Subgroups — #14573084

Order 2432.

496 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597396	C₂	1	2	yes	maximal, complemented
2	C₄×#64541	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926763	C₂	1	2	yes	maximal, complemented
4	#11926756	C₂	1	2	yes	maximal, complemented
5	#11926751	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926746	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926741	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926736	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926540	C₂	1	2	yes	maximal, complemented
11	C₁₉×#532	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₉	C₂²	1	2	yes	—
13	C₂×#64541	C₂²	1	1	yes	characteristic
14	C₁₉×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₉×#127	C₂²	1	1	yes	characteristic
16	C₁₉×#124	C₂²	1	2	yes	—
17	C₁₉×#121	—	2	1	no	—
18	C₁₉×#121	—	2	1	no	—
19	C₄×D₄×C₁₉	—	2	1	no	—
20	C₄×D₄×C₁₉	—	2	1	no	—
21	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
22	#10597396	—	2	8	no	—
23	C₂×#64528	C₂²	1	2	yes	—
24	C₂×#64527	C₂²	1	2	yes	—
25	C₂×#64526	C₂²	1	2	yes	—
26	C₂×#64525	C₂²	1	2	yes	—
27	C₄×C₂²⋊D₁₉	—	2	1	no	—
28	C₄×C₂²⋊D₁₉	—	2	1	no	—
29	#10597372	C₂²	1	1	yes	characteristic
30	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	—
31	C₄×D₇₆	C₂²	1	1	yes	characteristic
32	C₄×D₇₆	C₂²	1	1	yes	characteristic
33	#10597318	—	2	1	no	—
34	C₄²×D₁₉	—	2	1	no	—
35	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
36	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
37	#10597292	C₂²	1	2	yes	—
38	#10597292	C₂²	1	2	yes	—
39	#10597289	C₂²	1	2	yes	—
40	#10597289	C₂²	1	2	yes	—
41	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
42	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
43	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
44	#41×C₁₉	—	2	4	no	—
45	#41×C₁₉	—	2	4	no	—
46	C₂×Q₈×C₁₉	C₂³	1	2	yes	—
47	C₂×D₄×C₁₉	—	2	2	no	—
48	C₂×D₄×C₁₉	—	2	1	no	—
49	C₂×D₄×C₁₉	—	2	1	no	—
50	C₂×D₄×C₁₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₁₉	—	2	2	no	—
52	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₁₉	—	2	1	no	—
56	#64541	—	2	2	no	—
57	#64541	D₄	1	4	yes	complemented
58	C₂×D₇₆	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₁₉	—	2	1	no	—
60	C₂×C₄×D₁₉	—	2	1	no	—
61	C₂×C₄×D₁₉	—	2	1	no	—
62	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₉	—	2	1	no	—
64	C₄⋊C₄×C₁₉	—	2	1	no	—
65	C₂²⋊C₄×C₁₉	—	2	2	no	—
66	C₂²⋊C₄×C₁₉	—	2	2	no	—
67	C₄²×C₁₉	D₄	1	1	yes	characteristic
68	C₄²×C₁₉	D₄	1	1	yes	characteristic
69	C₄²×C₁₉	C₂³	1	1	yes	characteristic
70	C₄²×C₁₉	C₂³	1	1	yes	characteristic
71	C₂²⋊Dic₁₉	—	2	1	no	—
72	#64528	—	2	4	no	—
73	#64527	—	2	4	no	—
74	#64526	—	2	4	no	—
75	#64525	—	2	4	no	—
76	(C₂×D₁₉)⋊C₄	—	2	1	no	—
77	(C₂×D₁₉)⋊C₄	—	2	1	no	—
78	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
80	Dic₁₉⋊C₄	—	2	1	no	—
81	Dic₁₉⋊C₄	—	2	1	no	—
82	C₄×Dic₁₉	—	2	1	no	—
83	C₄×Dic₁₉	—	2	1	no	—
84	C₄×Dic₁₉	—	2	1	no	—
85	C₂×C₁₉⋊C₈	—	2	4	no	—
86	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
87	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
88	C₂³×C₁₉	—	2	2	no	—
89	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×D₁₉	—	2	1	no	—
91	Q₈×C₁₉	—	2	2	no	—
92	Q₈×C₁₉	—	2	2	no	—
93	D₄×C₁₉	—	2	2	no	—
94	D₄×C₁₉	—	2	2	no	—
95	D₄×C₁₉	—	4	2	no	—
96	D₄×C₁₉	—	4	4	no	—
97	D₄×C₁₉	—	4	2	no	—
98	C₂×C₄×C₁₉	—	2	2	no	—
99	C₂×C₄×C₁₉	—	2	2	no	—
100	C₂×C₄×C₁₉	—	4	1	no	—
101	C₂×C₄×C₁₉	—	4	4	no	—
102	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₉	#41	1	2	yes	—
109	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₁₉	—	2	4	no	—
111	C₂×Dic₁₉	—	2	1	no	—
112	C₂×Dic₁₉	—	2	1	no	—
113	C₂×Dic₁₉	—	2	1	no	—
114	D₇₆	—	2	1	no	—
115	D₇₆	C₂×D₄	1	2	yes	—
116	C₄×D₁₉	—	2	4	no	—
117	C₄×D₁₉	—	4	2	no	—
118	C₄×D₁₉	—	4	2	no	—
119	Dic₃₈	—	2	1	no	—
120	Dic₃₈	C₂×D₄	1	2	yes	—
121	C₁₉⋊C₈	—	2	2	no	—
122	C₁₉⋊C₈	—	2	2	no	—
123	#3035	—	19	1	no	maximal, Sylow
124	C₂²×C₁₉	—	2	1	no	—
125	C₂²×C₁₉	—	2	1	no	—
126	C₂²×C₁₉	—	4	2	no	—
127	C₂²×C₁₉	—	4	2	no	—
128	C₂²×C₁₉	—	4	2	no	—
129	C₂²×C₁₉	#121	1	2	yes	—
130	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×D₁₉	—	2	2	no	—
132	C₂×D₁₉	—	4	1	no	—
133	C₂×D₁₉	—	4	1	no	—
134	C₄×C₁₉	—	2	1	no	—
135	C₄×C₁₉	—	2	1	no	—
136	C₄×C₁₉	—	2	1	no	—
137	C₄×C₁₉	—	4	2	no	—
138	C₄×C₁₉	#121	1	2	yes	—
139	C₄×C₁₉	#135	1	2	yes	complemented
140	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
142	Dic₁₉	—	2	2	no	—
143	Dic₁₉	—	4	1	no	—
144	Dic₁₉	—	4	1	no	—
145	C₂×#135	—	19	2	no	—
146	#532	D₁₉	1	1	yes	characteristic, complemented
147	C₄×#41	—	19	1	no	—
148	#464	—	19	2	no	—
149	#463	—	19	2	no	—
150	#462	—	19	1	no	—
151	#461	—	19	1	no	—
152	#460	—	19	1	no	—
153	#459	—	19	1	no	—
154	C₂×C₄⋊C₈	—	19	1	no	—
155	#416	—	19	2	no	—
156	C₂×C₁₉	—	2	2	no	—
157	C₂×C₁₉	—	8	2	no	—
158	C₂×C₁₉	C₂×#121	1	1	yes	characteristic
159	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
160	C₂×C₁₉	#575	1	1	yes	characteristic
161	D₁₉	—	4	2	no	—
162	C₂×#41	—	19	1	no	—
163	C₂×#41	C₂×D₁₉	1	2	yes	—
164	#135	—	38	8	no	—
165	C₂×Q₁₆	—	19	2	no	—
166	C₂×SD₁₆	—	19	2	no	—
167	C₂×SD₁₆	—	19	2	no	—
168	C₂×D₈	—	19	2	no	—
169	C₂²×C₈	—	19	2	no	—
170	C₄⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
171	#127	C₂×D₁₉	1	1	yes	characteristic
172	#124	C₂×D₁₉	1	2	yes	—
173	#121	—	2	1	no	—
174	#121	—	2	1	no	—
175	C₄×Q₈	—	19	1	no	—
176	C₄×Q₈	—	19	1	no	—
177	C₄×D₄	—	2	1	no	—
178	C₄×D₄	—	2	1	no	—
179	C₄×D₄	—	19	1	no	—
180	C₄×D₄	—	19	1	no	—
181	C₄×D₄	—	38	1	no	—
182	C₄×D₄	—	38	1	no	—
183	#117	—	19	1	no	—
184	#117	—	38	1	no	—
185	C₂×C₄²	—	38	1	no	—
186	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
187	C₄⋊C₈	—	19	1	no	—
188	C₄⋊C₈	—	19	1	no	—
189	C₄⋊C₈	—	19	2	no	—
190	#103	—	19	2	no	—
191	#103	—	19	2	no	—
192	#102	—	19	2	no	—
193	#102	—	19	2	no	—
194	C₁₉	#3035	1	1	yes	characteristic, complemented, Sylow
195	#41	—	2	4	no	—
196	#41	—	2	4	no	—
197	#41	—	19	4	no	—
198	#41	—	38	2	no	—
199	C₂×Q₈	—	19	1	no	—
200	C₂×Q₈	C₂²×D₁₉	1	2	yes	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	—	2	1	no	—
204	C₂×D₄	—	19	1	no	—
205	C₂×D₄	—	38	1	no	—
206	C₂×D₄	C₂²×D₁₉	1	2	yes	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	38	1	no	—
209	C₂²×C₄	—	38	1	no	—
210	C₂²×C₄	—	38	1	no	—
211	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
212	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
213	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
214	Q₁₆	—	38	4	no	—
215	SD₁₆	—	38	4	no	—
216	SD₁₆	—	38	4	no	—
217	D₈	—	38	4	no	—
218	C₂×C₈	—	19	2	no	—
219	C₂×C₈	—	19	2	no	—
220	C₂×C₈	—	38	4	no	—
221	C₄⋊C₄	—	2	1	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₄⋊C₄	—	19	1	no	—
224	C₄⋊C₄	—	19	1	no	—
225	C₄⋊C₄	—	38	1	no	—
226	C₄⋊C₄	—	38	1	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	38	1	no	—
230	C₂²⋊C₄	—	38	1	no	—
231	C₂²⋊C₄	—	38	1	no	—
232	C₄²	—	38	1	no	—
233	C₄²	—	38	1	no	—
234	C₄²	—	38	1	no	—
235	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
236	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
237	C₄²	C₂²×D₁₉	1	1	yes	characteristic
238	C₄²	C₂²×D₁₉	1	1	yes	characteristic
239	C₂³	—	2	2	no	—
240	C₂³	—	38	1	no	—
241	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
242	Q₈	—	2	2	no	—
243	Q₈	—	2	2	no	—
244	Q₈	—	19	2	no	—
245	Q₈	—	38	1	no	—
246	D₄	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	2	no	—
251	D₄	—	19	2	no	—
252	D₄	—	38	4	no	—
253	D₄	—	38	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	38	1	no	—
259	C₂×C₄	—	38	4	no	—
260	C₂×C₄	—	38	1	no	—
261	C₂×C₄	—	38	1	no	—
262	C₂×C₄	—	76	2	no	—
263	C₂×C₄	—	76	2	no	—
264	C₂×C₄	D₄×D₁₉	1	2	yes	—
265	C₂×C₄	#64543	1	2	yes	—
266	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
267	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
268	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
269	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
270	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
271	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
272	C₈	—	38	2	no	—
273	C₈	—	38	2	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	38	2	no	—
280	C₂²	—	76	1	no	—
281	C₂²	—	76	1	no	—
282	C₂²	#10597384	1	2	yes	—
283	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	4	2	no	—
288	C₄	—	38	2	no	—
289	C₄	—	76	1	no	—
290	C₄	—	76	1	no	—
291	C₄	#10597384	1	2	yes	—
292	C₄	#10597396	1	2	yes	complemented
293	C₄	C₂×D₄×D₁₉	1	1	yes	characteristic
294	C₄	C₂×#64543	1	1	yes	characteristic
295	C₂	—	2	2	no	—
296	C₂	—	8	2	no	—
297	C₂	—	76	2	no	—
298	C₂	C₂×#10597384	1	1	yes	characteristic
299	C₂	C₂×#10597396	1	1	yes	characteristic
300	C₂	#11927342	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#532	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#532	D₁₉
hypocenter	C₁₉	#3035
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#10597396	C₂	semidirect
2. C₄×#64541	C₂	semidirect
3. #11926763	C₂	semidirect
4. #11926756	C₂	semidirect
5. #11926751	C₂	semidirect
6. #11926746	C₂	semidirect
7. #11926741	C₂	semidirect
8. #11926736	C₂	semidirect
9. C₂×C₄⋊(C₁₉⋊C₈)	C₂	semidirect
10. #11926540	C₂	semidirect
11. C₁₉×#532	C₂	semidirect
57. #64541	D₄	semidirect
139. C₄×C₁₉	#135	semidirect
146. #532	D₁₉	semidirect
194. C₁₉	#3035	semidirect
292. C₄	#10597396	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597396	2
C₄×#64541	2
#11926763	2
#11926756	2
#11926751	2
#11926746	2
#11926741	2
#11926736	2
C₂×C₄⋊(C₁₉⋊C₈)	2
#11926540	2
C₁₉×#532	2
#3035	19