Subgroups — #14576323

Order 2432.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927271	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927266	C₂	1	1	yes	maximal, characteristic
3	#11926901	C₂	1	1	yes	maximal, characteristic
4	#11926900	C₂	1	1	yes	maximal, characteristic
5	#11926893	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926854	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926823	C₂	1	1	yes	maximal, characteristic
8	#11926769	C₂	1	1	yes	maximal, characteristic
9	#11926767	C₂	1	1	yes	maximal, characteristic
10	#11926753	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926668	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926638	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#64528	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926631	C₂	1	1	yes	maximal, characteristic
15	C₁₉×#479	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₉×#125	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
18	#10597393	C₂²	1	1	yes	characteristic
19	#10597390	C₂²	1	1	yes	characteristic
20	C₂×#64528	C₂²	1	1	yes	characteristic
21	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
22	C₂×#64516	C₂²	1	1	yes	characteristic
23	#10597344	—	2	1	no	—
24	#10597342	—	2	1	no	—
25	#10597339	C₂²	1	1	yes	characteristic
26	#10597336	C₂²	1	1	yes	characteristic
27	#10597335	—	2	1	no	—
28	#10597335	—	2	1	no	—
29	#10597334	—	2	1	no	—
30	#10597334	C₂²	1	1	yes	characteristic
31	#10597333	—	2	1	no	—
32	#10597333	C₂²	1	1	yes	characteristic
33	#10597333	C₂²	1	1	yes	characteristic
34	#10597333	C₂²	1	1	yes	characteristic
35	#10597322	—	2	1	no	—
36	#10597321	C₂²	1	1	yes	characteristic
37	#10597318	—	2	1	no	—
38	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
40	C₁₉×#107	C₂²	1	1	yes	characteristic
41	C₁₉×#106	C₂²	1	1	yes	characteristic
42	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₉×#103	C₂²	1	1	yes	characteristic
44	C₁₉×#103	C₂²	1	1	yes	characteristic
45	#10597292	C₂²	1	1	yes	characteristic
46	#10597278	C₂²	1	1	yes	characteristic
47	#10597273	C₂²	1	1	yes	characteristic
48	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
49	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
52	#10597267	C₂²	1	1	yes	characteristic
53	#10597267	C₂²	1	1	yes	characteristic
54	#10597267	C₂²	1	1	yes	characteristic
55	#10597266	C₂²	1	1	yes	characteristic
56	#10597266	C₂²	1	1	yes	characteristic
57	#10597264	C₂²	1	1	yes	characteristic
58	C₄×C₁₉⋊C₈	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
60	C₂×D₇₆	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₁₉	—	2	1	no	—
62	C₂×Dic₃₈	—	2	1	no	—
63	C₂×Dic₃₈	D₄	1	1	yes	characteristic
64	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
65	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₁₉	—	2	1	no	—
69	C₄⋊C₄×C₁₉	—	2	1	no	—
70	C₄⋊C₄×C₁₉	—	2	1	no	—
71	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
74	C₄²×C₁₉	—	2	1	no	—
75	C₄²×C₁₉	C₂³	1	1	yes	characteristic
76	#64528	—	2	2	no	—
77	(C₂×D₁₉)⋊C₄	—	2	1	no	—
78	(C₂×D₁₉)⋊C₄	—	2	1	no	—
79	(C₂×D₁₉)⋊C₄	—	2	1	no	—
80	(C₂×D₁₉)⋊C₄	—	2	1	no	—
81	(C₂×D₁₉)⋊C₄	—	2	1	no	—
82	C₄⋊Dic₁₉	—	2	1	no	—
83	C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
84	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
85	Dic₁₉⋊C₄	—	2	1	no	—
86	Dic₁₉⋊C₄	—	2	1	no	—
87	Dic₁₉⋊C₄	—	2	1	no	—
88	Dic₁₉⋊C₄	—	2	1	no	—
89	Dic₁₉⋊C₄	—	2	1	no	—
90	Dic₁₉⋊C₄	—	2	1	no	—
91	Dic₁₉⋊C₄	—	2	1	no	—
92	Dic₁₉⋊C₄	—	2	1	no	—
93	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
94	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₁₉	—	2	1	no	—
96	C₄×Dic₁₉	—	2	1	no	—
97	C₄×Dic₁₉	—	2	1	no	—
98	C₄×Dic₁₉	—	2	1	no	—
99	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
100	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
101	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
103	Dic₇₆	—	2	2	no	—
104	#64516	—	2	2	no	—
105	C₂²×D₁₉	—	2	1	no	—
106	Q₈×C₁₉	—	2	1	no	—
107	Q₈×C₁₉	—	2	1	no	—
108	C₂×C₄×C₁₉	—	2	1	no	—
109	C₂×C₄×C₁₉	—	2	1	no	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₁₉	—	2	1	no	—
116	C₂×Dic₁₉	—	2	1	no	—
117	C₂×Dic₁₉	—	2	1	no	—
118	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
119	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
122	D₇₆	—	2	1	no	—
123	D₇₆	—	2	1	no	—
124	C₄×D₁₉	—	4	2	no	—
125	Dic₃₈	—	2	1	no	—
126	Dic₃₈	—	2	2	no	—
127	Dic₃₈	—	2	1	no	—
128	Dic₃₈	—	2	1	no	—
129	Dic₃₈	—	4	2	no	—
130	Dic₃₈	#41	1	2	yes	—
131	C₈×C₁₉	—	2	1	no	—
132	C₈×C₁₉	—	2	1	no	—
133	C₁₉⋊C₈	—	2	1	no	—
134	C₁₉⋊C₈	—	2	1	no	—
135	#3364	—	19	1	no	maximal, Sylow
136	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₁₉	—	4	1	no	—
138	C₂×D₁₉	—	4	1	no	—
139	C₂×D₁₉	—	4	1	no	—
140	C₄×C₁₉	—	2	1	no	—
141	C₄×C₁₉	—	2	1	no	—
142	C₄×C₁₉	—	4	1	no	—
143	C₄×C₁₉	—	4	1	no	—
144	C₄×C₁₉	—	4	1	no	—
145	C₄×C₁₉	—	4	1	no	—
146	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
147	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₁₉	—	2	2	no	—
149	Dic₁₉	—	2	1	no	—
150	Dic₁₉	—	2	1	no	—
151	Dic₁₉	—	2	1	no	—
152	Dic₁₉	—	4	1	no	—
153	Dic₁₉	—	4	1	no	—
154	Dic₁₉	#135	1	2	yes	—
155	#557	—	19	1	no	—
156	#556	—	19	1	no	—
157	#488	—	19	1	no	—
158	#488	—	19	1	no	—
159	#487	—	19	1	no	—
160	#479	—	19	1	no	—
161	#479	D₁₉	1	1	yes	characteristic, complemented
162	#474	—	19	1	no	—
163	#464	—	19	1	no	—
164	#464	—	19	1	no	—
165	#462	—	19	1	no	—
166	C₈×Q₈	—	19	1	no	—
167	C₄×Q₁₆	—	19	1	no	—
168	C₄×Q₁₆	—	19	1	no	—
169	C₄×SD₁₆	—	19	1	no	—
170	C₂×C₁₉	#548	1	1	yes	characteristic
171	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
172	C₂×C₁₉	#578	1	1	yes	characteristic
173	D₁₉	—	8	1	no	—
174	C₂×Q₁₆	—	19	1	no	—
175	C₂×Q₁₆	—	19	1	no	—
176	C₂×SD₁₆	—	19	1	no	—
177	C₄⋊Q₈	—	19	1	no	—
178	C₄⋊Q₈	—	19	1	no	—
179	C₄⋊Q₈	—	38	1	no	—
180	#126	—	38	1	no	—
181	#126	—	38	1	no	—
182	#125	—	19	1	no	—
183	#125	—	38	1	no	—
184	#125	—	38	1	no	—
185	#125	C₂×D₁₉	1	1	yes	characteristic
186	#124	—	19	1	no	—
187	#124	—	19	1	no	—
188	C₂²⋊Q₈	—	38	1	no	—
189	C₄×Q₈	—	19	1	no	—
190	C₄×Q₈	—	19	1	no	—
191	C₄×Q₈	—	19	1	no	—
192	C₄×Q₈	—	19	1	no	—
193	C₄×Q₈	—	19	1	no	—
194	C₄×Q₈	—	38	1	no	—
195	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
196	C₄×D₄	—	19	1	no	—
197	#117	—	38	1	no	—
198	#107	—	19	1	no	—
199	#107	C₂×D₁₉	1	1	yes	characteristic
200	#106	C₂×D₁₉	1	1	yes	characteristic
201	C₄⋊C₈	—	19	1	no	—
202	C₄⋊C₈	—	19	1	no	—
203	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
204	#103	—	19	1	no	—
205	#103	—	19	1	no	—
206	#103	—	19	1	no	—
207	#103	—	19	1	no	—
208	#103	—	19	1	no	—
209	#103	C₂×D₁₉	1	1	yes	characteristic
210	#103	C₂×D₁₉	1	1	yes	characteristic
211	#102	—	19	1	no	—
212	#102	—	19	1	no	—
213	#102	—	19	1	no	—
214	C₄×C₈	—	19	1	no	—
215	C₄×C₈	—	19	1	no	—
216	C₄×C₈	—	19	1	no	—
217	C₁₉	#3364	1	1	yes	characteristic, complemented, Sylow
218	C₂×Q₈	—	19	1	no	—
219	C₂×Q₈	—	19	1	no	—
220	C₂×Q₈	—	19	1	no	—
221	C₂×Q₈	—	38	1	no	—
222	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
223	C₂×D₄	—	19	1	no	—
224	C₂²×C₄	—	38	1	no	—
225	Q₁₆	—	38	2	no	—
226	Q₁₆	—	38	2	no	—
227	SD₁₆	—	38	2	no	—
228	C₂×C₈	—	19	1	no	—
229	C₂×C₈	—	19	1	no	—
230	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	19	1	no	—
236	C₄⋊C₄	—	19	1	no	—
237	C₄⋊C₄	—	19	1	no	—
238	C₄⋊C₄	—	19	1	no	—
239	C₄⋊C₄	—	38	1	no	—
240	C₄⋊C₄	—	38	1	no	—
241	C₄⋊C₄	—	38	1	no	—
242	C₄⋊C₄	—	38	1	no	—
243	C₄⋊C₄	—	38	1	no	—
244	C₄⋊C₄	—	38	1	no	—
245	C₄⋊C₄	—	38	1	no	—
246	C₄⋊C₄	—	38	1	no	—
247	C₄⋊C₄	—	38	1	no	—
248	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
251	C₂²⋊C₄	—	38	1	no	—
252	C₂²⋊C₄	—	38	1	no	—
253	C₂²⋊C₄	—	38	1	no	—
254	C₂²⋊C₄	—	38	1	no	—
255	C₂²⋊C₄	—	38	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	19	1	no	—
258	C₄²	—	19	1	no	—
259	C₄²	—	38	1	no	—
260	C₄²	—	38	1	no	—
261	C₄²	—	38	1	no	—
262	C₄²	—	38	1	no	—
263	C₄²	C₂²×D₁₉	1	1	yes	characteristic
264	C₂³	—	38	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	19	2	no	—
268	Q₈	—	38	2	no	—
269	Q₈	—	38	1	no	—
270	Q₈	—	38	1	no	—
271	Q₈	—	38	1	no	—
272	Q₈	—	76	2	no	—
273	D₄	—	38	1	no	—
274	D₄	—	38	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	19	1	no	—
280	C₂×C₄	—	19	1	no	—
281	C₂×C₄	—	19	1	no	—
282	C₂×C₄	—	19	1	no	—
283	C₂×C₄	—	38	1	no	—
284	C₂×C₄	—	38	1	no	—
285	C₂×C₄	—	38	1	no	—
286	C₂×C₄	—	76	2	no	—
287	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
288	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	38	1	no	—
293	C₈	—	38	1	no	—
294	C₂²	—	76	1	no	—
295	C₂²	—	76	1	no	—
296	C₂²	—	76	1	no	—
297	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	19	2	no	—
305	C₄	—	38	1	no	—
306	C₄	—	38	1	no	—
307	C₄	—	38	1	no	—
308	C₄	—	38	2	no	—
309	C₄	—	76	1	no	—
310	C₄	—	76	1	no	—
311	C₄	#10597433	1	1	yes	characteristic
312	C₄	#41×D₁₉	1	1	yes	characteristic
313	C₂	—	152	1	no	—
314	C₂	#11927204	1	1	yes	characteristic
315	C₂	#135×D₁₉	1	1	yes	characteristic
316	C₂	#11927360	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₉
commutator, cosocle	C₂×C₄×C₁₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉
Fitting	C₁₉×#479	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂×D₇₆	C₂³
hypercenter	#479	D₁₉
hypocenter	C₁₉	#3364
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
1. #11927271	C₂	semidirect
5. #11926893	C₂	semidirect
6. #11926854	C₂	semidirect
10. #11926753	C₂	semidirect
11. #11926668	C₂	semidirect
12. #11926638	C₂	semidirect
13. C₄×#64528	C₂	semidirect
15. C₁₉×#479	C₂	semidirect
161. #479	D₁₉	semidirect
217. C₁₉	#3364	semidirect

Maximal subgroups

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

subgroup	index
#11927271	2
#11927266	2
#11926901	2
#11926900	2
#11926893	2
#11926854	2
#11926823	2
#11926769	2
#11926767	2
#11926753	2
#11926668	2
#11926638	2
C₄×#64528	2
#11926631	2
C₁₉×#479	2
#3364	19