Subgroups — C₂×Q₈×C₄⋊C₄×C₁₉ #18040071

Order 4864.

3246 conjugacy classes of subgroups.

The subgroup lattice has 276 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	2	yes	maximal
2	C₂²×C₄×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₄⋊C₄×C₁₉	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₉×#389	C₂	1	6	yes	maximal
5	C₂×C₁₉×#384	C₂	1	3	yes	maximal
6	C₂×C₄×C₄⋊C₄×C₁₉	C₂	1	3	yes	maximal
7	C₂³×Q₈×C₁₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₉	—	2	16	no	—
9	C₂×C₄×Q₈×C₁₉	C₄	1	16	yes	complemented
10	C₂×C₄×Q₈×C₁₉	C₂²	1	16	yes	—
11	C₂×C₄×Q₈×C₁₉	C₂²	1	8	yes	—
12	C₂²×C₄⋊C₄×C₁₉	C₂²	1	6	yes	—
13	C₂²×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
14	C₂²×C₄⋊C₄×C₁₉	C₂²	1	6	yes	—
15	C₂²×C₄⋊C₄×C₁₉	C₂²	1	3	yes	—
16	C₂²×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₁₉	C₂²	1	6	yes	—
18	C₂²×C₄²×C₁₉	C₂²	1	3	yes	—
19	C₁₉×#389	C₂²	1	48	yes	—
20	C₁₉×#384	C₂²	1	24	yes	—
21	C₄×C₄⋊C₄×C₁₉	C₂²	1	24	yes	—
22	C₂×C₁₉×#95	C₂²	1	6	yes	—
23	C₂²×Q₈×C₁₉	C₂×C₄	1	8	yes	—
24	C₂²×Q₈×C₁₉	D₄	1	8	yes	—
25	C₂²×Q₈×C₁₉	Q₈	1	8	yes	—
26	C₂²×Q₈×C₁₉	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₉	C₂³	1	6	yes	—
28	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
29	C₂³×C₄×C₁₉	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₉	C₂³	1	3	yes	—
31	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
32	C₄×Q₈×C₁₉	—	2	64	no	—
33	C₄×Q₈×C₁₉	C₂×C₄	1	64	yes	complemented
34	C₂×C₄⋊C₄×C₁₉	—	2	24	no	—
35	C₂×C₄⋊C₄×C₁₉	—	2	12	no	—
36	C₂×C₄⋊C₄×C₁₉	—	4	12	no	—
37	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	24	yes	—
38	C₂×C₄⋊C₄×C₁₉	Q₈	1	8	yes	direct product factor
39	C₂×C₄⋊C₄×C₁₉	C₂³	1	12	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	12	yes	—
43	C₂×C₄⋊C₄×C₁₉	C₂³	1	4	yes	—
44	C₂×C₄²×C₁₉	—	2	24	no	—
45	C₂×C₄²×C₁₉	C₂×C₄	1	24	yes	—
46	C₂×C₄²×C₁₉	C₂³	1	24	yes	—
47	C₂×C₄²×C₁₉	C₂³	1	12	yes	—
48	C₁₉×#95	C₂³	1	24	yes	—
49	C₂⁴×C₁₉	C₂⁴	1	1	yes	characteristic
50	C₂×Q₈×C₁₉	C₄⋊C₄	1	64	yes	direct product factor
51	C₂×Q₈×C₁₉	C₂²×C₄	1	16	yes	—
52	C₂×Q₈×C₁₉	C₂×D₄	1	16	yes	—
53	C₂×Q₈×C₁₉	C₂×Q₈	1	16	yes	—
54	C₂²×C₄×C₁₉	—	2	4	no	—
55	C₂²×C₄×C₁₉	—	2	6	no	—
56	C₂²×C₄×C₁₉	—	2	6	no	—
57	C₂²×C₄×C₁₉	—	2	4	no	—
58	C₂²×C₄×C₁₉	—	2	12	no	—
59	C₂²×C₄×C₁₉	—	2	12	no	—
60	C₂²×C₄×C₁₉	—	2	12	no	—
61	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
62	C₂²×C₄×C₁₉	C₂²×C₄	1	12	yes	—
63	C₂²×C₄×C₁₉	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₁₉	C₂×D₄	1	12	yes	—
65	C₂²×C₄×C₁₉	C₂×Q₈	1	8	yes	—
66	C₂²×C₄×C₁₉	C₂×Q₈	1	12	yes	—
67	C₂²×C₄×C₁₉	C₂×Q₈	1	4	yes	—
68	C₂²×C₄×C₁₉	#41	1	4	yes	—
69	C₂²×C₄×C₁₉	C₂⁴	1	12	yes	—
70	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
71	C₂²×C₄×C₁₉	C₂⁴	1	4	yes	—
72	C₂²×C₄×C₁₉	C₂⁴	1	6	yes	—
73	C₂²×C₄×C₁₉	C₂⁴	1	2	yes	—
74	C₄⋊C₄×C₁₉	—	2	48	no	—
75	C₄⋊C₄×C₁₉	—	2	24	no	—
76	C₄⋊C₄×C₁₉	—	4	24	no	—
77	C₄⋊C₄×C₁₉	C₂²×C₄	1	48	yes	—
78	C₄⋊C₄×C₁₉	C₂×Q₈	1	16	yes	direct product factor
79	C₄²×C₁₉	—	2	48	no	—
80	C₄²×C₁₉	C₂²×C₄	1	48	yes	—
81	C₂×Q₈×C₄⋊C₄	C₁₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₁₉	C₂³×C₄	1	2	yes	—
83	C₂³×C₁₉	C₂²×D₄	1	2	yes	—
84	C₂³×C₁₉	C₂²×Q₈	1	2	yes	—
85	C₂³×C₁₉	C₂²×Q₈	1	2	yes	—
86	C₂³×C₁₉	C₂×#41	1	2	yes	—
87	C₂³×C₁₉	H(5,𝔽₂)	1	2	yes	—
88	C₂³×C₁₉	2₋¹⁺⁴	1	2	yes	—
89	C₂³×C₁₉	C₂⁵	1	1	yes	characteristic
90	Q₈×C₁₉	C₂×C₄⋊C₄	1	64	yes	direct product factor
91	C₂×C₄×C₁₉	—	2	24	no	—
92	C₂×C₄×C₁₉	—	2	16	no	—
93	C₂×C₄×C₁₉	—	2	4	no	—
94	C₂×C₄×C₁₉	—	2	6	no	—
95	C₂×C₄×C₁₉	—	2	6	no	—
96	C₂×C₄×C₁₉	—	2	4	no	—
97	C₂×C₄×C₁₉	—	2	12	no	—
98	C₂×C₄×C₁₉	—	2	12	no	—
99	C₂×C₄×C₁₉	—	2	12	no	—
100	C₂×C₄×C₁₉	—	4	24	no	—
101	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	48	yes	—
102	C₂×C₄×C₁₉	C₄×Q₈	1	16	yes	complemented
103	C₂×C₄×C₁₉	C₂³×C₄	1	12	yes	—
104	C₂×C₄×C₁₉	C₂³×C₄	1	12	yes	—
105	C₂×C₄×C₁₉	C₂³×C₄	1	4	yes	—
106	C₂×C₄×C₁₉	C₂²×D₄	1	12	yes	—
107	C₂×C₄×C₁₉	C₂²×Q₈	1	8	yes	—
108	C₂×C₄×C₁₉	C₂²×Q₈	1	12	yes	—
109	C₂×C₄×C₁₉	C₂²×Q₈	1	4	yes	—
110	C₂×C₄×C₁₉	C₂×#41	1	4	yes	—
111	C₂²×C₄×Q₈	C₂×C₁₉	1	2	yes	—
112	C₂²×C₄×Q₈	C₂×C₁₉	1	1	yes	characteristic
113	Q₈×C₄⋊C₄	C₂×C₁₉	1	16	yes	direct product factor
114	C₂×#389	C₂×C₁₉	1	6	yes	—
115	C₂×#384	C₂×C₁₉	1	3	yes	—
116	C₂×C₄×C₄⋊C₄	C₂×C₁₉	1	3	yes	—
117	C₂²×C₁₉	C₂²×C₄⋊C₄	1	4	yes	—
118	C₂²×C₁₉	C₂×C₄×Q₈	1	4	yes	—
119	C₂²×C₁₉	#520	1	4	yes	—
120	C₂²×C₁₉	D₄×Q₈	1	4	yes	—
121	C₂²×C₁₉	#550	1	4	yes	—
122	C₂²×C₁₉	#557	1	4	yes	—
123	C₂²×C₁₉	Q₈²	1	4	yes	—
124	C₂²×C₁₉	C₂⁴×C₄	1	1	yes	characteristic
125	C₂²×C₁₉	C₂³×D₄	1	1	yes	characteristic
126	C₂²×C₁₉	C₂³×Q₈	1	1	yes	characteristic
127	C₂²×C₁₉	C₂³×Q₈	1	1	yes	characteristic
128	C₂²×C₁₉	C₂²×#41	1	1	yes	characteristic
129	C₂²×C₁₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
130	C₂²×C₁₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
131	C₄×C₁₉	—	2	12	no	—
132	C₄×C₁₉	—	2	8	no	—
133	C₄×C₁₉	—	4	12	no	—
134	C₄×C₁₉	C₂²×C₄⋊C₄	1	24	yes	—
135	C₄×C₁₉	C₂×C₄×Q₈	1	8	yes	complemented
136	C₂³×Q₈	C₂²×C₁₉	1	1	yes	characteristic
137	C₂×C₄×Q₈	—	2	16	no	—
138	C₂×C₄×Q₈	C₄×C₁₉	1	16	yes	complemented
139	C₂×C₄×Q₈	C₂²×C₁₉	1	16	yes	—
140	C₂×C₄×Q₈	C₂²×C₁₉	1	8	yes	—
141	C₂²×C₄⋊C₄	C₂²×C₁₉	1	6	yes	—
142	C₂²×C₄⋊C₄	C₂²×C₁₉	1	3	yes	—
143	C₂²×C₄⋊C₄	C₂²×C₁₉	1	6	yes	—
144	C₂²×C₄⋊C₄	C₂²×C₁₉	1	3	yes	—
145	C₂²×C₄⋊C₄	C₂²×C₁₉	1	1	yes	characteristic
146	C₂²×C₄²	C₂²×C₁₉	1	6	yes	—
147	C₂²×C₄²	C₂²×C₁₉	1	3	yes	—
148	#389	C₂²×C₁₉	1	48	yes	—
149	#384	C₂²×C₁₉	1	24	yes	—
150	C₄×C₄⋊C₄	C₂²×C₁₉	1	24	yes	—
151	C₂×#95	C₂²×C₁₉	1	6	yes	—
152	C₂×C₁₉	Q₈×C₄⋊C₄	1	8	yes	direct product factor
153	C₂×C₁₉	C₂³×C₄⋊C₄	1	1	yes	characteristic
154	C₂×C₁₉	C₂²×C₄×Q₈	1	1	yes	characteristic
155	C₂×C₁₉	C₂×#520	1	1	yes	characteristic
156	C₂×C₁₉	C₂×D₄×Q₈	1	1	yes	characteristic
157	C₂×C₁₉	C₂×#550	1	1	yes	characteristic
158	C₂×C₁₉	C₂×#557	1	1	yes	characteristic
159	C₂×C₁₉	C₂×Q₈²	1	1	yes	characteristic
160	C₂²×Q₈	C₂×C₄×C₁₉	1	8	yes	—
161	C₂²×Q₈	D₄×C₁₉	1	8	yes	—
162	C₂²×Q₈	Q₈×C₁₉	1	8	yes	—
163	C₂²×Q₈	C₂³×C₁₉	1	4	yes	—
164	C₂³×C₄	C₂³×C₁₉	1	6	yes	—
165	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
166	C₂³×C₄	C₂³×C₁₉	1	2	yes	—
167	C₂³×C₄	C₂³×C₁₉	1	3	yes	—
168	C₂³×C₄	C₂³×C₁₉	1	1	yes	characteristic
169	C₄×Q₈	—	2	64	no	—
170	C₄×Q₈	C₂×C₄×C₁₉	1	64	yes	complemented
171	C₂×C₄⋊C₄	—	2	24	no	—
172	C₂×C₄⋊C₄	—	2	12	no	—
173	C₂×C₄⋊C₄	—	4	12	no	—
174	C₂×C₄⋊C₄	C₂×C₄×C₁₉	1	24	yes	—
175	C₂×C₄⋊C₄	Q₈×C₁₉	1	8	yes	direct product factor
176	C₂×C₄⋊C₄	C₂³×C₁₉	1	24	yes	—
177	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₉	1	24	yes	—
179	C₂×C₄⋊C₄	C₂³×C₁₉	1	12	yes	—
180	C₂×C₄⋊C₄	C₂³×C₁₉	1	4	yes	—
181	C₂×C₄²	—	2	24	no	—
182	C₂×C₄²	C₂×C₄×C₁₉	1	24	yes	—
183	C₂×C₄²	C₂³×C₁₉	1	24	yes	—
184	C₂×C₄²	C₂³×C₁₉	1	12	yes	—
185	#95	C₂³×C₁₉	1	24	yes	—
186	C₁₉	C₂×Q₈×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
187	C₂⁴	C₂⁴×C₁₉	1	1	yes	characteristic
188	C₂×Q₈	C₄⋊C₄×C₁₉	1	64	yes	direct product factor
189	C₂×Q₈	C₂²×C₄×C₁₉	1	16	yes	—
190	C₂×Q₈	C₂×D₄×C₁₉	1	16	yes	—
191	C₂×Q₈	C₂×Q₈×C₁₉	1	16	yes	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	6	no	—
194	C₂²×C₄	—	2	6	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	12	no	—
197	C₂²×C₄	—	2	12	no	—
198	C₂²×C₄	—	2	12	no	—
199	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
200	C₂²×C₄	C₂²×C₄×C₁₉	1	12	yes	—
201	C₂²×C₄	C₂²×C₄×C₁₉	1	4	yes	—
202	C₂²×C₄	C₂×D₄×C₁₉	1	12	yes	—
203	C₂²×C₄	C₂×Q₈×C₁₉	1	8	yes	—
204	C₂²×C₄	C₂×Q₈×C₁₉	1	12	yes	—
205	C₂²×C₄	C₂×Q₈×C₁₉	1	4	yes	—
206	C₂²×C₄	#41×C₁₉	1	4	yes	—
207	C₂²×C₄	C₂⁴×C₁₉	1	12	yes	—
208	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
209	C₂²×C₄	C₂⁴×C₁₉	1	4	yes	—
210	C₂²×C₄	C₂⁴×C₁₉	1	6	yes	—
211	C₂²×C₄	C₂⁴×C₁₉	1	2	yes	—
212	C₄⋊C₄	—	2	48	no	—
213	C₄⋊C₄	—	2	24	no	—
214	C₄⋊C₄	—	4	24	no	—
215	C₄⋊C₄	C₂²×C₄×C₁₉	1	48	yes	—
216	C₄⋊C₄	C₂×Q₈×C₁₉	1	16	yes	direct product factor
217	C₄²	—	2	48	no	—
218	C₄²	C₂²×C₄×C₁₉	1	48	yes	—
219	C₂³	C₂³×C₄×C₁₉	1	2	yes	—
220	C₂³	C₂²×D₄×C₁₉	1	2	yes	—
221	C₂³	C₂²×Q₈×C₁₉	1	2	yes	—
222	C₂³	C₂²×Q₈×C₁₉	1	2	yes	—
223	C₂³	C₂×#41×C₁₉	1	2	yes	—
224	C₂³	C₁₉×H(5,𝔽₂)	1	2	yes	—
225	C₂³	C₁₉×2₋¹⁺⁴	1	2	yes	—
226	C₂³	C₂⁵×C₁₉	1	1	yes	characteristic
227	Q₈	C₂×C₄⋊C₄×C₁₉	1	64	yes	direct product factor
228	C₂×C₄	—	2	24	no	—
229	C₂×C₄	—	2	16	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	6	no	—
232	C₂×C₄	—	2	6	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	12	no	—
235	C₂×C₄	—	2	12	no	—
236	C₂×C₄	—	2	12	no	—
237	C₂×C₄	—	4	24	no	—
238	C₂×C₄	C₂×C₄⋊C₄×C₁₉	1	48	yes	—
239	C₂×C₄	C₄×Q₈×C₁₉	1	16	yes	complemented
240	C₂×C₄	C₂³×C₄×C₁₉	1	12	yes	—
241	C₂×C₄	C₂³×C₄×C₁₉	1	12	yes	—
242	C₂×C₄	C₂³×C₄×C₁₉	1	4	yes	—
243	C₂×C₄	C₂²×D₄×C₁₉	1	12	yes	—
244	C₂×C₄	C₂²×Q₈×C₁₉	1	8	yes	—
245	C₂×C₄	C₂²×Q₈×C₁₉	1	12	yes	—
246	C₂×C₄	C₂²×Q₈×C₁₉	1	4	yes	—
247	C₂×C₄	C₂×#41×C₁₉	1	4	yes	—
248	C₂²	C₂²×C₄⋊C₄×C₁₉	1	4	yes	—
249	C₂²	C₂×C₄×Q₈×C₁₉	1	4	yes	—
250	C₂²	C₁₉×#520	1	4	yes	—
251	C₂²	D₄×Q₈×C₁₉	1	4	yes	—
252	C₂²	C₁₉×#550	1	4	yes	—
253	C₂²	C₁₉×#557	1	4	yes	—
254	C₂²	Q₈²×C₁₉	1	4	yes	—
255	C₂²	C₂⁴×C₄×C₁₉	1	1	yes	characteristic
256	C₂²	C₂³×D₄×C₁₉	1	1	yes	characteristic
257	C₂²	C₂³×Q₈×C₁₉	1	1	yes	characteristic
258	C₂²	C₂³×Q₈×C₁₉	1	1	yes	characteristic
259	C₂²	C₂²×#41×C₁₉	1	1	yes	characteristic
260	C₂²	C₂×C₁₉×H(5,𝔽₂)	1	1	yes	characteristic
261	C₂²	C₂×C₁₉×2₋¹⁺⁴	1	1	yes	characteristic
262	C₄	—	2	12	no	—
263	C₄	—	2	8	no	—
264	C₄	—	4	12	no	—
265	C₄	C₂²×C₄⋊C₄×C₁₉	1	24	yes	—
266	C₄	C₂×C₄×Q₈×C₁₉	1	8	yes	complemented
267	C₂	Q₈×C₄⋊C₄×C₁₉	1	8	yes	direct product factor
268	C₂	C₂³×C₄⋊C₄×C₁₉	1	1	yes	characteristic
269	C₂	C₂²×C₄×Q₈×C₁₉	1	1	yes	characteristic
270	C₂	C₂×C₁₉×#520	1	1	yes	characteristic
271	C₂	C₂×D₄×Q₈×C₁₉	1	1	yes	characteristic
272	C₂	C₂×C₁₉×#550	1	1	yes	characteristic
273	C₂	C₂×C₁₉×#557	1	1	yes	characteristic
274	C₂	C₂×Q₈²×C₁₉	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₁₉	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₁₉
Frattini, cosocle	C₂³	C₂⁵×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
3. Q₈×C₄⋊C₄×C₁₉	C₂	direct (N × H)
9. C₂×C₄×Q₈×C₁₉	C₄	semidirect
33. C₄×Q₈×C₁₉	C₂×C₄	semidirect
38. C₂×C₄⋊C₄×C₁₉	Q₈	direct (N × H)
50. C₂×Q₈×C₁₉	C₄⋊C₄	direct (N × H)
78. C₄⋊C₄×C₁₉	C₂×Q₈	direct (N × H)
81. C₂×Q₈×C₄⋊C₄	C₁₉	direct (N × H)
90. Q₈×C₁₉	C₂×C₄⋊C₄	direct (N × H)
102. C₂×C₄×C₁₉	C₄×Q₈	semidirect
113. Q₈×C₄⋊C₄	C₂×C₁₉	direct (N × H)
135. C₄×C₁₉	C₂×C₄×Q₈	semidirect
138. C₂×C₄×Q₈	C₄×C₁₉	semidirect
152. C₂×C₁₉	Q₈×C₄⋊C₄	direct (N × H)
170. C₄×Q₈	C₂×C₄×C₁₉	semidirect
175. C₂×C₄⋊C₄	Q₈×C₁₉	direct (N × H)
186. C₁₉	C₂×Q₈×C₄⋊C₄	direct (N × H)
188. C₂×Q₈	C₄⋊C₄×C₁₉	direct (N × H)
216. C₄⋊C₄	C₂×Q₈×C₁₉	direct (N × H)
227. Q₈	C₂×C₄⋊C₄×C₁₉	direct (N × H)
239. C₂×C₄	C₄×Q₈×C₁₉	semidirect
266. C₄	C₂×C₄×Q₈×C₁₉	semidirect
267. C₂	Q₈×C₄⋊C₄×C₁₉	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
Q₈×C₄⋊C₄×C₁₉	2
C₂×C₁₉×#389	2
C₂×C₁₉×#384	2
C₂×C₄×C₄⋊C₄×C₁₉	2
C₂×Q₈×C₄⋊C₄	19

Subgroups — C2×Q8×C4⋊C4×C19 #18040071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈×C₄⋊C₄×C₁₉ #18040071