Subgroups — #14573720

Order 2432.

390 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927079	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#11926876	C₂	1	1	yes	maximal, characteristic, complemented
4	#11926859	C₂	1	1	yes	maximal, characteristic
5	#11926848	C₂	1	1	yes	maximal, characteristic, complemented
6	#11926767	C₂	1	1	yes	maximal, characteristic, complemented
7	#11926766	C₂	1	1	yes	maximal, characteristic, complemented
8	#11926701	C₂	1	1	yes	maximal, characteristic, complemented
9	#11926684	C₂	1	1	yes	maximal, characteristic
10	#11926638	C₂	1	1	yes	maximal, characteristic, complemented
11	#11926636	C₂	1	1	yes	maximal, characteristic, complemented
12	#11926631	C₂	1	1	yes	maximal, characteristic, complemented
13	#11926629	C₂	1	1	yes	maximal, characteristic, complemented
14	#11926596	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₉×#479	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#125	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
19	Q₈×Dic₁₉	C₂²	1	1	yes	characteristic
20	C₂×#64528	C₂²	1	1	yes	characteristic
21	C₂×#64527	C₂²	1	1	yes	characteristic
22	C₂×Dic₇₆	C₂²	1	1	yes	characteristic
23	C₂×#64516	C₂²	1	1	yes	characteristic
24	#10597344	—	2	1	no	—
25	#10597343	C₂²	1	1	yes	characteristic
26	#10597341	C₂²	1	1	yes	characteristic
27	#10597340	—	2	1	no	—
28	#10597339	—	2	1	no	—
29	#10597339	C₂²	1	1	yes	characteristic
30	#10597338	—	2	1	no	—
31	C₄⋊C₄×D₁₉	—	2	1	no	—
32	C₄⋊C₄×D₁₉	C₂²	1	1	yes	characteristic
33	#10597333	—	2	1	no	—
34	#10597333	C₂²	1	1	yes	characteristic
35	#10597333	C₂²	1	1	yes	characteristic
36	#10597321	C₂²	1	1	yes	characteristic
37	C₄²×D₁₉	—	2	1	no	—
38	C₄²×D₁₉	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₈	—	2	1	no	—
40	C₄×Dic₃₈	C₂²	1	1	yes	characteristic
41	C₁₉×#107	C₂²	1	1	yes	characteristic
42	C₁₉×#106	C₂²	1	1	yes	characteristic
43	C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₉×#103	C₂²	1	1	yes	characteristic
45	C₁₉×#103	C₂²	1	1	yes	characteristic
46	#10597278	C₂²	1	1	yes	characteristic
47	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
48	(C₂×D₁₉)⋊C₈	C₂²	1	1	yes	characteristic
49	#10597273	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	#10597265	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	Q₈×D₁₉	—	2	2	no	—
61	Q₈×D₁₉	—	2	2	no	—
62	C₂×D₇₆	—	2	1	no	—
63	C₂×D₇₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₉	—	2	1	no	—
65	C₂×C₄×D₁₉	—	2	1	no	—
66	C₂×C₄×D₁₉	—	2	1	no	—
67	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
70	C₂×Dic₃₈	—	2	1	no	—
71	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
72	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₉	—	2	1	no	—
76	C₄⋊C₄×C₁₉	—	2	1	no	—
77	C₄⋊C₄×C₁₉	—	2	1	no	—
78	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
81	C₄²×C₁₉	—	2	1	no	—
82	C₄²×C₁₉	C₂³	1	1	yes	characteristic
83	#64528	—	2	2	no	—
84	#64527	—	2	2	no	—
85	(C₂×D₁₉)⋊C₄	—	2	1	no	—
86	(C₂×D₁₉)⋊C₄	—	2	1	no	—
87	(C₂×D₁₉)⋊C₄	—	2	1	no	—
88	(C₂×D₁₉)⋊C₄	—	2	1	no	—
89	(C₂×D₁₉)⋊C₄	—	2	1	no	—
90	C₄⋊Dic₁₉	—	2	1	no	—
91	C₄⋊Dic₁₉	—	2	1	no	—
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	C₄×Dic₁₉	—	2	1	no	—
99	C₄×Dic₁₉	—	2	1	no	—
100	C₄×Dic₁₉	—	2	1	no	—
101	C₄×Dic₁₉	D₄	1	1	yes	characteristic
102	C₄×Dic₁₉	D₄	1	1	yes	characteristic
103	C₄×Dic₁₉	C₂³	1	1	yes	characteristic
104	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
106	Dic₇₆	—	2	2	no	—
107	#64516	—	2	2	no	—
108	C₂²×D₁₉	—	2	1	no	—
109	C₂²×D₁₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₁₉	—	2	1	no	—
111	Q₈×C₁₉	#41	1	2	yes	—
112	C₂×C₄×C₁₉	—	2	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	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
119	C₂×Dic₁₉	—	2	1	no	—
120	C₂×Dic₁₉	—	2	1	no	—
121	C₂×Dic₁₉	—	2	1	no	—
122	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₉	C₂×D₄	1	1	yes	characteristic
125	D₇₆	—	2	1	no	—
126	D₇₆	—	2	1	no	—
127	D₇₆	—	4	2	no	—
128	C₄×D₁₉	—	2	4	no	—
129	C₄×D₁₉	—	2	2	no	—
130	C₄×D₁₉	—	2	1	no	—
131	C₄×D₁₉	—	2	1	no	—
132	C₄×D₁₉	—	2	2	no	—
133	C₄×D₁₉	—	4	2	no	—
134	C₄×D₁₉	—	4	2	no	—
135	Dic₃₈	—	2	4	no	—
136	Dic₃₈	—	2	1	no	—
137	Dic₃₈	—	2	1	no	—
138	Dic₃₈	—	2	1	no	—
139	Dic₃₈	#41	1	2	yes	—
140	C₈×C₁₉	—	2	1	no	—
141	C₈×C₁₉	—	2	1	no	—
142	C₁₉⋊C₈	—	2	1	no	—
143	C₁₉⋊C₈	—	2	1	no	—
144	#3108	—	19	1	no	maximal, Sylow
145	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₉	—	2	1	no	—
147	C₂×D₁₉	—	2	1	no	—
148	C₂×D₁₉	—	4	1	no	—
149	C₂×D₁₉	—	4	1	no	—
150	C₂×D₁₉	—	4	1	no	—
151	C₂×D₁₉	#135	1	2	yes	—
152	C₄×C₁₉	—	2	2	no	—
153	C₄×C₁₉	—	2	1	no	—
154	C₄×C₁₉	—	2	1	no	—
155	C₄×C₁₉	—	4	1	no	—
156	C₄×C₁₉	—	4	1	no	—
157	C₄×C₁₉	—	4	1	no	—
158	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
159	C₄×C₁₉	C₂×#41	1	1	yes	characteristic
160	Dic₁₉	—	2	1	no	—
161	Dic₁₉	—	2	2	no	—
162	Dic₁₉	—	2	1	no	—
163	Dic₁₉	—	4	1	no	—
164	Dic₁₉	—	4	1	no	—
165	Dic₁₉	#137	1	2	yes	—
166	#529	—	19	1	no	—
167	C₂×C₄×Q₈	—	19	1	no	—
168	#483	—	19	1	no	—
169	#480	—	19	1	no	—
170	#479	—	19	1	no	—
171	#479	D₁₉	1	1	yes	characteristic, complemented
172	#464	—	19	1	no	—
173	#464	—	19	1	no	—
174	#451	—	19	1	no	—
175	#448	—	19	1	no	—
176	C₄×Q₁₆	—	19	1	no	—
177	C₄×Q₁₆	—	19	1	no	—
178	C₄×SD₁₆	—	19	1	no	—
179	C₄×SD₁₆	—	19	1	no	—
180	#431	—	19	1	no	—
181	C₂×C₁₉	#525	1	1	yes	characteristic
182	C₂×C₁₉	C₂×#135	1	1	yes	characteristic
183	C₂×C₁₉	C₂×#137	1	1	yes	characteristic
184	D₁₉	—	2	2	no	—
185	D₁₉	—	8	1	no	—
186	C₂²×Q₈	—	19	1	no	—
187	C₂×Q₁₆	—	19	1	no	—
188	C₂×Q₁₆	—	19	1	no	—
189	C₂×SD₁₆	—	19	1	no	—
190	C₂×SD₁₆	—	19	1	no	—
191	#126	—	38	1	no	—
192	#125	C₂×D₁₉	1	1	yes	characteristic
193	#124	—	19	1	no	—
194	#123	—	38	1	no	—
195	C₂²⋊Q₈	—	19	1	no	—
196	#121	—	19	1	no	—
197	C₄×Q₈	—	19	1	no	—
198	C₄×Q₈	—	19	1	no	—
199	C₄×Q₈	—	19	1	no	—
200	C₄×Q₈	—	19	1	no	—
201	C₄×Q₈	—	38	1	no	—
202	C₄×Q₈	—	38	1	no	—
203	C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
204	C₄×D₄	—	19	1	no	—
205	C₄×D₄	—	38	1	no	—
206	#117	—	38	1	no	—
207	C₂×C₄⋊C₄	—	19	1	no	—
208	C₂×C₄⋊C₄	—	38	1	no	—
209	C₂×C₄²	—	19	1	no	—
210	C₂×C₄²	—	38	1	no	—
211	#107	—	19	1	no	—
212	#107	C₂×D₁₉	1	1	yes	characteristic
213	#106	—	19	1	no	—
214	#106	C₂×D₁₉	1	1	yes	characteristic
215	C₄⋊C₈	—	19	1	no	—
216	C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
217	#103	—	19	1	no	—
218	#103	—	19	1	no	—
219	#103	—	19	1	no	—
220	#103	—	19	1	no	—
221	#103	C₂×D₁₉	1	1	yes	characteristic
222	#103	C₂×D₁₉	1	1	yes	characteristic
223	#102	—	19	1	no	—
224	#102	—	19	1	no	—
225	C₂²⋊C₈	—	19	1	no	—
226	C₂²⋊C₈	—	19	1	no	—
227	C₄×C₈	—	19	1	no	—
228	C₄×C₈	—	19	1	no	—
229	C₁₉	#3108	1	1	yes	characteristic, complemented, Sylow
230	C₂×Q₈	—	19	1	no	—
231	C₂×Q₈	—	19	1	no	—
232	C₂×Q₈	—	38	1	no	—
233	C₂×Q₈	—	38	2	no	—
234	C₂×Q₈	—	38	2	no	—
235	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
236	C₂×D₄	—	19	1	no	—
237	C₂×D₄	—	38	1	no	—
238	C₂²×C₄	—	19	1	no	—
239	C₂²×C₄	—	19	1	no	—
240	C₂²×C₄	—	19	1	no	—
241	C₂²×C₄	—	38	1	no	—
242	C₂²×C₄	—	38	1	no	—
243	C₂²×C₄	—	38	1	no	—
244	Q₁₆	—	38	2	no	—
245	Q₁₆	—	38	2	no	—
246	SD₁₆	—	38	2	no	—
247	SD₁₆	—	38	2	no	—
248	C₂×C₈	—	19	1	no	—
249	C₂×C₈	—	19	1	no	—
250	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₁₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	19	1	no	—
256	C₄⋊C₄	—	38	1	no	—
257	C₄⋊C₄	—	38	1	no	—
258	C₄⋊C₄	—	38	1	no	—
259	C₄⋊C₄	—	38	1	no	—
260	C₄⋊C₄	—	38	1	no	—
261	C₄⋊C₄	—	38	1	no	—
262	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	38	1	no	—
266	C₂²⋊C₄	—	38	1	no	—
267	C₂²⋊C₄	—	38	1	no	—
268	C₂²⋊C₄	—	38	1	no	—
269	C₂²⋊C₄	—	38	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	19	1	no	—
272	C₄²	—	19	1	no	—
273	C₄²	—	19	1	no	—
274	C₄²	—	38	1	no	—
275	C₄²	—	38	1	no	—
276	C₄²	—	38	1	no	—
277	C₄²	—	38	1	no	—
278	C₄²	C₂²×D₁₉	1	1	yes	characteristic
279	C₂³	—	19	1	no	—
280	C₂³	—	38	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	19	2	no	—
283	Q₈	—	38	4	no	—
284	Q₈	—	38	1	no	—
285	Q₈	—	38	1	no	—
286	Q₈	—	38	1	no	—
287	Q₈	#64545	1	2	yes	—
288	D₄	—	38	1	no	—
289	D₄	—	38	1	no	—
290	D₄	—	76	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	19	1	no	—
296	C₂×C₄	—	19	1	no	—
297	C₂×C₄	—	19	1	no	—
298	C₂×C₄	—	38	2	no	—
299	C₂×C₄	—	38	2	no	—
300	C₂×C₄	—	38	4	no	—
301	C₂×C₄	—	38	1	no	—
302	C₂×C₄	—	38	1	no	—
303	C₂×C₄	—	38	1	no	—
304	C₂×C₄	—	38	1	no	—
305	C₂×C₄	—	38	1	no	—
306	C₂×C₄	—	76	2	no	—
307	C₂×C₄	—	76	2	no	—
308	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
309	C₂×C₄	D₄×D₁₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	38	1	no	—
314	C₈	—	38	1	no	—
315	C₂²	—	19	2	no	—
316	C₂²	—	38	1	no	—
317	C₂²	—	38	1	no	—
318	C₂²	—	76	1	no	—
319	C₂²	—	76	1	no	—
320	C₂²	—	76	1	no	—
321	C₂²	C₂×D₄×D₁₉	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	19	2	no	—
329	C₄	—	38	1	no	—
330	C₄	—	38	2	no	—
331	C₄	—	38	1	no	—
332	C₄	—	76	1	no	—
333	C₄	—	76	1	no	—
334	C₄	C₂×#64545	1	1	yes	characteristic
335	C₄	#41×D₁₉	1	1	yes	characteristic
336	C₂	—	38	2	no	—
337	C₂	—	152	1	no	—
338	C₂	#11927057	1	1	yes	characteristic
339	C₂	#135×D₁₉	1	1	yes	characteristic
340	C₂	#137×D₁₉	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	#10597341	C₂²
hypercenter	#479	D₁₉
hypocenter	C₁₉	#3108
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
2. C₄×Q₈×D₁₉	C₂	semidirect
3. #11926876	C₂	semidirect
5. #11926848	C₂	semidirect
6. #11926767	C₂	semidirect
7. #11926766	C₂	semidirect
8. #11926701	C₂	semidirect
10. #11926638	C₂	semidirect
11. #11926636	C₂	semidirect
12. #11926631	C₂	semidirect
13. #11926629	C₂	semidirect
14. #11926596	C₂	semidirect
15. C₁₉×#479	C₂	semidirect
171. #479	D₁₉	semidirect
229. C₁₉	#3108	semidirect

Maximal subgroups

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

subgroup	index
#11927079	2
C₄×Q₈×D₁₉	2
#11926876	2
#11926859	2
#11926848	2
#11926767	2
#11926766	2
#11926701	2
#11926684	2
#11926638	2
#11926636	2
#11926631	2
#11926629	2
#11926596	2
C₁₉×#479	2
#3108	19