Subgroups — #14576376

Order 2432.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₃₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927266	C₂	1	1	yes	maximal, characteristic
3	#11926905	C₂	1	1	yes	maximal, characteristic
4	#11926904	C₂	1	1	yes	maximal, characteristic
5	#11926893	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926833	C₂	1	1	yes	maximal, characteristic
7	#11926829	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926828	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926768	C₂	1	1	yes	maximal, characteristic
10	#11926766	C₂	1	1	yes	maximal, characteristic
11	#11926752	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926672	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926648	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926629	C₂	1	1	yes	maximal, characteristic
15	C₁₉×#441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×C₁₉	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
19	#10597393	C₂²	1	1	yes	characteristic
20	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
21	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
22	#10597390	C₂²	1	1	yes	characteristic
23	C₂×#64528	C₂²	1	1	yes	characteristic
24	C₂×#64527	C₂²	1	1	yes	characteristic
25	#10597344	—	2	1	no	—
26	#10597342	—	2	1	no	—
27	#10597339	C₂²	1	1	yes	characteristic
28	#10597334	—	2	1	no	—
29	#10597334	—	2	1	no	—
30	#10597334	C₂²	1	1	yes	characteristic
31	#10597334	C₂²	1	1	yes	characteristic
32	#10597333	C₂²	1	1	yes	characteristic
33	#10597322	—	2	1	no	—
34	#10597321	C₂²	1	1	yes	characteristic
35	#10597318	—	2	1	no	—
36	#10597315	—	2	1	no	—
37	#10597315	C₂²	1	1	yes	characteristic
38	C₄×Dic₃₈	—	2	1	no	—
39	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
40	C₁₉×#106	C₂²	1	1	yes	characteristic
41	C₁₉×#103	C₂²	1	1	yes	characteristic
42	C₁₉×#103	C₂²	1	1	yes	characteristic
43	C₁₉×#97	C₂²	1	1	yes	characteristic
44	#10597292	C₂²	1	1	yes	characteristic
45	#10597292	C₂²	1	1	yes	characteristic
46	#10597278	C₂²	1	1	yes	characteristic
47	#10597278	C₂²	1	1	yes	characteristic
48	#10597273	C₂²	1	1	yes	characteristic
49	#10597273	C₂²	1	1	yes	characteristic
50	#10597272	C₂²	1	1	yes	characteristic
51	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
52	Dic₁₉⋊C₈	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₉	C₂²	1	1	yes	characteristic
54	#10597267	C₂²	1	1	yes	characteristic
55	#10597266	C₂²	1	1	yes	characteristic
56	#10597265	C₂²	1	1	yes	characteristic
57	#10597265	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×Q₈×C₁₉	C₂³	1	1	yes	characteristic
61	C₂×D₇₆	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₉	—	2	1	no	—
63	C₂×Dic₃₈	—	2	1	no	—
64	C₂×Dic₃₈	—	2	1	no	—
65	C₂×Dic₃₈	—	2	1	no	—
66	C₂×Dic₃₈	D₄	1	1	yes	characteristic
67	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
68	Q₁₆×C₁₉	—	2	2	no	—
69	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₉	—	2	1	no	—
72	C₄⋊C₄×C₁₉	—	2	1	no	—
73	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
75	C₄²×C₁₉	—	2	1	no	—
76	C₄²×C₁₉	—	2	1	no	—
77	C₄²×C₁₉	C₂³	1	1	yes	characteristic
78	#64528	—	2	2	no	—
79	#64527	—	2	2	no	—
80	(C₂×D₁₉)⋊C₄	—	2	1	no	—
81	(C₂×D₁₉)⋊C₄	—	2	1	no	—
82	(C₂×D₁₉)⋊C₄	—	2	1	no	—
83	(C₂×D₁₉)⋊C₄	—	2	1	no	—
84	(C₂×D₁₉)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₁₉	—	2	1	no	—
86	C₄⋊Dic₁₉	—	2	1	no	—
87	C₄⋊Dic₁₉	—	2	1	no	—
88	C₄⋊Dic₁₉	D₄	1	1	yes	characteristic
89	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₁₉	C₂³	1	1	yes	characteristic
91	Dic₁₉⋊C₄	—	2	1	no	—
92	Dic₁₉⋊C₄	—	2	1	no	—
93	Dic₁₉⋊C₄	—	2	1	no	—
94	Dic₁₉⋊C₄	—	2	1	no	—
95	Dic₁₉⋊C₄	—	2	1	no	—
96	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
97	Dic₁₉⋊C₄	D₄	1	1	yes	characteristic
98	C₄×Dic₁₉	—	2	1	no	—
99	C₄×Dic₁₉	—	2	1	no	—
100	C₄×Dic₁₉	—	2	1	no	—
101	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
102	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
103	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂²×D₁₉	—	2	1	no	—
106	Q₈×C₁₉	—	2	1	no	—
107	Q₈×C₁₉	—	2	1	no	—
108	Q₈×C₁₉	—	2	1	no	—
109	Q₈×C₁₉	#41	1	2	yes	—
110	C₂×C₄×C₁₉	—	2	1	no	—
111	C₂×C₄×C₁₉	—	2	1	no	—
112	C₂×C₄×C₁₉	—	2	1	no	—
113	C₂×C₄×C₁₉	—	2	1	no	—
114	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₁₉	—	2	1	no	—
118	C₂×Dic₁₉	—	2	1	no	—
119	C₂×Dic₁₉	—	2	1	no	—
120	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
124	D₇₆	—	2	1	no	—
125	D₇₆	—	2	1	no	—
126	C₄×D₁₉	—	4	2	no	—
127	Dic₃₈	—	2	2	no	—
128	Dic₃₈	—	2	1	no	—
129	Dic₃₈	—	2	1	no	—
130	Dic₃₈	—	8	1	no	—
131	Dic₃₈	—	8	1	no	—
132	Dic₃₈	—	8	1	no	—
133	C₈×C₁₉	—	2	1	no	—
134	C₈×C₁₉	—	2	1	no	—
135	C₁₉⋊C₈	—	2	1	no	—
136	C₁₉⋊C₈	—	2	1	no	—
137	#3369	—	19	1	no	maximal, Sylow
138	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₁₉	—	4	1	no	—
140	C₂×D₁₉	—	4	1	no	—
141	C₂×D₁₉	—	4	1	no	—
142	C₄×C₁₉	—	2	2	no	—
143	C₄×C₁₉	—	2	1	no	—
144	C₄×C₁₉	—	2	1	no	—
145	C₄×C₁₉	—	4	1	no	—
146	C₄×C₁₉	—	4	1	no	—
147	C₄×C₁₉	—	4	1	no	—
148	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
149	C₄×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
150	Dic₁₉	—	2	1	no	—
151	Dic₁₉	—	2	1	no	—
152	Dic₁₉	—	2	1	no	—
153	Dic₁₉	—	4	1	no	—
154	Dic₁₉	—	4	1	no	—
155	Dic₁₉	—	4	1	no	—
156	Dic₁₉	#137	1	2	yes	—
157	Q₈²	—	19	1	no	—
158	#556	—	19	1	no	—
159	#489	—	19	1	no	—
160	#489	—	19	1	no	—
161	#487	—	19	1	no	—
162	#476	—	19	1	no	—
163	#475	—	19	1	no	—
164	#475	—	19	1	no	—
165	#464	—	19	1	no	—
166	#464	—	19	1	no	—
167	#462	—	19	1	no	—
168	#446	—	19	1	no	—
169	#441	—	19	1	no	—
170	#441	D₁₉	1	1	yes	characteristic, complemented
171	C₄×SD₁₆	—	19	1	no	—
172	C₂×C₁₉	#548	1	1	yes	characteristic
173	C₂×C₁₉	C₂×#137	1	1	yes	characteristic
174	C₂×C₁₉	#577	1	1	yes	characteristic
175	D₁₉	—	8	1	no	—
176	C₂×Q₁₆	—	19	1	no	—
177	C₂×Q₁₆	C₂×D₁₉	1	1	yes	characteristic
178	C₂×SD₁₆	—	19	1	no	—
179	C₄⋊Q₈	—	19	1	no	—
180	C₄⋊Q₈	—	19	1	no	—
181	C₄⋊Q₈	—	19	1	no	—
182	C₄⋊Q₈	—	19	1	no	—
183	C₄⋊Q₈	—	38	1	no	—
184	C₄⋊Q₈	—	38	1	no	—
185	C₄⋊Q₈	—	38	1	no	—
186	#126	—	38	1	no	—
187	#126	—	38	1	no	—
188	#124	—	19	1	no	—
189	#124	—	19	1	no	—
190	C₂²⋊Q₈	—	38	1	no	—
191	C₄×Q₈	—	19	1	no	—
192	C₄×Q₈	—	19	1	no	—
193	C₄×Q₈	—	19	1	no	—
194	C₄×Q₈	—	19	1	no	—
195	C₄×Q₈	—	38	1	no	—
196	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
197	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
198	C₄×D₄	—	19	1	no	—
199	#117	—	38	1	no	—
200	#106	—	19	1	no	—
201	#106	—	19	1	no	—
202	#106	C₂×D₁₉	1	1	yes	characteristic
203	C₄⋊C₈	—	19	1	no	—
204	C₄⋊C₈	—	19	1	no	—
205	C₄⋊C₈	—	19	1	no	—
206	#103	—	19	1	no	—
207	#103	—	19	1	no	—
208	#103	—	19	1	no	—
209	#103	—	19	1	no	—
210	#103	—	19	1	no	—
211	#103	C₂×D₁₉	1	1	yes	characteristic
212	#103	C₂×D₁₉	1	1	yes	characteristic
213	#102	—	19	1	no	—
214	#102	—	19	1	no	—
215	#102	—	19	1	no	—
216	#97	—	19	1	no	—
217	#97	C₂×D₁₉	1	1	yes	characteristic
218	C₄×C₈	—	19	1	no	—
219	C₁₉	#3369	1	1	yes	characteristic, complemented, Sylow
220	C₂×Q₈	—	19	1	no	—
221	C₂×Q₈	—	19	1	no	—
222	C₂×Q₈	—	38	1	no	—
223	C₂×Q₈	—	38	1	no	—
224	C₂×Q₈	—	38	1	no	—
225	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
226	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
227	C₂×D₄	—	19	1	no	—
228	C₂²×C₄	—	38	1	no	—
229	Q₁₆	—	2	2	no	—
230	Q₁₆	—	38	2	no	—
231	SD₁₆	—	38	2	no	—
232	C₂×C₈	—	19	1	no	—
233	C₂×C₈	—	19	1	no	—
234	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	19	1	no	—
239	C₄⋊C₄	—	19	1	no	—
240	C₄⋊C₄	—	19	1	no	—
241	C₄⋊C₄	—	19	1	no	—
242	C₄⋊C₄	—	19	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	—	38	1	no	—
249	C₄⋊C₄	—	38	1	no	—
250	C₄⋊C₄	—	38	1	no	—
251	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
253	C₂²⋊C₄	—	38	1	no	—
254	C₂²⋊C₄	—	38	1	no	—
255	C₂²⋊C₄	—	38	1	no	—
256	C₂²⋊C₄	—	38	1	no	—
257	C₂²⋊C₄	—	38	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	19	1	no	—
261	C₄²	—	19	1	no	—
262	C₄²	—	38	1	no	—
263	C₄²	—	38	1	no	—
264	C₄²	—	38	1	no	—
265	C₄²	C₂²×D₁₉	1	1	yes	characteristic
266	C₂³	—	38	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	38	2	no	—
271	Q₈	—	38	1	no	—
272	Q₈	—	38	1	no	—
273	Q₈	—	152	1	no	—
274	Q₈	—	152	1	no	—
275	Q₈	—	152	1	no	—
276	Q₈	#64541	1	2	yes	—
277	D₄	—	38	1	no	—
278	D₄	—	38	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	19	1	no	—
284	C₂×C₄	—	19	1	no	—
285	C₂×C₄	—	19	1	no	—
286	C₂×C₄	—	19	1	no	—
287	C₂×C₄	—	38	1	no	—
288	C₂×C₄	—	38	1	no	—
289	C₂×C₄	—	38	1	no	—
290	C₂×C₄	—	76	2	no	—
291	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
292	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	38	1	no	—
297	C₈	—	38	1	no	—
298	C₂²	—	76	1	no	—
299	C₂²	—	76	1	no	—
300	C₂²	—	76	1	no	—
301	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	19	2	no	—
309	C₄	—	38	1	no	—
310	C₄	—	38	1	no	—
311	C₄	—	38	1	no	—
312	C₄	—	76	1	no	—
313	C₄	—	76	1	no	—
314	C₄	—	76	1	no	—
315	C₄	C₂×#64541	1	1	yes	characteristic
316	C₄	#10597436	1	1	yes	characteristic
317	C₂	—	152	1	no	—
318	C₂	#11927203	1	1	yes	characteristic
319	C₂	#137×D₁₉	1	1	yes	characteristic
320	C₂	#11927353	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×#441	C₂
socle	C₂²×C₁₉	C₂²×D₄
prime radical	C₂×D₇₆	C₂³
hypercenter	#441	D₁₉
hypocenter	C₁₉	#3369
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₈	C₂	semidirect
5. #11926893	C₂	semidirect
7. #11926829	C₂	semidirect
8. #11926828	C₂	semidirect
11. #11926752	C₂	semidirect
12. #11926672	C₂	semidirect
13. #11926648	C₂	semidirect
15. C₁₉×#441	C₂	semidirect
170. #441	D₁₉	semidirect
219. C₁₉	#3369	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃₈	2
#11927266	2
#11926905	2
#11926904	2
#11926893	2
#11926833	2
#11926829	2
#11926828	2
#11926768	2
#11926766	2
#11926752	2
#11926672	2
#11926648	2
#11926629	2
C₁₉×#441	2
#3369	19