Subgroups — #18081081

Order 4864.

398 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564353	C₂	1	2	yes	maximal
2	C₄×#10597275	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926297	C₂	1	1	yes	maximal, characteristic
4	#14561475	C₂	1	2	yes	maximal
5	C₂×C₈²×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597275	C₄	1	1	yes	characteristic
7	C₂×#10597275	C₄	1	1	yes	characteristic
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×Dic₁₉⋊C₈	—	2	2	no	—
10	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
11	#11926307	C₄	1	2	yes	—
12	#11926298	—	2	2	no	—
13	#11926297	—	2	2	no	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₈²×C₁₉	—	2	2	no	—
18	C₂²×C₈×C₁₉	—	2	1	no	—
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
24	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
25	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
26	C₂×C₄×Dic₁₉	—	2	2	no	—
27	C₂²×C₁₉⋊C₈	—	2	2	no	—
28	C₄×C₈×C₁₉	—	2	1	no	—
29	C₄×C₈×C₁₉	—	2	1	no	—
30	C₄×C₈×C₁₉	—	2	1	no	—
31	C₄×C₈×C₁₉	D₄	1	2	yes	—
32	C₄×C₈×C₁₉	Q₈	1	2	yes	—
33	C₄×C₈×C₁₉	Q₈	1	2	yes	—
34	C₁₉⋊#95	—	2	2	no	—
35	#10597275	—	2	2	no	—
36	#10597275	C₈	1	4	yes	complemented
37	Dic₁₉⋊C₈	—	4	4	no	—
38	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
39	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
40	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
43	C₂²×Dic₁₉	—	2	2	no	—
44	C₂²×Dic₁₉	—	2	2	no	—
45	C₂×C₈×C₁₉	—	2	2	no	—
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₂×C₈×C₁₉	—	2	1	no	—
48	C₂×C₈×C₁₉	—	2	1	no	—
49	C₂×C₈×C₁₉	—	2	2	no	—
50	C₂×C₈×C₁₉	—	2	1	no	—
51	C₂×C₈×C₁₉	—	2	1	no	—
52	C₂×C₈×C₁₉	—	2	1	no	—
53	C₂×C₈×C₁₉	—	2	1	no	—
54	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₁₉	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₁₉	M₁₆	1	2	yes	—
58	C₂×C₈×C₁₉	D₈	1	2	yes	—
59	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
60	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₁₉	#41	1	1	yes	characteristic
64	C₄⋊Dic₁₉	—	2	2	no	—
65	C₄⋊Dic₁₉	—	2	2	no	—
66	C₄⋊Dic₁₉	—	2	2	no	—
67	C₄⋊Dic₁₉	C₂×C₈	1	4	yes	—
68	C₄×Dic₁₉	—	4	2	no	—
69	C₄×Dic₁₉	—	4	2	no	—
70	C₂×C₁₉⋊C₈	—	4	2	no	—
71	C₂×C₁₉⋊C₈	—	4	2	no	—
72	C₂×C₁₉⋊C₈	—	4	2	no	—
73	#9594	—	19	1	no	maximal, Sylow
74	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₁₉	—	2	1	no	—
76	C₂×C₄×C₁₉	—	2	1	no	—
77	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
79	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
80	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
82	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₁₉	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₄×C₁₉	C₂×M₁₆	1	1	yes	characteristic
85	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
86	C₂×C₄×C₁₉	#131	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
90	C₂×C₄×C₁₉	#135	1	1	yes	characteristic
91	C₂×Dic₁₉	—	2	4	no	—
92	C₂×Dic₁₉	—	4	2	no	—
93	C₂×Dic₁₉	—	4	2	no	—
94	C₂×Dic₁₉	—	4	2	no	—
95	C₂×Dic₁₉	—	4	2	no	—
96	C₂×Dic₁₉	—	4	2	no	—
97	C₈×C₁₉	—	2	4	no	—
98	C₈×C₁₉	—	2	2	no	—
99	C₈×C₁₉	—	2	2	no	—
100	C₈×C₁₉	—	2	2	no	—
101	C₈×C₁₉	C₄⋊C₈	1	4	yes	complemented
102	C₁₉⋊C₈	—	8	2	no	—
103	#1939	—	19	2	no	—
104	C₄×#107	—	19	1	no	—
105	C₂×#335	—	19	1	no	—
106	C₂×C₈²	D₁₉	1	1	yes	characteristic
107	#1279	—	19	2	no	—
108	C₂²×C₁₉	#389	1	1	yes	characteristic
109	C₂²×C₁₉	C₈×D₄	1	1	yes	characteristic
110	C₂²×C₁₉	#436	1	1	yes	characteristic
111	C₂²×C₁₉	C₄×D₈	1	1	yes	characteristic
112	C₂²×C₁₉	C₄×Q₁₆	1	1	yes	characteristic
113	C₂²×C₁₉	#443	1	1	yes	characteristic
114	C₂²×C₁₉	#443	1	1	yes	characteristic
115	C₄×C₁₉	—	2	1	no	—
116	C₄×C₁₉	—	2	1	no	—
117	C₄×C₁₉	C₂×C₄⋊C₈	1	1	yes	characteristic
118	C₄×C₁₉	#424	1	1	yes	characteristic
119	C₄×C₁₉	C₈×Q₈	1	1	yes	characteristic
120	C₄×C₁₉	#446	1	1	yes	characteristic
121	C₄×C₁₉	#466	1	1	yes	characteristic
122	C₄×C₁₉	#467	1	1	yes	characteristic
123	C₄×C₁₉	#499	1	1	yes	characteristic
124	C₄×C₁₉	#500	1	1	yes	characteristic
125	Dic₁₉	—	4	4	no	—
126	Dic₁₉	—	8	2	no	—
127	C₂×#107	—	19	1	no	—
128	C₂×#107	—	19	1	no	—
129	C₂×C₄⋊C₈	—	19	2	no	—
130	C₂×C₄⋊C₈	—	38	2	no	—
131	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
134	C₄×C₄⋊C₄	—	19	2	no	—
135	#340	—	19	2	no	—
136	#336	—	38	2	no	—
137	#335	—	38	2	no	—
138	C₈²	—	2	2	no	—
139	C₂×C₁₉	C₈×D₈	1	1	yes	characteristic
140	C₂×C₁₉	C₈×Q₁₆	1	1	yes	characteristic
141	C₂×C₁₉	#1589	1	1	yes	characteristic
142	C₂×C₁₉	#1591	1	1	yes	characteristic
143	C₂×C₁₉	#1939	1	1	yes	characteristic
144	C₂×C₁₉	#1942	1	1	yes	characteristic
145	C₂×C₁₉	#1945	1	1	yes	characteristic
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	38	2	no	—
148	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
149	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
150	C₂²×C₈	D₇₆	1	1	yes	characteristic
151	C₂²×C₈	D₇₆	1	1	yes	characteristic
152	C₂×C₄⋊C₄	—	19	2	no	—
153	C₂×C₄⋊C₄	—	19	2	no	—
154	C₂×C₄²	—	38	2	no	—
155	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
156	#107	—	19	4	no	—
157	#107	—	38	2	no	—
158	C₄⋊C₈	—	38	2	no	—
159	C₄⋊C₈	—	38	2	no	—
160	C₄⋊C₈	—	76	4	no	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	Dic₃₈	1	2	yes	—
165	C₄×C₈	Dic₃₈	1	2	yes	—
166	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
167	#95	—	38	2	no	—
168	C₁₉	#9594	1	1	yes	characteristic, complemented, Sylow
169	C₂²×C₄	—	38	2	no	—
170	C₂²×C₄	—	38	2	no	—
171	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
173	C₂²×C₄	#64541	1	1	yes	characteristic
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	76	2	no	—
184	C₂×C₈	—	76	2	no	—
185	C₂×C₈	—	76	2	no	—
186	C₂×C₈	C₈×D₁₉	1	2	yes	—
187	C₂×C₈	#64515	1	2	yes	—
188	C₂×C₈	D₁₅₂	1	2	yes	—
189	C₂×C₈	Dic₇₆	1	2	yes	—
190	C₂×C₈	Dic₁₉⋊C₄	1	2	yes	—
191	C₂×C₈	Dic₁₉⋊C₄	1	2	yes	—
192	C₄⋊C₄	—	19	4	no	—
193	C₄⋊C₄	—	38	2	no	—
194	C₄⋊C₄	—	38	2	no	—
195	C₄⋊C₄	—	38	2	no	—
196	C₄²	—	76	2	no	—
197	C₄²	—	76	2	no	—
198	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
199	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
200	C₄²	#64541	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
202	C₂³	C₄×D₇₆	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	38	4	no	—
206	C₂×C₄	—	76	2	no	—
207	C₂×C₄	—	76	2	no	—
208	C₂×C₄	—	76	2	no	—
209	C₂×C₄	—	76	2	no	—
210	C₂×C₄	—	76	2	no	—
211	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
212	C₂×C₄	#10597315	1	1	yes	characteristic
213	C₂×C₄	#10597316	1	1	yes	characteristic
214	C₂×C₄	C₂×C₈×D₁₉	1	1	yes	characteristic
215	C₂×C₄	C₂×#64515	1	1	yes	characteristic
216	C₂×C₄	#10597347	1	1	yes	characteristic
217	C₂×C₄	#10597347	1	1	yes	characteristic
218	C₂×C₄	C₂×D₁₅₂	1	1	yes	characteristic
219	C₂×C₄	#10597350	1	1	yes	characteristic
220	C₂×C₄	#10597350	1	1	yes	characteristic
221	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
222	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
223	C₂×C₄	#10597370	1	1	yes	characteristic
224	C₂×C₄	#10597376	1	1	yes	characteristic
225	C₈	—	2	4	no	—
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	152	2	no	—
230	C₈	Dic₁₉⋊C₈	1	4	yes	complemented
231	C₂²	#11926442	1	1	yes	characteristic
232	C₂²	C₈×D₇₆	1	1	yes	characteristic
233	C₂²	#11926623	1	1	yes	characteristic
234	C₂²	C₄×D₁₅₂	1	1	yes	characteristic
235	C₂²	C₄×Dic₇₆	1	1	yes	characteristic
236	C₂²	#11926662	1	1	yes	characteristic
237	C₂²	#11926662	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	76	4	no	—
241	C₄	—	152	2	no	—
242	C₄	C₂×Dic₁₉⋊C₈	1	1	yes	characteristic
243	C₄	#11926570	1	1	yes	characteristic
244	C₄	C₈×Dic₃₈	1	1	yes	characteristic
245	C₄	#11926674	1	1	yes	characteristic
246	C₄	#11926780	1	1	yes	characteristic
247	C₄	#11926785	1	1	yes	characteristic
248	C₄	#11926946	1	1	yes	characteristic
249	C₄	#11926949	1	1	yes	characteristic
250	C₂	C₈×D₁₅₂	1	1	yes	characteristic
251	C₂	C₈×Dic₇₆	1	1	yes	characteristic
252	C₂	#14562521	1	1	yes	characteristic
253	C₂	#14562533	1	1	yes	characteristic
254	C₂	#14564353	1	1	yes	characteristic
255	C₂	#14564368	1	1	yes	characteristic
256	C₂	#14564383	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₈²×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₈²	D₁₉
hypocenter	C₁₉	#9594
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
36. #10597275	C₈	semidirect
101. C₈×C₁₉	C₄⋊C₈	semidirect
168. C₁₉	#9594	semidirect
230. C₈	Dic₁₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#14564353	2
C₄×#10597275	2
C₂×#11926297	2
#14561475	2
C₂×C₈²×C₁₉	2
#9594	19