Subgroups — C₂²×C₁₉×Q₈⋊C₈ #18023391

Order 4864.

1454 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₉×Q₈⋊C₈	C₂	1	12	yes	maximal, direct product factor
5	C₂³×Q₈×C₁₉	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₉	—	2	1	no	—
7	C₂³×C₈×C₁₉	—	2	1	no	—
8	C₂×C₄×Q₈×C₁₉	—	2	3	no	—
9	C₂×C₄×Q₈×C₁₉	—	2	3	no	—
10	C₂×C₄×Q₈×C₁₉	C₄	1	6	yes	—
11	C₂×C₄×Q₈×C₁₉	C₂²	1	6	yes	—
12	C₂²×C₄⋊C₄×C₁₉	—	2	1	no	—
13	C₂²×C₄⋊C₄×C₁₉	C₄	1	1	yes	characteristic
14	C₂²×C₄²×C₁₉	—	2	1	no	—
15	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₁₉×C₄⋊C₈	—	2	3	no	—
17	C₂×C₁₉×C₄⋊C₈	C₂²	1	6	yes	—
18	C₂×C₄×C₈×C₁₉	—	2	3	no	—
19	C₂×C₄×C₈×C₁₉	C₂²	1	6	yes	—
20	C₁₉×Q₈⋊C₈	C₂²	1	16	yes	direct product factor
21	C₂²×Q₈×C₁₉	—	2	3	no	—
22	C₂²×Q₈×C₁₉	—	2	4	no	—
23	C₂²×Q₈×C₁₉	C₈	1	8	yes	complemented
24	C₂²×Q₈×C₁₉	C₂×C₄	1	6	yes	—
25	C₂³×C₄×C₁₉	—	2	1	no	—
26	C₂³×C₄×C₁₉	—	2	1	no	—
27	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₉	—	2	8	no	—
31	C₂²×C₈×C₁₉	—	2	6	no	—
32	C₂²×C₈×C₁₉	—	2	6	no	—
33	C₂²×C₈×C₁₉	—	4	4	no	—
34	C₄×Q₈×C₁₉	—	2	12	no	—
35	C₄×Q₈×C₁₉	—	2	6	no	—
36	C₄×Q₈×C₁₉	—	2	6	no	—
37	C₄×Q₈×C₁₉	C₂×C₄	1	12	yes	—
38	C₄×Q₈×C₁₉	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₁₉	—	2	6	no	—
40	C₂×C₄⋊C₄×C₁₉	—	2	3	no	—
41	C₂×C₄⋊C₄×C₁₉	—	2	6	no	—
42	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	6	yes	—
43	C₂×C₄²×C₁₉	—	2	6	no	—
44	C₂×C₄²×C₁₉	—	2	6	no	—
45	C₂×C₄²×C₁₉	C₂×C₄	1	3	yes	—
46	C₂×C₄²×C₁₉	D₄	1	3	yes	—
47	C₂×C₄²×C₁₉	D₄	1	3	yes	—
48	C₂×C₄²×C₁₉	C₂³	1	3	yes	—
49	C₁₉×C₄⋊C₈	—	2	6	no	—
50	C₁₉×C₄⋊C₈	C₂³	1	4	yes	—
51	C₄×C₈×C₁₉	—	2	6	no	—
52	C₄×C₈×C₁₉	C₂³	1	4	yes	—
53	C₂⁴×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Q₈×C₁₉	—	2	24	no	—
55	C₂×Q₈×C₁₉	—	2	6	no	—
56	C₂×Q₈×C₁₉	—	2	12	no	—
57	C₂×Q₈×C₁₉	C₂×C₈	1	24	yes	complemented
58	C₂×Q₈×C₁₉	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₉	—	2	8	no	—
60	C₂²×C₄×C₁₉	—	2	4	no	—
61	C₂²×C₄×C₁₉	—	2	6	no	—
62	C₂²×C₄×C₁₉	—	2	6	no	—
63	C₂²×C₄×C₁₉	—	4	4	no	—
64	C₂²×C₄×C₁₉	C₂²⋊C₄	1	3	yes	—
65	C₂²×C₄×C₁₉	C₂²⋊C₄	1	3	yes	—
66	C₂²×C₄×C₁₉	C₂²⋊C₄	1	3	yes	—
67	C₂²×C₄×C₁₉	C₂×C₈	1	4	yes	—
68	C₂²×C₄×C₁₉	M₁₆	1	4	yes	—
69	C₂²×C₄×C₁₉	SD₁₆	1	4	yes	—
70	C₂²×C₄×C₁₉	Q₁₆	1	4	yes	—
71	C₂²×C₄×C₁₉	C₂²×C₄	1	3	yes	—
72	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
73	C₂²×C₄×C₁₉	C₂×D₄	1	3	yes	—
74	C₂×C₈×C₁₉	—	2	24	no	—
75	C₂×C₈×C₁₉	—	2	4	no	—
76	C₂×C₈×C₁₉	—	2	4	no	—
77	C₂×C₈×C₁₉	—	4	12	no	—
78	C₄⋊C₄×C₁₉	—	2	12	no	—
79	C₄⋊C₄×C₁₉	—	2	6	no	—
80	C₄⋊C₄×C₁₉	—	2	4	no	—
81	C₄⋊C₄×C₁₉	C₂²×C₄	1	4	yes	—
82	C₄²×C₁₉	—	2	12	no	—
83	C₄²×C₁₉	—	2	4	no	—
84	C₄²×C₁₉	C₂²⋊C₄	1	6	yes	—
85	C₄²×C₁₉	C₂²×C₄	1	3	yes	—
86	C₄²×C₁₉	C₂×D₄	1	3	yes	—
87	C₄²×C₁₉	C₂×D₄	1	3	yes	—
88	C₄²×C₁₉	C₂⁴	1	1	yes	characteristic
89	C₂²×Q₈⋊C₈	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₁₉	C₂²⋊C₈	1	4	yes	—
91	C₂³×C₁₉	#103	1	4	yes	—
92	C₂³×C₁₉	#104	1	4	yes	—
93	C₂³×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
94	Q₈×C₁₉	—	2	4	no	—
95	Q₈×C₁₉	—	2	24	no	—
96	Q₈×C₁₉	C₂²×C₈	1	8	yes	complemented
97	C₂×C₄×C₁₉	—	2	4	no	—
98	C₂×C₄×C₁₉	—	2	24	no	—
99	C₂×C₄×C₁₉	—	2	6	no	—
100	C₂×C₄×C₁₉	—	2	6	no	—
101	C₂×C₄×C₁₉	—	2	4	no	—
102	C₂×C₄×C₁₉	—	4	12	no	—
103	C₂×C₄×C₁₉	C₂²⋊C₈	1	12	yes	—
104	C₂×C₄×C₁₉	#103	1	12	yes	—
105	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
106	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
107	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	3	yes	—
108	C₂×C₄×C₁₉	C₂²×C₈	1	6	yes	—
109	C₂×C₄×C₁₉	C₂×M₁₆	1	6	yes	—
110	C₂×C₄×C₁₉	C₂×SD₁₆	1	6	yes	—
111	C₂×C₄×C₁₉	C₂×Q₁₆	1	6	yes	—
112	C₂×C₄×C₁₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₉	C₂²×D₄	1	1	yes	characteristic
115	C₈×C₁₉	—	2	8	no	—
116	C₈×C₁₉	—	4	4	no	—
117	C₂²×C₄×Q₈	C₂×C₁₉	1	1	yes	characteristic
118	C₂²×C₄⋊C₈	C₂×C₁₉	1	1	yes	characteristic
119	C₂²×C₄×C₈	C₂×C₁₉	1	1	yes	characteristic
120	C₂×Q₈⋊C₈	C₂×C₁₉	1	12	yes	direct product factor
121	C₂²×C₁₉	Q₈⋊C₈	1	16	yes	direct product factor
122	C₂²×C₁₉	C₂×C₂²⋊C₈	1	6	yes	—
123	C₂²×C₁₉	C₂×#103	1	6	yes	—
124	C₂²×C₁₉	C₂×#104	1	6	yes	—
125	C₂²×C₁₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
126	C₄×C₁₉	—	2	8	no	—
127	C₄×C₁₉	—	2	1	no	—
128	C₄×C₁₉	—	2	3	no	—
129	C₄×C₁₉	—	4	4	no	—
130	C₄×C₁₉	C₂×C₂²⋊C₈	1	6	yes	—
131	C₄×C₁₉	C₂×#103	1	6	yes	—
132	C₄×C₁₉	C₂³×C₈	1	1	yes	characteristic
133	C₄×C₁₉	C₂²×M₁₆	1	1	yes	characteristic
134	C₄×C₁₉	C₂²×SD₁₆	1	1	yes	characteristic
135	C₄×C₁₉	C₂²×Q₁₆	1	1	yes	characteristic
136	C₂³×Q₈	C₄×C₁₉	1	1	yes	characteristic
137	C₂³×C₈	—	2	1	no	—
138	C₂³×C₈	—	2	1	no	—
139	C₂×C₄×Q₈	—	2	3	no	—
140	C₂×C₄×Q₈	—	2	3	no	—
141	C₂×C₄×Q₈	C₄×C₁₉	1	6	yes	—
142	C₂×C₄×Q₈	C₂²×C₁₉	1	6	yes	—
143	C₂²×C₄⋊C₄	—	2	1	no	—
144	C₂²×C₄⋊C₄	C₄×C₁₉	1	1	yes	characteristic
145	C₂²×C₄²	—	2	1	no	—
146	C₂²×C₄²	C₂²×C₁₉	1	1	yes	characteristic
147	C₂×C₄⋊C₈	—	2	3	no	—
148	C₂×C₄⋊C₈	C₂²×C₁₉	1	6	yes	—
149	C₂×C₄×C₈	—	2	3	no	—
150	C₂×C₄×C₈	C₂²×C₁₉	1	6	yes	—
151	Q₈⋊C₈	C₂²×C₁₉	1	16	yes	direct product factor
152	C₂×C₁₉	C₂×Q₈⋊C₈	1	12	yes	direct product factor
153	C₂×C₁₉	C₂²×C₂²⋊C₈	1	1	yes	characteristic
154	C₂×C₁₉	C₂²×#103	1	1	yes	characteristic
155	C₂×C₁₉	C₂²×#104	1	1	yes	characteristic
156	C₂²×Q₈	—	2	3	no	—
157	C₂²×Q₈	—	2	4	no	—
158	C₂²×Q₈	C₈×C₁₉	1	8	yes	complemented
159	C₂²×Q₈	C₂×C₄×C₁₉	1	6	yes	—
160	C₂³×C₄	—	2	1	no	—
161	C₂³×C₄	—	2	1	no	—
162	C₂³×C₄	C₂×C₄×C₁₉	1	1	yes	characteristic
163	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
164	C₂³×C₄	D₄×C₁₉	1	1	yes	characteristic
165	C₂²×C₈	—	2	8	no	—
166	C₂²×C₈	—	2	6	no	—
167	C₂²×C₈	—	2	6	no	—
168	C₂²×C₈	—	4	4	no	—
169	C₄×Q₈	—	2	12	no	—
170	C₄×Q₈	—	2	6	no	—
171	C₄×Q₈	—	2	6	no	—
172	C₄×Q₈	C₂×C₄×C₁₉	1	12	yes	—
173	C₄×Q₈	C₂³×C₁₉	1	4	yes	—
174	C₂×C₄⋊C₄	—	2	6	no	—
175	C₂×C₄⋊C₄	—	2	3	no	—
176	C₂×C₄⋊C₄	—	2	6	no	—
177	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	6	yes	—
178	C₂×C₄²	—	2	6	no	—
179	C₂×C₄²	—	2	6	no	—
180	C₂×C₄²	C₂×C₄×C₁₉	1	3	yes	—
181	C₂×C₄²	D₄×C₁₉	1	3	yes	—
182	C₂×C₄²	D₄×C₁₉	1	3	yes	—
183	C₂×C₄²	C₂³×C₁₉	1	3	yes	—
184	C₄⋊C₈	—	2	6	no	—
185	C₄⋊C₈	C₂³×C₁₉	1	4	yes	—
186	C₄×C₈	—	2	6	no	—
187	C₄×C₈	C₂³×C₁₉	1	4	yes	—
188	C₁₉	C₂²×Q₈⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
189	C₂⁴	C₂²⋊C₄×C₁₉	1	1	yes	characteristic
190	C₂×Q₈	—	2	6	no	—
191	C₂×Q₈	—	2	12	no	—
192	C₂×Q₈	—	2	24	no	—
193	C₂×Q₈	C₂×C₈×C₁₉	1	24	yes	complemented
194	C₂×Q₈	C₂²×C₄×C₁₉	1	4	yes	—
195	C₂²×C₄	—	2	6	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	6	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	C₂²⋊C₄×C₁₉	1	3	yes	—
201	C₂²×C₄	C₂²⋊C₄×C₁₉	1	3	yes	—
202	C₂²×C₄	C₂²⋊C₄×C₁₉	1	3	yes	—
203	C₂²×C₄	C₂×C₈×C₁₉	1	4	yes	—
204	C₂²×C₄	M₁₆×C₁₉	1	4	yes	—
205	C₂²×C₄	SD₁₆×C₁₉	1	4	yes	—
206	C₂²×C₄	Q₁₆×C₁₉	1	4	yes	—
207	C₂²×C₄	C₂²×C₄×C₁₉	1	3	yes	—
208	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
209	C₂²×C₄	C₂×D₄×C₁₉	1	3	yes	—
210	C₂×C₈	—	2	24	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	12	no	—
214	C₄⋊C₄	—	2	12	no	—
215	C₄⋊C₄	—	2	6	no	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	C₂²×C₄×C₁₉	1	4	yes	—
218	C₄²	—	2	12	no	—
219	C₄²	—	2	4	no	—
220	C₄²	C₂²⋊C₄×C₁₉	1	6	yes	—
221	C₄²	C₂²×C₄×C₁₉	1	3	yes	—
222	C₄²	C₂×D₄×C₁₉	1	3	yes	—
223	C₄²	C₂×D₄×C₁₉	1	3	yes	—
224	C₄²	C₂⁴×C₁₉	1	1	yes	characteristic
225	C₂³	C₁₉×C₂²⋊C₈	1	4	yes	—
226	C₂³	C₁₉×#103	1	4	yes	—
227	C₂³	C₁₉×#104	1	4	yes	—
228	C₂³	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
229	Q₈	—	2	24	no	—
230	Q₈	—	2	4	no	—
231	Q₈	C₂²×C₈×C₁₉	1	8	yes	complemented
232	C₂×C₄	—	2	24	no	—
233	C₂×C₄	—	2	6	no	—
234	C₂×C₄	—	2	6	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	12	no	—
238	C₂×C₄	C₁₉×C₂²⋊C₈	1	12	yes	—
239	C₂×C₄	C₁₉×#103	1	12	yes	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
241	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₁₉	1	3	yes	—
243	C₂×C₄	C₂²×C₈×C₁₉	1	6	yes	—
244	C₂×C₄	C₂×M₁₆×C₁₉	1	6	yes	—
245	C₂×C₄	C₂×SD₁₆×C₁₉	1	6	yes	—
246	C₂×C₄	C₂×Q₁₆×C₁₉	1	6	yes	—
247	C₂×C₄	C₂³×C₄×C₁₉	1	1	yes	characteristic
248	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
249	C₂×C₄	C₂²×D₄×C₁₉	1	1	yes	characteristic
250	C₈	—	2	8	no	—
251	C₈	—	4	4	no	—
252	C₂²	C₁₉×Q₈⋊C₈	1	16	yes	direct product factor
253	C₂²	C₂×C₁₉×C₂²⋊C₈	1	6	yes	—
254	C₂²	C₂×C₁₉×#103	1	6	yes	—
255	C₂²	C₂×C₁₉×#104	1	6	yes	—
256	C₂²	C₂²×C₂²⋊C₄×C₁₉	1	1	yes	characteristic
257	C₄	—	2	8	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	3	no	—
260	C₄	—	4	4	no	—
261	C₄	C₂×C₁₉×C₂²⋊C₈	1	6	yes	—
262	C₄	C₂×C₁₉×#103	1	6	yes	—
263	C₄	C₂³×C₈×C₁₉	1	1	yes	characteristic
264	C₄	C₂²×M₁₆×C₁₉	1	1	yes	characteristic
265	C₄	C₂²×SD₁₆×C₁₉	1	1	yes	characteristic
266	C₄	C₂²×Q₁₆×C₁₉	1	1	yes	characteristic
267	C₂	C₂×C₁₉×Q₈⋊C₈	1	12	yes	direct product factor
268	C₂	C₂²×C₁₉×C₂²⋊C₈	1	1	yes	characteristic
269	C₂	C₂²×C₁₉×#103	1	1	yes	characteristic
270	C₂	C₂²×C₁₉×#104	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄×C₁₉	D₄
commutator	C₄	C₂³×C₈×C₁₉
Frattini, cosocle	C₄²	C₂⁴×C₁₉
socle, prime radical	C₂⁴×C₁₉	C₂²⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₂×C₁₉×Q₈⋊C₈	C₂	direct (N × H)
20. C₁₉×Q₈⋊C₈	C₂²	direct (N × H)
23. C₂²×Q₈×C₁₉	C₈	semidirect
57. C₂×Q₈×C₁₉	C₂×C₈	semidirect
89. C₂²×Q₈⋊C₈	C₁₉	direct (N × H)
96. Q₈×C₁₉	C₂²×C₈	semidirect
120. C₂×Q₈⋊C₈	C₂×C₁₉	direct (N × H)
121. C₂²×C₁₉	Q₈⋊C₈	direct (N × H)
151. Q₈⋊C₈	C₂²×C₁₉	direct (N × H)
152. C₂×C₁₉	C₂×Q₈⋊C₈	direct (N × H)
158. C₂²×Q₈	C₈×C₁₉	semidirect
188. C₁₉	C₂²×Q₈⋊C₈	direct (N × H)
193. C₂×Q₈	C₂×C₈×C₁₉	semidirect
231. Q₈	C₂²×C₈×C₁₉	semidirect
252. C₂²	C₁₉×Q₈⋊C₈	direct (N × H)
267. C₂	C₂×C₁₉×Q₈⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₁₉	2
C₂²×C₁₉×C₄⋊C₈	2
C₂²×C₄×C₈×C₁₉	2
C₂×C₁₉×Q₈⋊C₈	2
C₂²×Q₈⋊C₈	19

Subgroups — C22×C19×Q8⋊C8 #18023391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₉×Q₈⋊C₈ #18023391