Subgroups — C₂²×C₁₉×#533 #18068626

Order 4864.

8238 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₉×#533	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₁₉×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₂²×C₁₉×#125	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₉×C₂²⋊Q₈	C₂	1	4	yes	maximal
5	C₂²×C₄×Q₈×C₁₉	C₂	1	4	yes	maximal, complemented
6	C₂²×C₁₉×#117	C₂	1	2	yes	maximal
7	C₂³×C₄²×C₁₉	C₂	1	1	yes	maximal, characteristic
8	C₂³×Q₈×C₁₉	C₂²	1	4	yes	—
9	C₂⁴×C₄×C₁₉	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₁₉	C₂²	1	1	yes	characteristic
11	C₁₉×#533	C₂²	1	256	yes	direct product factor
12	C₂×C₁₉×C₄⋊Q₈	C₂²	1	48	yes	complemented
13	C₂×C₁₉×#125	C₂²	1	48	yes	complemented
14	C₂×C₁₉×C₂²⋊Q₈	C₂²	1	96	yes	—
15	C₂×C₄×Q₈×C₁₉	C₂²	1	96	yes	complemented
16	C₂×C₁₉×#117	C₂²	1	48	yes	—
17	C₂²×C₄⋊C₄×C₁₉	C₂²	1	8	yes	—
18	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
19	C₂²×C₄⋊C₄×C₁₉	C₂²	1	4	yes	—
20	C₂²×C₂²⋊C₄×C₁₉	C₂²	1	4	yes	—
21	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₁₉	C₂²	1	24	yes	—
24	C₂²×C₄²×C₁₉	C₂²	1	2	yes	—
25	C₂²×C₄²×C₁₉	C₂²	1	4	yes	—
26	C₂⁵×C₁₉	C₂³	1	1	yes	characteristic
27	C₂²×Q₈×C₁₉	—	2	32	no	—
28	C₂²×Q₈×C₁₉	C₂³	1	48	yes	—
29	C₂³×C₄×C₁₉	—	2	16	no	—
30	C₂³×C₄×C₁₉	Q₈	1	16	yes	complemented
31	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
32	C₂³×C₄×C₁₉	C₂³	1	24	yes	—
33	C₂³×C₄×C₁₉	C₂³	1	8	yes	—
34	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₁₉	C₂³	1	12	yes	—
36	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
37	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
38	C₁₉×C₄⋊Q₈	C₂³	1	128	yes	complemented
39	C₁₉×#125	C₂³	1	128	yes	complemented
40	C₁₉×C₂²⋊Q₈	C₂³	1	256	yes	—
41	C₄×Q₈×C₁₉	C₂³	1	256	yes	complemented
42	C₁₉×#117	C₂³	1	128	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	48	yes	—
44	C₂×C₄⋊C₄×C₁₉	C₂³	1	96	yes	—
45	C₂×C₄⋊C₄×C₁₉	C₂³	1	48	yes	—
46	C₂×C₂²⋊C₄×C₁₉	C₂³	1	48	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	24	yes	—
48	C₂×C₄²×C₁₉	C₂³	1	12	yes	—
49	C₂×C₄²×C₁₉	C₂³	1	12	yes	—
50	C₂×C₄²×C₁₉	C₂³	1	64	yes	—
51	C₂×C₄²×C₁₉	C₂³	1	48	yes	—
52	C₂⁴×C₁₉	—	2	8	no	—
53	C₂⁴×C₁₉	C₂×Q₈	1	8	yes	—
54	C₂⁴×C₁₉	C₂⁴	1	6	yes	—
55	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
56	C₂×Q₈×C₁₉	—	2	192	no	—
57	C₂×Q₈×C₁₉	C₂⁴	1	64	yes	—
58	C₂²×C₄×C₁₉	—	2	8	no	—
59	C₂²×C₄×C₁₉	—	2	32	no	—
60	C₂²×C₄×C₁₉	—	2	96	no	—
61	C₂²×C₄×C₁₉	C₂×Q₈	1	8	yes	—
62	C₂²×C₄×C₁₉	C₂×Q₈	1	8	yes	—
63	C₂²×C₄×C₁₉	C₂×Q₈	1	96	yes	complemented
64	C₂²×C₄×C₁₉	#41	1	16	yes	—
65	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
66	C₂²×C₄×C₁₉	C₂⁴	1	32	yes	—
67	C₂²×C₄×C₁₉	C₂⁴	1	48	yes	—
68	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
69	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
70	C₂²×C₄×C₁₉	C₂⁴	1	16	yes	—
71	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
72	C₄⋊C₄×C₁₉	C₂⁴	1	64	yes	—
73	C₄⋊C₄×C₁₉	C₂⁴	1	128	yes	—
74	C₄⋊C₄×C₁₉	C₂⁴	1	64	yes	—
75	C₂²⋊C₄×C₁₉	C₂⁴	1	64	yes	—
76	C₄²×C₁₉	C₂⁴	1	32	yes	—
77	C₄²×C₁₉	C₂⁴	1	16	yes	—
78	C₄²×C₁₉	C₂⁴	1	16	yes	—
79	C₄²×C₁₉	C₂⁴	1	64	yes	—
80	C₂²×#533	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₁₉	—	2	32	no	—
82	C₂³×C₁₉	—	2	24	no	—
83	C₂³×C₁₉	C₂²×Q₈	1	24	yes	—
84	C₂³×C₁₉	C₂²×Q₈	1	4	yes	—
85	C₂³×C₁₉	C₂×#41	1	8	yes	—
86	C₂³×C₁₉	C₂⁵	1	4	yes	—
87	C₂³×C₁₉	C₂⁵	1	3	yes	—
88	Q₈×C₁₉	—	2	128	no	—
89	C₂×C₄×C₁₉	—	2	24	no	—
90	C₂×C₄×C₁₉	—	2	96	no	—
91	C₂×C₄×C₁₉	—	2	64	no	—
92	C₂×C₄×C₁₉	C₂²×Q₈	1	24	yes	—
93	C₂×C₄×C₁₉	C₂²×Q₈	1	24	yes	—
94	C₂×C₄×C₁₉	C₂²×Q₈	1	64	yes	complemented
95	C₂×C₄×C₁₉	C₂×#41	1	48	yes	—
96	C₂×C₄×C₁₉	C₂⁵	1	8	yes	—
97	C₂×C₄×C₁₉	C₂⁵	1	32	yes	—
98	C₂×C₄×C₁₉	C₂⁵	1	4	yes	—
99	C₂×C₄×C₁₉	C₂⁵	1	4	yes	—
100	C₂×C₄×C₁₉	C₂⁵	1	8	yes	—
101	C₂×#533	C₂×C₁₉	1	48	yes	direct product factor
102	C₂²×C₄⋊Q₈	C₂×C₁₉	1	2	yes	complemented
103	C₂²×#125	C₂×C₁₉	1	2	yes	complemented
104	C₂²×C₂²⋊Q₈	C₂×C₁₉	1	4	yes	—
105	C₂²×C₄×Q₈	C₂×C₁₉	1	4	yes	complemented
106	C₂²×#117	C₂×C₁₉	1	2	yes	—
107	C₂³×C₄²	C₂×C₁₉	1	1	yes	characteristic
108	C₂²×C₁₉	—	2	8	no	—
109	C₂²×C₁₉	—	2	48	no	—
110	C₂²×C₁₉	#533	1	16	yes	direct product factor
111	C₂²×C₁₉	C₂³×Q₈	1	8	yes	—
112	C₂²×C₁₉	C₂³×Q₈	1	6	yes	—
113	C₂²×C₁₉	C₂²×#41	1	12	yes	—
114	C₂²×C₁₉	C₂⁶	1	1	yes	characteristic
115	C₄×C₁₉	—	2	8	no	—
116	C₄×C₁₉	—	2	32	no	—
117	C₄×C₁₉	C₂³×Q₈	1	8	yes	—
118	C₄×C₁₉	C₂³×Q₈	1	8	yes	—
119	C₄×C₁₉	C₂²×#41	1	16	yes	—
120	C₂³×Q₈	C₂²×C₁₉	1	4	yes	—
121	C₂⁴×C₄	C₂²×C₁₉	1	2	yes	—
122	C₂⁴×C₄	C₂²×C₁₉	1	1	yes	characteristic
123	#533	C₂²×C₁₉	1	256	yes	direct product factor
124	C₂×C₄⋊Q₈	C₂²×C₁₉	1	48	yes	complemented
125	C₂×#125	C₂²×C₁₉	1	48	yes	complemented
126	C₂×C₂²⋊Q₈	C₂²×C₁₉	1	96	yes	—
127	C₂×C₄×Q₈	C₂²×C₁₉	1	96	yes	complemented
128	C₂×#117	C₂²×C₁₉	1	48	yes	—
129	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
130	C₂²×C₄⋊C₄	C₂²×C₁₉	1	8	yes	—
131	C₂²×C₄⋊C₄	C₂²×C₁₉	1	4	yes	—
132	C₂²×C₂²⋊C₄	C₂²×C₁₉	1	4	yes	—
133	C₂²×C₄²	C₂²×C₁₉	1	2	yes	—
134	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
135	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
136	C₂²×C₄²	C₂²×C₁₉	1	24	yes	—
137	C₂²×C₄²	C₂²×C₁₉	1	4	yes	—
138	C₂×C₁₉	—	2	8	no	—
139	C₂×C₁₉	C₂×#533	1	12	yes	direct product factor
140	C₂×C₁₉	C₂⁴×Q₈	1	1	yes	characteristic
141	C₂×C₁₉	C₂³×#41	1	2	yes	—
142	C₂⁵	C₂³×C₁₉	1	1	yes	characteristic
143	C₂²×Q₈	—	2	32	no	—
144	C₂²×Q₈	C₂³×C₁₉	1	48	yes	—
145	C₂³×C₄	—	2	16	no	—
146	C₂³×C₄	Q₈×C₁₉	1	16	yes	complemented
147	C₂³×C₄	C₂³×C₁₉	1	8	yes	—
148	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
149	C₂³×C₄	C₂³×C₁₉	1	24	yes	—
150	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
151	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
152	C₂³×C₄	C₂³×C₁₉	1	12	yes	—
153	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
154	C₄⋊Q₈	C₂³×C₁₉	1	128	yes	complemented
155	#125	C₂³×C₁₉	1	128	yes	complemented
156	C₂²⋊Q₈	C₂³×C₁₉	1	256	yes	—
157	C₄×Q₈	C₂³×C₁₉	1	256	yes	complemented
158	#117	C₂³×C₁₉	1	128	yes	—
159	C₂×C₄⋊C₄	C₂³×C₁₉	1	48	yes	—
160	C₂×C₄⋊C₄	C₂³×C₁₉	1	96	yes	—
161	C₂×C₄⋊C₄	C₂³×C₁₉	1	48	yes	—
162	C₂×C₂²⋊C₄	C₂³×C₁₉	1	48	yes	—
163	C₂×C₄²	C₂³×C₁₉	1	48	yes	—
164	C₂×C₄²	C₂³×C₁₉	1	24	yes	—
165	C₂×C₄²	C₂³×C₁₉	1	12	yes	—
166	C₂×C₄²	C₂³×C₁₉	1	12	yes	—
167	C₂×C₄²	C₂³×C₁₉	1	64	yes	—
168	C₁₉	C₂²×#533	1	1	yes	characteristic, direct product factor, Sylow
169	C₂⁴	—	2	8	no	—
170	C₂⁴	C₂×Q₈×C₁₉	1	8	yes	—
171	C₂⁴	C₂⁴×C₁₉	1	6	yes	—
172	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
173	C₂×Q₈	—	2	192	no	—
174	C₂×Q₈	C₂⁴×C₁₉	1	64	yes	—
175	C₂²×C₄	—	2	8	no	—
176	C₂²×C₄	—	2	32	no	—
177	C₂²×C₄	—	2	96	no	—
178	C₂²×C₄	C₂×Q₈×C₁₉	1	8	yes	—
179	C₂²×C₄	C₂×Q₈×C₁₉	1	8	yes	—
180	C₂²×C₄	C₂×Q₈×C₁₉	1	96	yes	complemented
181	C₂²×C₄	#41×C₁₉	1	16	yes	—
182	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
183	C₂²×C₄	C₂⁴×C₁₉	1	32	yes	—
184	C₂²×C₄	C₂⁴×C₁₉	1	48	yes	—
185	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
186	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
187	C₂²×C₄	C₂⁴×C₁₉	1	16	yes	—
188	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
189	C₄⋊C₄	C₂⁴×C₁₉	1	64	yes	—
190	C₄⋊C₄	C₂⁴×C₁₉	1	128	yes	—
191	C₄⋊C₄	C₂⁴×C₁₉	1	64	yes	—
192	C₂²⋊C₄	C₂⁴×C₁₉	1	64	yes	—
193	C₄²	C₂⁴×C₁₉	1	32	yes	—
194	C₄²	C₂⁴×C₁₉	1	16	yes	—
195	C₄²	C₂⁴×C₁₉	1	16	yes	—
196	C₄²	C₂⁴×C₁₉	1	64	yes	—
197	C₂³	—	2	32	no	—
198	C₂³	—	2	24	no	—
199	C₂³	C₂²×Q₈×C₁₉	1	24	yes	—
200	C₂³	C₂²×Q₈×C₁₉	1	4	yes	—
201	C₂³	C₂×#41×C₁₉	1	8	yes	—
202	C₂³	C₂⁵×C₁₉	1	4	yes	—
203	C₂³	C₂⁵×C₁₉	1	3	yes	—
204	Q₈	—	2	128	no	—
205	C₂×C₄	—	2	24	no	—
206	C₂×C₄	—	2	96	no	—
207	C₂×C₄	—	2	64	no	—
208	C₂×C₄	C₂²×Q₈×C₁₉	1	24	yes	—
209	C₂×C₄	C₂²×Q₈×C₁₉	1	24	yes	—
210	C₂×C₄	C₂²×Q₈×C₁₉	1	64	yes	complemented
211	C₂×C₄	C₂×#41×C₁₉	1	48	yes	—
212	C₂×C₄	C₂⁵×C₁₉	1	8	yes	—
213	C₂×C₄	C₂⁵×C₁₉	1	32	yes	—
214	C₂×C₄	C₂⁵×C₁₉	1	4	yes	—
215	C₂×C₄	C₂⁵×C₁₉	1	4	yes	—
216	C₂×C₄	C₂⁵×C₁₉	1	8	yes	—
217	C₂²	—	2	48	no	—
218	C₂²	—	2	8	no	—
219	C₂²	C₁₉×#533	1	16	yes	direct product factor
220	C₂²	C₂³×Q₈×C₁₉	1	8	yes	—
221	C₂²	C₂³×Q₈×C₁₉	1	6	yes	—
222	C₂²	C₂²×#41×C₁₉	1	12	yes	—
223	C₂²	C₂⁶×C₁₉	1	1	yes	characteristic
224	C₄	—	2	8	no	—
225	C₄	—	2	32	no	—
226	C₄	C₂³×Q₈×C₁₉	1	8	yes	—
227	C₄	C₂³×Q₈×C₁₉	1	8	yes	—
228	C₄	C₂²×#41×C₁₉	1	16	yes	—
229	C₂	—	2	8	no	—
230	C₂	C₂×C₁₉×#533	1	12	yes	direct product factor
231	C₂	C₂⁴×Q₈×C₁₉	1	1	yes	characteristic
232	C₂	C₂³×#41×C₁₉	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₁₉	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶×C₁₉
socle	C₂⁴×C₁₉	C₂⁴
prime radical	C₂⁵×C₁₉	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₉×#533	C₂	direct (N × H)
2. C₂²×C₁₉×C₄⋊Q₈	C₂	semidirect
3. C₂²×C₁₉×#125	C₂	semidirect
5. C₂²×C₄×Q₈×C₁₉	C₂	semidirect
11. C₁₉×#533	C₂²	direct (N × H)
12. C₂×C₁₉×C₄⋊Q₈	C₂²	semidirect
13. C₂×C₁₉×#125	C₂²	semidirect
15. C₂×C₄×Q₈×C₁₉	C₂²	semidirect
30. C₂³×C₄×C₁₉	Q₈	semidirect
38. C₁₉×C₄⋊Q₈	C₂³	semidirect
39. C₁₉×#125	C₂³	semidirect
41. C₄×Q₈×C₁₉	C₂³	semidirect
63. C₂²×C₄×C₁₉	C₂×Q₈	semidirect
80. C₂²×#533	C₁₉	direct (N × H)
94. C₂×C₄×C₁₉	C₂²×Q₈	semidirect
101. C₂×#533	C₂×C₁₉	direct (N × H)
102. C₂²×C₄⋊Q₈	C₂×C₁₉	semidirect
103. C₂²×#125	C₂×C₁₉	semidirect
105. C₂²×C₄×Q₈	C₂×C₁₉	semidirect
110. C₂²×C₁₉	#533	direct (N × H)
123. #533	C₂²×C₁₉	direct (N × H)
124. C₂×C₄⋊Q₈	C₂²×C₁₉	semidirect
125. C₂×#125	C₂²×C₁₉	semidirect
127. C₂×C₄×Q₈	C₂²×C₁₉	semidirect
139. C₂×C₁₉	C₂×#533	direct (N × H)
146. C₂³×C₄	Q₈×C₁₉	semidirect
154. C₄⋊Q₈	C₂³×C₁₉	semidirect
155. #125	C₂³×C₁₉	semidirect
157. C₄×Q₈	C₂³×C₁₉	semidirect
168. C₁₉	C₂²×#533	direct (N × H)
180. C₂²×C₄	C₂×Q₈×C₁₉	semidirect
210. C₂×C₄	C₂²×Q₈×C₁₉	semidirect
219. C₂²	C₁₉×#533	direct (N × H)
230. C₂	C₂×C₁₉×#533	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×#533	2
C₂²×C₁₉×C₄⋊Q₈	2
C₂²×C₁₉×#125	2
C₂²×C₁₉×C₂²⋊Q₈	2
C₂²×C₄×Q₈×C₁₉	2
C₂²×C₁₉×#117	2
C₂³×C₄²×C₁₉	2
C₂²×#533	19

Subgroups — C22×C19×#533 #18068626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₉×#533 #18068626