Subgroups — #14577279

Order 2432.

1560 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10597430	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927193	C₂	1	8	yes	maximal, complemented
3	#11927182	C₂	1	8	yes	maximal, complemented
4	#11927118	C₂	1	2	yes	maximal, complemented
5	#11927103	C₂	1	4	yes	maximal, complemented
6	#11927055	C₂	1	2	yes	maximal, complemented
7	C₂×#10597375	C₂	1	4	yes	maximal, complemented
8	#11927019	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₉	—	2	8	no	—
11	C₂×#41×C₁₉	C₂²	1	8	yes	complemented
12	C₂²×Q₈×C₁₉	C₂²	1	1	yes	characteristic, complemented
13	C₂²×D₄×C₁₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₉	C₂²	1	2	yes	complemented
15	C₂³×C₄×C₁₉	C₂²	1	2	yes	complemented
16	C₂³×C₄×C₁₉	C₂²	1	1	yes	characteristic, complemented
17	C₂²×C₂²⋊D₁₉	C₂²	1	4	yes	—
18	#10597430	—	2	8	no	—
19	C₂×#64543	C₂²	1	4	yes	complemented
20	C₂×D₄×D₁₉	C₂²	1	4	yes	complemented
21	C₂×#64541	C₂²	1	2	yes	—
22	#10597399	C₂²	1	4	yes	complemented
23	C₂×C₂²⋊Dic₁₉	C₂²	1	4	yes	complemented
24	#10597393	C₂²	1	4	yes	complemented
25	#10597392	C₂²	1	8	yes	complemented
26	#10597390	C₂²	1	4	yes	complemented
27	#10597385	C₂²	1	16	yes	complemented
28	#10597383	C₂²	1	8	yes	complemented
29	#10597381	C₂²	1	8	yes	complemented
30	D₄×Dic₁₉	C₂²	1	8	yes	complemented
31	#10597376	C₂²	1	4	yes	complemented
32	#10597375	C₂²	1	4	yes	complemented
33	#10597375	C₂²	1	8	yes	complemented
34	#10597375	C₂²	1	16	yes	complemented
35	C₄×C₂²⋊D₁₉	C₂²	1	8	yes	complemented
36	C₂×(C₂×D₁₉)⋊C₄	C₂²	1	8	yes	complemented
37	#10597372	C₂²	1	2	yes	complemented
38	#10597370	C₂²	1	4	yes	complemented
39	C₂×Dic₁₉⋊C₄	C₂²	1	8	yes	complemented
40	C₂⁴×C₁₉	C₂³	1	1	yes	characteristic
41	C₂⁴×C₁₉	C₂³	1	2	yes	—
42	C₂³×D₁₉	C₂³	1	4	yes	—
43	#41×C₁₉	—	2	32	no	—
44	C₂×Q₈×C₁₉	—	2	4	no	—
45	C₂×Q₈×C₁₉	C₂³	1	4	yes	—
46	C₂×D₄×C₁₉	—	2	8	no	—
47	C₂×D₄×C₁₉	D₄	1	8	yes	complemented
48	C₂×D₄×C₁₉	C₂³	1	8	yes	—
49	C₂×D₄×C₁₉	C₂³	1	4	yes	—
50	C₂²×C₄×C₁₉	—	2	8	no	—
51	C₂²×C₄×C₁₉	—	2	4	no	—
52	C₂²×C₄×C₁₉	C₂³	1	4	yes	—
53	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₁₉	C₂³	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂³	1	8	yes	—
56	C₂²×C₄×C₁₉	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₉	—	2	16	no	—
58	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
59	C₂×C₂²⋊D₁₉	C₂³	1	4	yes	—
60	C₂×C₂²⋊D₁₉	C₂³	1	8	yes	—
61	C₂²×Dic₁₉	C₂³	1	4	yes	—
62	C₂²×Dic₁₉	C₂³	1	8	yes	—
63	#64543	—	2	16	no	—
64	D₄×D₁₉	—	2	16	no	—
65	#64541	—	2	8	no	—
66	C₂×D₇₆	C₂³	1	2	yes	—
67	C₂×C₄×D₁₉	C₂³	1	4	yes	—
68	C₂×Dic₃₈	C₂³	1	2	yes	—
69	C₂²⋊Dic₁₉	C₂³	1	8	yes	—
70	C₂²⋊Dic₁₉	C₂³	1	8	yes	—
71	C₂²⋊Dic₁₉	C₂³	1	4	yes	—
72	(C₂×D₁₉)⋊C₄	C₂³	1	8	yes	—
73	(C₂×D₁₉)⋊C₄	C₂³	1	16	yes	—
74	C₄⋊Dic₁₉	C₂³	1	4	yes	—
75	Dic₁₉⋊C₄	C₂³	1	16	yes	—
76	Dic₁₉⋊C₄	C₂³	1	8	yes	—
77	C₄×Dic₁₉	C₂³	1	4	yes	—
78	C₂³×C₁₉	—	2	2	no	—
79	C₂³×C₁₉	—	4	4	no	—
80	C₂³×C₁₉	—	4	2	no	—
81	C₂³×C₁₉	C₂×D₄	1	8	yes	—
82	C₂³×C₁₉	C₂⁴	1	2	yes	—
83	C₂³×C₁₉	C₂⁴	1	1	yes	characteristic
84	C₂³×C₁₉	C₂⁴	1	4	yes	—
85	C₂²×D₁₉	—	2	8	no	—
86	C₂²×D₁₉	—	4	8	no	—
87	C₂²×D₁₉	C₂⁴	1	4	yes	—
88	Q₈×C₁₉	—	2	8	no	—
89	D₄×C₁₉	—	2	16	no	—
90	D₄×C₁₉	C₂×D₄	1	16	yes	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	—	2	4	no	—
93	C₂×C₄×C₁₉	—	2	16	no	—
94	C₂×C₄×C₁₉	—	2	8	no	—
95	C₂×C₄×C₁₉	C₂×D₄	1	4	yes	—
96	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
97	C₂×C₄×C₁₉	C₂⁴	1	4	yes	—
98	C₂×C₄×C₁₉	C₂⁴	1	2	yes	—
99	C₂²⋊D₁₉	—	2	8	no	—
100	C₂²⋊D₁₉	—	2	16	no	—
101	C₂²⋊D₁₉	—	4	8	no	—
102	C₂×Dic₁₉	—	2	8	no	—
103	C₂×Dic₁₉	—	4	8	no	—
104	C₂×Dic₁₉	C₂⁴	1	8	yes	—
105	C₂×Dic₁₉	C₂⁴	1	4	yes	—
106	D₇₆	—	2	4	no	—
107	C₄×D₁₉	—	2	8	no	—
108	Dic₃₈	—	2	4	no	—
109	#3485	—	19	1	no	maximal, Sylow
110	C₂²×C₁₉	—	2	8	no	—
111	C₂²×C₁₉	—	2	2	no	—
112	C₂²×C₁₉	—	2	2	no	—
113	C₂²×C₁₉	—	2	4	no	—
114	C₂²×C₁₉	—	4	8	no	—
115	C₂²×C₁₉	—	4	4	no	—
116	C₂²×C₁₉	C₂²×D₄	1	8	yes	—
117	C₂²×C₁₉	C₂²×D₄	1	2	yes	—
118	C₂²×C₁₉	C₂⁵	1	1	yes	characteristic
119	C₂×D₁₉	—	2	4	no	—
120	C₂×D₁₉	—	2	8	no	—
121	C₂×D₁₉	—	8	8	no	—
122	C₄×C₁₉	—	2	2	no	—
123	C₄×C₁₉	—	2	4	no	—
124	C₄×C₁₉	C₂²×D₄	1	4	yes	—
125	Dic₁₉	—	2	4	no	—
126	Dic₁₉	—	2	8	no	—
127	C₂×H(5,𝔽₂)	—	19	1	no	—
128	C₂²×#41	D₁₉	1	1	yes	characteristic, complemented
129	#547	—	19	8	no	—
130	#546	—	19	8	no	—
131	#536	—	19	2	no	—
132	#534	—	19	4	no	—
133	#525	—	19	2	no	—
134	C₂×#123	—	19	4	no	—
135	#518	—	19	1	no	—
136	C₂×C₁₉	—	2	8	no	—
137	C₂×C₁₉	—	2	2	no	—
138	C₂×C₁₉	—	4	2	no	—
139	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
140	C₂×C₁₉	#585	1	2	yes	—
141	D₁₉	—	4	4	no	—
142	H(5,𝔽₂)	—	38	8	no	—
143	C₂×#41	—	2	8	no	—
144	C₂×#41	—	19	4	no	—
145	C₂×#41	—	19	2	no	—
146	C₂×#41	C₂×D₁₉	1	8	yes	complemented
147	C₂²×Q₈	C₂×D₁₉	1	1	yes	characteristic, complemented
148	C₂²×D₄	—	19	4	no	—
149	C₂²×D₄	—	19	4	no	—
150	C₂²×D₄	C₂×D₁₉	1	1	yes	characteristic
151	C₂²×D₄	C₂×D₁₉	1	2	yes	complemented
152	C₂³×C₄	C₂×D₁₉	1	2	yes	complemented
153	C₂³×C₄	C₂×D₁₉	1	1	yes	characteristic, complemented
154	C₄⋊Q₈	—	19	4	no	—
155	#124	—	19	4	no	—
156	#123	—	19	8	no	—
157	#123	—	19	4	no	—
158	#123	—	19	8	no	—
159	#123	—	19	16	no	—
160	C₂²⋊Q₈	—	19	8	no	—
161	C₂²⋊Q₈	—	19	4	no	—
162	#121	—	19	16	no	—
163	#121	—	19	4	no	—
164	#120	—	19	4	no	—
165	#120	—	19	8	no	—
166	C₄×D₄	—	19	8	no	—
167	C₄×D₄	—	19	8	no	—
168	#117	—	19	2	no	—
169	C₂×C₄⋊C₄	—	19	8	no	—
170	C₂×C₂²⋊C₄	—	19	4	no	—
171	C₂×C₂²⋊C₄	—	19	8	no	—
172	C₁₉	#3485	1	1	yes	characteristic, complemented, Sylow
173	C₂⁴	—	19	4	no	—
174	C₂⁴	C₂²×D₁₉	1	1	yes	characteristic
175	C₂⁴	C₂²×D₁₉	1	2	yes	—
176	#41	—	2	32	no	—
177	#41	—	38	16	no	—
178	#41	—	38	8	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	19	2	no	—
181	C₂×Q₈	C₂²×D₁₉	1	4	yes	—
182	C₂×D₄	—	2	8	no	—
183	C₂×D₄	—	19	4	no	—
184	C₂×D₄	—	19	8	no	—
185	C₂×D₄	—	19	8	no	—
186	C₂×D₄	—	19	2	no	—
187	C₂×D₄	—	38	16	no	—
188	C₂×D₄	—	38	16	no	—
189	C₂×D₄	C₂²⋊D₁₉	1	8	yes	complemented
190	C₂×D₄	C₂²×D₁₉	1	8	yes	—
191	C₂×D₄	C₂²×D₁₉	1	4	yes	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	19	4	no	—
195	C₂²×C₄	—	19	8	no	—
196	C₂²×C₄	—	19	4	no	—
197	C₂²×C₄	C₂²×D₁₉	1	8	yes	—
198	C₂²×C₄	C₂²×D₁₉	1	4	yes	—
199	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₁₉	1	2	yes	—
201	C₂²×C₄	C₂²×D₁₉	1	1	yes	characteristic
202	C₄⋊C₄	—	19	8	no	—
203	C₄⋊C₄	—	19	16	no	—
204	C₄⋊C₄	—	19	4	no	—
205	C₂²⋊C₄	—	19	4	no	—
206	C₂²⋊C₄	—	19	8	no	—
207	C₂²⋊C₄	—	19	8	no	—
208	C₂²⋊C₄	—	19	8	no	—
209	C₂²⋊C₄	—	19	16	no	—
210	C₄²	—	19	4	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	4	4	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	19	4	no	—
215	C₂³	—	38	8	no	—
216	C₂³	—	76	8	no	—
217	C₂³	C₂×C₂²⋊D₁₉	1	8	yes	—
218	C₂³	C₂³×D₁₉	1	2	yes	—
219	C₂³	C₂³×D₁₉	1	1	yes	characteristic
220	C₂³	C₂³×D₁₉	1	4	yes	—
221	Q₈	—	2	8	no	—
222	Q₈	—	38	4	no	—
223	D₄	—	2	16	no	—
224	D₄	—	38	8	no	—
225	D₄	—	38	16	no	—
226	D₄	—	38	4	no	—
227	D₄	—	76	8	no	—
228	D₄	C₂×C₂²⋊D₁₉	1	16	yes	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	19	8	no	—
234	C₂×C₄	—	19	4	no	—
235	C₂×C₄	—	38	8	no	—
236	C₂×C₄	—	38	8	no	—
237	C₂×C₄	—	76	8	no	—
238	C₂×C₄	C₂×C₂²⋊D₁₉	1	4	yes	—
239	C₂×C₄	C₂³×D₁₉	1	2	yes	—
240	C₂×C₄	C₂³×D₁₉	1	4	yes	—
241	C₂×C₄	C₂³×D₁₉	1	2	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	8	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	38	4	no	—
249	C₂²	—	38	8	no	—
250	C₂²	—	152	8	no	—
251	C₂²	C₂²×C₂²⋊D₁₉	1	2	yes	—
252	C₂²	C₂²×C₂²⋊D₁₉	1	8	yes	—
253	C₂²	C₂⁴×D₁₉	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	4	no	—
256	C₄	—	38	4	no	—
257	C₄	—	38	8	no	—
258	C₄	C₂²×C₂²⋊D₁₉	1	4	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	2	8	no	—
261	C₂	—	4	2	no	—
262	C₂	—	76	4	no	—
263	C₂	C₂³×C₂²⋊D₁₉	1	1	yes	characteristic
264	C₂	#11927380	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₉
commutator, socle, cosocle	C₂²×C₁₉	C₂⁵
Fitting	C₂²×#41×C₁₉	C₂
hypercenter	C₂²×#41	D₁₉
hypocenter	C₁₉	#3485
LCS	G ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#10597430	C₂	semidirect
2. #11927193	C₂	semidirect
3. #11927182	C₂	semidirect
4. #11927118	C₂	semidirect
5. #11927103	C₂	semidirect
6. #11927055	C₂	semidirect
7. C₂×#10597375	C₂	semidirect
8. #11927019	C₂	semidirect
9. C₂²×#41×C₁₉	C₂	semidirect
11. C₂×#41×C₁₉	C₂²	semidirect
12. C₂²×Q₈×C₁₉	C₂²	semidirect
14. C₂²×D₄×C₁₉	C₂²	semidirect
15. C₂³×C₄×C₁₉	C₂²	semidirect
16. C₂³×C₄×C₁₉	C₂²	semidirect
19. C₂×#64543	C₂²	semidirect
20. C₂×D₄×D₁₉	C₂²	semidirect
22. #10597399	C₂²	semidirect
23. C₂×C₂²⋊Dic₁₉	C₂²	semidirect
24. #10597393	C₂²	semidirect
25. #10597392	C₂²	semidirect
26. #10597390	C₂²	semidirect
27. #10597385	C₂²	semidirect
28. #10597383	C₂²	semidirect
29. #10597381	C₂²	semidirect
30. D₄×Dic₁₉	C₂²	semidirect
31. #10597376	C₂²	semidirect
32. #10597375	C₂²	semidirect
33. #10597375	C₂²	semidirect
34. #10597375	C₂²	semidirect
35. C₄×C₂²⋊D₁₉	C₂²	semidirect
36. C₂×(C₂×D₁₉)⋊C₄	C₂²	semidirect
37. #10597372	C₂²	semidirect
38. #10597370	C₂²	semidirect
39. C₂×Dic₁₉⋊C₄	C₂²	semidirect
47. C₂×D₄×C₁₉	D₄	semidirect
128. C₂²×#41	D₁₉	semidirect
146. C₂×#41	C₂×D₁₉	semidirect
147. C₂²×Q₈	C₂×D₁₉	semidirect
151. C₂²×D₄	C₂×D₁₉	semidirect
152. C₂³×C₄	C₂×D₁₉	semidirect
153. C₂³×C₄	C₂×D₁₉	semidirect
172. C₁₉	#3485	semidirect
189. C₂×D₄	C₂²⋊D₁₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10597430	2
#11927193	2
#11927182	2
#11927118	2
#11927103	2
#11927055	2
C₂×#10597375	2
#11927019	2
C₂²×#41×C₁₉	2
#3485	19