Subgroups — #14577874

Order 2432.

1270 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	#11927284	C₂	1	4	yes	maximal, complemented
2	#11927240	C₂	1	4	yes	maximal, complemented
3	D₄×D₇₆	C₂	1	4	yes	maximal, complemented
4	#11927176	C₂	1	4	yes	maximal, complemented
5	#11927149	C₂	1	2	yes	maximal, complemented
6	#11927134	C₂	1	4	yes	maximal, complemented
7	#11927108	C₂	1	2	yes	maximal, complemented
8	#11927102	C₂	1	4	yes	maximal, complemented
9	C₂×#10597376	C₂	1	2	yes	maximal, complemented
10	C₁₉×#525	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic, complemented
12	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₂²⋊D₁₉	C₂²	1	4	yes	complemented
14	C₂×D₄×D₁₉	C₂²	1	8	yes	complemented
15	C₂²×D₇₆	C₂²	1	2	yes	complemented
16	C₂²×D₇₆	C₂²	1	2	yes	complemented
17	C₁₉×#123	C₂²	1	2	yes	complemented
18	C₁₉×C₂²⋊Q₈	C₂²	1	2	yes	complemented
19	C₁₉×#121	C₂²	1	2	yes	complemented
20	C₁₉×#120	C₂²	1	2	yes	complemented
21	C₄×D₄×C₁₉	C₂²	1	4	yes	complemented
22	C₁₉×#117	C₂²	1	1	yes	characteristic, complemented
23	#10597393	C₂²	1	4	yes	complemented
24	#10597386	C₂²	1	4	yes	complemented
25	#10597385	C₂²	1	8	yes	complemented
26	#10597383	C₂²	1	4	yes	complemented
27	#10597383	C₂²	1	4	yes	complemented
28	#10597376	C₂²	1	4	yes	complemented
29	#10597376	C₂²	1	4	yes	complemented
30	#10597376	C₂²	1	4	yes	complemented
31	#10597376	C₂²	1	4	yes	complemented
32	#10597376	C₂²	1	4	yes	complemented
33	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	4	yes	complemented
34	#10597372	C₂²	1	2	yes	complemented
35	C₂×C₄⋊Dic₁₉	C₂²	1	2	yes	complemented
36	#10597344	C₂²	1	8	yes	complemented
37	#10597341	C₂²	1	8	yes	complemented
38	#10597340	C₂²	1	8	yes	complemented
39	#10597330	C₂²	1	8	yes	complemented
40	#10597329	C₂²	1	8	yes	complemented
41	#10597328	C₂²	1	8	yes	complemented
42	#10597328	C₂²	1	8	yes	complemented
43	(C₂²⋊D₁₉)⋊C₄	C₂²	1	8	yes	complemented
44	#10597320	C₂²	1	2	yes	complemented
45	C₄×D₇₆	C₂²	1	4	yes	complemented
46	#10597316	C₂²	1	2	yes	complemented
47	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
48	C₂³×D₁₉	C₂³	1	4	yes	—
49	C₂³×D₁₉	C₂³	1	4	yes	—
50	#41×C₁₉	—	4	2	no	—
51	C₂×Q₈×C₁₉	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₁₉	C₂³	1	4	yes	—
53	C₂×D₄×C₁₉	C₂³	1	1	yes	characteristic
54	C₂×D₄×C₁₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₉	—	4	2	no	—
56	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
61	C₂×C₂²⋊D₁₉	—	4	8	no	—
62	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
63	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
64	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
65	C₂²×Dic₁₉	C₂³	1	4	yes	—
66	D₄×D₁₉	—	4	16	no	—
67	C₂×D₇₆	—	4	4	no	—
68	C₂×D₇₆	—	4	4	no	—
69	C₂×D₇₆	C₂³	1	4	yes	—
70	C₂×D₇₆	C₂³	1	4	yes	—
71	C₂×D₇₆	C₂³	1	4	yes	—
72	C₂×D₇₆	C₂³	1	8	yes	—
73	C₂×C₄×D₁₉	C₂³	1	8	yes	—
74	C₄⋊C₄×C₁₉	C₂³	1	2	yes	—
75	C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
76	C₂²⋊C₄×C₁₉	C₂³	1	2	yes	—
77	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
78	C₂²⋊C₄×C₁₉	C₂³	1	4	yes	—
79	C₄²×C₁₉	C₂³	1	2	yes	—
80	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
81	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
82	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
83	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
84	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
85	C₄⋊Dic₁₉	C₂³	1	4	yes	—
86	C₄⋊Dic₁₉	C₂³	1	4	yes	—
87	Dic₁₉⋊C₄	C₂³	1	8	yes	—
88	C₄×Dic₁₉	C₂³	1	4	yes	—
89	C₂³×C₁₉	—	2	4	no	—
90	C₂³×C₁₉	—	4	1	no	—
91	C₂³×C₁₉	C₂⁴	1	2	yes	—
92	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
93	C₂³×C₁₉	C₂⁴	1	2	yes	—
94	C₂²×D₁₉	—	2	8	no	—
95	C₂²×D₁₉	—	4	4	no	—
96	C₂²×D₁₉	—	4	4	no	—
97	C₂²×D₁₉	—	4	8	no	—
98	C₂²×D₁₉	—	4	4	no	—
99	C₂²×D₁₉	C₂⁴	1	8	yes	—
100	Q₈×C₁₉	—	4	1	no	—
101	D₄×C₁₉	—	2	8	no	—
102	D₄×C₁₉	—	4	2	no	—
103	D₄×C₁₉	—	4	1	no	—
104	C₂×C₄×C₁₉	—	2	4	no	—
105	C₂×C₄×C₁₉	—	2	4	no	—
106	C₂×C₄×C₁₉	—	2	2	no	—
107	C₂×C₄×C₁₉	—	4	1	no	—
108	C₂×C₄×C₁₉	—	4	2	no	—
109	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
110	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
111	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
112	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₁₉	—	2	16	no	—
115	C₂²⋊D₁₉	—	4	8	no	—
116	C₂²⋊D₁₉	—	4	8	no	—
117	C₂×Dic₁₉	—	4	4	no	—
118	C₂×Dic₁₉	C₂⁴	1	8	yes	—
119	D₇₆	—	2	8	no	—
120	D₇₆	—	4	4	no	—
121	D₇₆	—	4	4	no	—
122	D₇₆	—	4	8	no	—
123	C₄×D₁₉	—	4	8	no	—
124	#3537	—	19	1	no	maximal, Sylow
125	C₂²×C₁₉	—	2	2	no	—
126	C₂²×C₁₉	—	2	2	no	—
127	C₂²×C₁₉	—	2	2	no	—
128	C₂²×C₁₉	—	2	4	no	—
129	C₂²×C₁₉	—	2	1	no	—
130	C₂²×C₁₉	—	2	2	no	—
131	C₂²×C₁₉	—	4	4	no	—
132	C₂²×C₁₉	H(5,𝔽₂)	1	4	yes	—
133	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
134	C₂×D₁₉	—	2	8	no	—
135	C₂×D₁₉	—	2	8	no	—
136	C₂×D₁₉	—	2	8	no	—
137	C₂×D₁₉	—	8	8	no	—
138	C₂×D₁₉	—	8	8	no	—
139	C₄×C₁₉	—	2	4	no	—
140	C₄×C₁₉	—	2	2	no	—
141	C₄×C₁₉	—	2	2	no	—
142	C₄×C₁₉	—	2	1	no	—
143	C₄×C₁₉	H(5,𝔽₂)	1	2	yes	—
144	Dic₁₉	—	2	8	no	—
145	#560	—	19	4	no	—
146	#553	—	19	4	no	—
147	D₄²	—	19	4	no	—
148	D₄²	—	19	4	no	—
149	#540	—	19	2	no	—
150	#538	—	19	4	no	—
151	#535	—	19	2	no	—
152	#534	—	19	4	no	—
153	#525	D₁₉	1	1	yes	characteristic, complemented
154	C₂×#121	—	19	2	no	—
155	C₂×C₁₉	—	2	4	no	—
156	C₂×C₁₉	—	4	2	no	—
157	C₂×C₁₉	—	4	1	no	—
158	C₂×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₁₉	C₂×H(5,𝔽₂)	1	2	yes	—
160	D₁₉	—	4	8	no	—
161	C₂×#41	C₂×D₁₉	1	1	yes	characteristic, complemented
162	C₂²×D₄	—	19	4	no	—
163	C₂²×D₄	—	19	8	no	—
164	C₂²×D₄	—	19	2	no	—
165	C₂²×D₄	—	19	2	no	—
166	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic, complemented
167	#127	—	19	4	no	—
168	#127	—	19	2	no	—
169	#126	—	19	8	no	—
170	#125	—	19	2	no	—
171	#124	—	19	4	no	—
172	#123	—	19	8	no	—
173	#123	—	19	8	no	—
174	#123	C₂×D₁₉	1	2	yes	complemented
175	C₂²⋊Q₈	C₂×D₁₉	1	2	yes	complemented
176	#121	—	19	8	no	—
177	#121	—	19	4	no	—
178	#121	—	19	4	no	—
179	#121	—	19	4	no	—
180	#121	—	19	4	no	—
181	#121	—	19	4	no	—
182	#121	—	19	8	no	—
183	#121	—	19	8	no	—
184	#121	C₂×D₁₉	1	2	yes	complemented
185	#120	—	19	4	no	—
186	#120	—	19	4	no	—
187	#120	—	19	8	no	—
188	#120	—	19	8	no	—
189	#120	C₂×D₁₉	1	2	yes	complemented
190	C₄×D₄	—	19	8	no	—
191	C₄×D₄	—	19	4	no	—
192	C₄×D₄	C₂×D₁₉	1	4	yes	complemented
193	#117	—	19	2	no	—
194	#117	C₂×D₁₉	1	1	yes	characteristic, complemented
195	C₂×C₄⋊C₄	—	19	2	no	—
196	C₂×C₂²⋊C₄	—	19	4	no	—
197	C₁₉	#3537	1	1	yes	characteristic, complemented, Sylow
198	C₂⁴	—	19	4	no	—
199	C₂⁴	—	19	4	no	—
200	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
201	#41	—	4	2	no	—
202	C₂×Q₈	C₂²×D₁₉	1	1	yes	characteristic
203	C₂×D₄	—	19	8	no	—
204	C₂×D₄	—	19	8	no	—
205	C₂×D₄	—	19	8	no	—
206	C₂×D₄	—	19	4	no	—
207	C₂×D₄	—	19	4	no	—
208	C₂×D₄	—	19	8	no	—
209	C₂×D₄	—	19	4	no	—
210	C₂×D₄	—	76	4	no	—
211	C₂×D₄	—	76	4	no	—
212	C₂×D₄	—	76	16	no	—
213	C₂×D₄	—	76	8	no	—
214	C₂×D₄	C₂²×D₁₉	1	4	yes	—
215	C₂×D₄	C₂²×D₁₉	1	1	yes	characteristic
216	C₂×D₄	C₂²×D₁₉	1	2	yes	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	19	4	no	—
219	C₂²×C₄	—	19	8	no	—
220	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
221	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
222	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
223	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
224	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
225	C₄⋊C₄	—	19	8	no	—
226	C₄⋊C₄	—	19	4	no	—
227	C₄⋊C₄	—	19	4	no	—
228	C₄⋊C₄	C₂²×D₁₉	1	2	yes	—
229	C₄⋊C₄	C₂²×D₁₉	1	4	yes	—
230	C₂²⋊C₄	—	19	4	no	—
231	C₂²⋊C₄	—	19	8	no	—
232	C₂²⋊C₄	—	19	8	no	—
233	C₂²⋊C₄	—	19	8	no	—
234	C₂²⋊C₄	—	19	8	no	—
235	C₂²⋊C₄	C₂²×D₁₉	1	2	yes	—
236	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
237	C₂²⋊C₄	C₂²×D₁₉	1	4	yes	—
238	C₄²	—	19	4	no	—
239	C₄²	C₂²×D₁₉	1	2	yes	—
240	C₂³	—	2	4	no	—
241	C₂³	—	4	1	no	—
242	C₂³	—	19	8	no	—
243	C₂³	—	38	8	no	—
244	C₂³	—	76	4	no	—
245	C₂³	—	76	4	no	—
246	C₂³	—	76	4	no	—
247	C₂³	—	76	8	no	—
248	C₂³	C₂³×D₁₉	1	2	yes	—
249	C₂³	C₂³×D₁₉	1	1	yes	characteristic
250	C₂³	C₂³×D₁₉	1	2	yes	—
251	Q₈	—	4	1	no	—
252	D₄	—	2	8	no	—
253	D₄	—	4	1	no	—
254	D₄	—	4	2	no	—
255	D₄	—	38	16	no	—
256	D₄	—	38	8	no	—
257	D₄	—	76	4	no	—
258	D₄	—	76	4	no	—
259	D₄	—	76	8	no	—
260	D₄	—	76	8	no	—
261	D₄	—	76	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	19	8	no	—
268	C₂×C₄	—	76	8	no	—
269	C₂×C₄	—	76	4	no	—
270	C₂×C₄	C₂³×D₁₉	1	2	yes	—
271	C₂×C₄	C₂³×D₁₉	1	4	yes	—
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	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	38	8	no	—
283	C₂²	—	38	8	no	—
284	C₂²	—	38	8	no	—
285	C₂²	—	152	8	no	—
286	C₂²	—	152	8	no	—
287	C₂²	#10597435	1	4	yes	—
288	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	4	no	—
293	C₄	—	38	8	no	—
294	C₄	#10597430	1	2	yes	—
295	C₂	—	2	4	no	—
296	C₂	—	4	1	no	—
297	C₂	—	4	2	no	—
298	C₂	—	76	8	no	—
299	C₂	C₂×#10597430	1	1	yes	characteristic
300	C₂	C₂×#10597435	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₁₉×#525	C₂
hypercenter	#525	D₁₉
hypocenter	C₁₉	#3537
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ #525

Complements

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

N	H ≅ G/N	type
1. #11927284	C₂	semidirect
2. #11927240	C₂	semidirect
3. D₄×D₇₆	C₂	semidirect
4. #11927176	C₂	semidirect
5. #11927149	C₂	semidirect
6. #11927134	C₂	semidirect
7. #11927108	C₂	semidirect
8. #11927102	C₂	semidirect
9. C₂×#10597376	C₂	semidirect
10. C₁₉×#525	C₂	semidirect
11. C₂×#41×C₁₉	C₂²	semidirect
12. C₂³×C₄×C₁₉	C₂²	semidirect
13. C₂²×C₂²⋊D₁₉	C₂²	semidirect
14. C₂×D₄×D₁₉	C₂²	semidirect
15. C₂²×D₇₆	C₂²	semidirect
16. C₂²×D₇₆	C₂²	semidirect
17. C₁₉×#123	C₂²	semidirect
18. C₁₉×C₂²⋊Q₈	C₂²	semidirect
19. C₁₉×#121	C₂²	semidirect
20. C₁₉×#120	C₂²	semidirect
21. C₄×D₄×C₁₉	C₂²	semidirect
22. C₁₉×#117	C₂²	semidirect
23. #10597393	C₂²	semidirect
24. #10597386	C₂²	semidirect
25. #10597385	C₂²	semidirect
26. #10597383	C₂²	semidirect
27. #10597383	C₂²	semidirect
28. #10597376	C₂²	semidirect
29. #10597376	C₂²	semidirect
30. #10597376	C₂²	semidirect
31. #10597376	C₂²	semidirect
32. #10597376	C₂²	semidirect
33. C₂×(C₂×D₁₉)⋊C₄	C₂²	semidirect
34. #10597372	C₂²	semidirect
35. C₂×C₄⋊Dic₁₉	C₂²	semidirect
36. #10597344	C₂²	semidirect
37. #10597341	C₂²	semidirect
38. #10597340	C₂²	semidirect
39. #10597330	C₂²	semidirect
40. #10597329	C₂²	semidirect
41. #10597328	C₂²	semidirect
42. #10597328	C₂²	semidirect
43. (C₂²⋊D₁₉)⋊C₄	C₂²	semidirect
44. #10597320	C₂²	semidirect
45. C₄×D₇₆	C₂²	semidirect
46. #10597316	C₂²	semidirect
153. #525	D₁₉	semidirect
161. C₂×#41	C₂×D₁₉	semidirect
166. C₂³×C₄	C₂×D₁₉	semidirect
174. #123	C₂×D₁₉	semidirect
175. C₂²⋊Q₈	C₂×D₁₉	semidirect
184. #121	C₂×D₁₉	semidirect
189. #120	C₂×D₁₉	semidirect
192. C₄×D₄	C₂×D₁₉	semidirect
194. #117	C₂×D₁₉	semidirect
197. C₁₉	#3537	semidirect

Maximal subgroups

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

subgroup	index
#11927284	2
#11927240	2
D₄×D₇₆	2
#11927176	2
#11927149	2
#11927134	2
#11927108	2
#11927102	2
C₂×#10597376	2
C₁₉×#525	2
#3537	19