Subgroups — #14577305

Order 2432.

1658 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	C₂×#10597435	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#41×D₁₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#11927216	C₂	1	2	yes	maximal, complemented
4	#11927206	C₂	1	2	yes	maximal, complemented
5	#11927189	C₂	1	6	yes	maximal, complemented
6	D₄×D₇₆	C₂	1	6	yes	maximal, complemented
7	#11927125	C₂	1	3	yes	maximal, complemented
8	#11927113	C₂	1	3	yes	maximal, complemented
9	#11927056	C₂	1	3	yes	maximal, complemented
10	C₂×#10597341	C₂	1	3	yes	maximal, complemented
11	C₁₉×#520	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
13	#10597435	—	2	8	no	—
14	#41×D₁₉	—	2	8	no	—
15	C₂×#64545	C₂²	1	1	yes	characteristic, complemented
16	C₂×#64545	C₂²	1	2	yes	complemented
17	C₂×Q₈×D₁₉	C₂²	1	1	yes	characteristic, complemented
18	C₂×#64543	C₂²	1	3	yes	complemented
19	C₂×D₄×D₁₉	C₂²	1	3	yes	complemented
20	C₂×D₄×D₁₉	C₂²	1	6	yes	complemented
21	C₂×#64541	C₂²	1	3	yes	complemented
22	C₂×#64541	C₂²	1	3	yes	complemented
23	C₂²×D₇₆	C₂²	1	3	yes	complemented
24	C₂²×D₇₆	C₂²	1	6	yes	complemented
25	C₂²×C₄×D₁₉	C₂²	1	3	yes	complemented
26	C₄×Q₈×C₁₉	C₂²	1	2	yes	complemented
27	C₄×D₄×C₁₉	C₂²	1	6	yes	complemented
28	C₁₉×#117	C₂²	1	3	yes	complemented
29	C₂×C₄⋊C₄×C₁₉	C₂²	1	3	yes	complemented
30	#10597376	C₂²	1	12	yes	complemented
31	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	6	yes	complemented
32	#10597343	C₂²	1	1	yes	characteristic, complemented
33	#10597343	C₂²	1	3	yes	complemented
34	#10597343	C₂²	1	6	yes	complemented
35	#10597341	C₂²	1	3	yes	complemented
36	#10597341	C₂²	1	3	yes	complemented
37	#10597341	C₂²	1	6	yes	complemented
38	#10597341	C₂²	1	6	yes	complemented
39	#10597341	C₂²	1	6	yes	complemented
40	#10597341	C₂²	1	6	yes	complemented
41	#10597332	C₂²	1	6	yes	complemented
42	#10597328	C₂²	1	6	yes	complemented
43	#10597328	C₂²	1	12	yes	complemented
44	#10597321	C₂²	1	6	yes	complemented
45	#10597320	C₂²	1	6	yes	complemented
46	C₄×D₇₆	C₂²	1	6	yes	complemented
47	C₄×D₇₆	C₂²	1	6	yes	complemented
48	C₂³×D₁₉	C₂³	1	3	yes	—
49	C₂³×D₁₉	C₂³	1	6	yes	—
50	#41×C₁₉	D₄	1	8	yes	complemented
51	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₁₉	C₂³	1	3	yes	—
53	C₂²×C₄×C₁₉	C₂³	1	6	yes	—
54	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
55	C₂×C₂²⋊D₁₉	C₂³	1	6	yes	—
56	C₂×C₂²⋊D₁₉	C₂³	1	6	yes	—
57	C₂²×Dic₁₉	C₂³	1	3	yes	—
58	#64545	—	2	4	no	—
59	#64545	—	2	8	no	—
60	Q₈×D₁₉	—	2	4	no	—
61	#64543	—	2	12	no	—
62	D₄×D₁₉	—	2	12	no	—
63	D₄×D₁₉	—	2	24	no	—
64	#64541	—	2	12	no	—
65	#64541	—	2	12	no	—
66	C₂×D₇₆	—	2	12	no	—
67	C₂×D₇₆	—	4	12	no	—
68	C₂×D₇₆	C₂³	1	3	yes	—
69	C₂×D₇₆	C₂³	1	6	yes	—
70	C₂×D₇₆	C₂³	1	6	yes	—
71	C₂×D₇₆	C₂³	1	3	yes	—
72	C₂×D₇₆	C₂³	1	12	yes	—
73	C₂×D₇₆	C₂³	1	6	yes	—
74	C₂×C₄×D₁₉	—	2	12	no	—
75	C₂×C₄×D₁₉	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₁₉	C₂³	1	3	yes	—
77	C₂×C₄×D₁₉	C₂³	1	3	yes	—
78	C₂×C₄×D₁₉	C₂³	1	3	yes	—
79	C₂×C₄×D₁₉	C₂³	1	6	yes	—
80	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
81	C₂×Dic₃₈	C₂³	1	3	yes	—
82	C₄⋊C₄×C₁₉	C₂³	1	3	yes	—
83	C₄⋊C₄×C₁₉	C₂³	1	6	yes	—
84	C₄⋊C₄×C₁₉	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₁₉	C₂³	1	6	yes	—
86	C₄²×C₁₉	C₂³	1	6	yes	—
87	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
88	(C₂×D₁₉)⋊C₄	C₂³	1	12	yes	—
89	(C₂×D₁₉)⋊C₄	C₂³	1	2	yes	—
90	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
91	(C₂×D₁₉)⋊C₄	C₂³	1	6	yes	—
92	C₄⋊Dic₁₉	C₂³	1	6	yes	—
93	C₄⋊Dic₁₉	C₂³	1	2	yes	—
94	C₂³×C₁₉	C₂⁴	1	3	yes	—
95	C₂²×D₁₉	—	2	6	no	—
96	C₂²×D₁₉	—	2	12	no	—
97	C₂²×D₁₉	—	2	6	no	—
98	C₂²×D₁₉	—	4	3	no	—
99	C₂²×D₁₉	—	4	6	no	—
100	C₂²×D₁₉	—	4	6	no	—
101	C₂²×D₁₉	C₂⁴	1	6	yes	—
102	C₂²×D₁₉	C₂⁴	1	3	yes	—
103	C₂²×D₁₉	C₂⁴	1	1	yes	characteristic
104	Q₈×C₁₉	C₂×D₄	1	4	yes	complemented
105	D₄×C₁₉	C₂×D₄	1	12	yes	complemented
106	C₂×C₄×C₁₉	—	4	6	no	—
107	C₂×C₄×C₁₉	C₂×D₄	1	12	yes	complemented
108	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
109	C₂×C₄×C₁₉	C₂⁴	1	6	yes	—
110	C₂×C₄×C₁₉	C₂⁴	1	3	yes	—
111	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₁₉	—	2	12	no	—
113	C₂²⋊D₁₉	—	2	12	no	—
114	C₂×Dic₁₉	—	2	6	no	—
115	C₂×Dic₁₉	C₂⁴	1	2	yes	—
116	C₂×Dic₁₉	C₂⁴	1	3	yes	—
117	C₂×Dic₁₉	C₂⁴	1	1	yes	characteristic
118	D₇₆	—	2	12	no	—
119	D₇₆	—	2	6	no	—
120	D₇₆	—	2	12	no	—
121	D₇₆	—	2	6	no	—
122	D₇₆	—	4	12	no	—
123	D₇₆	—	4	6	no	—
124	C₄×D₁₉	—	2	2	no	—
125	C₄×D₁₉	—	2	6	no	—
126	C₄×D₁₉	—	2	6	no	—
127	C₄×D₁₉	—	2	6	no	—
128	C₄×D₁₉	—	2	12	no	—
129	Dic₃₈	—	2	2	no	—
130	Dic₃₈	—	2	6	no	—
131	#3488	—	19	1	no	maximal, Sylow
132	C₂²×C₁₉	—	2	3	no	—
133	C₂²×C₁₉	—	2	3	no	—
134	C₂²×C₁₉	C₂²×D₄	1	6	yes	—
135	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
136	C₂×D₁₉	—	2	1	no	—
137	C₂×D₁₉	—	2	1	no	—
138	C₂×D₁₉	—	2	3	no	—
139	C₂×D₁₉	—	2	6	no	—
140	C₂×D₁₉	—	2	6	no	—
141	C₂×D₁₉	—	2	3	no	—
142	C₂×D₁₉	—	2	3	no	—
143	C₂×D₁₉	—	2	6	no	—
144	C₂×D₁₉	—	4	12	no	—
145	C₂×D₁₉	—	8	12	no	—
146	C₂×D₁₉	H(5,𝔽₂)	1	2	yes	complemented
147	C₄×C₁₉	—	2	6	no	—
148	C₄×C₁₉	—	2	2	no	—
149	C₄×C₁₉	C₂²×D₄	1	2	yes	—
150	C₄×C₁₉	C₂²×D₄	1	6	yes	—
151	Dic₁₉	—	2	3	no	—
152	Dic₁₉	—	2	2	no	—
153	Dic₁₉	—	2	1	no	—
154	C₂×H(5,𝔽₂)	—	19	1	no	—
155	C₂²×#41	—	19	1	no	—
156	#550	—	19	2	no	—
157	#548	—	19	2	no	—
158	#546	—	19	6	no	—
159	D₄²	—	19	6	no	—
160	#537	—	19	3	no	—
161	#535	—	19	3	no	—
162	#525	—	19	3	no	—
163	C₂×#121	—	19	3	no	—
164	#520	D₁₉	1	1	yes	characteristic, complemented
165	C₂×C₁₉	—	2	6	no	—
166	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
167	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×C₁₉	#585	1	1	yes	characteristic
169	D₁₉	—	2	2	no	—
170	D₁₉	—	4	3	no	—
171	D₁₉	—	4	6	no	—
172	H(5,𝔽₂)	—	38	8	no	—
173	C₂×#41	—	19	2	no	—
174	C₂×#41	—	19	1	no	—
175	C₂×#41	—	19	3	no	—
176	C₂×#41	—	19	3	no	—
177	C₂×#41	—	19	3	no	—
178	C₂×#41	—	38	8	no	—
179	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
180	C₂²×Q₈	—	19	1	no	—
181	C₂²×D₄	—	19	3	no	—
182	C₂²×D₄	—	19	6	no	—
183	C₂²×D₄	—	19	3	no	—
184	C₂²×D₄	—	19	6	no	—
185	C₂³×C₄	—	19	3	no	—
186	#127	—	19	6	no	—
187	#124	—	19	6	no	—
188	#123	—	19	6	no	—
189	C₂²⋊Q₈	—	19	1	no	—
190	C₂²⋊Q₈	—	19	3	no	—
191	C₂²⋊Q₈	—	19	6	no	—
192	#121	—	19	12	no	—
193	#121	—	19	3	no	—
194	#121	—	19	6	no	—
195	#121	—	19	6	no	—
196	#121	—	19	6	no	—
197	#121	—	19	3	no	—
198	#121	—	19	6	no	—
199	#120	—	19	6	no	—
200	#120	—	19	12	no	—
201	C₄×Q₈	C₂×D₁₉	1	2	yes	complemented
202	C₄×D₄	—	19	6	no	—
203	C₄×D₄	—	19	6	no	—
204	C₄×D₄	C₂×D₁₉	1	6	yes	complemented
205	#117	C₂×D₁₉	1	3	yes	complemented
206	C₂×C₄⋊C₄	C₂×D₁₉	1	3	yes	complemented
207	C₂×C₂²⋊C₄	—	19	6	no	—
208	C₁₉	#3488	1	1	yes	characteristic, complemented, Sylow
209	C₂⁴	—	19	3	no	—
210	C₂⁴	—	19	6	no	—
211	#41	—	38	4	no	—
212	#41	—	38	8	no	—
213	#41	—	38	12	no	—
214	#41	—	38	12	no	—
215	#41	—	38	12	no	—
216	#41	D₇₆	1	8	yes	complemented
217	C₂×Q₈	—	19	1	no	—
218	C₂×Q₈	—	19	3	no	—
219	C₂×Q₈	—	38	4	no	—
220	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
221	C₂×D₄	—	19	6	no	—
222	C₂×D₄	—	19	6	no	—
223	C₂×D₄	—	19	6	no	—
224	C₂×D₄	—	19	3	no	—
225	C₂×D₄	—	19	6	no	—
226	C₂×D₄	—	19	3	no	—
227	C₂×D₄	—	19	6	no	—
228	C₂×D₄	—	19	12	no	—
229	C₂×D₄	—	38	12	no	—
230	C₂×D₄	—	38	24	no	—
231	C₂×D₄	—	38	12	no	—
232	C₂×D₄	—	76	12	no	—
233	C₂×D₄	C₂²×D₁₉	1	3	yes	—
234	C₂²×C₄	—	19	3	no	—
235	C₂²×C₄	—	19	3	no	—
236	C₂²×C₄	—	19	3	no	—
237	C₂²×C₄	—	19	6	no	—
238	C₂²×C₄	—	19	1	no	—
239	C₂²×C₄	—	19	3	no	—
240	C₂²×C₄	—	38	12	no	—
241	C₂²×C₄	C₂²×D₁₉	1	6	yes	—
242	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
243	C₄⋊C₄	—	19	2	no	—
244	C₄⋊C₄	—	19	6	no	—
245	C₄⋊C₄	C₂²×D₁₉	1	3	yes	—
246	C₄⋊C₄	C₂²×D₁₉	1	6	yes	—
247	C₄⋊C₄	C₂²×D₁₉	1	1	yes	characteristic
248	C₂²⋊C₄	—	19	6	no	—
249	C₂²⋊C₄	—	19	12	no	—
250	C₂²⋊C₄	—	19	2	no	—
251	C₂²⋊C₄	—	19	6	no	—
252	C₂²⋊C₄	—	19	6	no	—
253	C₂²⋊C₄	C₂²×D₁₉	1	6	yes	—
254	C₄²	C₂²×D₁₉	1	6	yes	—
255	C₂³	—	19	1	no	—
256	C₂³	—	19	6	no	—
257	C₂³	—	19	3	no	—
258	C₂³	—	38	6	no	—
259	C₂³	—	38	6	no	—
260	C₂³	—	38	12	no	—
261	C₂³	—	76	3	no	—
262	C₂³	—	76	6	no	—
263	C₂³	—	76	6	no	—
264	C₂³	C₂³×D₁₉	1	3	yes	—
265	Q₈	—	38	6	no	—
266	Q₈	—	38	2	no	—
267	Q₈	C₂×D₇₆	1	4	yes	complemented
268	D₄	—	38	12	no	—
269	D₄	—	38	12	no	—
270	D₄	—	38	12	no	—
271	D₄	—	38	6	no	—
272	D₄	—	38	12	no	—
273	D₄	—	38	6	no	—
274	D₄	—	76	12	no	—
275	D₄	—	76	6	no	—
276	D₄	C₂×D₇₆	1	12	yes	complemented
277	C₂×C₄	—	4	6	no	—
278	C₂×C₄	—	19	1	no	—
279	C₂×C₄	—	19	3	no	—
280	C₂×C₄	—	19	2	no	—
281	C₂×C₄	—	38	6	no	—
282	C₂×C₄	—	38	2	no	—
283	C₂×C₄	—	38	6	no	—
284	C₂×C₄	—	38	6	no	—
285	C₂×C₄	—	38	6	no	—
286	C₂×C₄	—	38	12	no	—
287	C₂×C₄	C₂×D₇₆	1	12	yes	complemented
288	C₂×C₄	C₂³×D₁₉	1	2	yes	—
289	C₂×C₄	C₂³×D₁₉	1	6	yes	—
290	C₂×C₄	C₂³×D₁₉	1	3	yes	—
291	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
292	C₂²	—	2	3	no	—
293	C₂²	—	2	3	no	—
294	C₂²	—	19	2	no	—
295	C₂²	—	38	1	no	—
296	C₂²	—	38	1	no	—
297	C₂²	—	38	3	no	—
298	C₂²	—	38	6	no	—
299	C₂²	—	38	6	no	—
300	C₂²	—	38	3	no	—
301	C₂²	—	38	3	no	—
302	C₂²	—	38	6	no	—
303	C₂²	—	76	12	no	—
304	C₂²	—	152	12	no	—
305	C₂²	C₂²×D₇₆	1	6	yes	—
306	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
307	C₄	—	2	6	no	—
308	C₄	—	2	2	no	—
309	C₄	—	38	1	no	—
310	C₄	—	38	2	no	—
311	C₄	—	38	3	no	—
312	C₄	C₂²×D₇₆	1	2	yes	—
313	C₄	C₂²×D₇₆	1	6	yes	—
314	C₂	—	2	6	no	—
315	C₂	—	38	2	no	—
316	C₂	—	76	3	no	—
317	C₂	—	76	6	no	—
318	C₂	C₂³×D₇₆	1	1	yes	characteristic
319	C₂	H(5,𝔽₂)×D₁₉	1	1	yes	characteristic
320	C₂	#11927381	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#520	C₂
hypercenter	#520	D₁₉
hypocenter	C₁₉	#3488
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #520

Complements

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

N	H ≅ G/N	type
1. C₂×#10597435	C₂	semidirect
2. C₂×#41×D₁₉	C₂	semidirect
3. #11927216	C₂	semidirect
4. #11927206	C₂	semidirect
5. #11927189	C₂	semidirect
6. D₄×D₇₆	C₂	semidirect
7. #11927125	C₂	semidirect
8. #11927113	C₂	semidirect
9. #11927056	C₂	semidirect
10. C₂×#10597341	C₂	semidirect
11. C₁₉×#520	C₂	semidirect
15. C₂×#64545	C₂²	semidirect
16. C₂×#64545	C₂²	semidirect
17. C₂×Q₈×D₁₉	C₂²	semidirect
18. C₂×#64543	C₂²	semidirect
19. C₂×D₄×D₁₉	C₂²	semidirect
20. C₂×D₄×D₁₉	C₂²	semidirect
21. C₂×#64541	C₂²	semidirect
22. C₂×#64541	C₂²	semidirect
23. C₂²×D₇₆	C₂²	semidirect
24. C₂²×D₇₆	C₂²	semidirect
25. C₂²×C₄×D₁₉	C₂²	semidirect
26. C₄×Q₈×C₁₉	C₂²	semidirect
27. C₄×D₄×C₁₉	C₂²	semidirect
28. C₁₉×#117	C₂²	semidirect
29. C₂×C₄⋊C₄×C₁₉	C₂²	semidirect
30. #10597376	C₂²	semidirect
31. C₂×(C₂×D₁₉)⋊C₄	C₂²	semidirect
32. #10597343	C₂²	semidirect
33. #10597343	C₂²	semidirect
34. #10597343	C₂²	semidirect
35. #10597341	C₂²	semidirect
36. #10597341	C₂²	semidirect
37. #10597341	C₂²	semidirect
38. #10597341	C₂²	semidirect
39. #10597341	C₂²	semidirect
40. #10597341	C₂²	semidirect
41. #10597332	C₂²	semidirect
42. #10597328	C₂²	semidirect
43. #10597328	C₂²	semidirect
44. #10597321	C₂²	semidirect
45. #10597320	C₂²	semidirect
46. C₄×D₇₆	C₂²	semidirect
47. C₄×D₇₆	C₂²	semidirect
50. #41×C₁₉	D₄	semidirect
104. Q₈×C₁₉	C₂×D₄	semidirect
105. D₄×C₁₉	C₂×D₄	semidirect
107. C₂×C₄×C₁₉	C₂×D₄	semidirect
146. C₂×D₁₉	H(5,𝔽₂)	semidirect
164. #520	D₁₉	semidirect
201. C₄×Q₈	C₂×D₁₉	semidirect
204. C₄×D₄	C₂×D₁₉	semidirect
205. #117	C₂×D₁₉	semidirect
206. C₂×C₄⋊C₄	C₂×D₁₉	semidirect
208. C₁₉	#3488	semidirect
216. #41	D₇₆	semidirect
267. Q₈	C₂×D₇₆	semidirect
276. D₄	C₂×D₇₆	semidirect
287. C₂×C₄	C₂×D₇₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597435	2
C₂×#41×D₁₉	2
#11927216	2
#11927206	2
#11927189	2
D₄×D₇₆	2
#11927125	2
#11927113	2
#11927056	2
C₂×#10597341	2
C₁₉×#520	2
#3488	19