Subgroups — #18073166

Order 4864.

578 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563488	C₂	1	1	yes	maximal, characteristic
2	#14563381	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₁₉⋊C₈)	C₂	1	2	yes	maximal
4	#14561476	C₂	1	2	yes	maximal
5	C₄²×Q₈×C₁₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
7	C₂×#10597260	—	2	1	no	—
8	C₂×C₄×C₁₉⋊C₈	C₂²	1	2	yes	—
9	#11926299	—	2	2	no	—
10	#11926299	—	2	1	no	—
11	Q₈⋊(C₁₉⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₁₉	C₄	1	2	yes	—
13	C₂×C₄×Q₈×C₁₉	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₁₉	—	2	2	no	—
15	C₄×C₄⋊C₄×C₁₉	—	2	1	no	—
16	C₄×C₄⋊C₄×C₁₉	C₄	1	2	yes	—
17	C₄×C₄⋊C₄×C₁₉	C₂²	1	1	yes	characteristic
18	C₄³×C₁₉	—	2	1	no	—
19	C₄³×C₁₉	C₂²	1	1	yes	characteristic
20	C₂²×Q₈×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₉	—	2	4	no	—
22	C₄×Q₈×C₁₉	—	2	4	no	—
23	C₄×Q₈×C₁₉	—	2	2	no	—
24	C₄×Q₈×C₁₉	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₁₉	—	2	2	no	—
26	C₂×C₄⋊C₄×C₁₉	—	2	1	no	—
27	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₁₉	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	—	2	1	no	—
31	C₂×C₄²×C₁₉	—	2	2	no	—
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₉	D₄	1	2	yes	—
36	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
38	C₂²×C₁₉⋊C₈	—	2	2	no	—
39	C₂²×C₁₉⋊C₈	—	2	2	no	—
40	C₁₉×#95	—	2	2	no	—
41	C₁₉×#95	—	2	1	no	—
42	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
43	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	4	yes	—
44	#10597260	—	4	2	no	—
45	C₄×C₁₉⋊C₈	—	2	2	no	—
46	C₄×C₁₉⋊C₈	C₂×C₄	1	4	yes	—
47	C₂×Q₈×C₁₉	—	2	1	no	—
48	C₂×Q₈×C₁₉	—	2	2	no	—
49	C₂×Q₈×C₁₉	C₄²	1	2	yes	—
50	C₂×Q₈×C₁₉	M₁₆	1	4	yes	—
51	C₂²×C₄×C₁₉	—	2	1	no	—
52	C₂²×C₄×C₁₉	—	2	1	no	—
53	C₂²×C₄×C₁₉	—	2	2	no	—
54	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
59	C₄⋊C₄×C₁₉	—	2	8	no	—
60	C₄⋊C₄×C₁₉	—	2	1	no	—
61	C₄⋊C₄×C₁₉	—	2	4	no	—
62	C₄⋊C₄×C₁₉	—	2	2	no	—
63	C₄⋊C₄×C₁₉	C₄²	1	2	yes	—
64	C₄⋊C₄×C₁₉	M₁₆	1	4	yes	—
65	C₄²×C₁₉	—	2	8	no	—
66	C₄²×C₁₉	—	2	4	no	—
67	C₄²×C₁₉	—	2	2	no	—
68	C₄²×C₁₉	—	2	2	no	—
69	C₄²×C₁₉	—	2	2	no	—
70	C₄²×C₁₉	—	4	2	no	—
71	C₄²×C₁₉	—	4	4	no	—
72	C₄²×C₁₉	—	4	2	no	—
73	C₄²×C₁₉	SD₁₆	1	2	yes	—
74	C₄²×C₁₉	Q₁₆	1	2	yes	—
75	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
76	C₄²×C₁₉	#41	1	2	yes	—
77	C₂×C₁₉⋊C₈	—	2	4	no	—
78	C₂×C₁₉⋊C₈	—	2	4	no	—
79	C₂×C₁₉⋊C₈	—	4	4	no	—
80	C₂×C₁₉⋊C₈	—	4	2	no	—
81	C₂×C₁₉⋊C₈	—	4	2	no	—
82	#8175	—	19	1	no	maximal, Sylow
83	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	Q₈×C₁₉	—	2	2	no	—
85	Q₈×C₁₉	—	2	4	no	—
86	Q₈×C₁₉	#97	1	4	yes	complemented
87	C₂×C₄×C₁₉	—	2	2	no	—
88	C₂×C₄×C₁₉	—	2	2	no	—
89	C₂×C₄×C₁₉	—	2	4	no	—
90	C₂×C₄×C₁₉	—	2	2	no	—
91	C₂×C₄×C₁₉	—	2	2	no	—
92	C₂×C₄×C₁₉	—	2	2	no	—
93	C₂×C₄×C₁₉	—	2	2	no	—
94	C₂×C₄×C₁₉	—	2	4	no	—
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	2	2	no	—
97	C₂×C₄×C₁₉	—	4	2	no	—
98	C₂×C₄×C₁₉	—	4	2	no	—
99	C₂×C₄×C₁₉	—	4	2	no	—
100	C₂×C₄×C₁₉	#103	1	4	yes	—
101	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
108	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
109	C₂×C₄×C₁₉	#135	1	2	yes	—
110	C₁₉⋊C₈	—	4	4	no	—
111	C₁₉⋊C₈	—	4	4	no	—
112	C₄²×Q₈	D₁₉	1	1	yes	characteristic
113	#1767	—	19	1	no	—
114	#1744	—	19	1	no	—
115	C₂×Q₈⋊C₈	—	19	2	no	—
116	#1279	—	19	2	no	—
117	C₂²×C₁₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₁₉	#407	1	2	yes	—
119	C₂²×C₁₉	C₂×#103	1	1	yes	characteristic
120	C₂²×C₁₉	#416	1	1	yes	characteristic
121	C₂²×C₁₉	#421	1	2	yes	—
122	C₄×C₁₉	—	2	4	no	—
123	C₄×C₁₉	—	2	4	no	—
124	C₄×C₁₉	—	2	2	no	—
125	C₄×C₁₉	—	4	2	no	—
126	C₄×C₁₉	—	4	4	no	—
127	C₄×C₁₉	—	4	2	no	—
128	C₄×C₁₉	C₂×#97	1	1	yes	characteristic
129	C₄×C₁₉	C₄×M₁₆	1	1	yes	characteristic
130	C₄×C₁₉	#432	1	2	yes	—
131	C₄×C₁₉	C₄×SD₁₆	1	2	yes	—
132	C₄×C₁₉	C₄×Q₁₆	1	2	yes	—
133	C₂×C₄×Q₈	Dic₁₉	1	2	yes	—
134	C₂×C₄×Q₈	C₂×D₁₉	1	1	yes	characteristic
135	C₂×C₄⋊C₈	—	19	2	no	—
136	C₂×#97	—	38	1	no	—
137	C₂×C₄×C₈	—	19	2	no	—
138	C₄×C₄⋊C₄	—	2	1	no	—
139	C₄×C₄⋊C₄	—	2	2	no	—
140	C₄×C₄⋊C₄	Dic₁₉	1	2	yes	—
141	C₄×C₄⋊C₄	C₂×D₁₉	1	1	yes	characteristic
142	C₄³	—	2	1	no	—
143	C₄³	C₂×D₁₉	1	1	yes	characteristic
144	#336	—	38	2	no	—
145	#336	—	38	1	no	—
146	Q₈⋊C₈	—	19	8	no	—
147	C₂×C₁₉	#1487	1	2	yes	—
148	C₂×C₁₉	#1488	1	2	yes	—
149	C₂×C₁₉	#1749	1	1	yes	characteristic
150	C₂×C₁₉	#1759	1	1	yes	characteristic
151	C₂×C₁₉	C₄×#103	1	1	yes	characteristic
152	C₂²×Q₈	C₂×Dic₁₉	1	1	yes	characteristic
153	C₂²×C₈	—	38	2	no	—
154	C₂²×C₈	—	38	2	no	—
155	C₄×Q₈	—	2	4	no	—
156	C₄×Q₈	—	2	4	no	—
157	C₄×Q₈	—	2	2	no	—
158	C₄×Q₈	C₄×D₁₉	1	4	yes	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	C₂×Dic₁₉	1	2	yes	—
162	C₂×C₄⋊C₄	C₂×Dic₁₉	1	1	yes	characteristic
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	—	2	1	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
168	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
169	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
170	C₂×C₄²	C₂²⋊D₁₉	1	1	yes	characteristic
171	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
172	C₄⋊C₈	—	19	4	no	—
173	C₄⋊C₈	—	38	2	no	—
174	#97	—	76	2	no	—
175	C₄×C₈	—	19	4	no	—
176	C₄×C₈	—	38	2	no	—
177	#95	—	2	2	no	—
178	#95	—	2	1	no	—
179	C₁₉	#8175	1	1	yes	characteristic, complemented, Sylow
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	#64520	1	4	yes	—
183	C₂×Q₈	C₄×Dic₁₉	1	2	yes	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
188	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
189	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
190	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
191	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
192	C₂×C₈	—	38	4	no	—
193	C₂×C₈	—	38	4	no	—
194	C₂×C₈	—	76	4	no	—
195	C₂×C₈	—	76	2	no	—
196	C₂×C₈	—	76	2	no	—
197	C₄⋊C₄	—	2	8	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	#64520	1	4	yes	—
202	C₄⋊C₄	C₄×Dic₁₉	1	2	yes	—
203	C₄²	—	2	8	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	4	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	4	no	—
210	C₄²	—	4	2	no	—
211	C₄²	#64527	1	2	yes	—
212	C₄²	#64528	1	2	yes	—
213	C₄²	C₂×C₄×D₁₉	1	2	yes	—
214	C₄²	#64541	1	2	yes	—
215	C₂³	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
216	Q₈	—	2	4	no	—
217	Q₈	—	2	2	no	—
218	Q₈	#10597260	1	4	yes	complemented
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	#10597292	1	4	yes	—
233	C₂×C₄	C₂×#64520	1	2	yes	—
234	C₂×C₄	C₂×#64520	1	2	yes	—
235	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
236	C₂×C₄	#10597372	1	1	yes	characteristic
237	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
238	C₂×C₄	C₄×C₂²⋊D₁₉	1	2	yes	—
239	C₂×C₄	C₂×#64527	1	1	yes	characteristic
240	C₂×C₄	C₂×#64528	1	1	yes	characteristic
241	C₂×C₄	#10597396	1	2	yes	—
242	C₈	—	76	4	no	—
243	C₈	—	76	4	no	—
244	C₂²	C₄×C₂²⋊Dic₁₉	1	1	yes	characteristic
245	C₂²	#11926508	1	2	yes	—
246	C₂²	C₂×#10597292	1	1	yes	characteristic
247	C₂²	#11926540	1	1	yes	characteristic
248	C₂²	#11926561	1	2	yes	—
249	C₄	—	2	4	no	—
250	C₄	—	2	4	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	4	no	—
254	C₄	—	4	2	no	—
255	C₄	C₂×#10597260	1	1	yes	characteristic
256	C₄	C₄×#64520	1	1	yes	characteristic
257	C₄	#11926601	1	2	yes	—
258	C₄	C₄×#64527	1	2	yes	—
259	C₄	C₄×#64528	1	2	yes	—
260	C₂	#14562064	1	2	yes	—
261	C₂	#14562069	1	2	yes	—
262	C₂	#14563397	1	1	yes	characteristic
263	C₂	#14563445	1	1	yes	characteristic
264	C₂	C₄×#10597292	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₉
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₄²×Q₈×C₁₉	C₂
socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₄²×Q₈	D₁₉
hypocenter	C₁₉	#8175
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
11. Q₈⋊(C₁₉⋊C₈)	C₄	semidirect
86. Q₈×C₁₉	#97	semidirect
179. C₁₉	#8175	semidirect
218. Q₈	#10597260	semidirect

Maximal subgroups

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

subgroup	index
#14563488	2
#14563381	2
C₂×Q₈⋊(C₁₉⋊C₈)	2
#14561476	2
C₄²×Q₈×C₁₉	2
#8175	19