Subgroups — #14566644

Order 2432.

620 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
3	#11926451	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926443	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926443	C₂	1	2	yes	maximal, complemented
6	#11926430	C₂	1	2	yes	maximal, complemented
7	#11926419	C₂	1	2	yes	maximal
8	C₄⋊C₄×Dic₁₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊C₄×Dic₁₉	C₂	1	2	yes	maximal
10	C₄×C₂²⋊Dic₁₉	C₂	1	1	yes	maximal, characteristic
11	C₂×C₁₉×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂³×Dic₁₉	C₂²	1	2	yes	—
15	C₁₉×C₂²⋊Q₈	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
18	C₂×C₂²⋊C₄×C₁₉	C₂²	1	2	yes	—
19	C₂×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊Dic₁₉	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₉	—	2	4	no	—
22	C₂×C₄⋊Dic₁₉	—	2	4	no	—
23	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
24	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
25	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	—
27	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₁₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
31	C₂×C₄×Dic₁₉	C₂²	1	2	yes	—
32	C₁₉⋊#95	C₂²	1	1	yes	characteristic
33	C₁₉⋊#95	C₂²	1	1	yes	characteristic
34	C₁₉⋊#95	C₂²	1	2	yes	—
35	C₁₉⋊#95	C₂²	1	2	yes	—
36	C₁₉⋊#95	C₂²	1	2	yes	—
37	C₁₉⋊#95	C₂²	1	2	yes	—
38	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₁₉	—	2	2	no	—
40	C₂×Q₈×C₁₉	—	2	2	no	—
41	C₂×Q₈×C₁₉	C₂×C₄	1	4	yes	complemented
42	C₂²×C₄×C₁₉	—	2	2	no	—
43	C₂²×C₄×C₁₉	—	2	2	no	—
44	C₂²×C₄×C₁₉	C₂×C₄	1	4	yes	—
45	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
47	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
49	C₂²×Dic₁₉	—	2	4	no	—
50	C₂²×Dic₁₉	—	2	4	no	—
51	C₂²×Dic₁₉	—	4	2	no	—
52	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
53	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
54	C₂²×Dic₁₉	C₂³	1	2	yes	—
55	C₂²×Dic₁₉	C₂³	1	2	yes	—
56	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
57	C₂²×Dic₁₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₁₉	C₂×C₄	1	4	yes	complemented
59	C₄⋊C₄×C₁₉	C₂×C₄	1	8	yes	complemented
60	C₂²⋊C₄×C₁₉	C₂×C₄	1	8	yes	—
61	C₂²⋊Dic₁₉	—	2	2	no	—
62	C₂²⋊Dic₁₉	Q₈	1	4	yes	complemented
63	C₄⋊Dic₁₉	—	2	2	no	—
64	C₄⋊Dic₁₉	—	2	4	no	—
65	C₄⋊Dic₁₉	—	2	4	no	—
66	C₄⋊Dic₁₉	—	4	2	no	—
67	C₄⋊Dic₁₉	D₄	1	4	yes	—
68	C₄×Dic₁₉	—	2	2	no	—
69	C₄×Dic₁₉	—	2	2	no	—
70	C₄×Dic₁₉	—	2	4	no	—
71	C₄×Dic₁₉	—	4	2	no	—
72	C₂³×C₁₉	—	2	1	no	—
73	C₂³×C₁₉	—	2	1	no	—
74	C₂³×C₁₉	—	2	1	no	—
75	C₂³×C₁₉	—	2	1	no	—
76	C₂³×C₁₉	C₂²×C₄	1	2	yes	—
77	C₂³×C₁₉	C₂×Q₈	1	2	yes	—
78	C₂³×C₁₉	#41	1	2	yes	—
79	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
80	Q₈×C₁₉	—	2	8	no	—
81	C₂×C₄×C₁₉	—	2	2	no	—
82	C₂×C₄×C₁₉	—	2	1	no	—
83	C₂×C₄×C₁₉	—	2	1	no	—
84	C₂×C₄×C₁₉	—	2	2	no	—
85	C₂×C₄×C₁₉	—	2	2	no	—
86	C₂×C₄×C₁₉	—	2	2	no	—
87	C₂×C₄×C₁₉	—	2	8	no	—
88	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
89	C₂×C₄×C₁₉	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₉	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₉	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₉	#41	1	2	yes	—
94	C₂×Dic₁₉	—	2	1	no	—
95	C₂×Dic₁₉	—	2	1	no	—
96	C₂×Dic₁₉	—	2	2	no	—
97	C₂×Dic₁₉	—	2	1	no	—
98	C₂×Dic₁₉	—	2	2	no	—
99	C₂×Dic₁₉	—	2	1	no	—
100	C₂×Dic₁₉	—	2	1	no	—
101	C₂×Dic₁₉	—	2	2	no	—
102	C₂×Dic₁₉	—	2	1	no	—
103	C₂×Dic₁₉	—	2	1	no	—
104	C₂×Dic₁₉	—	2	1	no	—
105	C₂×Dic₁₉	—	2	2	no	—
106	C₂×Dic₁₉	—	2	1	no	—
107	C₂×Dic₁₉	—	2	2	no	—
108	C₂×Dic₁₉	—	2	1	no	—
109	C₂×Dic₁₉	—	4	8	no	—
110	C₂×Dic₁₉	C₂×D₄	1	4	yes	—
111	C₂×Dic₁₉	C₂×Q₈	1	2	yes	—
112	C₂×Dic₁₉	C₂×Q₈	1	2	yes	—
113	#2360	—	19	1	no	maximal, Sylow
114	C₂²×C₁₉	—	2	2	no	—
115	C₂²×C₁₉	—	2	2	no	—
116	C₂²×C₁₉	—	2	2	no	—
117	C₂²×C₁₉	—	2	2	no	—
118	C₂²×C₁₉	—	2	2	no	—
119	C₂²×C₁₉	—	2	2	no	—
120	C₂²×C₁₉	C₄×Q₈	1	4	yes	complemented
121	C₂²×C₁₉	C₂³×C₄	1	1	yes	characteristic
122	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
123	C₂²×C₁₉	C₂²×Q₈	1	1	yes	characteristic
124	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
125	C₂²×C₁₉	C₂×#41	1	1	yes	characteristic
126	C₂²×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂²×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
128	C₄×C₁₉	—	2	2	no	—
129	C₄×C₁₉	—	2	4	no	—
130	C₄×C₁₉	—	2	4	no	—
131	C₄×C₁₉	C₄×D₄	1	4	yes	—
132	Dic₁₉	—	2	4	no	—
133	Dic₁₉	—	2	2	no	—
134	Dic₁₉	—	2	2	no	—
135	Dic₁₉	—	4	1	no	—
136	Dic₁₉	—	4	2	no	—
137	Dic₁₉	—	4	1	no	—
138	C₂×C₂²⋊Q₈	D₁₉	1	1	yes	characteristic
139	C₂×C₄×Q₈	—	19	1	no	—
140	C₂²×C₄⋊C₄	—	19	1	no	—
141	#391	—	19	1	no	—
142	#389	—	19	1	no	—
143	#389	—	19	2	no	—
144	#387	—	19	2	no	—
145	#385	—	19	2	no	—
146	C₄×C₄⋊C₄	—	19	1	no	—
147	C₄×C₂²⋊C₄	—	19	1	no	—
148	C₄×C₂²⋊C₄	—	19	2	no	—
149	C₂×C₁₉	—	2	4	no	—
150	C₂×C₁₉	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂×C₁₉	C₂×C₄×Q₈	1	1	yes	characteristic
152	C₂×C₁₉	#520	1	1	yes	characteristic
153	C₂×C₁₉	#546	1	1	yes	characteristic
154	C₂×C₁₉	D₄×Q₈	1	1	yes	characteristic
155	C₂×C₁₉	#552	1	1	yes	characteristic
156	C₂×C₁₉	#554	1	1	yes	characteristic
157	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic
158	C₂³×C₄	—	19	2	no	—
159	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic
160	C₂²⋊Q₈	Dic₁₉	1	8	yes	complemented
161	C₄×Q₈	—	38	4	no	—
162	C₂×C₄⋊C₄	—	19	1	no	—
163	C₂×C₄⋊C₄	—	19	2	no	—
164	C₂×C₄⋊C₄	—	19	1	no	—
165	C₂×C₄⋊C₄	—	19	2	no	—
166	C₂×C₄⋊C₄	—	19	2	no	—
167	C₂×C₄⋊C₄	—	38	4	no	—
168	C₂×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
169	C₂×C₄⋊C₄	C₂×D₁₉	1	2	yes	—
170	C₂×C₂²⋊C₄	—	19	1	no	—
171	C₂×C₂²⋊C₄	—	19	1	no	—
172	C₂×C₂²⋊C₄	C₂×D₁₉	1	2	yes	—
173	C₂×C₄²	—	19	2	no	—
174	C₂×C₄²	—	19	1	no	—
175	C₂×C₄²	—	19	1	no	—
176	C₂×C₄²	—	19	2	no	—
177	#95	—	19	1	no	—
178	#95	—	19	2	no	—
179	#95	—	19	1	no	—
180	#95	—	19	2	no	—
181	#95	—	19	2	no	—
182	#95	—	19	2	no	—
183	C₁₉	#2360	1	1	yes	characteristic, complemented, Sylow
184	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
185	C₂×Q₈	—	2	2	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×Q₈	C₂×Dic₁₉	1	4	yes	complemented
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	19	1	no	—
191	C₂²×C₄	—	19	1	no	—
192	C₂²×C₄	—	19	2	no	—
193	C₂²×C₄	—	19	1	no	—
194	C₂²×C₄	—	19	2	no	—
195	C₂²×C₄	—	19	1	no	—
196	C₂²×C₄	—	38	4	no	—
197	C₂²×C₄	—	38	4	no	—
198	C₂²×C₄	—	76	2	no	—
199	C₂²×C₄	C₂×Dic₁₉	1	4	yes	—
200	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
202	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
204	C₄⋊C₄	—	19	4	no	—
205	C₄⋊C₄	—	38	4	no	—
206	C₄⋊C₄	—	38	4	no	—
207	C₄⋊C₄	—	38	2	no	—
208	C₄⋊C₄	—	76	2	no	—
209	C₄⋊C₄	C₂×Dic₁₉	1	4	yes	complemented
210	C₄⋊C₄	C₂×Dic₁₉	1	8	yes	complemented
211	C₂²⋊C₄	—	19	4	no	—
212	C₂²⋊C₄	—	38	2	no	—
213	C₂²⋊C₄	C₂×Dic₁₉	1	8	yes	—
214	C₄²	—	38	4	no	—
215	C₄²	—	38	2	no	—
216	C₄²	—	38	2	no	—
217	C₄²	—	76	2	no	—
218	C₂³	—	2	1	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	2	1	no	—
222	C₂³	Q₈×D₁₉	1	2	yes	—
223	C₂³	#64545	1	2	yes	—
224	C₂³	C₂²×Dic₁₉	1	2	yes	—
225	C₂³	C₂³×D₁₉	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	19	4	no	—
235	C₂×C₄	—	19	2	no	—
236	C₂×C₄	—	19	2	no	—
237	C₂×C₄	—	38	1	no	—
238	C₂×C₄	—	38	1	no	—
239	C₂×C₄	—	38	2	no	—
240	C₂×C₄	—	38	1	no	—
241	C₂×C₄	—	38	2	no	—
242	C₂×C₄	—	38	1	no	—
243	C₂×C₄	—	38	1	no	—
244	C₂×C₄	—	38	2	no	—
245	C₂×C₄	—	38	1	no	—
246	C₂×C₄	—	38	1	no	—
247	C₂×C₄	—	38	1	no	—
248	C₂×C₄	—	38	2	no	—
249	C₂×C₄	—	38	1	no	—
250	C₂×C₄	—	38	2	no	—
251	C₂×C₄	—	38	1	no	—
252	C₂×C₄	—	76	8	no	—
253	C₂×C₄	D₄×D₁₉	1	2	yes	—
254	C₂×C₄	#64543	1	2	yes	—
255	C₂×C₄	C₂²×Dic₁₉	1	4	yes	—
256	C₂×C₄	C₂²×Dic₁₉	1	4	yes	—
257	C₂×C₄	C₂²×Dic₁₉	1	2	yes	—
258	C₂×C₄	C₂²×Dic₁₉	1	2	yes	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	Q₈×Dic₁₉	1	4	yes	complemented
266	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
267	C₂²	C₂×#64543	1	1	yes	characteristic
268	C₂²	C₂×Q₈×D₁₉	1	1	yes	characteristic
269	C₂²	C₂×#64545	1	1	yes	characteristic
270	C₂²	#10597435	1	1	yes	characteristic
271	C₂²	#10597436	1	1	yes	characteristic
272	C₂²	C₂³×Dic₁₉	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	38	4	no	—
277	C₄	—	38	2	no	—
278	C₄	—	38	2	no	—
279	C₄	—	76	2	no	—
280	C₄	—	76	1	no	—
281	C₄	—	76	1	no	—
282	C₄	D₄×Dic₁₉	1	4	yes	—
283	C₂	—	2	4	no	—
284	C₂	C₂×D₄×Dic₁₉	1	1	yes	characteristic
285	C₂	C₂×Q₈×Dic₁₉	1	1	yes	characteristic
286	C₂	#11927027	1	1	yes	characteristic
287	C₂	#11927185	1	1	yes	characteristic
288	C₂	#11927207	1	1	yes	characteristic
289	C₂	#11927236	1	1	yes	characteristic
290	C₂	#11927252	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₉
commutator	C₂²×C₁₉	C₂³×C₄
Fitting	C₂×C₁₉×C₂²⋊Q₈	C₂
socle, cosocle	C₂³×C₁₉	C₂⁴
prime radical	C₂⁴×C₁₉	C₂³
hypercenter	C₂×C₂²⋊Q₈	D₁₉
hypocenter	C₁₉	#2360
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×Dic₁₉	C₂	semidirect
3. #11926451	C₂	semidirect
4. #11926443	C₂	semidirect
5. #11926443	C₂	semidirect
6. #11926430	C₂	semidirect
8. C₄⋊C₄×Dic₁₉	C₂	semidirect
15. C₁₉×C₂²⋊Q₈	C₄	semidirect
41. C₂×Q₈×C₁₉	C₂×C₄	semidirect
58. C₄⋊C₄×C₁₉	C₂×C₄	semidirect
59. C₄⋊C₄×C₁₉	C₂×C₄	semidirect
62. C₂²⋊Dic₁₉	Q₈	semidirect
120. C₂²×C₁₉	C₄×Q₈	semidirect
160. C₂²⋊Q₈	Dic₁₉	semidirect
183. C₁₉	#2360	semidirect
187. C₂×Q₈	C₂×Dic₁₉	semidirect
209. C₄⋊C₄	C₂×Dic₁₉	semidirect
210. C₄⋊C₄	C₂×Dic₁₉	semidirect
265. C₂²	Q₈×Dic₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×Dic₁₉	2
C₂²×C₄⋊Dic₁₉	2
#11926451	2
#11926443	2
#11926430	2
#11926419	2
C₄⋊C₄×Dic₁₉	2
C₂²⋊C₄×Dic₁₉	2
C₄×C₂²⋊Dic₁₉	2
C₂×C₁₉×C₂²⋊Q₈	2
#2360	19