Subgroups — C₂²×C₁₉×#544 #18068635

Order 4864.

7630 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#544	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₁₉×C₄⋊Q₈	C₂	1	4	yes	maximal, complemented
3	C₂²×C₁₉×#125	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₉×C₂²⋊Q₈	C₂	1	4	yes	maximal
5	C₂²×C₁₉×#117	C₂	1	2	yes	maximal
6	C₂³×C₄⋊C₄×C₁₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₉	C₂²	1	4	yes	—
8	C₂⁴×C₄×C₁₉	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
10	C₁₉×#544	C₂²	1	256	yes	direct product factor
11	C₂×C₁₉×C₄⋊Q₈	C₂²	1	96	yes	complemented
12	C₂×C₁₉×#125	C₂²	1	96	yes	complemented
13	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	96	yes	—
14	C₂×C₁₉×#117	C₂²	1	48	yes	—
15	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
16	C₂²×C₄⋊C₄×C₁₉	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
19	C₂²×C₄⋊C₄×C₁₉	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₁₉	C₂²	1	24	yes	—
21	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
22	C₂²×C₄²×C₁₉	C₂²	1	4	yes	—
23	C₂⁵×C₁₉	C₂³	1	1	yes	characteristic
24	C₂²×Q₈×C₁₉	—	4	16	no	—
25	C₂²×Q₈×C₁₉	C₂³	1	48	yes	—
26	C₂³×C₄×C₁₉	—	4	8	no	—
27	C₂³×C₄×C₁₉	Q₈	1	16	yes	complemented
28	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
29	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
30	C₂³×C₄×C₁₉	C₂³	1	4	yes	—
31	C₂³×C₄×C₁₉	C₂³	1	24	yes	—
32	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
33	C₂³×C₄×C₁₉	C₂³	1	12	yes	—
34	C₁₉×C₄⋊Q₈	C₂³	1	256	yes	complemented
35	C₁₉×#125	C₂³	1	256	yes	complemented
36	C₁₉×C₂²⋊Q₈	C₂³	1	256	yes	—
37	C₁₉×#117	C₂³	1	128	yes	—
38	C₂×C₄⋊C₄×C₁₉	C₂³	1	48	yes	—
39	C₂×C₄⋊C₄×C₁₉	C₂³	1	64	yes	—
40	C₂×C₄⋊C₄×C₁₉	C₂³	1	24	yes	—
41	C₂×C₄⋊C₄×C₁₉	C₂³	1	24	yes	—
42	C₂×C₄⋊C₄×C₁₉	C₂³	1	48	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	96	yes	—
44	C₂×C₂²⋊C₄×C₁₉	C₂³	1	48	yes	—
45	C₂×C₄²×C₁₉	C₂³	1	48	yes	—
46	C₂⁴×C₁₉	—	2	4	no	—
47	C₂⁴×C₁₉	—	2	4	no	—
48	C₂⁴×C₁₉	C₂×Q₈	1	8	yes	—
49	C₂⁴×C₁₉	C₂⁴	1	6	yes	—
50	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
51	C₂×Q₈×C₁₉	—	4	96	no	—
52	C₂×Q₈×C₁₉	C₂⁴	1	64	yes	—
53	C₂²×C₄×C₁₉	—	2	16	no	—
54	C₂²×C₄×C₁₉	—	2	16	no	—
55	C₂²×C₄×C₁₉	—	2	16	no	—
56	C₂²×C₄×C₁₉	—	4	48	no	—
57	C₂²×C₄×C₁₉	C₂×Q₈	1	16	yes	—
58	C₂²×C₄×C₁₉	C₂×Q₈	1	96	yes	complemented
59	C₂²×C₄×C₁₉	C₂⁴	1	24	yes	—
60	C₂²×C₄×C₁₉	C₂⁴	1	24	yes	—
61	C₂²×C₄×C₁₉	C₂⁴	1	24	yes	—
62	C₂²×C₄×C₁₉	C₂⁴	1	32	yes	—
63	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
64	C₂²×C₄×C₁₉	C₂⁴	1	16	yes	—
65	C₄⋊C₄×C₁₉	C₂⁴	1	64	yes	—
66	C₄⋊C₄×C₁₉	C₂⁴	1	32	yes	—
67	C₄⋊C₄×C₁₉	C₂⁴	1	32	yes	—
68	C₄⋊C₄×C₁₉	C₂⁴	1	64	yes	—
69	C₄⋊C₄×C₁₉	C₂⁴	1	128	yes	—
70	C₂²⋊C₄×C₁₉	C₂⁴	1	64	yes	—
71	C₄²×C₁₉	C₂⁴	1	64	yes	—
72	C₂²×#544	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₁₉	—	2	32	no	—
74	C₂³×C₁₉	—	2	12	no	—
75	C₂³×C₁₉	—	2	12	no	—
76	C₂³×C₁₉	C₂²×Q₈	1	24	yes	—
77	C₂³×C₁₉	C₂²×Q₈	1	4	yes	—
78	C₂³×C₁₉	H(5,𝔽₂)	1	4	yes	—
79	C₂³×C₁₉	2₋¹⁺⁴	1	4	yes	—
80	C₂³×C₁₉	C₂⁵	1	4	yes	—
81	C₂³×C₁₉	C₂⁵	1	3	yes	—
82	Q₈×C₁₉	—	4	64	no	—
83	C₂×C₄×C₁₉	—	2	48	no	—
84	C₂×C₄×C₁₉	—	2	48	no	—
85	C₂×C₄×C₁₉	—	2	48	no	—
86	C₂×C₄×C₁₉	—	4	32	no	—
87	C₂×C₄×C₁₉	C₂²×Q₈	1	48	yes	—
88	C₂×C₄×C₁₉	C₂²×Q₈	1	64	yes	complemented
89	C₂×C₄×C₁₉	C₂⁵	1	16	yes	—
90	C₂×C₄×C₁₉	C₂⁵	1	16	yes	—
91	C₂×C₄×C₁₉	C₂⁵	1	16	yes	—
92	C₂×C₄×C₁₉	C₂⁵	1	8	yes	—
93	C₂×#544	C₂×C₁₉	1	48	yes	direct product factor
94	C₂²×C₄⋊Q₈	C₂×C₁₉	1	4	yes	complemented
95	C₂²×#125	C₂×C₁₉	1	4	yes	complemented
96	C₂²×C₂²⋊Q₈	C₂×C₁₉	1	4	yes	—
97	C₂²×#117	C₂×C₁₉	1	2	yes	—
98	C₂³×C₄⋊C₄	C₂×C₁₉	1	1	yes	characteristic
99	C₂²×C₁₉	—	2	4	no	—
100	C₂²×C₁₉	—	2	4	no	—
101	C₂²×C₁₉	—	2	48	no	—
102	C₂²×C₁₉	#544	1	16	yes	direct product factor
103	C₂²×C₁₉	C₂³×Q₈	1	8	yes	—
104	C₂²×C₁₉	C₂³×Q₈	1	6	yes	—
105	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	6	yes	—
106	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	6	yes	—
107	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
108	C₄×C₁₉	—	2	16	no	—
109	C₄×C₁₉	—	2	16	no	—
110	C₄×C₁₉	—	2	16	no	—
111	C₄×C₁₉	C₂³×Q₈	1	16	yes	—
112	C₂³×Q₈	C₂²×C₁₉	1	4	yes	—
113	C₂⁴×C₄	C₂²×C₁₉	1	2	yes	—
114	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
115	#544	C₂²×C₁₉	1	256	yes	direct product factor
116	C₂×C₄⋊Q₈	C₂²×C₁₉	1	96	yes	complemented
117	C₂×#125	C₂²×C₁₉	1	96	yes	complemented
118	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	96	yes	—
119	C₂×#117	C₂²×C₁₉	1	48	yes	—
120	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
121	C₂²×C₄⋊C₄	C₂²×C₁₉	1	24	yes	—
122	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
123	C₂²×C₄⋊C₄	C₂²×C₁₉	1	2	yes	—
124	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
125	C₂²×C₄⋊C₄	C₂²×C₁₉	1	8	yes	—
126	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
127	C₂²×C₄²	C₂²×C₁₉	1	4	yes	—
128	C₂×C₁₉	—	2	8	no	—
129	C₂×C₁₉	C₂×#544	1	12	yes	direct product factor
130	C₂×C₁₉	C₂⁴×Q₈	1	1	yes	characteristic
131	C₂×C₁₉	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₁₉	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
133	C₂⁵	C₂³×C₁₉	1	1	yes	characteristic
134	C₂²×Q₈	—	4	16	no	—
135	C₂²×Q₈	C₂³×C₁₉	1	48	yes	—
136	C₂³×C₄	—	4	8	no	—
137	C₂³×C₄	Q₈×C₁₉	1	16	yes	complemented
138	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
139	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
140	C₂³×C₄	C₂³×C₁₉	1	4	yes	—
141	C₂³×C₄	C₂³×C₁₉	1	24	yes	—
142	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
143	C₂³×C₄	C₂³×C₁₉	1	12	yes	—
144	C₄⋊Q₈	C₂³×C₁₉	1	256	yes	complemented
145	#125	C₂³×C₁₉	1	256	yes	complemented
146	C₂²⋊Q₈	C₂³×C₁₉	1	256	yes	—
147	#117	C₂³×C₁₉	1	128	yes	—
148	C₂×C₄⋊C₄	C₂³×C₁₉	1	48	yes	—
149	C₂×C₄⋊C₄	C₂³×C₁₉	1	64	yes	—
150	C₂×C₄⋊C₄	C₂³×C₁₉	1	24	yes	—
151	C₂×C₄⋊C₄	C₂³×C₁₉	1	24	yes	—
152	C₂×C₄⋊C₄	C₂³×C₁₉	1	48	yes	—
153	C₂×C₄⋊C₄	C₂³×C₁₉	1	96	yes	—
154	C₂×C₂²⋊C₄	C₂³×C₁₉	1	48	yes	—
155	C₂×C₄²	C₂³×C₁₉	1	48	yes	—
156	C₁₉	C₂²×#544	1	1	yes	characteristic, direct product factor, Sylow
157	C₂⁴	—	2	4	no	—
158	C₂⁴	—	2	4	no	—
159	C₂⁴	C₂×Q₈×C₁₉	1	8	yes	—
160	C₂⁴	C₂⁴×C₁₉	1	6	yes	—
161	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
162	C₂×Q₈	—	4	96	no	—
163	C₂×Q₈	C₂⁴×C₁₉	1	64	yes	—
164	C₂²×C₄	—	2	16	no	—
165	C₂²×C₄	—	2	16	no	—
166	C₂²×C₄	—	2	16	no	—
167	C₂²×C₄	—	4	48	no	—
168	C₂²×C₄	C₂×Q₈×C₁₉	1	16	yes	—
169	C₂²×C₄	C₂×Q₈×C₁₉	1	96	yes	complemented
170	C₂²×C₄	C₂⁴×C₁₉	1	24	yes	—
171	C₂²×C₄	C₂⁴×C₁₉	1	24	yes	—
172	C₂²×C₄	C₂⁴×C₁₉	1	24	yes	—
173	C₂²×C₄	C₂⁴×C₁₉	1	32	yes	—
174	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
175	C₂²×C₄	C₂⁴×C₁₉	1	16	yes	—
176	C₄⋊C₄	C₂⁴×C₁₉	1	64	yes	—
177	C₄⋊C₄	C₂⁴×C₁₉	1	32	yes	—
178	C₄⋊C₄	C₂⁴×C₁₉	1	32	yes	—
179	C₄⋊C₄	C₂⁴×C₁₉	1	64	yes	—
180	C₄⋊C₄	C₂⁴×C₁₉	1	128	yes	—
181	C₂²⋊C₄	C₂⁴×C₁₉	1	64	yes	—
182	C₄²	C₂⁴×C₁₉	1	64	yes	—
183	C₂³	—	2	32	no	—
184	C₂³	—	2	12	no	—
185	C₂³	—	2	12	no	—
186	C₂³	C₂²×Q₈×C₁₉	1	24	yes	—
187	C₂³	C₂²×Q₈×C₁₉	1	4	yes	—
188	C₂³	C₁₉×H(5,𝔽₂)	1	4	yes	—
189	C₂³	C₁₉×2₋¹⁺⁴	1	4	yes	—
190	C₂³	C₂⁵×C₁₉	1	4	yes	—
191	C₂³	C₂⁵×C₁₉	1	3	yes	—
192	Q₈	—	4	64	no	—
193	C₂×C₄	—	2	48	no	—
194	C₂×C₄	—	2	48	no	—
195	C₂×C₄	—	2	48	no	—
196	C₂×C₄	—	4	32	no	—
197	C₂×C₄	C₂²×Q₈×C₁₉	1	48	yes	—
198	C₂×C₄	C₂²×Q₈×C₁₉	1	64	yes	complemented
199	C₂×C₄	C₂⁵×C₁₉	1	16	yes	—
200	C₂×C₄	C₂⁵×C₁₉	1	16	yes	—
201	C₂×C₄	C₂⁵×C₁₉	1	16	yes	—
202	C₂×C₄	C₂⁵×C₁₉	1	8	yes	—
203	C₂²	—	2	48	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	C₁₉×#544	1	16	yes	direct product factor
207	C₂²	C₂³×Q₈×C₁₉	1	8	yes	—
208	C₂²	C₂³×Q₈×C₁₉	1	6	yes	—
209	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	6	yes	—
210	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	6	yes	—
211	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
212	C₄	—	2	16	no	—
213	C₄	—	2	16	no	—
214	C₄	—	2	16	no	—
215	C₄	C₂³×Q₈×C₁₉	1	16	yes	—
216	C₂	—	2	8	no	—
217	C₂	C₂×C₁₉×#544	1	12	yes	direct product factor
218	C₂	C₂⁴×Q₈×C₁₉	1	1	yes	characteristic
219	C₂	C₂²×C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
220	C₂	C₂²×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₉
prime radical	C₂⁵×C₁₉	C₂³
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. C₂×C₁₉×#544	C₂	direct (N × H)
2. C₂²×C₁₉×C₄⋊Q₈	C₂	semidirect
3. C₂²×C₁₉×#125	C₂	semidirect
10. C₁₉×#544	C₂²	direct (N × H)
11. C₂×C₁₉×C₄⋊Q₈	C₂²	semidirect
12. C₂×C₁₉×#125	C₂²	semidirect
27. C₂³×C₄×C₁₉	Q₈	semidirect
34. C₁₉×C₄⋊Q₈	C₂³	semidirect
35. C₁₉×#125	C₂³	semidirect
58. C₂²×C₄×C₁₉	C₂×Q₈	semidirect
72. C₂²×#544	C₁₉	direct (N × H)
88. C₂×C₄×C₁₉	C₂²×Q₈	semidirect
93. C₂×#544	C₂×C₁₉	direct (N × H)
94. C₂²×C₄⋊Q₈	C₂×C₁₉	semidirect
95. C₂²×#125	C₂×C₁₉	semidirect
102. C₂²×C₁₉	#544	direct (N × H)
115. #544	C₂²×C₁₉	direct (N × H)
116. C₂×C₄⋊Q₈	C₂²×C₁₉	semidirect
117. C₂×#125	C₂²×C₁₉	semidirect
129. C₂×C₁₉	C₂×#544	direct (N × H)
137. C₂³×C₄	Q₈×C₁₉	semidirect
144. C₄⋊Q₈	C₂³×C₁₉	semidirect
145. #125	C₂³×C₁₉	semidirect
156. C₁₉	C₂²×#544	direct (N × H)
169. C₂²×C₄	C₂×Q₈×C₁₉	semidirect
198. C₂×C₄	C₂²×Q₈×C₁₉	semidirect
206. C₂²	C₁₉×#544	direct (N × H)
217. C₂	C₂×C₁₉×#544	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₉×#544	2
C₂²×C₁₉×C₄⋊Q₈	2
C₂²×C₁₉×#125	2
C₂²×C₁₉×C₂²⋊Q₈	2
C₂²×C₁₉×#117	2
C₂³×C₄⋊C₄×C₁₉	2
C₂²×#544	19

Subgroups — C22×C19×#544 #18068635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₉×#544 #18068635