Subgroups — #18088456

Order 4864.

746 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926582	C₂	1	2	yes	maximal
2	#14563368	C₂	1	2	yes	maximal
3	C₂×#11926301	C₂	1	2	yes	maximal
4	C₂×C₁₉×#405	C₂	1	1	yes	maximal, characteristic
5	C₂×#10597372	C₂²	1	4	yes	—
6	#11926582	—	2	4	no	—
7	#11926582	C₄	1	8	yes	complemented
8	C₂×#10597275	—	2	2	no	—
9	C₂×#10597274	—	2	2	no	—
10	#11926311	—	2	4	no	—
11	#11926301	—	2	4	no	—
12	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₉×#117	C₂²	1	1	yes	characteristic
15	C₁₉×#405	—	2	2	no	—
16	C₁₉×#405	—	2	2	no	—
17	C₂×C₁₉×#97	C₄	1	2	yes	complemented
18	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	complemented
19	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
20	C₂×M₁₆×C₁₉	—	2	2	no	—
21	C₂×M₁₆×C₁₉	—	2	2	no	—
22	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
23	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
24	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₉	D₄	1	2	yes	—
29	C₂²×C₈×C₁₉	D₄	1	2	yes	—
30	C₂²×C₈×C₁₉	D₄	1	2	yes	—
31	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
32	C₁₉×#117	—	2	2	no	—
33	C₁₉×#117	—	2	2	no	—
34	C₂×C₄⋊C₄×C₁₉	Q₈	1	2	yes	—
35	C₂×C₂²⋊C₄×C₁₉	D₄	1	2	yes	—
36	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊Dic₁₉	—	2	4	no	—
38	#10597372	—	2	4	no	—
39	#10597372	—	2	4	no	—
40	#10597372	—	2	4	no	—
41	#10597372	C₂×C₄	1	8	yes	—
42	C₂×C₄⋊Dic₁₉	—	2	4	no	—
43	C₂×C₄×Dic₁₉	—	2	4	no	—
44	C₁₉×#97	—	2	4	no	—
45	C₄×C₈×C₁₉	—	2	2	no	—
46	C₄×C₈×C₁₉	—	2	2	no	—
47	#10597275	—	2	8	no	—
48	#10597274	—	2	8	no	—
49	C₂⁴×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
51	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	#41	1	2	yes	—
60	C₂²×C₄×C₁₉	#41	1	2	yes	—
61	C₂²×Dic₁₉	—	4	4	no	—
62	M₁₆×C₁₉	—	2	4	no	—
63	M₁₆×C₁₉	—	2	4	no	—
64	C₂×C₈×C₁₉	—	2	2	no	—
65	C₂×C₈×C₁₉	—	2	2	no	—
66	C₂×C₈×C₁₉	C₄²	1	2	yes	—
67	C₂×C₈×C₁₉	C₄²	1	2	yes	—
68	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
69	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
70	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
74	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
75	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
76	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
77	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
78	C₄⋊C₄×C₁₉	—	2	4	no	—
79	C₂²⋊C₄×C₁₉	—	2	4	no	—
80	C₄²×C₁₉	—	2	2	no	—
81	C₄²×C₁₉	—	2	2	no	—
82	C₂²⋊Dic₁₉	—	2	8	no	—
83	C₂²⋊Dic₁₉	—	4	4	no	—
84	C₄⋊Dic₁₉	—	2	8	no	—
85	C₄⋊Dic₁₉	—	4	4	no	—
86	C₄×Dic₁₉	—	2	8	no	—
87	C₄×Dic₁₉	—	4	4	no	—
88	#10866	—	19	1	no	maximal, Sylow
89	C₂³×C₁₉	—	4	2	no	—
90	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂³×C₁₉	C₄×D₄	1	2	yes	—
92	C₂³×C₁₉	#121	1	2	yes	—
93	C₂³×C₁₉	#127	1	1	yes	characteristic
94	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₁₉	—	2	4	no	—
96	C₂×C₄×C₁₉	—	4	4	no	—
97	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
100	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
108	C₂×C₄×C₁₉	#121	1	2	yes	—
109	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₁₉	#124	1	2	yes	—
112	C₂×C₄×C₁₉	#125	1	2	yes	—
113	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
115	C₂×Dic₁₉	—	4	8	no	—
116	C₂×Dic₁₉	—	8	8	no	—
117	C₈×C₁₉	—	2	4	no	—
118	C₈×C₁₉	#95	1	2	yes	—
119	C₈×C₁₉	#95	1	4	yes	—
120	C₈×C₁₉	#95	1	2	yes	—
121	C₂×#427	—	19	2	no	—
122	C₂×#405	D₁₉	1	1	yes	characteristic
123	#1740	—	19	2	no	—
124	C₂×H(3,ℤ/4)	—	19	2	no	—
125	C₂²×C₁₉	—	2	4	no	—
126	C₂²×C₁₉	—	2	2	no	—
127	C₂²×C₁₉	—	4	2	no	—
128	C₂²×C₁₉	—	4	1	no	—
129	C₂²×C₁₉	—	4	1	no	—
130	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₁₉	#380	1	1	yes	characteristic
132	C₂²×C₁₉	#384	1	1	yes	characteristic
133	C₂²×C₁₉	#389	1	2	yes	—
134	C₂²×C₁₉	#390	1	2	yes	—
135	C₄×C₁₉	—	4	4	no	—
136	C₄×C₁₉	C₂×#95	1	1	yes	characteristic
137	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₁₉	#380	1	1	yes	characteristic
139	C₄×C₁₉	#382	1	1	yes	characteristic
140	C₄×C₁₉	#389	1	2	yes	—
141	C₄×C₁₉	#391	1	2	yes	—
142	Dic₁₉	—	8	8	no	—
143	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
144	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
145	C₂×#117	—	19	4	no	—
146	C₂×#117	C₂×D₁₉	1	1	yes	characteristic
147	#427	—	19	8	no	—
148	#427	—	38	4	no	—
149	C₂×#107	—	38	2	no	—
150	C₂×#106	—	38	2	no	—
151	#405	—	2	2	no	—
152	#405	—	2	2	no	—
153	C₂×#97	Dic₁₉	1	2	yes	complemented
154	C₂×C₄×C₈	Dic₁₉	1	2	yes	complemented
155	#342	—	38	4	no	—
156	H(3,ℤ/4)	—	38	4	no	—
157	C₂×C₁₉	—	2	2	no	—
158	C₂×C₁₉	—	2	1	no	—
159	C₂×C₁₉	—	2	1	no	—
160	C₂×C₁₉	—	2	2	no	—
161	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
162	C₂×C₁₉	#1812	1	2	yes	—
163	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	Dic₃₈	1	2	yes	—
167	C₂×M₁₆	D₇₆	1	2	yes	—
168	C₂²×C₈	Dic₃₈	1	2	yes	—
169	C₂²×C₈	C₄×D₁₉	1	4	yes	—
170	C₂²×C₈	D₇₆	1	2	yes	—
171	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
172	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
173	C₂²×C₈	C₂×Dic₁₉	1	1	yes	characteristic
174	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
175	C₂²×C₈	C₂²⋊D₁₉	1	2	yes	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	19	8	no	—
179	#117	—	38	4	no	—
180	#117	—	38	4	no	—
181	#117	—	38	4	no	—
182	C₂×C₄⋊C₄	—	38	4	no	—
183	C₂×C₄⋊C₄	Dic₃₈	1	2	yes	—
184	C₂×C₂²⋊C₄	—	38	4	no	—
185	C₂×C₂²⋊C₄	D₇₆	1	2	yes	—
186	C₂×C₄²	—	38	4	no	—
187	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
188	#107	—	38	8	no	—
189	#106	—	38	8	no	—
190	#97	—	2	4	no	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	—	2	2	no	—
193	C₁₉	#10866	1	1	yes	characteristic, complemented, Sylow
194	C₂⁴	C₂×D₇₆	1	1	yes	characteristic
195	C₂²×C₄	—	76	4	no	—
196	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
197	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
198	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
199	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
201	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
203	C₂²×C₄	#64541	1	2	yes	—
204	C₂²×C₄	#64541	1	2	yes	—
205	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
207	M₁₆	—	2	4	no	—
208	M₁₆	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
212	C₂×C₈	C₄×Dic₁₉	1	2	yes	—
213	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
214	C₂×C₈	Dic₁₉⋊C₄	1	4	yes	—
215	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
216	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
217	C₂×C₈	C₄⋊Dic₁₉	1	2	yes	—
218	C₂×C₈	(C₂×D₁₉)⋊C₄	1	4	yes	—
219	C₂×C₈	(C₂×D₁₉)⋊C₄	1	4	yes	—
220	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
221	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
222	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	38	8	no	—
225	C₄⋊C₄	—	76	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	38	8	no	—
228	C₂²⋊C₄	—	76	4	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	2	2	no	—
231	C₄²	—	38	8	no	—
232	C₄²	—	76	4	no	—
233	C₂³	—	4	2	no	—
234	C₂³	#10597315	1	1	yes	characteristic
235	C₂³	C₄×D₇₆	1	2	yes	—
236	C₂³	#10597320	1	1	yes	characteristic
237	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
238	C₂³	#10597376	1	2	yes	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	76	8	no	—
242	C₂×C₄	—	152	8	no	—
243	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
244	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
245	C₂×C₄	#10597315	1	1	yes	characteristic
246	C₂×C₄	#10597315	1	1	yes	characteristic
247	C₂×C₄	#10597316	1	2	yes	—
248	C₂×C₄	C₄×D₇₆	1	2	yes	—
249	C₂×C₄	#10597321	1	2	yes	—
250	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
251	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
252	C₂×C₄	#10597370	1	2	yes	—
253	C₂×C₄	#10597370	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
256	C₂×C₄	#10597372	1	1	yes	characteristic
257	C₂×C₄	#10597372	1	1	yes	characteristic
258	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
259	C₂×C₄	#10597376	1	2	yes	—
260	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
261	C₈	—	2	4	no	—
262	C₈	C₁₉⋊#95	1	2	yes	—
263	C₈	C₁₉⋊#95	1	4	yes	—
264	C₈	C₁₉⋊#95	1	2	yes	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
271	C₂²	#11926404	1	1	yes	characteristic
272	C₂²	#11926414	1	1	yes	characteristic
273	C₂²	#11926442	1	2	yes	—
274	C₂²	#11926446	1	2	yes	—
275	C₄	—	4	4	no	—
276	C₄	—	152	8	no	—
277	C₄	C₂×C₁₉⋊#95	1	1	yes	characteristic
278	C₄	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
279	C₄	#11926404	1	1	yes	characteristic
280	C₄	#11926410	1	1	yes	characteristic
281	C₄	#11926442	1	2	yes	—
282	C₄	#11926450	1	2	yes	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	2	no	—
287	C₂	#14561865	1	1	yes	characteristic
288	C₂	#14563681	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597372
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₁₉×#405	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂⁴×C₁₉	C₂×D₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×#405	D₁₉
hypocenter	C₁₉	#10866
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
7. #11926582	C₄	semidirect
17. C₂×C₁₉×#97	C₄	semidirect
18. C₂×C₄×C₈×C₁₉	C₄	semidirect
153. C₂×#97	Dic₁₉	semidirect
154. C₂×C₄×C₈	Dic₁₉	semidirect
193. C₁₉	#10866	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926582	2
#14563368	2
C₂×#11926301	2
C₂×C₁₉×#405	2
#10866	19