Subgroups — #14578406

Order 2432.

1424 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11927380	C₂	1	1	yes	maximal, characteristic, complemented
2	#11927359	C₂	1	4	yes	maximal, complemented
3	#11927355	C₂	1	4	yes	maximal, complemented
4	#11927352	C₂	1	4	yes	maximal, complemented
5	#11927347	C₂	1	4	yes	maximal, complemented
6	#11927342	C₂	1	6	yes	maximal, complemented
7	C₂×#10597396	C₂	1	6	yes	maximal, complemented
8	C₂×#10597394	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₉×#585	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₉×2₋¹⁺⁴	—	2	1	no	—
11	C₁₉×2₋¹⁺⁴	C₂²	1	4	yes	complemented
12	C₁₉×H(5,𝔽₂)	—	2	3	no	—
13	C₁₉×H(5,𝔽₂)	C₂²	1	4	yes	complemented
14	C₂×#41×C₁₉	—	2	4	no	—
15	C₂×#41×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×#41×C₁₉	C₂²	1	6	yes	complemented
17	#10597436	C₂²	1	4	yes	complemented
18	#10597435	C₂²	1	4	yes	complemented
19	#41×D₁₉	—	2	4	no	—
20	#10597433	—	2	1	no	—
21	#10597430	—	2	3	no	—
22	C₂×#64541	C₂²	1	6	yes	complemented
23	#10597397	C₂²	1	12	yes	complemented
24	#10597396	C₂²	1	4	yes	complemented
25	#10597396	C₂²	1	12	yes	complemented
26	#10597396	C₂²	1	12	yes	complemented
27	#10597396	C₂²	1	12	yes	complemented
28	#10597395	C₂²	1	12	yes	complemented
29	#10597394	C₂²	1	4	yes	complemented
30	#10597394	C₂²	1	4	yes	complemented
31	C₂×#64528	C₂²	1	6	yes	complemented
32	#10597388	C₂²	1	12	yes	complemented
33	C₂×#64527	C₂²	1	6	yes	complemented
34	C₂×#64526	C₂²	1	6	yes	complemented
35	#10597378	C₂²	1	12	yes	complemented
36	C₂×#64525	C₂²	1	6	yes	complemented
37	C₂×#64520	C₂²	1	3	yes	complemented
38	C₂²×C₁₉⋊C₈	C₂²	1	3	yes	complemented
39	#41×C₁₉	—	2	4	no	—
40	#41×C₁₉	—	2	12	no	—
41	#41×C₁₉	—	2	6	no	—
42	#41×C₁₉	—	2	6	no	—
43	#41×C₁₉	D₄	1	4	yes	—
44	#41×C₁₉	C₂³	1	12	yes	—
45	#41×C₁₉	C₂³	1	4	yes	—
46	#41×C₁₉	C₂³	1	4	yes	—
47	C₂×Q₈×C₁₉	—	2	4	no	—
48	C₂×Q₈×C₁₉	D₄	1	1	yes	characteristic
49	C₂×Q₈×C₁₉	C₂³	1	6	yes	—
50	C₂×D₄×C₁₉	—	2	6	no	—
51	C₂×D₄×C₁₉	—	2	12	no	—
52	C₂×D₄×C₁₉	D₄	1	3	yes	—
53	C₂×D₄×C₁₉	C₂³	1	6	yes	—
54	C₂²×C₄×C₁₉	—	2	6	no	—
55	C₂²×C₄×C₁₉	C₂³	1	3	yes	—
56	C₂×C₂²⋊D₁₉	—	2	6	no	—
57	#64545	—	2	4	no	—
58	Q₈×D₁₉	—	2	4	no	—
59	#64543	—	2	12	no	—
60	D₄×D₁₉	—	2	12	no	—
61	#64541	—	2	6	no	—
62	#64541	—	2	6	no	—
63	#64541	C₂³	1	12	yes	—
64	#64541	C₂³	1	4	yes	—
65	C₂×D₇₆	C₂³	1	6	yes	—
66	C₂×C₄×D₁₉	—	2	6	no	—
67	C₂×Dic₃₈	C₂³	1	6	yes	—
68	#64528	C₂³	1	12	yes	—
69	#64528	C₂³	1	4	yes	—
70	#64527	C₂³	1	4	yes	—
71	#64527	C₂³	1	12	yes	—
72	#64526	C₂³	1	12	yes	—
73	#64526	C₂³	1	4	yes	—
74	#64525	C₂³	1	4	yes	—
75	#64525	C₂³	1	12	yes	—
76	#64520	C₂³	1	12	yes	—
77	C₂×C₁₉⋊C₈	C₂³	1	6	yes	—
78	C₂×C₁₉⋊C₈	C₂³	1	6	yes	—
79	C₂×C₁₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₂×C₁₉⋊C₈	C₂³	1	3	yes	—
81	C₂³×C₁₉	—	2	6	no	—
82	C₂³×C₁₉	C₂×D₄	1	3	yes	—
83	C₂²×D₁₉	—	2	6	no	—
84	Q₈×C₁₉	—	2	6	no	—
85	Q₈×C₁₉	C₂×D₄	1	4	yes	—
86	Q₈×C₁₉	C₂⁴	1	4	yes	—
87	D₄×C₁₉	—	2	12	no	—
88	D₄×C₁₉	—	2	4	no	—
89	D₄×C₁₉	—	2	6	no	—
90	D₄×C₁₉	C₂×D₄	1	12	yes	—
91	D₄×C₁₉	C₂⁴	1	4	yes	—
92	C₂×C₄×C₁₉	—	2	6	no	—
93	C₂×C₄×C₁₉	—	2	4	no	—
94	C₂×C₄×C₁₉	—	2	12	no	—
95	C₂×C₄×C₁₉	C₂×D₄	1	6	yes	—
96	C₂×C₄×C₁₉	C₂×D₄	1	3	yes	—
97	C₂×C₄×C₁₉	C₂⁴	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₂⁴	1	6	yes	—
99	C₂²⋊D₁₉	—	2	4	no	—
100	C₂²⋊D₁₉	—	2	12	no	—
101	C₂×Dic₁₉	—	2	6	no	—
102	D₇₆	—	2	6	no	—
103	D₇₆	C₂⁴	1	4	yes	—
104	C₄×D₁₉	—	2	4	no	—
105	C₄×D₁₉	—	2	12	no	—
106	Dic₃₈	—	2	6	no	—
107	Dic₃₈	C₂⁴	1	4	yes	—
108	C₁₉⋊C₈	C₂⁴	1	1	yes	characteristic
109	C₁₉⋊C₈	C₂⁴	1	1	yes	characteristic
110	C₁₉⋊C₈	C₂⁴	1	6	yes	—
111	#3584	—	19	1	no	maximal, Sylow
112	C₂²×C₁₉	—	2	4	no	—
113	C₂²×C₁₉	—	4	3	no	—
114	C₂²×C₁₉	—	8	6	no	—
115	C₂²×C₁₉	C₂²×D₄	1	6	yes	—
116	C₂²×C₁₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×D₁₉	—	2	4	no	—
118	C₂×D₁₉	—	8	6	no	—
119	C₄×C₁₉	—	2	4	no	—
120	C₄×C₁₉	C₂²×D₄	1	6	yes	—
121	C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
122	C₄×C₁₉	C₂⁵	1	1	yes	characteristic
123	Dic₁₉	—	2	4	no	—
124	#585	—	19	1	no	—
125	#585	D₁₉	1	1	yes	characteristic, complemented
126	#578	—	19	4	no	—
127	#577	—	19	4	no	—
128	#577	—	19	4	no	—
129	#576	—	19	4	no	—
130	#575	—	19	6	no	—
131	C₂×#135	—	19	6	no	—
132	C₂×#131	—	19	1	no	—
133	C₂×C₁₉	—	2	6	no	—
134	C₂×C₁₉	—	2	1	no	—
135	C₂×C₁₉	—	4	4	no	—
136	C₂×C₁₉	C₂³×D₄	1	1	yes	characteristic
137	D₁₉	—	4	4	no	—
138	2₋¹⁺⁴	—	2	1	no	—
139	2₋¹⁺⁴	—	19	4	no	—
140	2₋¹⁺⁴	—	38	1	no	—
141	2₋¹⁺⁴	C₂×D₁₉	1	4	yes	complemented
142	H(5,𝔽₂)	—	2	3	no	—
143	H(5,𝔽₂)	—	19	4	no	—
144	H(5,𝔽₂)	—	38	3	no	—
145	H(5,𝔽₂)	C₂×D₁₉	1	4	yes	complemented
146	C₂×#41	—	2	4	no	—
147	C₂×#41	—	19	6	no	—
148	C₂×#41	—	38	4	no	—
149	C₂×#41	C₂×D₁₉	1	1	yes	characteristic
150	C₂×#41	C₂×D₁₉	1	6	yes	complemented
151	#137	—	19	12	no	—
152	#137	—	19	12	no	—
153	AGL(1,ℤ/8)	—	19	12	no	—
154	AGL(1,ℤ/8)	—	19	12	no	—
155	#135	—	19	4	no	—
156	#135	—	19	12	no	—
157	#135	—	19	12	no	—
158	#135	—	19	12	no	—
159	C₂×Q₁₆	—	19	6	no	—
160	C₂×SD₁₆	—	19	6	no	—
161	C₂×SD₁₆	—	19	6	no	—
162	C₂×D₈	—	19	6	no	—
163	#131	—	19	4	no	—
164	#131	—	19	4	no	—
165	C₂×M₁₆	—	19	3	no	—
166	C₂²×C₈	—	19	3	no	—
167	C₁₉	#3584	1	1	yes	characteristic, complemented, Sylow
168	#41	—	2	4	no	—
169	#41	—	2	12	no	—
170	#41	—	2	6	no	—
171	#41	—	2	6	no	—
172	#41	—	19	4	no	—
173	#41	—	19	12	no	—
174	#41	—	38	4	no	—
175	#41	—	38	12	no	—
176	#41	—	38	6	no	—
177	#41	—	38	6	no	—
178	#41	C₂²⋊D₁₉	1	4	yes	—
179	#41	C₂²×D₁₉	1	4	yes	—
180	#41	C₂²×D₁₉	1	12	yes	—
181	#41	C₂²×D₁₉	1	4	yes	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	19	6	no	—
184	C₂×Q₈	—	38	4	no	—
185	C₂×Q₈	C₂²⋊D₁₉	1	1	yes	characteristic
186	C₂×Q₈	C₂²×D₁₉	1	6	yes	—
187	C₂×D₄	—	2	6	no	—
188	C₂×D₄	—	2	12	no	—
189	C₂×D₄	—	19	6	no	—
190	C₂×D₄	—	38	6	no	—
191	C₂×D₄	—	38	12	no	—
192	C₂×D₄	C₂²⋊D₁₉	1	3	yes	—
193	C₂×D₄	C₂²×D₁₉	1	6	yes	—
194	C₂²×C₄	—	2	6	no	—
195	C₂²×C₄	—	38	6	no	—
196	C₂²×C₄	C₂²×D₁₉	1	3	yes	—
197	Q₁₆	—	19	4	no	—
198	Q₁₆	—	19	12	no	—
199	SD₁₆	—	19	4	no	—
200	SD₁₆	—	19	12	no	—
201	SD₁₆	—	19	4	no	—
202	SD₁₆	—	19	12	no	—
203	D₈	—	19	4	no	—
204	D₈	—	19	12	no	—
205	M₁₆	—	19	12	no	—
206	C₂×C₈	—	19	1	no	—
207	C₂×C₈	—	19	3	no	—
208	C₂×C₈	—	19	6	no	—
209	C₂×C₈	—	19	6	no	—
210	C₂³	—	2	6	no	—
211	C₂³	—	38	6	no	—
212	C₂³	C₂×C₂²⋊D₁₉	1	3	yes	—
213	Q₈	—	2	6	no	—
214	Q₈	—	19	4	no	—
215	Q₈	—	38	6	no	—
216	Q₈	C₂×C₂²⋊D₁₉	1	4	yes	—
217	Q₈	C₂³×D₁₉	1	4	yes	—
218	D₄	—	2	12	no	—
219	D₄	—	2	4	no	—
220	D₄	—	2	6	no	—
221	D₄	—	19	4	no	—
222	D₄	—	38	12	no	—
223	D₄	—	38	4	no	—
224	D₄	—	38	6	no	—
225	D₄	C₂×C₂²⋊D₁₉	1	12	yes	—
226	D₄	C₂³×D₁₉	1	4	yes	—
227	C₂×C₄	—	2	6	no	—
228	C₂×C₄	—	2	12	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	38	6	no	—
231	C₂×C₄	—	38	12	no	—
232	C₂×C₄	—	38	4	no	—
233	C₂×C₄	C₂×C₂²⋊D₁₉	1	3	yes	—
234	C₂×C₄	C₂×C₂²⋊D₁₉	1	6	yes	—
235	C₂×C₄	C₂³×D₁₉	1	6	yes	—
236	C₂×C₄	C₂³×D₁₉	1	1	yes	characteristic
237	C₈	—	19	1	no	—
238	C₈	—	19	1	no	—
239	C₈	—	19	6	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	4	3	no	—
242	C₂²	—	8	6	no	—
243	C₂²	—	38	4	no	—
244	C₂²	—	152	6	no	—
245	C₂²	C₂²×C₂²⋊D₁₉	1	6	yes	—
246	C₂²	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	38	4	no	—
249	C₄	C₂²×C₂²⋊D₁₉	1	6	yes	—
250	C₄	C₂²×C₂²⋊D₁₉	1	1	yes	characteristic
251	C₄	C₂⁴×D₁₉	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	2	6	no	—
254	C₂	—	4	4	no	—
255	C₂	—	76	4	no	—
256	C₂	C₂³×C₂²⋊D₁₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₁₉
commutator, cosocle	C₄×C₁₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₉
Fitting	C₁₉×#585	C₂
socle	C₂×C₁₉	C₂³×D₄
hypercenter	#585	D₁₉
hypocenter	C₁₉	#3584
LCS	G ▷ C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #585

Complements

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

N	H ≅ G/N	type
1. #11927380	C₂	semidirect
2. #11927359	C₂	semidirect
3. #11927355	C₂	semidirect
4. #11927352	C₂	semidirect
5. #11927347	C₂	semidirect
6. #11927342	C₂	semidirect
7. C₂×#10597396	C₂	semidirect
8. C₂×#10597394	C₂	semidirect
9. C₁₉×#585	C₂	semidirect
11. C₁₉×2₋¹⁺⁴	C₂²	semidirect
13. C₁₉×H(5,𝔽₂)	C₂²	semidirect
16. C₂×#41×C₁₉	C₂²	semidirect
17. #10597436	C₂²	semidirect
18. #10597435	C₂²	semidirect
22. C₂×#64541	C₂²	semidirect
23. #10597397	C₂²	semidirect
24. #10597396	C₂²	semidirect
25. #10597396	C₂²	semidirect
26. #10597396	C₂²	semidirect
27. #10597396	C₂²	semidirect
28. #10597395	C₂²	semidirect
29. #10597394	C₂²	semidirect
30. #10597394	C₂²	semidirect
31. C₂×#64528	C₂²	semidirect
32. #10597388	C₂²	semidirect
33. C₂×#64527	C₂²	semidirect
34. C₂×#64526	C₂²	semidirect
35. #10597378	C₂²	semidirect
36. C₂×#64525	C₂²	semidirect
37. C₂×#64520	C₂²	semidirect
38. C₂²×C₁₉⋊C₈	C₂²	semidirect
125. #585	D₁₉	semidirect
141. 2₋¹⁺⁴	C₂×D₁₉	semidirect
145. H(5,𝔽₂)	C₂×D₁₉	semidirect
150. C₂×#41	C₂×D₁₉	semidirect
167. C₁₉	#3584	semidirect

Maximal subgroups

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

subgroup	index
#11927380	2
#11927359	2
#11927355	2
#11927352	2
#11927347	2
#11927342	2
C₂×#10597396	2
C₂×#10597394	2
C₁₉×#585	2
#3584	19