Subgroups — #14574882

Order 2432.

332 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927172	C₂	1	1	yes	maximal, characteristic, complemented
2	#11926954	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926943	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926908	C₂	1	1	yes	maximal, characteristic, complemented
5	#11926890	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926865	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926865	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926860	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926855	C₂	1	1	yes	maximal, characteristic, complemented
10	#11926844	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926844	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926834	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926824	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926605	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#540	C₂	1	1	yes	maximal, characteristic
16	C₁₉×#127	C₂²	1	1	yes	characteristic
17	C₁₉×#125	C₂²	1	1	yes	characteristic
18	C₁₉×#123	—	2	1	no	—
19	C₁₉×#123	—	2	1	no	—
20	C₁₉×#121	—	2	1	no	—
21	C₁₉×#121	—	2	1	no	—
22	C₁₉×#121	C₂²	1	1	yes	characteristic
23	C₁₉×#121	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
26	C₁₉×#117	C₂²	1	1	yes	characteristic
27	#10597372	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₁₉	C₂²	1	1	yes	characteristic
29	#10597336	—	2	1	no	—
30	#10597334	—	2	1	no	—
31	#10597324	—	2	1	no	—
32	#10597324	—	2	1	no	—
33	#10597316	C₂²	1	1	yes	characteristic
34	#10597316	C₂²	1	1	yes	characteristic
35	#10597315	C₂²	1	1	yes	characteristic
36	#10597315	C₂²	1	1	yes	characteristic
37	#10597289	C₂²	1	1	yes	characteristic
38	#10597289	C₂²	1	1	yes	characteristic
39	#10597289	C₂²	1	1	yes	characteristic
40	#10597289	C₂²	1	1	yes	characteristic
41	#10597289	C₂²	1	1	yes	characteristic
42	#10597289	C₂²	1	1	yes	characteristic
43	#10597289	C₂²	1	1	yes	characteristic
44	#10597289	C₂²	1	1	yes	characteristic
45	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
46	C₂²⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
47	#10597265	C₂²	1	1	yes	characteristic
48	#10597265	C₂²	1	1	yes	characteristic
49	#10597265	C₂²	1	1	yes	characteristic
50	#10597265	C₂²	1	1	yes	characteristic
51	#10597264	C₂²	1	1	yes	characteristic
52	#10597264	C₂²	1	1	yes	characteristic
53	#10597264	C₂²	1	1	yes	characteristic
54	#10597264	C₂²	1	1	yes	characteristic
55	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
56	C₄⋊(C₁₉⋊C₈)	C₂²	1	1	yes	characteristic
57	#10597260	C₂²	1	1	yes	characteristic
58	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₉	—	2	1	no	—
60	C₂×D₄×C₁₉	—	2	1	no	—
61	C₂×D₄×C₁₉	—	2	1	no	—
62	C₂×D₄×C₁₉	—	2	1	no	—
63	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₁₉	—	2	1	no	—
66	C₂²×C₄×C₁₉	—	2	1	no	—
67	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
68	C₂²×Dic₁₉	—	2	1	no	—
69	C₂×Dic₃₈	—	2	1	no	—
70	C₂×Dic₃₈	—	2	1	no	—
71	C₄⋊C₄×C₁₉	—	2	1	no	—
72	C₄⋊C₄×C₁₉	—	2	1	no	—
73	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₉	—	2	1	no	—
78	C₂²⋊C₄×C₁₉	—	2	1	no	—
79	C₂²⋊C₄×C₁₉	—	2	1	no	—
80	C₂²⋊C₄×C₁₉	—	2	1	no	—
81	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₉	D₄	1	1	yes	characteristic
83	C₄²×C₁₉	C₂³	1	1	yes	characteristic
84	C₄²×C₁₉	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₁₉	—	2	1	no	—
86	C₄⋊Dic₁₉	—	2	1	no	—
87	C₄⋊Dic₁₉	—	2	1	no	—
88	C₄⋊Dic₁₉	—	2	1	no	—
89	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
93	Dic₁₉⋊C₄	—	2	1	no	—
94	Dic₁₉⋊C₄	—	2	1	no	—
95	Dic₁₉⋊C₄	—	2	1	no	—
96	Dic₁₉⋊C₄	—	2	1	no	—
97	C₄×Dic₁₉	—	2	1	no	—
98	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
99	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
100	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂³×C₁₉	—	2	1	no	—
103	C₂³×C₁₉	—	2	1	no	—
104	C₂³×C₁₉	C₂×D₄	1	1	yes	characteristic
105	D₄×C₁₉	—	2	1	no	—
106	D₄×C₁₉	—	2	1	no	—
107	D₄×C₁₉	—	2	1	no	—
108	D₄×C₁₉	—	2	1	no	—
109	D₄×C₁₉	—	8	1	no	—
110	D₄×C₁₉	—	8	1	no	—
111	D₄×C₁₉	—	8	1	no	—
112	D₄×C₁₉	—	8	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	—	2	1	no	—
115	C₂×C₄×C₁₉	—	2	1	no	—
116	C₂×C₄×C₁₉	—	2	1	no	—
117	C₂×C₄×C₁₉	—	8	1	no	—
118	C₂×C₄×C₁₉	—	8	1	no	—
119	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₁₉	—	2	1	no	—
126	C₂×Dic₁₉	—	2	1	no	—
127	C₂×Dic₁₉	—	2	1	no	—
128	C₂×Dic₁₉	—	2	1	no	—
129	C₂×Dic₁₉	—	8	1	no	—
130	Dic₃₈	—	8	1	no	—
131	Dic₃₈	—	8	1	no	—
132	C₁₉⋊C₈	—	2	1	no	—
133	C₁₉⋊C₈	—	2	1	no	—
134	C₁₉⋊C₈	—	2	1	no	—
135	C₁₉⋊C₈	—	2	1	no	—
136	#3241	—	19	1	no	maximal, Sylow
137	C₂²×C₁₉	—	2	1	no	—
138	C₂²×C₁₉	—	2	1	no	—
139	C₂²×C₁₉	—	2	1	no	—
140	C₂²×C₁₉	—	4	1	no	—
141	C₂²×C₁₉	—	4	1	no	—
142	C₂²×C₁₉	—	4	1	no	—
143	C₂²×C₁₉	—	4	1	no	—
144	C₂²×C₁₉	—	4	1	no	—
145	C₂²×C₁₉	—	4	1	no	—
146	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
147	C₄×C₁₉	—	2	1	no	—
148	C₄×C₁₉	—	2	1	no	—
149	C₄×C₁₉	—	2	1	no	—
150	C₄×C₁₉	—	2	1	no	—
151	C₄×C₁₉	—	2	1	no	—
152	C₄×C₁₉	—	4	1	no	—
153	C₄×C₁₉	—	4	1	no	—
154	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
155	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
156	Dic₁₉	—	4	1	no	—
157	Dic₁₉	—	4	1	no	—
158	Dic₁₉	—	4	1	no	—
159	Dic₁₉	—	4	1	no	—
160	#544	—	19	1	no	—
161	#540	D₁₉	1	1	yes	characteristic
162	#501	—	19	1	no	—
163	#499	—	19	1	no	—
164	#490	—	19	1	no	—
165	#486	—	19	1	no	—
166	#482	—	19	1	no	—
167	#482	—	19	1	no	—
168	#481	—	19	1	no	—
169	#480	—	19	1	no	—
170	#478	—	19	1	no	—
171	#478	—	19	1	no	—
172	#476	—	19	1	no	—
173	#474	—	19	1	no	—
174	#433	—	19	1	no	—
175	C₂×C₁₉	—	4	1	no	—
176	C₂×C₁₉	—	8	1	no	—
177	C₂×C₁₉	—	8	1	no	—
178	C₂×C₁₉	#536	1	1	yes	characteristic
179	C₂×C₁₉	#576	1	1	yes	characteristic
180	C₂×C₁₉	#577	1	1	yes	characteristic
181	C₄⋊Q₈	—	19	1	no	—
182	C₄⋊Q₈	—	19	1	no	—
183	C₄⋊Q₈	—	38	1	no	—
184	#127	C₂×D₁₉	1	1	yes	characteristic
185	#125	—	19	1	no	—
186	#125	—	19	1	no	—
187	#125	—	38	1	no	—
188	#125	C₂×D₁₉	1	1	yes	characteristic
189	#123	—	2	1	no	—
190	#123	—	2	1	no	—
191	C₂²⋊Q₈	—	38	1	no	—
192	C₂²⋊Q₈	—	38	1	no	—
193	#121	—	2	1	no	—
194	#121	—	2	1	no	—
195	#121	C₂×D₁₉	1	1	yes	characteristic
196	#121	C₂×D₁₉	1	1	yes	characteristic
197	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
198	C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
199	#117	—	19	1	no	—
200	#117	C₂×D₁₉	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	19	1	no	—
202	#107	—	19	1	no	—
203	#107	—	19	1	no	—
204	#107	—	19	1	no	—
205	#107	—	19	1	no	—
206	#106	—	19	1	no	—
207	#106	—	19	1	no	—
208	#106	—	19	1	no	—
209	#106	—	19	1	no	—
210	C₄⋊C₈	—	19	1	no	—
211	C₄⋊C₈	—	19	1	no	—
212	#102	—	19	1	no	—
213	#102	—	19	1	no	—
214	#102	—	19	1	no	—
215	#102	—	19	1	no	—
216	#102	—	19	1	no	—
217	#102	—	19	1	no	—
218	#102	—	19	1	no	—
219	#102	—	19	1	no	—
220	C₂²⋊C₈	—	19	1	no	—
221	C₂²⋊C₈	—	19	1	no	—
222	#97	—	19	1	no	—
223	C₄×C₈	—	19	1	no	—
224	C₁₉	#3241	1	1	yes	characteristic, complemented, Sylow
225	C₂×Q₈	—	38	1	no	—
226	C₂×Q₈	—	38	1	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	2	1	no	—
230	C₂×D₄	—	2	1	no	—
231	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
232	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
233	C₂²×C₄	—	2	1	no	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	38	1	no	—
236	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
237	C₂×C₈	—	19	1	no	—
238	C₂×C₈	—	19	1	no	—
239	C₂×C₈	—	19	1	no	—
240	C₂×C₈	—	19	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	19	1	no	—
244	C₄⋊C₄	—	19	1	no	—
245	C₄⋊C₄	—	19	1	no	—
246	C₄⋊C₄	—	19	1	no	—
247	C₄⋊C₄	—	38	1	no	—
248	C₄⋊C₄	—	38	1	no	—
249	C₄⋊C₄	—	38	1	no	—
250	C₄⋊C₄	—	38	1	no	—
251	C₄⋊C₄	—	38	1	no	—
252	C₄⋊C₄	—	38	1	no	—
253	C₄⋊C₄	—	38	1	no	—
254	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	38	1	no	—
263	C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
264	C₂²⋊C₄	C₂²⋊D₁₉	1	1	yes	characteristic
265	C₄²	—	38	1	no	—
266	C₄²	C₂²×D₁₉	1	1	yes	characteristic
267	C₄²	C₂²×D₁₉	1	1	yes	characteristic
268	C₂³	—	2	1	no	—
269	C₂³	—	2	1	no	—
270	C₂³	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
271	Q₈	—	152	1	no	—
272	Q₈	—	152	1	no	—
273	D₄	—	2	1	no	—
274	D₄	—	2	1	no	—
275	D₄	—	2	1	no	—
276	D₄	—	2	1	no	—
277	D₄	—	8	1	no	—
278	D₄	—	8	1	no	—
279	D₄	—	8	1	no	—
280	D₄	—	8	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	8	1	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	38	1	no	—
288	C₂×C₄	—	38	1	no	—
289	C₂×C₄	—	38	1	no	—
290	C₂×C₄	—	38	1	no	—
291	C₂×C₄	—	152	1	no	—
292	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
298	C₈	—	38	1	no	—
299	C₈	—	38	1	no	—
300	C₈	—	38	1	no	—
301	C₈	—	38	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	76	1	no	—
320	C₄	—	76	1	no	—
321	C₄	—	76	1	no	—
322	C₄	—	76	1	no	—
323	C₄	#10597436	1	1	yes	characteristic
324	C₄	#10597436	1	1	yes	characteristic
325	C₂	—	4	1	no	—
326	C₂	—	8	1	no	—
327	C₂	—	8	1	no	—
328	C₂	#11927115	1	1	yes	characteristic
329	C₂	#11927347	1	1	yes	characteristic
330	C₂	#11927355	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting, prime radical	C₁₉×#540	C₂
socle	C₂²×C₁₉	C₂²×D₄
hypercenter	#540	D₁₉
hypocenter	C₁₉	#3241
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #540

Complements

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

N	H ≅ G/N	type
1. #11927172	C₂	semidirect
2. #11926954	C₂	semidirect
3. #11926943	C₂	semidirect
4. #11926908	C₂	semidirect
5. #11926890	C₂	semidirect
6. #11926865	C₂	semidirect
7. #11926865	C₂	semidirect
8. #11926860	C₂	semidirect
9. #11926855	C₂	semidirect
10. #11926844	C₂	semidirect
11. #11926844	C₂	semidirect
12. #11926834	C₂	semidirect
13. #11926824	C₂	semidirect
14. #11926605	C₂	semidirect
224. C₁₉	#3241	semidirect

Maximal subgroups

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

subgroup	index
#11927172	2
#11926954	2
#11926943	2
#11926908	2
#11926890	2
#11926865	2
#11926860	2
#11926855	2
#11926844	2
#11926834	2
#11926824	2
#11926605	2
C₁₉×#540	2
#3241	19