Subgroups — Q₈×C₁₉×SD₃₂ #18039905

Order 4864.

494 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Q₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₈×C₁₉	C₂	1	1	yes	maximal, characteristic
3	C₁₉×#2254	C₂	1	3	yes	maximal, complemented
4	C₁₉×#2226	C₂	1	3	yes	maximal
5	C₁₉×#2225	C₂	1	3	yes	maximal, complemented
6	Q₈×C₁₆×C₁₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₁₉×SD₃₂	C₂	1	3	yes	maximal
8	Q₈²×C₁₉	—	2	1	no	—
9	D₄×Q₈×C₁₉	—	2	1	no	—
10	C₂×C₁₉×SD₃₂	C₂²	1	1	yes	characteristic
11	C₁₉×#500	C₂²	1	3	yes	—
12	C₁₉×#500	C₂²	1	3	yes	—
13	C₁₉×#477	—	2	3	no	—
14	C₁₉×#474	—	2	3	no	—
15	C₈×Q₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₄×Q₁₆×C₁₉	C₂²	1	3	yes	—
17	C₄×D₈×C₁₉	C₂²	1	3	yes	—
18	C₁₉×#367	C₂²	1	3	yes	—
19	C₁₉×#367	C₂²	1	6	yes	—
20	C₁₉×C₄⋊C₁₆	C₂²	1	3	yes	—
21	C₁₉×#358	C₂²	1	3	yes	—
22	C₁₉×#357	C₂²	1	3	yes	—
23	C₄×C₁₆×C₁₉	C₂²	1	3	yes	—
24	C₂²×Q₈×C₁₉	—	4	1	no	—
25	C₂×Q₁₆×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×D₈×C₁₉	C₂³	1	1	yes	characteristic
27	C₁₉×C₄⋊Q₈	—	2	3	no	—
28	C₁₉×C₄⋊Q₈	—	2	3	no	—
29	C₁₉×C₄⋊Q₈	—	4	3	no	—
30	C₁₉×C₂²⋊Q₈	—	4	3	no	—
31	C₄×Q₈×C₁₉	—	2	3	no	—
32	C₄×Q₈×C₁₉	—	4	1	no	—
33	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
34	C₄×D₄×C₁₉	—	2	3	no	—
35	C₁₉×SD₃₂	Q₈	1	4	yes	direct product factor
36	C₂×C₁₆×C₁₉	C₂³	1	3	yes	—
37	C₂×C₁₆×C₁₉	C₂³	1	1	yes	characteristic
38	C₁₉×#107	—	2	3	no	—
39	C₁₉×#107	—	2	3	no	—
40	C₁₉×#107	C₂³	1	3	yes	—
41	C₁₉×#107	C₂³	1	3	yes	—
42	C₁₉×C₄⋊C₈	D₄	1	3	yes	—
43	C₁₉×#103	—	2	3	no	—
44	C₁₉×#102	—	2	3	no	—
45	C₄×C₈×C₁₉	C₂³	1	3	yes	—
46	C₂×Q₈×C₁₉	—	2	1	no	—
47	C₂×Q₈×C₁₉	—	4	3	no	—
48	C₂×Q₈×C₁₉	—	4	3	no	—
49	C₂×Q₈×C₁₉	—	8	4	no	—
50	C₂×Q₈×C₁₉	D₈	1	1	yes	characteristic
51	C₂×D₄×C₁₉	—	2	1	no	—
52	C₂²×C₄×C₁₉	—	4	3	no	—
53	Q₁₆×C₁₉	—	2	1	no	—
54	Q₁₆×C₁₉	C₂×Q₈	1	2	yes	complemented
55	D₈×C₁₉	—	2	1	no	—
56	D₈×C₁₉	C₂×Q₈	1	2	yes	—
57	C₂×C₈×C₁₉	C₂×D₄	1	3	yes	—
58	C₂×C₈×C₁₉	C₂⁴	1	1	yes	characteristic
59	C₄⋊C₄×C₁₉	—	2	3	no	—
60	C₄⋊C₄×C₁₉	—	2	3	no	—
61	C₄⋊C₄×C₁₉	—	4	3	no	—
62	C₄⋊C₄×C₁₉	—	4	3	no	—
63	C₄⋊C₄×C₁₉	—	4	3	no	—
64	C₄⋊C₄×C₁₉	D₈	1	3	yes	—
65	C₂²⋊C₄×C₁₉	—	4	3	no	—
66	C₄²×C₁₉	—	4	3	no	—
67	C₄²×C₁₉	C₂×D₄	1	3	yes	—
68	C₁₆×C₁₉	—	2	3	no	—
69	C₁₆×C₁₉	C₂×Q₈	1	2	yes	complemented
70	Q₈×SD₃₂	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₁₉	—	4	1	no	—
72	Q₈×C₁₉	—	2	2	no	—
73	Q₈×C₁₉	—	4	1	no	—
74	Q₈×C₁₉	—	8	6	no	—
75	Q₈×C₁₉	—	16	3	no	—
76	Q₈×C₁₉	SD₃₂	1	4	yes	direct product factor
77	D₄×C₁₉	—	2	2	no	—
78	D₄×C₁₉	—	4	1	no	—
79	C₂×C₄×C₁₉	—	4	1	no	—
80	C₂×C₄×C₁₉	—	4	3	no	—
81	C₂×C₄×C₁₉	—	4	3	no	—
82	C₂×C₄×C₁₉	—	8	6	no	—
83	C₂×C₄×C₁₉	C₂×D₈	1	3	yes	—
84	C₂×C₄×C₁₉	C₂×D₈	1	3	yes	—
85	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
86	C₈×C₁₉	—	2	3	no	—
87	C₈×C₁₉	C₂²×Q₈	1	1	yes	characteristic
88	C₈×C₁₉	2₋¹⁺⁴	1	1	yes	characteristic
89	Q₈×Q₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
90	Q₈×D₈	C₂×C₁₉	1	1	yes	characteristic
91	#2254	C₂×C₁₉	1	3	yes	complemented
92	#2226	C₂×C₁₉	1	3	yes	—
93	#2225	C₂×C₁₉	1	3	yes	complemented
94	Q₈×C₁₆	C₂×C₁₉	1	1	yes	characteristic, complemented
95	C₄×SD₃₂	C₂×C₁₉	1	3	yes	—
96	C₂²×C₁₉	—	4	2	no	—
97	C₂²×C₁₉	—	8	1	no	—
98	C₂²×C₁₉	—	8	1	no	—
99	C₂²×C₁₉	C₂²×D₈	1	1	yes	characteristic
100	C₄×C₁₉	—	2	3	no	—
101	C₄×C₁₉	—	4	2	no	—
102	C₄×C₁₉	—	8	3	no	—
103	C₄×C₁₉	—	8	3	no	—
104	C₄×C₁₉	C₂×SD₃₂	1	6	yes	—
105	C₄×C₁₉	D₄×Q₈	1	1	yes	characteristic
106	C₄×C₁₉	#578	1	1	yes	characteristic
107	Q₈²	—	2	1	no	—
108	D₄×Q₈	—	2	1	no	—
109	C₂×SD₃₂	C₂²×C₁₉	1	1	yes	characteristic
110	#500	C₂²×C₁₉	1	3	yes	—
111	#500	C₂²×C₁₉	1	3	yes	—
112	#477	—	2	3	no	—
113	#474	—	2	3	no	—
114	C₈×Q₈	C₂²×C₁₉	1	1	yes	characteristic
115	C₄×Q₁₆	C₂²×C₁₉	1	3	yes	—
116	C₄×D₈	C₂²×C₁₉	1	3	yes	—
117	#367	C₂²×C₁₉	1	3	yes	—
118	#367	C₂²×C₁₉	1	6	yes	—
119	C₄⋊C₁₆	C₂²×C₁₉	1	3	yes	—
120	#358	C₂²×C₁₉	1	3	yes	—
121	#357	C₂²×C₁₉	1	3	yes	—
122	C₄×C₁₆	C₂²×C₁₉	1	3	yes	—
123	C₂×C₁₉	—	8	2	no	—
124	C₂×C₁₉	Q₈×D₈	1	1	yes	characteristic
125	C₂×C₁₉	C₂²×SD₃₂	1	1	yes	characteristic
126	C₂×C₁₉	#3416	1	1	yes	characteristic
127	C₂²×Q₈	—	4	1	no	—
128	C₂×Q₁₆	C₂³×C₁₉	1	1	yes	characteristic
129	C₂×D₈	C₂³×C₁₉	1	1	yes	characteristic
130	C₄⋊Q₈	—	2	3	no	—
131	C₄⋊Q₈	—	2	3	no	—
132	C₄⋊Q₈	—	4	3	no	—
133	C₂²⋊Q₈	—	4	3	no	—
134	C₄×Q₈	—	2	3	no	—
135	C₄×Q₈	—	4	1	no	—
136	C₄×Q₈	D₄×C₁₉	1	1	yes	characteristic
137	C₄×D₄	—	2	3	no	—
138	SD₃₂	Q₈×C₁₉	1	4	yes	direct product factor
139	C₂×C₁₆	C₂³×C₁₉	1	3	yes	—
140	C₂×C₁₆	C₂³×C₁₉	1	1	yes	characteristic
141	#107	—	2	3	no	—
142	#107	—	2	3	no	—
143	#107	C₂³×C₁₉	1	3	yes	—
144	#107	C₂³×C₁₉	1	3	yes	—
145	C₄⋊C₈	D₄×C₁₉	1	3	yes	—
146	#103	—	2	3	no	—
147	#102	—	2	3	no	—
148	C₄×C₈	C₂³×C₁₉	1	3	yes	—
149	C₁₉	Q₈×SD₃₂	1	1	yes	characteristic, direct product factor, Sylow
150	C₂×Q₈	—	2	1	no	—
151	C₂×Q₈	—	4	3	no	—
152	C₂×Q₈	—	4	3	no	—
153	C₂×Q₈	—	8	4	no	—
154	C₂×Q₈	D₈×C₁₉	1	1	yes	characteristic
155	C₂×D₄	—	2	1	no	—
156	C₂²×C₄	—	4	3	no	—
157	Q₁₆	—	2	1	no	—
158	Q₁₆	C₂×Q₈×C₁₉	1	2	yes	complemented
159	D₈	—	2	1	no	—
160	D₈	C₂×Q₈×C₁₉	1	2	yes	—
161	C₂×C₈	C₂×D₄×C₁₉	1	3	yes	—
162	C₂×C₈	C₂⁴×C₁₉	1	1	yes	characteristic
163	C₄⋊C₄	—	2	3	no	—
164	C₄⋊C₄	—	2	3	no	—
165	C₄⋊C₄	—	4	3	no	—
166	C₄⋊C₄	—	4	3	no	—
167	C₄⋊C₄	—	4	3	no	—
168	C₄⋊C₄	D₈×C₁₉	1	3	yes	—
169	C₂²⋊C₄	—	4	3	no	—
170	C₄²	—	4	3	no	—
171	C₄²	C₂×D₄×C₁₉	1	3	yes	—
172	C₁₆	—	2	3	no	—
173	C₁₆	C₂×Q₈×C₁₉	1	2	yes	complemented
174	C₂³	—	4	1	no	—
175	Q₈	—	2	2	no	—
176	Q₈	—	4	1	no	—
177	Q₈	—	8	6	no	—
178	Q₈	—	16	3	no	—
179	Q₈	C₁₉×SD₃₂	1	4	yes	direct product factor
180	D₄	—	2	2	no	—
181	D₄	—	4	1	no	—
182	C₂×C₄	—	4	1	no	—
183	C₂×C₄	—	4	3	no	—
184	C₂×C₄	—	4	3	no	—
185	C₂×C₄	—	8	6	no	—
186	C₂×C₄	C₂×D₈×C₁₉	1	3	yes	—
187	C₂×C₄	C₂×D₈×C₁₉	1	3	yes	—
188	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
189	C₈	—	2	3	no	—
190	C₈	C₂²×Q₈×C₁₉	1	1	yes	characteristic
191	C₈	C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
192	C₂²	—	4	2	no	—
193	C₂²	—	8	1	no	—
194	C₂²	—	8	1	no	—
195	C₂²	C₂²×D₈×C₁₉	1	1	yes	characteristic
196	C₄	—	2	3	no	—
197	C₄	—	4	2	no	—
198	C₄	—	8	3	no	—
199	C₄	—	8	3	no	—
200	C₄	C₂×C₁₉×SD₃₂	1	6	yes	—
201	C₄	D₄×Q₈×C₁₉	1	1	yes	characteristic
202	C₄	C₁₉×#578	1	1	yes	characteristic
203	C₂	—	8	2	no	—
204	C₂	Q₈×D₈×C₁₉	1	1	yes	characteristic
205	C₂	C₂²×C₁₉×SD₃₂	1	1	yes	characteristic
206	C₂	C₁₉×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₉
prime radical	C₂×D₈×C₁₉	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄×Q₈×C₁₉ ◁ C₈×Q₈×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×Q₁₆×C₁₉	C₂	semidirect
3. C₁₉×#2254	C₂	semidirect
5. C₁₉×#2225	C₂	semidirect
6. Q₈×C₁₆×C₁₉	C₂	semidirect
35. C₁₉×SD₃₂	Q₈	direct (N × H)
54. Q₁₆×C₁₉	C₂×Q₈	semidirect
69. C₁₆×C₁₉	C₂×Q₈	semidirect
70. Q₈×SD₃₂	C₁₉	direct (N × H)
76. Q₈×C₁₉	SD₃₂	direct (N × H)
89. Q₈×Q₁₆	C₂×C₁₉	semidirect
91. #2254	C₂×C₁₉	semidirect
93. #2225	C₂×C₁₉	semidirect
94. Q₈×C₁₆	C₂×C₁₉	semidirect
138. SD₃₂	Q₈×C₁₉	direct (N × H)
149. C₁₉	Q₈×SD₃₂	direct (N × H)
158. Q₁₆	C₂×Q₈×C₁₉	semidirect
173. C₁₆	C₂×Q₈×C₁₉	semidirect
179. Q₈	C₁₉×SD₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×Q₁₆×C₁₉	2
Q₈×D₈×C₁₉	2
C₁₉×#2254	2
C₁₉×#2226	2
C₁₉×#2225	2
Q₈×C₁₆×C₁₉	2
C₄×C₁₉×SD₃₂	2
Q₈×SD₃₂	19

Subgroups — Q8×C19×SD32 #18039905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₁₉×SD₃₂ #18039905