Subgroups — #18088271

Order 4864.

854 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	C₂×#11926307	C₂	1	2	yes	maximal
2	C₂×#11926307	C₂	1	2	yes	maximal
3	#14561865	C₂	1	2	yes	maximal
4	C₂²×C₁₉×#97	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₁₉	C₂²	1	4	yes	—
6	C₂×C₁₉⋊#95	—	2	4	no	—
7	#11926307	—	2	4	no	—
8	#11926307	C₄	1	8	yes	complemented
9	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
10	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
12	C₂×C₁₉×#97	—	2	4	no	—
13	C₂×C₁₉×#97	C₄	1	4	yes	complemented
14	C₂³×C₄×C₁₉	D₄	1	2	yes	—
15	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
16	C₂³×Dic₁₉	—	2	4	no	—
17	C₂²×C₈×C₁₉	—	2	2	no	—
18	C₂²×C₈×C₁₉	—	2	2	no	—
19	C₂²×C₈×C₁₉	—	2	2	no	—
20	C₂²×C₈×C₁₉	—	2	2	no	—
21	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₉	D₄	1	4	yes	—
25	C₂×C₄²×C₁₉	—	2	2	no	—
26	C₂×C₄²×C₁₉	—	2	2	no	—
27	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₁₉	Q₈	1	2	yes	—
29	C₂×C₄⋊Dic₁₉	—	2	4	no	—
30	C₂×C₄⋊Dic₁₉	—	2	4	no	—
31	C₂×C₄⋊Dic₁₉	—	2	4	no	—
32	C₂×C₄⋊Dic₁₉	—	2	4	no	—
33	C₂×C₄⋊Dic₁₉	—	2	4	no	—
34	C₂×C₄⋊Dic₁₉	C₂×C₄	1	8	yes	—
35	C₁₉×#97	—	2	8	no	—
36	C₁₉⋊#95	—	4	8	no	—
37	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₉	—	2	2	no	—
39	C₂²×C₄×C₁₉	—	2	2	no	—
40	C₂²×C₄×C₁₉	—	2	4	no	—
41	C₂²×C₄×C₁₉	—	2	4	no	—
42	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	#41	1	2	yes	—
48	C₂²×C₄×C₁₉	#41	1	2	yes	—
49	C₂²×C₄×C₁₉	#41	1	2	yes	—
50	C₂²×C₄×C₁₉	#41	1	2	yes	—
51	C₂²×Dic₁₉	—	2	8	no	—
52	C₂²×Dic₁₉	—	4	4	no	—
53	C₂²×Dic₁₉	—	4	4	no	—
54	C₂²×Dic₁₉	—	4	4	no	—
55	C₂²×Dic₁₉	—	4	4	no	—
56	C₂²×Dic₁₉	—	4	4	no	—
57	C₂²×Dic₁₉	—	4	4	no	—
58	C₂×C₈×C₁₉	—	2	2	no	—
59	C₂×C₈×C₁₉	—	2	2	no	—
60	C₂×C₈×C₁₉	—	2	2	no	—
61	C₂×C₈×C₁₉	—	2	2	no	—
62	C₂×C₈×C₁₉	—	2	8	no	—
63	C₂×C₈×C₁₉	—	2	8	no	—
64	C₂×C₈×C₁₉	C₄²	1	4	yes	—
65	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
66	C₄²×C₁₉	—	2	2	no	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	—	2	4	no	—
69	C₄⋊Dic₁₉	—	2	8	no	—
70	C₄⋊Dic₁₉	—	2	8	no	—
71	C₄⋊Dic₁₉	—	4	4	no	—
72	C₄⋊Dic₁₉	—	4	4	no	—
73	#10821	—	19	1	no	maximal, Sylow
74	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂³×C₁₉	C₄×D₄	1	2	yes	—
76	C₂³×C₁₉	#121	1	2	yes	—
77	C₂³×C₁₉	#124	1	2	yes	—
78	C₂³×C₁₉	AGL(1,ℤ/8)	1	4	yes	—
79	C₂³×C₁₉	#137	1	4	yes	—
80	C₂×C₄×C₁₉	—	2	4	no	—
81	C₂×C₄×C₁₉	—	2	8	no	—
82	C₂×C₄×C₁₉	—	4	4	no	—
83	C₂×C₄×C₁₉	—	4	2	no	—
84	C₂×C₄×C₁₉	—	4	2	no	—
85	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₉	#117	1	4	yes	—
88	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
89	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
90	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
91	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
92	C₂×C₄×C₁₉	C₄×D₄	1	4	yes	—
93	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
94	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
95	C₂×C₄×C₁₉	#123	1	4	yes	—
96	C₂×C₄×C₁₉	#125	1	2	yes	—
97	C₂×C₄×C₁₉	#126	1	4	yes	—
98	C₂×Dic₁₉	—	4	8	no	—
99	C₂×Dic₁₉	—	4	8	no	—
100	C₂×Dic₁₉	—	4	8	no	—
101	C₂×Dic₁₉	—	8	4	no	—
102	C₂×Dic₁₉	—	8	4	no	—
103	C₂×Dic₁₉	—	8	4	no	—
104	C₂×Dic₁₉	—	8	4	no	—
105	C₈×C₁₉	—	2	4	no	—
106	C₈×C₁₉	—	2	4	no	—
107	C₂²×#97	D₁₉	1	1	yes	characteristic
108	C₂×#340	—	19	2	no	—
109	C₂×#340	—	19	2	no	—
110	C₄⋊#95	—	19	2	no	—
111	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₁₉	#380	1	1	yes	characteristic
113	C₂²×C₁₉	#382	1	1	yes	characteristic
114	C₂²×C₁₉	#388	1	4	yes	—
115	C₂²×C₁₉	#428	1	4	yes	—
116	C₂²×C₁₉	#440	1	4	yes	—
117	C₂²×C₁₉	#441	1	2	yes	—
118	C₂²×C₁₉	#442	1	2	yes	—
119	C₂²×C₁₉	#468	1	4	yes	—
120	C₂²×C₁₉	#469	1	2	yes	—
121	C₂²×C₁₉	#470	1	2	yes	—
122	C₂²×C₁₉	#489	1	4	yes	—
123	C₂²×C₁₉	#490	1	2	yes	—
124	C₂²×C₁₉	#491	1	2	yes	—
125	C₄×C₁₉	—	4	4	no	—
126	C₄×C₁₉	#376	1	1	yes	characteristic
127	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
128	C₄×C₁₉	#381	1	1	yes	characteristic
129	C₄×C₁₉	#383	1	1	yes	characteristic
130	C₄×C₁₉	#387	1	4	yes	—
131	Dic₁₉	—	8	8	no	—
132	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
133	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
134	C₂²×C₄⋊C₄	—	19	4	no	—
135	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
136	C₂×#97	—	2	4	no	—
137	C₂×#97	Dic₁₉	1	4	yes	complemented
138	C₂×#95	—	38	4	no	—
139	#340	—	19	8	no	—
140	#340	—	38	4	no	—
141	C₂×C₁₉	#1434	1	1	yes	characteristic
142	C₂×C₁₉	#1776	1	2	yes	—
143	C₂×C₁₉	#1809	1	2	yes	—
144	C₂×C₁₉	#1836	1	2	yes	—
145	C₂×C₁₉	#1935	1	4	yes	—
146	C₂×C₁₉	#1936	1	4	yes	—
147	C₂³×C₄	—	38	4	no	—
148	C₂³×C₄	D₇₆	1	2	yes	—
149	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
150	C₂²×C₈	—	2	2	no	—
151	C₂²×C₈	—	2	2	no	—
152	C₂²×C₈	—	2	2	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	C₄×D₁₉	1	4	yes	—
155	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
156	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
157	C₂²×C₈	C₂²⋊D₁₉	1	4	yes	—
158	C₂×C₄⋊C₄	—	19	8	no	—
159	C₂×C₄⋊C₄	—	38	4	no	—
160	C₂×C₄⋊C₄	—	38	4	no	—
161	C₂×C₄⋊C₄	—	38	4	no	—
162	C₂×C₄⋊C₄	—	38	4	no	—
163	C₂×C₄⋊C₄	—	38	4	no	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	Dic₃₈	1	2	yes	—
167	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
168	#97	—	2	8	no	—
169	#95	—	76	8	no	—
170	C₁₉	#10821	1	1	yes	characteristic, complemented, Sylow
171	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	38	8	no	—
177	C₂²×C₄	—	76	4	no	—
178	C₂²×C₄	—	76	4	no	—
179	C₂²×C₄	—	76	4	no	—
180	C₂²×C₄	—	76	4	no	—
181	C₂²×C₄	—	76	4	no	—
182	C₂²×C₄	—	76	4	no	—
183	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
184	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
186	C₂²×C₄	#64541	1	2	yes	—
187	C₂²×C₄	#64541	1	2	yes	—
188	C₂²×C₄	#64541	1	2	yes	—
189	C₂²×C₄	#64541	1	2	yes	—
190	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	—	2	8	no	—
198	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
199	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
200	C₄⋊C₄	—	38	8	no	—
201	C₄⋊C₄	—	38	8	no	—
202	C₄⋊C₄	—	76	4	no	—
203	C₄⋊C₄	—	76	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	4	no	—
206	C₄²	—	2	2	no	—
207	C₂³	C₄×D₇₆	1	2	yes	—
208	C₂³	#10597321	1	2	yes	—
209	C₂³	#10597354	1	4	yes	—
210	C₂³	#10597355	1	4	yes	—
211	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
212	C₂³	#10597376	1	2	yes	—
213	C₂×C₄	—	2	8	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	76	8	no	—
219	C₂×C₄	—	76	8	no	—
220	C₂×C₄	—	76	8	no	—
221	C₂×C₄	—	152	4	no	—
222	C₂×C₄	—	152	4	no	—
223	C₂×C₄	—	152	4	no	—
224	C₂×C₄	—	152	4	no	—
225	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
226	C₂×C₄	#10597316	1	2	yes	—
227	C₂×C₄	#10597318	1	4	yes	—
228	C₂×C₄	#10597322	1	4	yes	—
229	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
230	C₂×C₄	#10597370	1	2	yes	—
231	C₂×C₄	#10597372	1	1	yes	characteristic
232	C₂×C₄	#10597372	1	1	yes	characteristic
233	C₂×C₄	#10597372	1	1	yes	characteristic
234	C₂×C₄	#10597372	1	1	yes	characteristic
235	C₂×C₄	C₄×C₂²⋊D₁₉	1	4	yes	—
236	C₂×C₄	#10597375	1	4	yes	—
237	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
238	C₈	—	2	4	no	—
239	C₈	—	2	4	no	—
240	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
241	C₂²	#11926404	1	1	yes	characteristic
242	C₂²	#11926410	1	1	yes	characteristic
243	C₂²	#11926435	1	4	yes	—
244	C₂²	#11926586	1	4	yes	—
245	C₂²	#11926646	1	4	yes	—
246	C₂²	#11926652	1	2	yes	—
247	C₂²	#11926657	1	2	yes	—
248	C₂²	#11926790	1	4	yes	—
249	C₂²	#11926797	1	2	yes	—
250	C₂²	#11926802	1	2	yes	—
251	C₂²	#11926904	1	4	yes	—
252	C₂²	#11926909	1	2	yes	—
253	C₂²	#11926913	1	2	yes	—
254	C₄	—	4	4	no	—
255	C₄	—	152	8	no	—
256	C₄	#11926390	1	1	yes	characteristic
257	C₄	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
258	C₄	#11926407	1	1	yes	characteristic
259	C₄	#11926412	1	1	yes	characteristic
260	C₄	#11926428	1	4	yes	—
261	C₂	#14561862	1	1	yes	characteristic
262	C₂	#14563532	1	2	yes	—
263	C₂	#14563668	1	2	yes	—
264	C₂	#14563769	1	2	yes	—
265	C₂	#14564325	1	4	yes	—
266	C₂	#14564332	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂²×C₁₉×#97	C₂
socle, prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂²×#97	D₁₉
hypocenter	C₁₉	#10821
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
8. #11926307	C₄	semidirect
13. C₂×C₁₉×#97	C₄	semidirect
137. C₂×#97	Dic₁₉	semidirect
170. C₁₉	#10821	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926307	2
#14561865	2
C₂²×C₁₉×#97	2
#10821	19