Subgroups — #18088234

Order 4864.

674 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926315	C₂	1	2	yes	maximal
2	C₂×#11926304	C₂	1	2	yes	maximal
3	C₂×C₁₉⋊#338	C₂	1	1	yes	maximal, characteristic
4	C₂×#11926301	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×M₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64520	C₂²	1	2	yes	—
7	C₂×#10597372	C₂²	1	2	yes	—
8	#11926315	—	2	4	no	—
9	#11926304	—	2	4	no	—
10	C₁₉⋊#338	C₄	1	4	yes	complemented
11	#11926301	C₄	1	4	yes	complemented
12	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₁₉	—	2	4	no	—
16	C₂×C₁₉×#97	C₄	1	2	yes	complemented
17	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	complemented
18	C₂³×C₄×C₁₉	—	2	1	no	—
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	—	2	2	no	—
23	C₂×M₁₆×C₁₉	—	2	2	no	—
24	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
25	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
26	C₂×M₁₆×C₁₉	D₄	1	2	yes	—
27	C₂×M₁₆×C₁₉	Q₈	1	2	yes	—
28	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₉	—	2	2	no	—
31	C₂×C₄²×C₁₉	—	2	1	no	—
32	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₉	D₄	1	4	yes	—
35	C₂×C₂²⋊Dic₁₉	—	2	2	no	—
36	#10597372	—	2	2	no	—
37	#10597372	—	2	2	no	—
38	#10597372	—	2	2	no	—
39	#10597372	C₂×C₄	1	4	yes	—
40	C₂×C₄⋊Dic₁₉	—	2	2	no	—
41	C₂×C₄×Dic₁₉	—	2	2	no	—
42	C₂×#64520	—	2	2	no	—
43	C₂×#64520	—	2	2	no	—
44	C₂×#64520	—	2	2	no	—
45	C₂×#64520	—	2	2	no	—
46	C₂×#64520	—	2	2	no	—
47	C₂×#64520	C₂×C₄	1	4	yes	—
48	C₂²×C₁₉⋊C₈	—	2	2	no	—
49	C₁₉×#97	—	2	4	no	—
50	C₄×C₈×C₁₉	—	2	4	no	—
51	C₂⁴×C₁₉	#41	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	—	2	1	no	—
53	C₂²×C₄×C₁₉	—	2	1	no	—
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	—	4	4	no	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	#41	1	2	yes	—
63	C₂²×C₄×C₁₉	#41	1	2	yes	—
64	C₂²×C₄×C₁₉	#41	1	2	yes	—
65	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
66	C₂²×Dic₁₉	—	4	2	no	—
67	M₁₆×C₁₉	—	2	4	no	—
68	M₁₆×C₁₉	—	2	4	no	—
69	M₁₆×C₁₉	—	2	4	no	—
70	M₁₆×C₁₉	—	2	4	no	—
71	C₂×C₈×C₁₉	—	2	2	no	—
72	C₂×C₈×C₁₉	—	2	2	no	—
73	C₂×C₈×C₁₉	—	2	2	no	—
74	C₂×C₈×C₁₉	—	2	2	no	—
75	C₂×C₈×C₁₉	C₄²	1	4	yes	—
76	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
77	C₄²×C₁₉	—	2	1	no	—
78	C₄²×C₁₉	—	2	1	no	—
79	C₄²×C₁₉	—	2	4	no	—
80	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
81	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
82	C₂²⋊Dic₁₉	—	2	4	no	—
83	C₂²⋊Dic₁₉	—	4	2	no	—
84	C₄⋊Dic₁₉	—	2	4	no	—
85	C₄⋊Dic₁₉	—	4	2	no	—
86	C₄×Dic₁₉	—	2	4	no	—
87	C₄×Dic₁₉	—	4	2	no	—
88	#64520	—	2	4	no	—
89	#64520	—	2	4	no	—
90	#64520	—	4	2	no	—
91	#64520	—	4	2	no	—
92	C₂×C₁₉⋊C₈	—	2	4	no	—
93	C₂×C₁₉⋊C₈	—	4	2	no	—
94	C₂×C₁₉⋊C₈	—	4	2	no	—
95	#10813	—	19	1	no	maximal, Sylow
96	C₂³×C₁₉	—	4	2	no	—
97	C₂³×C₁₉	#117	1	1	yes	characteristic
98	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
99	C₂³×C₁₉	C₄×Q₈	1	1	yes	characteristic
100	C₂³×C₁₉	#123	1	2	yes	—
101	C₂³×C₁₉	#126	1	2	yes	—
102	C₂×C₄×C₁₉	—	2	4	no	—
103	C₂×C₄×C₁₉	—	2	2	no	—
104	C₂×C₄×C₁₉	—	2	2	no	—
105	C₂×C₄×C₁₉	—	4	8	no	—
106	C₂×C₄×C₁₉	—	4	2	no	—
107	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₉	#117	1	2	yes	—
111	C₂×C₄×C₁₉	#117	1	2	yes	—
112	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
113	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
114	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
118	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₉	#121	1	2	yes	—
120	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₁₉	#123	1	2	yes	—
122	C₂×C₄×C₁₉	#124	1	2	yes	—
123	C₂×C₄×C₁₉	#125	1	2	yes	—
124	C₂×C₄×C₁₉	#126	1	2	yes	—
125	C₂×Dic₁₉	—	4	4	no	—
126	C₂×Dic₁₉	—	8	4	no	—
127	C₈×C₁₉	—	2	4	no	—
128	C₈×C₁₉	—	2	4	no	—
129	C₁₉⋊C₈	—	4	4	no	—
130	C₂×C₄×M₁₆	D₁₉	1	1	yes	characteristic
131	C₂×#344	—	19	2	no	—
132	C₂×#339	—	19	2	no	—
133	C₂×#338	—	19	1	no	—
134	C₂×H(3,ℤ/4)	—	19	1	no	—
135	C₂²×C₁₉	—	2	4	no	—
136	C₂²×C₁₉	—	2	2	no	—
137	C₂²×C₁₉	—	4	2	no	—
138	C₂²×C₁₉	—	4	1	no	—
139	C₂²×C₁₉	—	4	1	no	—
140	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
141	C₂²×C₁₉	#381	1	1	yes	characteristic
142	C₂²×C₁₉	#383	1	1	yes	characteristic
143	C₂²×C₁₉	#387	1	2	yes	—
144	C₂²×C₁₉	#388	1	2	yes	—
145	C₄×C₁₉	—	2	4	no	—
146	C₄×C₁₉	—	4	2	no	—
147	C₄×C₁₉	#376	1	1	yes	characteristic
148	C₄×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₁₉	#380	1	1	yes	characteristic
150	C₄×C₁₉	#382	1	1	yes	characteristic
151	C₄×C₁₉	#387	1	2	yes	—
152	C₄×C₁₉	#388	1	2	yes	—
153	Dic₁₉	—	8	4	no	—
154	C₂²×M₁₆	—	19	2	no	—
155	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
156	C₂²×M₁₆	C₂×D₁₉	1	1	yes	characteristic
157	C₂×#117	—	19	2	no	—
158	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
159	C₄×M₁₆	—	2	4	no	—
160	C₂×#97	Dic₁₉	1	2	yes	complemented
161	C₂×C₄×C₈	Dic₁₉	1	2	yes	complemented
162	#344	—	38	4	no	—
163	#339	—	38	4	no	—
164	#338	—	19	4	no	—
165	H(3,ℤ/4)	—	19	4	no	—
166	C₂×C₁₉	—	2	2	no	—
167	C₂×C₁₉	—	2	1	no	—
168	C₂×C₁₉	—	2	1	no	—
169	C₂×C₁₉	—	2	2	no	—
170	C₂×C₁₉	#1434	1	1	yes	characteristic
171	C₂×C₁₉	#1841	1	2	yes	—
172	C₂³×C₄	—	2	1	no	—
173	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	19	4	no	—
179	C₂×M₁₆	—	38	2	no	—
180	C₂×M₁₆	—	38	2	no	—
181	C₂×M₁₆	—	38	2	no	—
182	C₂×M₁₆	—	38	2	no	—
183	C₂×M₁₆	—	38	2	no	—
184	C₂×M₁₆	Dic₃₈	1	2	yes	—
185	C₂×M₁₆	C₄×D₁₉	1	2	yes	—
186	C₂×M₁₆	C₄×D₁₉	1	2	yes	—
187	C₂×M₁₆	D₇₆	1	2	yes	—
188	C₂²×C₈	—	38	2	no	—
189	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
190	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
191	#117	—	19	4	no	—
192	#117	—	38	2	no	—
193	#117	—	38	2	no	—
194	#117	—	38	2	no	—
195	C₂×C₄⋊C₄	—	38	2	no	—
196	C₂×C₂²⋊C₄	—	38	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	—	38	2	no	—
200	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
201	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₁₉	1	4	yes	—
203	#97	—	2	4	no	—
204	C₄×C₈	—	2	4	no	—
205	C₁₉	#10813	1	1	yes	characteristic, complemented, Sylow
206	C₂⁴	#64541	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	76	2	no	—
212	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
216	C₂²×C₄	#64541	1	2	yes	—
217	C₂²×C₄	#64541	1	1	yes	characteristic
218	C₂²×C₄	#64541	1	2	yes	—
219	C₂²×C₄	#64541	1	2	yes	—
220	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	38	4	no	—
227	M₁₆	—	38	4	no	—
228	M₁₆	—	76	2	no	—
229	M₁₆	—	76	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	38	4	no	—
235	C₂×C₈	—	76	2	no	—
236	C₂×C₈	—	76	2	no	—
237	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
238	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
239	C₄⋊C₄	—	38	4	no	—
240	C₄⋊C₄	—	76	2	no	—
241	C₂²⋊C₄	—	38	4	no	—
242	C₂²⋊C₄	—	76	2	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	2	4	no	—
246	C₄²	—	38	4	no	—
247	C₄²	—	76	2	no	—
248	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
249	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
250	C₂³	—	4	2	no	—
251	C₂³	C₄×Dic₃₈	1	1	yes	characteristic
252	C₂³	C₄×D₇₆	1	1	yes	characteristic
253	C₂³	#10597322	1	2	yes	—
254	C₂³	#10597372	1	1	yes	characteristic
255	C₂³	#10597375	1	2	yes	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	4	8	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	76	4	no	—
262	C₂×C₄	—	152	4	no	—
263	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
264	C₂×C₄	#10597316	1	2	yes	—
265	C₂×C₄	#10597318	1	2	yes	—
266	C₂×C₄	#10597318	1	2	yes	—
267	C₂×C₄	C₄×D₇₆	1	1	yes	characteristic
268	C₂×C₄	#10597321	1	2	yes	—
269	C₂×C₄	#10597322	1	2	yes	—
270	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
271	C₂×C₄	#10597370	1	2	yes	—
272	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
273	C₂×C₄	#10597372	1	1	yes	characteristic
274	C₂×C₄	#10597372	1	1	yes	characteristic
275	C₂×C₄	#10597372	1	1	yes	characteristic
276	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
277	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
278	C₂×C₄	#10597375	1	2	yes	—
279	C₂×C₄	#10597376	1	2	yes	—
280	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₈	—	76	4	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
290	C₂²	#11926407	1	1	yes	characteristic
291	C₂²	#11926412	1	1	yes	characteristic
292	C₂²	#11926428	1	2	yes	—
293	C₂²	#11926435	1	2	yes	—
294	C₄	—	2	4	no	—
295	C₄	—	4	2	no	—
296	C₄	—	152	4	no	—
297	C₄	#11926390	1	1	yes	characteristic
298	C₄	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
299	C₄	#11926404	1	1	yes	characteristic
300	C₄	#11926410	1	1	yes	characteristic
301	C₄	#11926428	1	2	yes	—
302	C₄	#11926435	1	2	yes	—
303	C₂	—	2	1	no	—
304	C₂	—	2	2	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	#14561862	1	1	yes	characteristic
308	C₂	#14563784	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#10597372
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂×C₄×M₁₆×C₁₉	C₂
socle	C₂²×C₁₉	#383
prime radical	C₂⁴×C₁₉	#41
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂×C₄×M₁₆	D₁₉
hypocenter	C₁₉	#10813
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
10. C₁₉⋊#338	C₄	semidirect
11. #11926301	C₄	semidirect
16. C₂×C₁₉×#97	C₄	semidirect
17. C₂×C₄×C₈×C₁₉	C₄	semidirect
160. C₂×#97	Dic₁₉	semidirect
161. C₂×C₄×C₈	Dic₁₉	semidirect
205. C₁₉	#10813	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926315	2
C₂×#11926304	2
C₂×C₁₉⋊#338	2
C₂×#11926301	2
C₂×C₄×M₁₆×C₁₉	2
#10813	19