Subgroups — #14574961

Order 2432.

398 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927094	C₂	1	1	yes	maximal, characteristic, complemented
2	#11926841	C₂	1	2	yes	maximal, complemented
3	#11926839	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926834	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926778	C₂	1	2	yes	maximal, complemented
6	#11926775	C₂	1	2	yes	maximal, complemented
7	#11926756	C₂	1	2	yes	maximal, complemented
8	#11926744	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926739	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄⋊(C₁₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₉×#529	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#64541	—	2	1	no	—
13	C₁₉×#126	—	2	1	no	—
14	C₁₉×#125	C₂²	1	1	yes	characteristic
15	C₁₉×#124	C₂²	1	1	yes	characteristic
16	C₁₉×#123	—	2	1	no	—
17	C₁₉×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₉×#121	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₁₉	—	2	1	no	—
21	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
22	C₁₉×#117	—	2	1	no	—
23	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
24	C₂×#64527	C₂²	1	2	yes	—
25	C₂×#64525	C₂²	1	2	yes	—
26	#10597376	—	2	1	no	—
27	#10597376	C₂²	1	2	yes	—
28	C₂²×C₁₉⋊C₈	C₂²	1	2	yes	—
29	#10597321	—	2	1	no	—
30	#10597320	C₂²	1	1	yes	characteristic
31	C₄×D₇₆	—	2	1	no	—
32	C₄×D₇₆	C₂²	1	2	yes	—
33	#10597315	C₂²	1	1	yes	characteristic
34	#10597292	C₂²	1	2	yes	—
35	#10597289	C₂²	1	2	yes	—
36	#10597266	C₂²	1	2	yes	—
37	#10597266	C₂²	1	2	yes	—
38	#10597265	C₂²	1	2	yes	—
39	#10597264	C₂²	1	2	yes	—
40	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
42	C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
43	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₉	—	2	1	no	—
45	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	—	2	1	no	—
47	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
50	C₂×C₂²⋊D₁₉	—	2	2	no	—
51	#64541	—	4	4	no	—
52	C₂×D₇₆	—	2	1	no	—
53	C₂×D₇₆	—	2	1	no	—
54	C₂×D₇₆	C₂³	1	2	yes	—
55	C₂×C₄×D₁₉	—	2	2	no	—
56	C₂×Dic₃₈	—	2	1	no	—
57	C₄⋊C₄×C₁₉	—	2	1	no	—
58	C₄⋊C₄×C₁₉	—	2	1	no	—
59	C₄⋊C₄×C₁₉	—	2	1	no	—
60	C₄⋊C₄×C₁₉	—	2	1	no	—
61	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
63	C₂²⋊C₄×C₁₉	—	2	1	no	—
64	C₂²⋊C₄×C₁₉	—	2	1	no	—
65	C₂²⋊C₄×C₁₉	—	2	1	no	—
66	C₂²⋊C₄×C₁₉	—	2	1	no	—
67	C₂²⋊C₄×C₁₉	—	2	1	no	—
68	C₄²×C₁₉	—	2	1	no	—
69	C₄²×C₁₉	D₄	1	1	yes	characteristic
70	C₄²×C₁₉	D₄	1	1	yes	characteristic
71	C₄²×C₁₉	C₂³	1	1	yes	characteristic
72	C₄²×C₁₉	C₂³	1	1	yes	characteristic
73	#64527	—	4	2	no	—
74	#64525	—	4	2	no	—
75	(C₂×D₁₉)⋊C₄	—	2	2	no	—
76	(C₂×D₁₉)⋊C₄	—	2	2	no	—
77	C₄⋊Dic₁₉	—	2	1	no	—
78	C₄⋊Dic₁₉	C₂³	1	2	yes	—
79	C₂×C₁₉⋊C₈	—	4	2	no	—
80	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
81	C₂×C₁₉⋊C₈	C₂³	1	2	yes	—
82	C₂³×C₁₉	—	2	1	no	—
83	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×D₁₉	—	2	2	no	—
85	Q₈×C₁₉	—	2	1	no	—
86	Q₈×C₁₉	—	2	1	no	—
87	D₄×C₁₉	—	2	1	no	—
88	D₄×C₁₉	—	2	1	no	—
89	D₄×C₁₉	—	4	2	no	—
90	C₂×C₄×C₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	—	2	1	no	—
93	C₂×C₄×C₁₉	—	2	1	no	—
94	C₂×C₄×C₁₉	—	2	1	no	—
95	C₂×C₄×C₁₉	—	2	1	no	—
96	C₂×C₄×C₁₉	—	2	1	no	—
97	C₂×C₄×C₁₉	—	4	2	no	—
98	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₁₉	—	4	4	no	—
105	C₂×Dic₁₉	—	2	2	no	—
106	D₇₆	—	2	2	no	—
107	D₇₆	—	2	2	no	—
108	D₇₆	—	4	2	no	—
109	D₇₆	—	4	2	no	—
110	C₄×D₁₉	—	4	4	no	—
111	Dic₃₈	—	4	2	no	—
112	C₁₉⋊C₈	—	2	2	no	—
113	C₁₉⋊C₈	—	2	2	no	—
114	#3247	—	19	1	no	maximal, Sylow
115	C₂²×C₁₉	—	2	1	no	—
116	C₂²×C₁₉	—	2	1	no	—
117	C₂²×C₁₉	—	4	1	no	—
118	C₂²×C₁₉	—	4	1	no	—
119	C₂²×C₁₉	—	4	1	no	—
120	C₂²×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
121	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×D₁₉	—	4	2	no	—
123	C₂×D₁₉	—	4	2	no	—
124	C₂×D₁₉	—	4	2	no	—
125	C₄×C₁₉	—	2	1	no	—
126	C₄×C₁₉	—	2	1	no	—
127	C₄×C₁₉	—	2	1	no	—
128	C₄×C₁₉	—	2	1	no	—
129	C₄×C₁₉	—	4	1	no	—
130	C₄×C₁₉	—	4	1	no	—
131	C₄×C₁₉	—	4	1	no	—
132	C₄×C₁₉	#135	1	2	yes	—
133	C₄×C₁₉	H(5,𝔽₂)	1	1	yes	characteristic
134	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
135	Dic₁₉	—	4	2	no	—
136	#532	—	19	1	no	—
137	#529	D₁₉	1	1	yes	characteristic, complemented
138	#478	—	19	2	no	—
139	#477	—	19	1	no	—
140	#476	—	19	1	no	—
141	#466	—	19	2	no	—
142	#465	—	19	2	no	—
143	#463	—	19	2	no	—
144	#460	—	19	1	no	—
145	#459	—	19	1	no	—
146	C₂×C₄⋊C₈	—	19	1	no	—
147	C₂×C₁₉	—	2	2	no	—
148	C₂×C₁₉	—	8	1	no	—
149	C₂×C₁₉	#537	1	1	yes	characteristic
150	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
151	C₂×C₁₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	D₁₉	—	8	2	no	—
153	C₂×#41	—	38	1	no	—
154	C₂×SD₁₆	—	19	2	no	—
155	C₂×D₈	—	19	2	no	—
156	C₂²×C₈	—	19	2	no	—
157	C₄⋊Q₈	—	19	1	no	—
158	#127	—	19	1	no	—
159	#126	—	2	1	no	—
160	#125	C₂×D₁₉	1	1	yes	characteristic
161	#124	—	38	1	no	—
162	#124	C₂×D₁₉	1	1	yes	characteristic
163	#123	—	2	1	no	—
164	C₂²⋊Q₈	C₂×D₁₉	1	1	yes	characteristic
165	#121	—	19	2	no	—
166	#121	—	38	1	no	—
167	#121	C₂×D₁₉	1	1	yes	characteristic
168	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	—	19	2	no	—
171	C₄×D₄	—	38	1	no	—
172	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
173	#117	—	2	1	no	—
174	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
175	#107	—	19	2	no	—
176	#106	—	19	2	no	—
177	C₄⋊C₈	—	19	1	no	—
178	C₄⋊C₈	—	19	1	no	—
179	C₄⋊C₈	—	19	2	no	—
180	#103	—	19	2	no	—
181	#102	—	19	2	no	—
182	#102	—	19	2	no	—
183	#102	—	19	2	no	—
184	C₁₉	#3247	1	1	yes	characteristic, complemented, Sylow
185	#41	—	76	4	no	—
186	C₂×Q₈	—	38	1	no	—
187	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	19	2	no	—
190	C₂×D₄	—	38	2	no	—
191	C₂×D₄	—	38	1	no	—
192	C₂×D₄	—	38	1	no	—
193	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	38	2	no	—
196	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
197	C₂²×C₄	C₂²⋊D₁₉	1	1	yes	characteristic
198	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
199	SD₁₆	—	76	2	no	—
200	D₈	—	76	2	no	—
201	C₂×C₈	—	19	2	no	—
202	C₂×C₈	—	19	2	no	—
203	C₂×C₈	—	76	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	19	2	no	—
209	C₄⋊C₄	—	38	1	no	—
210	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
211	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
212	C₂²⋊C₄	—	2	1	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₂²⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	38	2	no	—
218	C₂²⋊C₄	—	38	2	no	—
219	C₄²	—	2	1	no	—
220	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
221	C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
222	C₄²	C₂²×D₁₉	1	1	yes	characteristic
223	C₄²	C₂²×D₁₉	1	1	yes	characteristic
224	C₂³	—	2	1	no	—
225	C₂³	—	38	2	no	—
226	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
227	Q₈	—	2	1	no	—
228	Q₈	—	2	1	no	—
229	Q₈	—	76	2	no	—
230	D₄	—	2	1	no	—
231	D₄	—	2	1	no	—
232	D₄	—	4	2	no	—
233	D₄	—	38	2	no	—
234	D₄	—	38	2	no	—
235	D₄	—	76	2	no	—
236	D₄	—	76	4	no	—
237	D₄	—	76	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	38	2	no	—
247	C₂×C₄	—	76	4	no	—
248	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
253	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
254	C₈	—	38	2	no	—
255	C₈	—	38	2	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	76	2	no	—
262	C₂²	—	76	2	no	—
263	C₂²	—	76	2	no	—
264	C₂²	#10597395	1	2	yes	—
265	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	76	2	no	—
274	C₄	#10597396	1	2	yes	—
275	C₄	#10597435	1	1	yes	characteristic
276	C₄	#10597436	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	8	1	no	—
279	C₂	—	152	2	no	—
280	C₂	#11927122	1	1	yes	characteristic
281	C₂	C₂×#10597396	1	1	yes	characteristic
282	C₂	C₂×#10597395	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#529	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#529	D₁₉
hypocenter	C₁₉	#3247
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #11927094	C₂	semidirect
2. #11926841	C₂	semidirect
3. #11926839	C₂	semidirect
4. #11926834	C₂	semidirect
5. #11926778	C₂	semidirect
6. #11926775	C₂	semidirect
7. #11926756	C₂	semidirect
8. #11926744	C₂	semidirect
9. #11926739	C₂	semidirect
10. C₂×C₄⋊(C₁₉⋊C₈)	C₂	semidirect
11. C₁₉×#529	C₂	semidirect
137. #529	D₁₉	semidirect
184. C₁₉	#3247	semidirect

Maximal subgroups

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

subgroup	index
#11927094	2
#11926841	2
#11926839	2
#11926834	2
#11926778	2
#11926775	2
#11926756	2
#11926744	2
#11926739	2
C₂×C₄⋊(C₁₉⋊C₈)	2
C₁₉×#529	2
#3247	19