Subgroups — #14577533

Order 2432.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927279	C₂	1	4	yes	maximal, complemented
2	#11927267	C₂	1	4	yes	maximal, complemented
3	#11927221	C₂	1	8	yes	maximal, complemented
4	#11927168	C₂	1	4	yes	maximal, complemented
5	#11927120	C₂	1	1	yes	maximal, characteristic, complemented
6	#11927113	C₂	1	1	yes	maximal, characteristic
7	#11927079	C₂	1	4	yes	maximal, complemented
8	C₂×#10597321	C₂	1	1	yes	maximal, characteristic, complemented
9	#11927021	C₂	1	1	yes	maximal, characteristic, complemented
10	#11927018	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#64541	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₉×#533	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#64541	C₂²	1	1	yes	characteristic
14	C₂²×D₇₆	C₂²	1	1	yes	characteristic
15	C₂²×Dic₃₈	C₂²	1	1	yes	characteristic
16	C₁₉×C₄⋊Q₈	C₂²	1	2	yes	complemented
17	C₁₉×#125	C₂²	1	2	yes	complemented
18	C₁₉×C₂²⋊Q₈	C₂²	1	4	yes	—
19	C₄×Q₈×C₁₉	C₂²	1	4	yes	complemented
20	C₁₉×#117	C₂²	1	1	yes	characteristic
21	C₁₉×#117	C₂²	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂²	1	1	yes	characteristic
23	#10597393	C₂²	1	8	yes	complemented
24	#10597392	C₂²	1	8	yes	complemented
25	C₄×C₂²⋊D₁₉	C₂²	1	4	yes	—
26	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	—
27	#10597372	C₂²	1	1	yes	characteristic
28	#10597344	C₂²	1	8	yes	complemented
29	#10597344	C₂²	1	8	yes	complemented
30	#10597343	C₂²	1	4	yes	complemented
31	#10597341	C₂²	1	4	yes	—
32	#10597340	C₂²	1	8	yes	complemented
33	#10597340	C₂²	1	8	yes	complemented
34	#10597339	C₂²	1	4	yes	—
35	#10597339	C₂²	1	8	yes	complemented
36	#10597338	C₂²	1	8	yes	complemented
37	#10597333	C₂²	1	4	yes	complemented
38	#10597332	C₂²	1	4	yes	—
39	#10597328	C₂²	1	4	yes	—
40	C₂²⋊C₄×D₁₉	C₂²	1	4	yes	—
41	#10597323	C₂²	1	4	yes	—
42	#10597321	C₂²	1	2	yes	—
43	#10597321	C₂²	1	2	yes	—
44	#10597321	C₂²	1	4	yes	—
45	#10597321	C₂²	1	2	yes	—
46	#10597321	C₂²	1	2	yes	complemented
47	#10597320	C₂²	1	2	yes	—
48	C₄×D₇₆	C₂²	1	4	yes	—
49	C₄×D₇₆	C₂²	1	2	yes	complemented
50	#10597318	C₂²	1	2	yes	complemented
51	C₄²×D₁₉	C₂²	1	2	yes	complemented
52	#10597315	C₂²	1	2	yes	complemented
53	C₄×Dic₃₈	C₂²	1	2	yes	complemented
54	C₄×Dic₃₈	C₂²	1	4	yes	complemented
55	C₂³×D₁₉	C₂³	1	2	yes	—
56	C₂×Q₈×C₁₉	C₂³	1	4	yes	—
57	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
59	C₂×C₂²⋊D₁₉	C₂³	1	2	yes	—
60	C₂²×Dic₁₉	C₂³	1	2	yes	—
61	#64541	—	2	4	no	—
62	C₂×D₇₆	—	2	4	no	—
63	C₂×D₇₆	C₂³	1	2	yes	—
64	C₂×D₇₆	C₂³	1	2	yes	—
65	C₂×D₇₆	C₂³	1	1	yes	characteristic
66	C₂×D₇₆	C₂³	1	8	yes	—
67	C₂×C₄×D₁₉	C₂³	1	2	yes	—
68	C₂×C₄×D₁₉	C₂³	1	8	yes	—
69	C₂×C₄×D₁₉	C₂³	1	4	yes	—
70	C₂×Dic₃₈	—	2	4	no	—
71	C₂×Dic₃₈	C₂³	1	2	yes	—
72	C₂×Dic₃₈	C₂³	1	2	yes	—
73	C₂×Dic₃₈	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
75	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
76	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
77	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
78	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
79	C₂²⋊C₄×C₁₉	C₂³	1	2	yes	—
80	C₂²⋊C₄×C₁₉	C₂³	1	2	yes	—
81	C₄²×C₁₉	C₂³	1	2	yes	—
82	C₄²×C₁₉	C₂³	1	1	yes	characteristic
83	C₄²×C₁₉	C₂³	1	1	yes	characteristic
84	C₄²×C₁₉	C₂³	1	2	yes	—
85	C₄²×C₁₉	C₂³	1	2	yes	—
86	C₂²⋊Dic₁₉	C₂³	1	2	yes	—
87	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
88	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
89	(C₂×D₁₉)⋊C₄	C₂³	1	4	yes	—
90	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
91	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
92	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
93	C₄⋊Dic₁₉	C₂³	1	8	yes	—
94	C₄⋊Dic₁₉	C₂³	1	2	yes	—
95	Dic₁₉⋊C₄	C₂³	1	4	yes	—
96	Dic₁₉⋊C₄	C₂³	1	8	yes	—
97	C₄×Dic₁₉	C₂³	1	8	yes	—
98	C₄×Dic₁₉	C₂³	1	4	yes	—
99	C₄×Dic₁₉	C₂³	1	2	yes	—
100	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
101	C₂²×D₁₉	—	2	4	no	—
102	C₂²×D₁₉	—	4	4	no	—
103	C₂²×D₁₉	C₂⁴	1	4	yes	—
104	C₂²×D₁₉	C₂⁴	1	2	yes	—
105	Q₈×C₁₉	—	4	4	no	—
106	C₂×C₄×C₁₉	—	4	2	no	—
107	C₂×C₄×C₁₉	#41	1	4	yes	—
108	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
109	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
110	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
111	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
112	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₁₉	—	2	4	no	—
115	C₂×Dic₁₉	—	2	4	no	—
116	C₂×Dic₁₉	C₂⁴	1	4	yes	—
117	C₂×Dic₁₉	C₂⁴	1	2	yes	—
118	C₂×Dic₁₉	C₂⁴	1	4	yes	—
119	D₇₆	—	2	4	no	—
120	D₇₆	—	2	4	no	—
121	D₇₆	—	2	2	no	—
122	D₇₆	—	4	8	no	—
123	C₄×D₁₉	—	2	4	no	—
124	C₄×D₁₉	—	4	4	no	—
125	C₄×D₁₉	—	4	8	no	—
126	Dic₃₈	—	2	4	no	—
127	Dic₃₈	—	2	4	no	—
128	Dic₃₈	—	2	2	no	—
129	#3510	—	19	1	no	maximal, Sylow
130	C₂²×C₁₉	—	2	2	no	—
131	C₂²×C₁₉	C₂×#41	1	2	yes	—
132	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
133	C₂×D₁₉	—	2	2	no	—
134	C₂×D₁₉	—	2	4	no	—
135	C₂×D₁₉	—	2	4	no	—
136	C₂×D₁₉	—	2	8	no	—
137	C₂×D₁₉	—	8	4	no	—
138	C₄×C₁₉	—	2	2	no	—
139	C₄×C₁₉	—	2	4	no	—
140	C₄×C₁₉	—	2	4	no	—
141	C₄×C₁₉	—	2	2	no	—
142	C₄×C₁₉	C₂×#41	1	2	yes	—
143	C₄×C₁₉	C₂×#41	1	2	yes	—
144	Dic₁₉	—	2	4	no	—
145	Dic₁₉	—	2	2	no	—
146	Dic₁₉	—	2	4	no	—
147	#559	—	19	4	no	—
148	#556	—	19	4	no	—
149	#551	—	19	8	no	—
150	#542	—	19	4	no	—
151	#536	—	19	1	no	—
152	#535	—	19	1	no	—
153	#533	D₁₉	1	1	yes	characteristic, complemented
154	#529	—	19	4	no	—
155	C₂×#124	—	19	1	no	—
156	#519	—	19	1	no	—
157	#518	—	19	1	no	—
158	C₄×#41	—	19	1	no	—
159	C₂×C₁₉	—	2	2	no	—
160	C₂×C₁₉	C₂²×#41	1	1	yes	characteristic
161	C₂×C₁₉	#585	1	2	yes	—
162	D₁₉	—	4	2	no	—
163	D₁₉	—	4	4	no	—
164	C₂×#41	—	19	1	no	—
165	C₂²×Q₈	—	19	1	no	—
166	C₂²×D₄	—	19	1	no	—
167	C₄⋊Q₈	—	19	2	no	—
168	C₄⋊Q₈	C₂×D₁₉	1	2	yes	complemented
169	#127	—	19	2	no	—
170	#126	—	19	8	no	—
171	#126	—	19	8	no	—
172	#125	C₂×D₁₉	1	2	yes	complemented
173	#124	—	19	8	no	—
174	#124	—	19	2	no	—
175	#124	—	19	2	no	—
176	#124	—	19	4	no	—
177	#124	—	19	2	no	—
178	#124	—	19	2	no	—
179	#123	—	19	8	no	—
180	#123	—	19	8	no	—
181	#123	—	19	4	no	—
182	C₂²⋊Q₈	—	19	8	no	—
183	C₂²⋊Q₈	—	19	4	no	—
184	C₂²⋊Q₈	C₂×D₁₉	1	4	yes	—
185	#121	—	19	4	no	—
186	#120	—	19	4	no	—
187	C₄×Q₈	—	19	4	no	—
188	C₄×Q₈	—	19	2	no	—
189	C₄×Q₈	—	19	4	no	—
190	C₄×Q₈	C₂×D₁₉	1	4	yes	complemented
191	C₄×D₄	—	19	4	no	—
192	C₄×D₄	—	19	4	no	—
193	C₄×D₄	—	19	8	no	—
194	C₄×D₄	—	19	2	no	—
195	C₄×D₄	—	19	4	no	—
196	#117	—	19	1	no	—
197	#117	—	19	8	no	—
198	#117	—	19	4	no	—
199	#117	—	19	2	no	—
200	#117	C₂×D₁₉	1	1	yes	characteristic
201	#117	C₂×D₁₉	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	—	19	4	no	—
203	C₂×C₂²⋊C₄	—	19	4	no	—
204	C₂×C₄²	—	19	2	no	—
205	C₂×C₄²	C₂×D₁₉	1	1	yes	characteristic
206	C₁₉	#3510	1	1	yes	characteristic, complemented, Sylow
207	C₂⁴	—	19	2	no	—
208	#41	—	38	4	no	—
209	C₂×Q₈	—	19	2	no	—
210	C₂×Q₈	—	19	2	no	—
211	C₂×Q₈	—	19	1	no	—
212	C₂×Q₈	—	38	4	no	—
213	C₂×Q₈	C₂²×D₁₉	1	4	yes	—
214	C₂×D₄	—	19	2	no	—
215	C₂×D₄	—	19	2	no	—
216	C₂×D₄	—	19	2	no	—
217	C₂×D₄	—	19	1	no	—
218	C₂×D₄	—	19	8	no	—
219	C₂×D₄	—	38	4	no	—
220	C₂²×C₄	—	19	2	no	—
221	C₂²×C₄	—	19	2	no	—
222	C₂²×C₄	—	19	4	no	—
223	C₂²×C₄	—	19	8	no	—
224	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
225	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
226	C₄⋊C₄	—	19	4	no	—
227	C₄⋊C₄	—	19	8	no	—
228	C₄⋊C₄	—	19	2	no	—
229	C₄⋊C₄	—	19	8	no	—
230	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
231	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
232	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
233	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
234	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
235	C₂²⋊C₄	—	19	2	no	—
236	C₂²⋊C₄	—	19	8	no	—
237	C₂²⋊C₄	—	19	8	no	—
238	C₂²⋊C₄	—	19	8	no	—
239	C₂²⋊C₄	—	19	4	no	—
240	C₂²⋊C₄	—	19	8	no	—
241	C₂²⋊C₄	—	19	8	no	—
242	C₂²⋊C₄	C₂²×D₁₉	1	2	yes	—
243	C₂²⋊C₄	C₂²×D₁₉	1	2	yes	—
244	C₄²	—	19	2	no	—
245	C₄²	—	19	4	no	—
246	C₄²	—	19	8	no	—
247	C₄²	C₂²×D₁₉	1	1	yes	characteristic
248	C₄²	C₂²×D₁₉	1	1	yes	characteristic
249	C₄²	C₂²×D₁₉	1	2	yes	—
250	C₄²	C₂²×D₁₉	1	2	yes	—
251	C₄²	C₂²×D₁₉	1	2	yes	—
252	C₂³	—	19	4	no	—
253	C₂³	—	19	2	no	—
254	C₂³	—	38	4	no	—
255	C₂³	—	76	4	no	—
256	C₂³	C₂³×D₁₉	1	1	yes	characteristic
257	Q₈	—	4	4	no	—
258	Q₈	—	38	4	no	—
259	Q₈	—	38	4	no	—
260	Q₈	—	38	2	no	—
261	D₄	—	38	4	no	—
262	D₄	—	38	4	no	—
263	D₄	—	38	4	no	—
264	D₄	—	38	2	no	—
265	D₄	—	76	8	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	19	4	no	—
268	C₂×C₄	—	19	4	no	—
269	C₂×C₄	—	19	2	no	—
270	C₂×C₄	—	38	4	no	—
271	C₂×C₄	—	38	4	no	—
272	C₂×C₄	—	76	4	no	—
273	C₂×C₄	—	76	8	no	—
274	C₂×C₄	#64545	1	4	yes	—
275	C₂×C₄	C₂³×D₁₉	1	2	yes	—
276	C₂×C₄	C₂³×D₁₉	1	4	yes	—
277	C₂×C₄	C₂³×D₁₉	1	2	yes	—
278	C₂×C₄	C₂³×D₁₉	1	4	yes	—
279	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	38	2	no	—
283	C₂²	—	38	4	no	—
284	C₂²	—	38	4	no	—
285	C₂²	—	38	8	no	—
286	C₂²	—	152	4	no	—
287	C₂²	C₂×#64545	1	2	yes	—
288	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	4	no	—
292	C₄	—	2	4	no	—
293	C₄	—	38	2	no	—
294	C₄	—	38	4	no	—
295	C₄	—	38	4	no	—
296	C₄	C₂×#64545	1	2	yes	—
297	C₄	C₂×#64545	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	76	4	no	—
300	C₂	—	76	2	no	—
301	C₂	C₂²×#64545	1	1	yes	characteristic
302	C₂	#11927380	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#533	C₂
prime radical	#11927113	C₂
hypercenter	#533	D₁₉
hypocenter	C₁₉	#3510
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. #11927279	C₂	semidirect
2. #11927267	C₂	semidirect
3. #11927221	C₂	semidirect
4. #11927168	C₂	semidirect
5. #11927120	C₂	semidirect
7. #11927079	C₂	semidirect
8. C₂×#10597321	C₂	semidirect
9. #11927021	C₂	semidirect
10. #11927018	C₂	semidirect
11. C₄×#64541	C₂	semidirect
12. C₁₉×#533	C₂	semidirect
16. C₁₉×C₄⋊Q₈	C₂²	semidirect
17. C₁₉×#125	C₂²	semidirect
19. C₄×Q₈×C₁₉	C₂²	semidirect
23. #10597393	C₂²	semidirect
24. #10597392	C₂²	semidirect
28. #10597344	C₂²	semidirect
29. #10597344	C₂²	semidirect
30. #10597343	C₂²	semidirect
32. #10597340	C₂²	semidirect
33. #10597340	C₂²	semidirect
35. #10597339	C₂²	semidirect
36. #10597338	C₂²	semidirect
37. #10597333	C₂²	semidirect
46. #10597321	C₂²	semidirect
49. C₄×D₇₆	C₂²	semidirect
50. #10597318	C₂²	semidirect
51. C₄²×D₁₉	C₂²	semidirect
52. #10597315	C₂²	semidirect
53. C₄×Dic₃₈	C₂²	semidirect
54. C₄×Dic₃₈	C₂²	semidirect
153. #533	D₁₉	semidirect
168. C₄⋊Q₈	C₂×D₁₉	semidirect
172. #125	C₂×D₁₉	semidirect
190. C₄×Q₈	C₂×D₁₉	semidirect
206. C₁₉	#3510	semidirect

Maximal subgroups

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

subgroup	index
#11927279	2
#11927267	2
#11927221	2
#11927168	2
#11927120	2
#11927113	2
#11927079	2
C₂×#10597321	2
#11927021	2
#11927018	2
C₄×#64541	2
C₁₉×#533	2
#3510	19