Subgroups — #18075673

Order 4864.

466 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14563773	C₂	1	1	yes	maximal, characteristic, complemented
2	#14562383	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562060	C₂	1	2	yes	maximal
4	#14561532	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₁₉	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	#11926568	C₂²	1	2	yes	—
8	#11926309	—	2	1	no	—
9	#11926309	—	2	1	no	—
10	#11926297	—	2	1	no	—
11	#11926295	—	2	1	no	—
12	D₄⋊(C₁₉⋊C₈)	—	2	4	no	—
13	C₂³×C₈×C₁₉	—	2	1	no	—
14	C₂×C₄×D₄×C₁₉	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₁₉	—	2	1	no	—
16	C₈×D₄×C₁₉	—	2	1	no	—
17	C₈×D₄×C₁₉	C₄	1	2	yes	—
18	C₈×D₄×C₁₉	C₄	1	2	yes	—
19	C₂×C₁₉×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₁₉×C₂²⋊C₈	—	2	1	no	—
21	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
22	C₂²×D₄×C₁₉	Q₈	1	1	yes	characteristic
23	C₂³×C₄×C₁₉	—	2	1	no	—
24	C₂²×C₈×C₁₉	—	2	1	no	—
25	C₂²×C₈×C₁₉	—	2	2	no	—
26	C₂²×C₈×C₁₉	—	2	2	no	—
27	C₂²×C₈×C₁₉	—	4	4	no	—
28	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
30	C₄×D₄×C₁₉	—	2	1	no	—
31	C₄×D₄×C₁₉	—	2	1	no	—
32	C₄×D₄×C₁₉	C₂×C₄	1	2	yes	—
33	C₄×D₄×C₁₉	D₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₁₉	Q₈	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₉	—	2	1	no	—
36	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
37	C₂×#64520	—	2	2	no	—
38	C₁₉×C₄⋊C₈	—	2	1	no	—
39	C₁₉×C₄⋊C₈	C₂×C₄	1	2	yes	—
40	C₁₉×C₂²⋊C₈	—	2	2	no	—
41	C₁₉×C₂²⋊C₈	—	2	2	no	—
42	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
45	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
46	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
49	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
50	C₄×C₁₉⋊C₈	—	2	2	no	—
51	C₂⁴×C₁₉	—	2	1	no	—
52	C₂×D₄×C₁₉	—	2	4	no	—
53	C₂×D₄×C₁₉	—	2	1	no	—
54	C₂×D₄×C₁₉	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₉	—	2	1	no	—
56	C₂²×C₄×C₁₉	—	2	2	no	—
57	C₂²×C₄×C₁₉	—	2	2	no	—
58	C₂²×C₄×C₁₉	—	4	2	no	—
59	C₂²×C₄×C₁₉	—	4	2	no	—
60	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
63	C₂×C₈×C₁₉	—	2	2	no	—
64	C₂×C₈×C₁₉	—	2	2	no	—
65	C₂×C₈×C₁₉	—	2	2	no	—
66	C₂×C₈×C₁₉	—	4	4	no	—
67	C₂×C₈×C₁₉	—	4	4	no	—
68	C₂×C₈×C₁₉	—	4	2	no	—
69	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₁₉	—	2	1	no	—
74	C₄⋊C₄×C₁₉	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₁₉	—	2	2	no	—
76	C₂²⋊C₄×C₁₉	—	2	2	no	—
77	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₁₉	#41	1	1	yes	characteristic
81	#64520	—	4	2	no	—
82	#64520	—	4	2	no	—
83	C₂×C₁₉⋊C₈	—	4	2	no	—
84	C₂×C₁₉⋊C₈	—	4	2	no	—
85	#8660	—	19	1	no	maximal, Sylow
86	C₂³×C₁₉	—	2	4	no	—
87	C₂³×C₁₉	—	2	2	no	—
88	C₂³×C₁₉	—	4	2	no	—
89	C₂³×C₁₉	—	4	2	no	—
90	C₂³×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
91	D₄×C₁₉	—	2	4	no	—
92	D₄×C₁₉	—	2	2	no	—
93	D₄×C₁₉	#108	1	4	yes	—
94	C₂×C₄×C₁₉	—	2	2	no	—
95	C₂×C₄×C₁₉	—	2	2	no	—
96	C₂×C₄×C₁₉	—	4	1	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	—	4	4	no	—
101	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
104	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
105	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₉	#135	1	2	yes	—
108	C₂×C₄×C₁₉	AGL(1,ℤ/8)	1	2	yes	—
109	C₈×C₁₉	—	2	1	no	—
110	C₈×C₁₉	—	2	1	no	—
111	C₈×C₁₉	—	2	1	no	—
112	C₈×C₁₉	—	2	1	no	—
113	C₈×C₁₉	—	4	2	no	—
114	C₁₉⋊C₈	—	8	2	no	—
115	C₁₉⋊C₈	—	8	2	no	—
116	C₂×C₈×D₄	D₁₉	1	1	yes	characteristic
117	#1837	—	19	1	no	—
118	#1564	—	19	1	no	—
119	#1486	—	19	2	no	—
120	#1301	—	19	2	no	—
121	C₂²×C₁₉	—	2	1	no	—
122	C₂²×C₁₉	—	2	1	no	—
123	C₂²×C₁₉	—	2	4	no	—
124	C₂²×C₁₉	—	4	8	no	—
125	C₂²×C₁₉	—	4	2	no	—
126	C₂²×C₁₉	—	4	2	no	—
127	C₂²×C₁₉	#380	1	1	yes	characteristic
128	C₂²×C₁₉	#478	1	2	yes	—
129	C₄×C₁₉	—	2	1	no	—
130	C₄×C₁₉	—	2	1	no	—
131	C₄×C₁₉	—	2	1	no	—
132	C₄×C₁₉	—	4	2	no	—
133	C₄×C₁₉	#416	1	1	yes	characteristic
134	C₄×C₁₉	#418	1	1	yes	characteristic
135	C₄×C₁₉	C₂×#108	1	2	yes	—
136	C₄×C₁₉	#463	1	2	yes	—
137	C₂³×C₈	—	2	1	no	—
138	C₂×C₄×D₄	C₂×D₁₉	1	1	yes	characteristic
139	C₈×D₄	—	2	1	no	—
140	C₈×D₄	—	2	1	no	—
141	C₈×D₄	Dic₁₉	1	2	yes	—
142	C₈×D₄	Dic₁₉	1	2	yes	—
143	#431	—	19	2	no	—
144	#423	—	19	2	no	—
145	C₂×C₄⋊C₈	C₂×D₁₉	1	1	yes	characteristic
146	C₂×C₂²⋊C₈	—	2	1	no	—
147	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
148	#341	—	38	1	no	—
149	#341	—	38	1	no	—
150	#335	—	38	1	no	—
151	#334	—	38	1	no	—
152	D₄⋊C₈	—	38	4	no	—
153	C₂×C₁₉	—	2	2	no	—
154	C₂×C₁₉	—	4	4	no	—
155	C₂×C₁₉	#1804	1	1	yes	characteristic
156	C₂×C₁₉	#1810	1	1	yes	characteristic
157	C₂×C₁₉	#1814	1	1	yes	characteristic
158	C₂²×D₄	Dic₃₈	1	1	yes	characteristic
159	C₂³×C₄	—	2	1	no	—
160	C₂×M₁₆	—	38	2	no	—
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	4	4	no	—
165	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
166	C₂²×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	D₇₆	1	2	yes	—
170	C₄×D₄	C₂×Dic₁₉	1	2	yes	—
171	C₂×C₄⋊C₄	Dic₃₈	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	2	1	no	—
173	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
174	C₄⋊C₈	—	2	1	no	—
175	C₄⋊C₈	—	38	2	no	—
176	C₄⋊C₈	—	38	2	no	—
177	C₄⋊C₈	—	38	2	no	—
178	C₄⋊C₈	C₂×Dic₁₉	1	2	yes	—
179	C₂²⋊C₈	—	2	2	no	—
180	C₂²⋊C₈	—	2	2	no	—
181	C₂²⋊C₈	—	38	2	no	—
182	C₄×C₈	—	38	2	no	—
183	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
184	C₄×C₈	C₂×Dic₁₉	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
187	C₁₉	#8660	1	1	yes	characteristic, complemented, Sylow
188	C₂⁴	—	2	1	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	C₄⋊Dic₁₉	1	2	yes	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
198	C₂²×C₄	#64541	1	1	yes	characteristic
199	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
200	M₁₆	—	76	2	no	—
201	M₁₆	—	76	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	76	2	no	—
209	C₂×C₈	—	76	2	no	—
210	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
213	C₂×C₈	C₂²⋊Dic₁₉	1	1	yes	characteristic
214	C₄⋊C₄	—	2	1	no	—
215	C₄⋊C₄	C₄⋊Dic₁₉	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₄²	C₂×D₇₆	1	1	yes	characteristic
219	C₄²	#64541	1	1	yes	characteristic
220	C₄²	C₂²×Dic₁₉	1	1	yes	characteristic
221	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
222	C₂³	—	2	4	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	4	2	no	—
226	C₂³	#10597370	1	1	yes	characteristic
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	#10597276	1	4	yes	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	#10597370	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
239	C₂×C₄	#10597372	1	1	yes	characteristic
240	C₂×C₄	#10597376	1	1	yes	characteristic
241	C₂×C₄	#10597376	1	1	yes	characteristic
242	C₂×C₄	#10597378	1	2	yes	—
243	C₂×C₄	#10597396	1	2	yes	—
244	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	—	2	1	no	—
248	C₈	—	2	1	no	—
249	C₈	—	4	2	no	—
250	C₈	—	152	2	no	—
251	C₈	—	152	2	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	8	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	#11926404	1	1	yes	characteristic
259	C₂²	#11926844	1	2	yes	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	#11926540	1	1	yes	characteristic
265	C₄	#11926549	1	1	yes	characteristic
266	C₄	C₂×#10597276	1	2	yes	—
267	C₄	#11926759	1	2	yes	—
268	C₂	—	2	2	no	—
269	C₂	—	4	4	no	—
270	C₂	#14563645	1	1	yes	characteristic
271	C₂	#14563673	1	1	yes	characteristic
272	C₂	#14563689	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597376
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₈×D₄×C₁₉	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂²×D₄×C₁₉	Q₈
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×C₈×D₄	D₁₉
hypocenter	C₁₉	#8660
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #14563773	C₂	semidirect
2. #14562383	C₂	semidirect
4. #14561532	C₂	semidirect
187. C₁₉	#8660	semidirect

Maximal subgroups

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

subgroup	index
#14563773	2
#14562383	2
#14562060	2
#14561532	2
C₂×C₈×D₄×C₁₉	2
#8660	19