C₂×C₁₉	#1280	1	1	yes	characteristic
133	C₂×C₁₉	#1294	1	2	yes	—
134	C₂×C₁₉	#1295	1	1	yes	characteristic
135	C₂×C₁₉	#1296	1	1	yes	characteristic
136	C₂³×C₄	—	2	2	no	—
137	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
138	C₂³×C₄	C₂²⋊D₁₉	1	2	yes	—
139	C₂²×C₈	—	38	8	no	—
140	C₂²×C₈	—	38	4	no	—
141	C₂²×C₈	—	76	4	no	—
142	C₂²×C₈	—	76	2	no	—
143	C₂²×C₈	—	76	2	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	4	2	no	—
146	C₂×C₄⋊C₄	—	4	2	no	—
147	C₂×C₄⋊C₄	C₁₉⋊C₈	1	4	yes	complemented
148	C₂×C₄²	—	4	2	no	—
149	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
150	C₂×C₄²	C₄×D₁₉	1	2	yes	—
151	C₂×C₄²	D₇₆	1	1	yes	characteristic
152	#95	—	4	2	no	—
153	#95	—	4	2	no	—
154	C₁₉	#7025	1	1	yes	characteristic, complemented, Sylow
155	C₂⁴	C₂²⋊Dic₁₉	1	1	yes	characteristic
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	4	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	—	4	4	no	—
166	C₂²×C₄	C₂×C₁₉⋊C₈	1	2	yes	—
167	C₂²×C₄	#64520	1	2	yes	—
168	C₂²×C₄	C₄×Dic₁₉	1	1	yes	characteristic
169	C₂²×C₄	Dic₁₉⋊C₄	1	2	yes	—
170	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
171	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	2	yes	—
172	C₂²×C₄	#64525	1	2	yes	—
173	C₂²×C₄	#64526	1	2	yes	—
174	C₂²×C₄	#64527	1	2	yes	—
175	C₂²×C₄	#64528	1	2	yes	—
176	C₂×C₈	—	38	8	no	—
177	C₂×C₈	—	76	8	no	—
178	C₂×C₈	—	76	8	no	—
179	C₂×C₈	—	76	4	no	—
180	C₂×C₈	—	76	4	no	—
181	C₄⋊C₄	—	4	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄²	—	4	1	no	—
184	C₄²	—	4	2	no	—
185	C₄²	—	4	1	no	—
186	C₂³	C₂²⋊(C₁₉⋊C₈)	1	2	yes	—
187	C₂³	C₁₉⋊#95	1	1	yes	characteristic
188	C₂³	#10597289	1	2	yes	—
189	C₂³	#10597290	1	1	yes	characteristic
190	C₂³	#10597291	1	2	yes	—
191	C₂³	#10597292	1	2	yes	—
192	C₂³	#10597293	1	1	yes	characteristic
193	C₂³	#10597294	1	4	yes	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	4	8	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	C₄×C₁₉⋊C₈	1	1	yes	characteristic
216	C₂×C₄	#10597260	1	1	yes	characteristic
217	C₂×C₄	C₄⋊(C₁₉⋊C₈)	1	2	yes	—
218	C₂×C₄	#10597264	1	2	yes	—
219	C₂×C₄	#10597265	1	2	yes	—
220	C₂×C₄	#10597266	1	2	yes	—
221	C₂×C₄	#10597267	1	2	yes	—
222	C₈	—	76	8	no	—
223	C₂²	#11926268	1	2	yes	—
224	C₂²	#11926270	1	2	yes	—
225	C₂²	D₄⋊(C₁₉⋊C₈)	1	4	yes	—
226	C₂²	Q₈⋊(C₁₉⋊C₈)	1	4	yes	—
227	C₂²	#11926278	1	4	yes	—
228	C₂²	#11926281	1	4	yes	—
229	C₂²	#11926284	1	2	yes	—
230	C₂²	#11926286	1	2	yes	—
231	C₂²	#11926288	1	1	yes	characteristic
232	C₂²	#11926290	1	2	yes	—
233	C₂²	#11926293	1	1	yes	characteristic
234	C₂²	#11926299	1	1	yes	characteristic
235	C₂²	#11926303	1	2	yes	—
236	C₂²	#11926306	1	2	yes	—
237	C₂²	#11926311	1	1	yes	characteristic
238	C₂²	#11926313	1	1	yes	characteristic
239	C₄	—	2	1	no	—
240	C₄	—	2	1	no	—
241	C₄	—	2	2	no	—
242	C₄	—	2	2	no	—
243	C₄	—	2	2	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	8	4	no	—
247	C₂	#14561456	1	2	yes	—
248	C₂	#14561458	1	4	yes	—
249	C₂	#14561461	1	2	yes	—
250	C₂	#14561476	1	2	yes	—
251	C₂	#14561478	1	1	yes	characteristic
252	C₂	#14561513	1	2	yes	—
253	C₂	#14561516	1	1	yes	characteristic
254	C₂	#14561518	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²⋊Dic₁₉
commutator	C₂×C₄×C₁₉	C₄×C₈
Frattini	C₂²×C₄²	C₂×D₁₉
Fitting	C₁₉×C₄⋊#95	C₂
socle, prime radical	C₂⁴×C₁₉	C₂²⋊C₄
cosocle	C₂²×C₄²×C₁₉	C₂²
hypercenter	C₄⋊#95	D₁₉
hypocenter	C₁₉	#7025
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₄⋊#95

Complements

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

N	H ≅ G/N	type
5. #11926299	C₄	semidirect
15. C₂×C₄⋊C₄×C₁₉	C₈	semidirect
147. C₂×C₄⋊C₄	C₁₉⋊C₈	semidirect
154. C₁₉	#7025	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926299	2
C₁₉×C₄⋊#95	2
#7025	19