Subgroups — C₂²×C₁₉×#537 #18068634

Order 4864.

12270 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₁₉	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₉×#537	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₁₉×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₉×#121	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₉×#121	C₂	1	4	yes	maximal, complemented
6	C₂³×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₁₉	—	2	32	no	—
8	C₂²×#41×C₁₉	C₂²	1	48	yes	complemented
9	C₂³×Q₈×C₁₉	C₂²	1	2	yes	—
10	C₂³×D₄×C₁₉	C₂²	1	4	yes	—
11	C₂³×D₄×C₁₉	C₂²	1	2	yes	—
12	C₂³×D₄×C₁₉	C₂²	1	4	yes	—
13	C₂⁴×C₄×C₁₉	C₂²	1	4	yes	—
14	C₂⁴×C₄×C₁₉	C₂²	1	2	yes	—
15	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
16	C₁₉×#537	C₂²	1	256	yes	direct product factor
17	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	96	yes	complemented
18	C₂×C₁₉×#121	C₂²	1	96	yes	complemented
19	C₂×C₁₉×#121	C₂²	1	96	yes	complemented
20	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
21	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
22	C₂²×C₄⋊C₄×C₁₉	C₂²	1	24	yes	complemented
23	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	8	yes	—
24	C₂⁵×C₁₉	C₂³	1	4	yes	—
25	C₂⁵×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×#41×C₁₉	—	2	384	no	—
27	C₂×#41×C₁₉	C₂³	1	128	yes	complemented
28	C₂²×Q₈×C₁₉	—	2	16	no	—
29	C₂²×Q₈×C₁₉	C₂³	1	24	yes	—
30	C₂²×D₄×C₁₉	—	2	32	no	—
31	C₂²×D₄×C₁₉	—	2	16	no	—
32	C₂²×D₄×C₁₉	—	4	16	no	—
33	C₂²×D₄×C₁₉	C₂³	1	48	yes	—
34	C₂²×D₄×C₁₉	C₂³	1	24	yes	—
35	C₂²×D₄×C₁₉	C₂³	1	48	yes	—
36	C₂³×C₄×C₁₉	—	2	32	no	—
37	C₂³×C₄×C₁₉	—	4	8	no	—
38	C₂³×C₄×C₁₉	D₄	1	16	yes	complemented
39	C₂³×C₄×C₁₉	C₂³	1	48	yes	—
40	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
41	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
42	C₂³×C₄×C₁₉	C₂³	1	24	yes	—
43	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
44	C₂³×C₄×C₁₉	C₂³	1	12	yes	—
45	C₁₉×C₂²⋊Q₈	C₂³	1	256	yes	complemented
46	C₁₉×#121	C₂³	1	256	yes	complemented
47	C₁₉×#121	C₂³	1	256	yes	complemented
48	C₂×C₄⋊C₄×C₁₉	C₂³	1	48	yes	—
49	C₂×C₄⋊C₄×C₁₉	C₂³	1	48	yes	—
50	C₂×C₄⋊C₄×C₁₉	C₂³	1	64	yes	complemented
51	C₂×C₂²⋊C₄×C₁₉	C₂³	1	96	yes	—
52	C₂⁴×C₁₉	—	2	4	no	—
53	C₂⁴×C₁₉	—	2	4	no	—
54	C₂⁴×C₁₉	—	2	16	no	—
55	C₂⁴×C₁₉	—	2	16	no	—
56	C₂⁴×C₁₉	—	2	16	no	—
57	C₂⁴×C₁₉	C₂×D₄	1	8	yes	—
58	C₂⁴×C₁₉	C₂⁴	1	24	yes	—
59	C₂⁴×C₁₉	C₂⁴	1	6	yes	—
60	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
61	#41×C₁₉	—	2	512	no	—
62	C₂×Q₈×C₁₉	—	2	96	no	—
63	C₂×Q₈×C₁₉	C₂⁴	1	32	yes	—
64	C₂×D₄×C₁₉	—	2	192	no	—
65	C₂×D₄×C₁₉	—	2	96	no	—
66	C₂×D₄×C₁₉	—	4	96	no	—
67	C₂×D₄×C₁₉	C₂⁴	1	64	yes	—
68	C₂×D₄×C₁₉	C₂⁴	1	32	yes	—
69	C₂×D₄×C₁₉	C₂⁴	1	64	yes	—
70	C₂²×C₄×C₁₉	—	2	16	no	—
71	C₂²×C₄×C₁₉	—	2	16	no	—
72	C₂²×C₄×C₁₉	—	2	192	no	—
73	C₂²×C₄×C₁₉	—	4	48	no	—
74	C₂²×C₄×C₁₉	C₂×D₄	1	16	yes	—
75	C₂²×C₄×C₁₉	C₂×D₄	1	96	yes	complemented
76	C₂²×C₄×C₁₉	C₂⁴	1	24	yes	—
77	C₂²×C₄×C₁₉	C₂⁴	1	24	yes	—
78	C₂²×C₄×C₁₉	C₂⁴	1	64	yes	—
79	C₂²×C₄×C₁₉	C₂⁴	1	32	yes	—
80	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
81	C₂²×C₄×C₁₉	C₂⁴	1	16	yes	—
82	C₄⋊C₄×C₁₉	C₂⁴	1	64	yes	—
83	C₄⋊C₄×C₁₉	C₂⁴	1	64	yes	—
84	C₂²⋊C₄×C₁₉	C₂⁴	1	128	yes	—
85	C₂²×#537	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₁₉	—	2	32	no	—
87	C₂³×C₁₉	—	2	12	no	—
88	C₂³×C₁₉	—	2	12	no	—
89	C₂³×C₁₉	—	2	48	no	—
90	C₂³×C₁₉	—	2	48	no	—
91	C₂³×C₁₉	—	2	48	no	—
92	C₂³×C₁₉	—	4	64	no	—
93	C₂³×C₁₉	C₂²×D₄	1	24	yes	—
94	C₂³×C₁₉	C₂²×D₄	1	4	yes	—
95	C₂³×C₁₉	H(5,𝔽₂)	1	4	yes	—
96	C₂³×C₁₉	2₋¹⁺⁴	1	4	yes	—
97	C₂³×C₁₉	C₂⁵	1	16	yes	—
98	C₂³×C₁₉	C₂⁵	1	4	yes	—
99	C₂³×C₁₉	C₂⁵	1	3	yes	—
100	Q₈×C₁₉	—	2	64	no	—
101	D₄×C₁₉	—	2	128	no	—
102	D₄×C₁₉	—	2	64	no	—
103	D₄×C₁₉	—	4	64	no	—
104	C₂×C₄×C₁₉	—	2	48	no	—
105	C₂×C₄×C₁₉	—	2	48	no	—
106	C₂×C₄×C₁₉	—	2	128	no	—
107	C₂×C₄×C₁₉	—	4	32	no	—
108	C₂×C₄×C₁₉	C₂²×D₄	1	48	yes	—
109	C₂×C₄×C₁₉	C₂²×D₄	1	64	yes	complemented
110	C₂×C₄×C₁₉	C₂⁵	1	16	yes	—
111	C₂×C₄×C₁₉	C₂⁵	1	16	yes	—
112	C₂×C₄×C₁₉	C₂⁵	1	8	yes	—
113	C₂³×#41	C₂×C₁₉	1	2	yes	complemented
114	C₂×#537	C₂×C₁₉	1	48	yes	direct product factor
115	C₂²×C₂²⋊Q₈	C₂×C₁₉	1	4	yes	complemented
116	C₂²×#121	C₂×C₁₉	1	4	yes	complemented
117	C₂²×#121	C₂×C₁₉	1	4	yes	complemented
118	C₂³×C₄⋊C₄	C₂×C₁₉	1	1	yes	characteristic, complemented
119	C₂²×C₁₉	—	2	4	no	—
120	C₂²×C₁₉	—	2	4	no	—
121	C₂²×C₁₉	—	2	48	no	—
122	C₂²×C₁₉	—	2	16	no	—
123	C₂²×C₁₉	—	2	16	no	—
124	C₂²×C₁₉	—	2	16	no	—
125	C₂²×C₁₉	—	4	96	no	—
126	C₂²×C₁₉	#537	1	16	yes	direct product factor
127	C₂²×C₁₉	C₂³×D₄	1	8	yes	—
128	C₂²×C₁₉	C₂³×D₄	1	6	yes	—
129	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	6	yes	—
130	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	6	yes	—
131	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
132	C₄×C₁₉	—	2	16	no	—
133	C₄×C₁₉	—	2	16	no	—
134	C₄×C₁₉	C₂³×D₄	1	16	yes	—
135	C₂²×#41	—	2	32	no	—
136	C₂²×#41	C₂²×C₁₉	1	48	yes	complemented
137	C₂³×Q₈	C₂²×C₁₉	1	2	yes	—
138	C₂³×D₄	C₂²×C₁₉	1	4	yes	—
139	C₂³×D₄	C₂²×C₁₉	1	2	yes	—
140	C₂³×D₄	C₂²×C₁₉	1	4	yes	—
141	C₂⁴×C₄	C₂²×C₁₉	1	4	yes	—
142	C₂⁴×C₄	C₂²×C₁₉	1	2	yes	—
143	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
144	#537	C₂²×C₁₉	1	256	yes	direct product factor
145	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	96	yes	complemented
146	C₂×#121	C₂²×C₁₉	1	96	yes	complemented
147	C₂×#121	C₂²×C₁₉	1	96	yes	complemented
148	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
149	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
150	C₂²×C₄⋊C₄	C₂²×C₁₉	1	24	yes	complemented
151	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	8	yes	—
152	C₂×C₁₉	—	2	8	no	—
153	C₂×C₁₉	—	4	16	no	—
154	C₂×C₁₉	C₂×#537	1	12	yes	direct product factor
155	C₂×C₁₉	C₂⁴×D₄	1	1	yes	characteristic
156	C₂×C₁₉	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×C₁₉	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
158	C₂⁵	C₂³×C₁₉	1	4	yes	—
159	C₂⁵	C₂³×C₁₉	1	1	yes	characteristic
160	C₂×#41	—	2	384	no	—
161	C₂×#41	C₂³×C₁₉	1	128	yes	complemented
162	C₂²×Q₈	—	2	16	no	—
163	C₂²×Q₈	C₂³×C₁₉	1	24	yes	—
164	C₂²×D₄	—	2	32	no	—
165	C₂²×D₄	—	2	16	no	—
166	C₂²×D₄	—	4	16	no	—
167	C₂²×D₄	C₂³×C₁₉	1	48	yes	—
168	C₂²×D₄	C₂³×C₁₉	1	24	yes	—
169	C₂²×D₄	C₂³×C₁₉	1	48	yes	—
170	C₂³×C₄	—	2	32	no	—
171	C₂³×C₄	—	4	8	no	—
172	C₂³×C₄	D₄×C₁₉	1	16	yes	complemented
173	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
174	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
175	C₂³×C₄	C₂³×C₁₉	1	48	yes	—
176	C₂³×C₄	C₂³×C₁₉	1	24	yes	—
177	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
178	C₂³×C₄	C₂³×C₁₉	1	12	yes	—
179	C₂²⋊Q₈	C₂³×C₁₉	1	256	yes	complemented
180	#121	C₂³×C₁₉	1	256	yes	complemented
181	#121	C₂³×C₁₉	1	256	yes	complemented
182	C₂×C₄⋊C₄	C₂³×C₁₉	1	48	yes	—
183	C₂×C₄⋊C₄	C₂³×C₁₉	1	48	yes	—
184	C₂×C₄⋊C₄	C₂³×C₁₉	1	64	yes	complemented
185	C₂×C₂²⋊C₄	C₂³×C₁₉	1	96	yes	—
186	C₁₉	C₂²×#537	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	—	2	4	no	—
188	C₂⁴	—	2	4	no	—
189	C₂⁴	—	2	16	no	—
190	C₂⁴	—	2	16	no	—
191	C₂⁴	—	2	16	no	—
192	C₂⁴	C₂×D₄×C₁₉	1	8	yes	—
193	C₂⁴	C₂⁴×C₁₉	1	24	yes	—
194	C₂⁴	C₂⁴×C₁₉	1	6	yes	—
195	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
196	#41	—	2	512	no	—
197	C₂×Q₈	—	2	96	no	—
198	C₂×Q₈	C₂⁴×C₁₉	1	32	yes	—
199	C₂×D₄	—	2	192	no	—
200	C₂×D₄	—	2	96	no	—
201	C₂×D₄	—	4	96	no	—
202	C₂×D₄	C₂⁴×C₁₉	1	64	yes	—
203	C₂×D₄	C₂⁴×C₁₉	1	32	yes	—
204	C₂×D₄	C₂⁴×C₁₉	1	64	yes	—
205	C₂²×C₄	—	2	16	no	—
206	C₂²×C₄	—	2	16	no	—
207	C₂²×C₄	—	2	192	no	—
208	C₂²×C₄	—	4	48	no	—
209	C₂²×C₄	C₂×D₄×C₁₉	1	16	yes	—
210	C₂²×C₄	C₂×D₄×C₁₉	1	96	yes	complemented
211	C₂²×C₄	C₂⁴×C₁₉	1	24	yes	—
212	C₂²×C₄	C₂⁴×C₁₉	1	24	yes	—
213	C₂²×C₄	C₂⁴×C₁₉	1	64	yes	—
214	C₂²×C₄	C₂⁴×C₁₉	1	32	yes	—
215	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
216	C₂²×C₄	C₂⁴×C₁₉	1	16	yes	—
217	C₄⋊C₄	C₂⁴×C₁₉	1	64	yes	—
218	C₄⋊C₄	C₂⁴×C₁₉	1	64	yes	—
219	C₂²⋊C₄	C₂⁴×C₁₉	1	128	yes	—
220	C₂³	—	2	32	no	—
221	C₂³	—	2	12	no	—
222	C₂³	—	2	12	no	—
223	C₂³	—	2	48	no	—
224	C₂³	—	2	48	no	—
225	C₂³	—	2	48	no	—
226	C₂³	—	4	64	no	—
227	C₂³	C₂²×D₄×C₁₉	1	24	yes	—
228	C₂³	C₂²×D₄×C₁₉	1	4	yes	—
229	C₂³	C₁₉×H(5,𝔽₂)	1	4	yes	—
230	C₂³	C₁₉×2₋¹⁺⁴	1	4	yes	—
231	C₂³	C₂⁵×C₁₉	1	16	yes	—
232	C₂³	C₂⁵×C₁₉	1	4	yes	—
233	C₂³	C₂⁵×C₁₉	1	3	yes	—
234	Q₈	—	2	64	no	—
235	D₄	—	2	128	no	—
236	D₄	—	2	64	no	—
237	D₄	—	4	64	no	—
238	C₂×C₄	—	2	48	no	—
239	C₂×C₄	—	2	48	no	—
240	C₂×C₄	—	2	128	no	—
241	C₂×C₄	—	4	32	no	—
242	C₂×C₄	C₂²×D₄×C₁₉	1	48	yes	—
243	C₂×C₄	C₂²×D₄×C₁₉	1	64	yes	complemented
244	C₂×C₄	C₂⁵×C₁₉	1	16	yes	—
245	C₂×C₄	C₂⁵×C₁₉	1	16	yes	—
246	C₂×C₄	C₂⁵×C₁₉	1	8	yes	—
247	C₂²	—	2	48	no	—
248	C₂²	—	2	4	no	—
249	C₂²	—	2	4	no	—
250	C₂²	—	2	16	no	—
251	C₂²	—	2	16	no	—
252	C₂²	—	2	16	no	—
253	C₂²	—	4	96	no	—
254	C₂²	C₁₉×#537	1	16	yes	direct product factor
255	C₂²	C₂³×D₄×C₁₉	1	8	yes	—
256	C₂²	C₂³×D₄×C₁₉	1	6	yes	—
257	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	6	yes	—
258	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	6	yes	—
259	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
260	C₄	—	2	16	no	—
261	C₄	—	2	16	no	—
262	C₄	C₂³×D₄×C₁₉	1	16	yes	—
263	C₂	—	2	8	no	—
264	C₂	—	4	16	no	—
265	C₂	C₂×C₁₉×#537	1	12	yes	direct product factor
266	C₂	C₂⁴×D₄×C₁₉	1	1	yes	characteristic
267	C₂	C₂²×C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
268	C₂	C₂²×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×#41×C₁₉	C₂	semidirect
2. C₂×C₁₉×#537	C₂	direct (N × H)
3. C₂²×C₁₉×C₂²⋊Q₈	C₂	semidirect
4. C₂²×C₁₉×#121	C₂	semidirect
5. C₂²×C₁₉×#121	C₂	semidirect
6. C₂³×C₄⋊C₄×C₁₉	C₂	semidirect
8. C₂²×#41×C₁₉	C₂²	semidirect
16. C₁₉×#537	C₂²	direct (N × H)
17. C₂×C₁₉×C₂²⋊Q₈	C₂²	semidirect
18. C₂×C₁₉×#121	C₂²	semidirect
19. C₂×C₁₉×#121	C₂²	semidirect
22. C₂²×C₄⋊C₄×C₁₉	C₂²	semidirect
27. C₂×#41×C₁₉	C₂³	semidirect
38. C₂³×C₄×C₁₉	D₄	semidirect
45. C₁₉×C₂²⋊Q₈	C₂³	semidirect
46. C₁₉×#121	C₂³	semidirect
47. C₁₉×#121	C₂³	semidirect
50. C₂×C₄⋊C₄×C₁₉	C₂³	semidirect
75. C₂²×C₄×C₁₉	C₂×D₄	semidirect
85. C₂²×#537	C₁₉	direct (N × H)
109. C₂×C₄×C₁₉	C₂²×D₄	semidirect
113. C₂³×#41	C₂×C₁₉	semidirect
114. C₂×#537	C₂×C₁₉	direct (N × H)
115. C₂²×C₂²⋊Q₈	C₂×C₁₉	semidirect
116. C₂²×#121	C₂×C₁₉	semidirect
117. C₂²×#121	C₂×C₁₉	semidirect
118. C₂³×C₄⋊C₄	C₂×C₁₉	semidirect
126. C₂²×C₁₉	#537	direct (N × H)
136. C₂²×#41	C₂²×C₁₉	semidirect
144. #537	C₂²×C₁₉	direct (N × H)
145. C₂×C₂²⋊Q₈	C₂²×C₁₉	semidirect
146. C₂×#121	C₂²×C₁₉	semidirect
147. C₂×#121	C₂²×C₁₉	semidirect
150. C₂²×C₄⋊C₄	C₂²×C₁₉	semidirect
154. C₂×C₁₉	C₂×#537	direct (N × H)
161. C₂×#41	C₂³×C₁₉	semidirect
172. C₂³×C₄	D₄×C₁₉	semidirect
179. C₂²⋊Q₈	C₂³×C₁₉	semidirect
180. #121	C₂³×C₁₉	semidirect
181. #121	C₂³×C₁₉	semidirect
184. C₂×C₄⋊C₄	C₂³×C₁₉	semidirect
186. C₁₉	C₂²×#537	direct (N × H)
210. C₂²×C₄	C₂×D₄×C₁₉	semidirect
243. C₂×C₄	C₂²×D₄×C₁₉	semidirect
254. C₂²	C₁₉×#537	direct (N × H)
265. C₂	C₂×C₁₉×#537	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₁₉	2
C₂×C₁₉×#537	2
C₂²×C₁₉×C₂²⋊Q₈	2
C₂²×C₁₉×#121	2
C₂³×C₄⋊C₄×C₁₉	2
C₂²×#537	19

Subgroups — C22×C19×#537 #18068634

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₉×#537 #18068634