Subgroups — #14577968

Order 2432.

1260 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927281	C₂	1	4	yes	maximal, complemented
2	#11927280	C₂	1	1	yes	maximal, characteristic, complemented
3	#11927214	C₂	1	2	yes	maximal, complemented
4	#11927188	C₂	1	4	yes	maximal, complemented
5	#11927179	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×C₂²⋊D₁₉	C₂	1	4	yes	maximal, complemented
7	#11927148	C₂	1	4	yes	maximal, complemented
8	#11927107	C₂	1	4	yes	maximal, complemented
9	#11927103	C₂	1	4	yes	maximal, complemented
10	#11927078	C₂	1	2	yes	maximal, complemented
11	C₂×C₁₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₉	C₂²	1	2	yes	complemented
13	C₂²×D₄×C₁₉	C₂²	1	4	yes	complemented
14	C₂²×C₂²⋊D₁₉	C₂²	1	4	yes	complemented
15	C₂×#64543	C₂²	1	4	yes	complemented
16	C₂×D₄×D₁₉	C₂²	1	4	yes	complemented
17	C₁₉×#127	C₂²	1	1	yes	characteristic
18	C₁₉×#127	C₂²	1	4	yes	complemented
19	C₁₉×#127	C₂²	1	1	yes	characteristic, complemented
20	C₁₉×#127	C₂²	1	2	yes	complemented
21	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic, complemented
22	#10597399	C₂²	1	8	yes	complemented
23	C₂×C₂²⋊Dic₁₉	C₂²	1	4	yes	complemented
24	#10597386	C₂²	1	4	yes	complemented
25	#10597385	C₂²	1	8	yes	complemented
26	#10597385	C₂²	1	8	yes	complemented
27	#10597385	C₂²	1	8	yes	complemented
28	#10597385	C₂²	1	8	yes	complemented
29	#10597384	C₂²	1	4	yes	complemented
30	#10597384	C₂²	1	4	yes	complemented
31	#10597383	C₂²	1	4	yes	complemented
32	#10597383	C₂²	1	4	yes	complemented
33	#10597383	C₂²	1	8	yes	complemented
34	#10597382	C₂²	1	4	yes	complemented
35	#10597381	C₂²	1	4	yes	complemented
36	#10597381	C₂²	1	4	yes	complemented
37	#10597381	C₂²	1	8	yes	complemented
38	D₄×Dic₁₉	C₂²	1	4	yes	complemented
39	D₄×Dic₁₉	C₂²	1	4	yes	complemented
40	#10597376	C₂²	1	2	yes	—
41	#10597375	C₂²	1	4	yes	complemented
42	C₄×C₂²⋊D₁₉	C₂²	1	4	yes	complemented
43	#10597370	C₂²	1	2	yes	complemented
44	#10597322	C₂²	1	4	yes	complemented
45	#10597321	C₂²	1	2	yes	complemented
46	C₄×D₇₆	C₂²	1	2	yes	complemented
47	#10597318	C₂²	1	4	yes	complemented
48	#10597316	C₂²	1	2	yes	complemented
49	C₄×Dic₃₈	C₂²	1	2	yes	complemented
50	C₂⁴×C₁₉	C₂³	1	4	yes	—
51	C₂³×D₁₉	C₂³	1	4	yes	—
52	C₂×D₄×C₁₉	—	4	8	no	—
53	C₂×D₄×C₁₉	—	4	4	no	—
54	C₂×D₄×C₁₉	C₂³	1	4	yes	—
55	C₂×D₄×C₁₉	C₂³	1	4	yes	—
56	C₂×D₄×C₁₉	C₂³	1	2	yes	—
57	C₂×D₄×C₁₉	C₂³	1	4	yes	—
58	C₂×D₄×C₁₉	C₂³	1	8	yes	—
59	C₂×D₄×C₁₉	C₂³	1	2	yes	—
60	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₉	—	4	8	no	—
63	C₂×C₂²⋊D₁₉	C₂³	1	4	yes	—
64	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
65	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
66	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
67	C₂²×Dic₁₉	C₂³	1	4	yes	—
68	C₂²×Dic₁₉	C₂³	1	8	yes	—
69	#64543	—	4	8	no	—
70	D₄×D₁₉	—	4	8	no	—
71	C₂×D₇₆	C₂³	1	2	yes	—
72	C₂×C₄×D₁₉	C₂³	1	4	yes	—
73	C₂×Dic₃₈	C₂³	1	2	yes	—
74	C₄²×C₁₉	C₂³	1	1	yes	characteristic
75	C₄²×C₁₉	C₂³	1	2	yes	—
76	C₄²×C₁₉	C₂³	1	1	yes	characteristic
77	C₂²⋊Dic₁₉	C₂³	1	8	yes	—
78	C₂²⋊Dic₁₉	C₂³	1	8	yes	—
79	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
80	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
81	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
82	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
83	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
84	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
85	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
86	C₄⋊Dic₁₉	C₂³	1	2	yes	—
87	C₄⋊Dic₁₉	C₂³	1	2	yes	—
88	Dic₁₉⋊C₄	C₂³	1	4	yes	—
89	Dic₁₉⋊C₄	C₂³	1	4	yes	—
90	Dic₁₉⋊C₄	C₂³	1	4	yes	—
91	Dic₁₉⋊C₄	C₂³	1	4	yes	—
92	C₄×Dic₁₉	C₂³	1	4	yes	—
93	C₂³×C₁₉	—	2	8	no	—
94	C₂³×C₁₉	—	4	8	no	—
95	C₂³×C₁₉	C₂⁴	1	4	yes	—
96	C₂³×C₁₉	C₂⁴	1	4	yes	—
97	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
98	C₂²×D₁₉	—	4	8	no	—
99	C₂²×D₁₉	—	4	4	no	—
100	C₂²×D₁₉	C₂⁴	1	4	yes	—
101	D₄×C₁₉	—	2	8	no	—
102	D₄×C₁₉	—	2	8	no	—
103	D₄×C₁₉	—	4	8	no	—
104	D₄×C₁₉	—	4	2	no	—
105	D₄×C₁₉	—	4	4	no	—
106	D₄×C₁₉	—	4	2	no	—
107	C₂×C₄×C₁₉	—	2	2	no	—
108	C₂×C₄×C₁₉	—	2	4	no	—
109	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
110	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
111	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
112	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
113	C₂²⋊D₁₉	—	2	8	no	—
114	C₂²⋊D₁₉	—	4	8	no	—
115	C₂²⋊D₁₉	—	4	8	no	—
116	C₂²⋊D₁₉	—	4	8	no	—
117	C₂×Dic₁₉	—	4	8	no	—
118	C₂×Dic₁₉	—	4	4	no	—
119	C₂×Dic₁₉	C₂⁴	1	4	yes	—
120	C₂×Dic₁₉	C₂⁴	1	4	yes	—
121	C₂×Dic₁₉	C₂⁴	1	4	yes	—
122	D₇₆	—	2	4	no	—
123	C₄×D₁₉	—	2	8	no	—
124	Dic₃₈	—	2	4	no	—
125	#3543	—	19	1	no	maximal, Sylow
126	C₂²×C₁₉	—	2	2	no	—
127	C₂²×C₁₉	—	2	8	no	—
128	C₂²×C₁₉	—	2	4	no	—
129	C₂²×C₁₉	—	2	4	no	—
130	C₂²×C₁₉	—	2	8	no	—
131	C₂²×C₁₉	—	8	8	no	—
132	C₂²×C₁₉	H(5,𝔽₂)	1	2	yes	—
133	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
134	C₂×D₁₉	—	2	4	no	—
135	C₂×D₁₉	—	2	4	no	—
136	C₂×D₁₉	—	2	4	no	—
137	C₂×D₁₉	—	8	8	no	—
138	C₄×C₁₉	—	2	1	no	—
139	C₄×C₁₉	—	2	1	no	—
140	C₄×C₁₉	—	2	2	no	—
141	C₄×C₁₉	H(5,𝔽₂)	1	4	yes	—
142	Dic₁₉	—	2	4	no	—
143	Dic₁₉	—	2	4	no	—
144	Dic₁₉	—	2	4	no	—
145	#560	—	19	4	no	—
146	#559	—	19	1	no	—
147	#550	—	19	2	no	—
148	#546	—	19	4	no	—
149	D₄²	—	19	4	no	—
150	D₄²	—	19	1	no	—
151	#540	—	19	4	no	—
152	#535	—	19	4	no	—
153	#534	—	19	4	no	—
154	C₂×#127	D₁₉	1	1	yes	characteristic, complemented
155	#529	—	19	2	no	—
156	C₂×C₁₉	—	2	2	no	—
157	C₂×C₁₉	—	4	4	no	—
158	C₂×C₁₉	—	4	4	no	—
159	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
161	D₁₉	—	4	4	no	—
162	C₂×#41	—	19	4	no	—
163	C₂²×D₄	—	19	4	no	—
164	C₂²×D₄	—	19	4	no	—
165	C₂²×D₄	C₂×D₁₉	1	2	yes	complemented
166	C₂²×D₄	C₂×D₁₉	1	4	yes	complemented
167	#127	—	19	4	no	—
168	#127	C₂×D₁₉	1	1	yes	characteristic
169	#127	C₂×D₁₉	1	4	yes	complemented
170	#127	C₂×D₁₉	1	1	yes	characteristic, complemented
171	#127	C₂×D₁₉	1	2	yes	complemented
172	#126	—	19	4	no	—
173	#125	—	19	2	no	—
174	#124	—	19	4	no	—
175	#124	—	19	2	no	—
176	#123	—	19	4	no	—
177	#123	—	19	4	no	—
178	#123	—	19	8	no	—
179	#123	—	19	4	no	—
180	C₂²⋊Q₈	—	19	2	no	—
181	#121	—	19	4	no	—
182	#121	—	19	4	no	—
183	#121	—	19	8	no	—
184	#121	—	19	8	no	—
185	#121	—	19	8	no	—
186	#121	—	19	8	no	—
187	#121	—	19	2	no	—
188	#120	—	19	8	no	—
189	#120	—	19	4	no	—
190	#120	—	19	4	no	—
191	#120	—	19	8	no	—
192	C₄×Q₈	—	19	2	no	—
193	C₄×D₄	—	19	4	no	—
194	C₄×D₄	—	19	4	no	—
195	C₄×D₄	—	19	4	no	—
196	C₄×D₄	—	19	2	no	—
197	#117	—	19	4	no	—
198	C₂×C₂²⋊C₄	—	19	4	no	—
199	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic, complemented
200	C₁₉	#3543	1	1	yes	characteristic, complemented, Sylow
201	C₂⁴	—	19	4	no	—
202	C₂⁴	C₂²×D₁₉	1	4	yes	—
203	#41	—	76	8	no	—
204	C₂×Q₈	—	19	2	no	—
205	C₂×D₄	—	4	8	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	19	4	no	—
208	C₂×D₄	—	19	8	no	—
209	C₂×D₄	—	19	8	no	—
210	C₂×D₄	—	19	8	no	—
211	C₂×D₄	—	19	2	no	—
212	C₂×D₄	—	76	8	no	—
213	C₂×D₄	—	76	8	no	—
214	C₂×D₄	C₂²×D₁₉	1	4	yes	—
215	C₂×D₄	C₂²×D₁₉	1	4	yes	—
216	C₂×D₄	C₂²×D₁₉	1	2	yes	—
217	C₂×D₄	C₂²×D₁₉	1	4	yes	—
218	C₂×D₄	C₂²×D₁₉	1	8	yes	—
219	C₂×D₄	C₂²×D₁₉	1	2	yes	—
220	C₂²×C₄	—	19	8	no	—
221	C₂²×C₄	—	19	4	no	—
222	C₂²×C₄	—	19	4	no	—
223	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
224	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
225	C₄⋊C₄	—	19	4	no	—
226	C₄⋊C₄	—	19	4	no	—
227	C₄⋊C₄	—	19	4	no	—
228	C₄⋊C₄	—	19	4	no	—
229	C₄⋊C₄	—	19	2	no	—
230	C₄⋊C₄	—	19	2	no	—
231	C₂²⋊C₄	—	19	4	no	—
232	C₂²⋊C₄	—	19	4	no	—
233	C₂²⋊C₄	—	19	4	no	—
234	C₂²⋊C₄	—	19	8	no	—
235	C₂²⋊C₄	—	19	8	no	—
236	C₂²⋊C₄	—	19	4	no	—
237	C₂²⋊C₄	—	19	4	no	—
238	C₂²⋊C₄	—	19	4	no	—
239	C₂²⋊C₄	—	19	4	no	—
240	C₄²	—	19	4	no	—
241	C₄²	C₂²×D₁₉	1	2	yes	—
242	C₄²	C₂²×D₁₉	1	1	yes	characteristic
243	C₄²	C₂²×D₁₉	1	1	yes	characteristic
244	C₂³	—	2	8	no	—
245	C₂³	—	4	8	no	—
246	C₂³	—	19	4	no	—
247	C₂³	—	76	4	no	—
248	C₂³	—	76	8	no	—
249	C₂³	C₂³×D₁₉	1	4	yes	—
250	C₂³	C₂³×D₁₉	1	4	yes	—
251	C₂³	C₂³×D₁₉	1	1	yes	characteristic
252	Q₈	—	38	4	no	—
253	D₄	—	2	8	no	—
254	D₄	—	2	8	no	—
255	D₄	—	4	8	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	38	8	no	—
260	D₄	—	38	4	no	—
261	D₄	—	76	8	no	—
262	D₄	—	76	8	no	—
263	D₄	—	76	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	19	4	no	—
267	C₂×C₄	—	19	4	no	—
268	C₂×C₄	—	19	4	no	—
269	C₂×C₄	—	38	8	no	—
270	C₂×C₄	—	76	4	no	—
271	C₂×C₄	—	76	8	no	—
272	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₁₉	1	2	yes	—
274	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
275	C₂×C₄	C₂³×D₁₉	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	8	8	no	—
282	C₂²	—	38	4	no	—
283	C₂²	—	38	4	no	—
284	C₂²	—	38	4	no	—
285	C₂²	—	152	8	no	—
286	C₂²	#10597430	1	2	yes	—
287	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	38	4	no	—
292	C₄	—	38	4	no	—
293	C₄	—	38	4	no	—
294	C₄	#10597430	1	4	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	—	4	4	no	—
298	C₂	—	76	4	no	—
299	C₂	C₂×#10597430	1	1	yes	characteristic
300	C₂	C₂×#10597430	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₂×C₁₉×#127	C₂
hypercenter	C₂×#127	D₁₉
hypocenter	C₁₉	#3543
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. #11927281	C₂	semidirect
2. #11927280	C₂	semidirect
3. #11927214	C₂	semidirect
4. #11927188	C₂	semidirect
5. #11927179	C₂	semidirect
6. D₄×C₂²⋊D₁₉	C₂	semidirect
7. #11927148	C₂	semidirect
8. #11927107	C₂	semidirect
9. #11927103	C₂	semidirect
10. #11927078	C₂	semidirect
11. C₂×C₁₉×#127	C₂	semidirect
12. C₂²×D₄×C₁₉	C₂²	semidirect
13. C₂²×D₄×C₁₉	C₂²	semidirect
14. C₂²×C₂²⋊D₁₉	C₂²	semidirect
15. C₂×#64543	C₂²	semidirect
16. C₂×D₄×D₁₉	C₂²	semidirect
18. C₁₉×#127	C₂²	semidirect
19. C₁₉×#127	C₂²	semidirect
20. C₁₉×#127	C₂²	semidirect
21. C₂×C₄²×C₁₉	C₂²	semidirect
22. #10597399	C₂²	semidirect
23. C₂×C₂²⋊Dic₁₉	C₂²	semidirect
24. #10597386	C₂²	semidirect
25. #10597385	C₂²	semidirect
26. #10597385	C₂²	semidirect
27. #10597385	C₂²	semidirect
28. #10597385	C₂²	semidirect
29. #10597384	C₂²	semidirect
30. #10597384	C₂²	semidirect
31. #10597383	C₂²	semidirect
32. #10597383	C₂²	semidirect
33. #10597383	C₂²	semidirect
34. #10597382	C₂²	semidirect
35. #10597381	C₂²	semidirect
36. #10597381	C₂²	semidirect
37. #10597381	C₂²	semidirect
38. D₄×Dic₁₉	C₂²	semidirect
39. D₄×Dic₁₉	C₂²	semidirect
41. #10597375	C₂²	semidirect
42. C₄×C₂²⋊D₁₉	C₂²	semidirect
43. #10597370	C₂²	semidirect
44. #10597322	C₂²	semidirect
45. #10597321	C₂²	semidirect
46. C₄×D₇₆	C₂²	semidirect
47. #10597318	C₂²	semidirect
48. #10597316	C₂²	semidirect
49. C₄×Dic₃₈	C₂²	semidirect
154. C₂×#127	D₁₉	semidirect
165. C₂²×D₄	C₂×D₁₉	semidirect
166. C₂²×D₄	C₂×D₁₉	semidirect
169. #127	C₂×D₁₉	semidirect
170. #127	C₂×D₁₉	semidirect
171. #127	C₂×D₁₉	semidirect
199. C₂×C₄²	C₂×D₁₉	semidirect
200. C₁₉	#3543	semidirect

Maximal subgroups

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

subgroup	index
#11927281	2
#11927280	2
#11927214	2
#11927188	2
#11927179	2
D₄×C₂²⋊D₁₉	2
#11927148	2
#11927107	2
#11927103	2
#11927078	2
C₂×C₁₉×#127	2
#3543	19