Subgroups — #18075423

Order 4864.

372 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14562383	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11926297	C₂	1	1	yes	maximal, characteristic, complemented
3	#14562022	C₂	1	2	yes	maximal
4	#14561486	C₂	1	2	yes	maximal, complemented
5	C₁₉×#2922	C₂	1	1	yes	maximal, characteristic
6	#11926597	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	#11926297	—	2	2	no	—
9	#11926297	—	2	1	no	—
10	#11926295	—	2	1	no	—
11	Q₈⋊(C₁₉⋊C₈)	—	2	2	no	—
12	D₄⋊(C₁₉⋊C₈)	—	2	2	no	—
13	C₄×#41×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#433	—	2	1	no	—
15	C₁₉×#433	—	2	1	no	—
16	C₁₉×#431	—	2	2	no	—
17	C₁₉×#424	C₄	1	2	yes	—
18	C₁₉×#408	C₄	1	2	yes	—
19	C₄×M₁₆×C₁₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
21	C₂×#41×C₁₉	C₂×C₄	1	1	yes	characteristic
22	C₂×M₁₆×C₁₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
24	C₄×Q₈×C₁₉	D₄	1	1	yes	characteristic
25	C₄×Q₈×C₁₉	Q₈	1	1	yes	characteristic
26	C₄×D₄×C₁₉	—	2	1	no	—
27	C₄×D₄×C₁₉	—	2	1	no	—
28	C₄×D₄×C₁₉	D₄	1	1	yes	characteristic
29	C₄×D₄×C₁₉	Q₈	1	1	yes	characteristic
30	C₁₉×#117	—	2	1	no	—
31	C₁₉×#117	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₉	—	2	1	no	—
33	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
34	C₂²×C₁₉⋊C₈	—	2	2	no	—
35	C₁₉×C₄⋊C₈	—	2	2	no	—
36	C₁₉×C₄⋊C₈	—	2	2	no	—
37	C₁₉×C₂²⋊C₈	—	2	2	no	—
38	C₁₉×C₂²⋊C₈	—	2	2	no	—
39	C₁₉×#97	—	2	1	no	—
40	C₄×C₈×C₁₉	—	2	1	no	—
41	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
42	C₄×C₈×C₁₉	D₄	1	2	yes	—
43	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
44	C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
45	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
46	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
47	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
48	C₄×C₁₉⋊C₈	—	2	2	no	—
49	#41×C₁₉	—	2	4	no	—
50	C₂×Q₈×C₁₉	C₄⋊C₄	1	1	yes	characteristic
51	C₂×D₄×C₁₉	—	2	1	no	—
52	C₂×D₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	—	2	1	no	—
54	C₂²×C₄×C₁₉	—	2	1	no	—
55	C₂²×C₄×C₁₉	—	2	1	no	—
56	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
58	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
59	M₁₆×C₁₉	—	2	4	no	—
60	C₂×C₈×C₁₉	—	2	4	no	—
61	C₂×C₈×C₁₉	—	2	2	no	—
62	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₉	C₂²⋊C₄	1	2	yes	—
64	C₄⋊C₄×C₁₉	—	2	1	no	—
65	C₄⋊C₄×C₁₉	—	2	1	no	—
66	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
67	C₄⋊C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
68	C₂²⋊C₄×C₁₉	—	2	1	no	—
69	C₂²⋊C₄×C₁₉	—	2	1	no	—
70	C₂²⋊C₄×C₁₉	—	2	1	no	—
71	C₄²×C₁₉	—	2	1	no	—
72	C₄²×C₁₉	—	2	1	no	—
73	C₄²×C₁₉	—	2	1	no	—
74	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
78	C₂×C₁₉⋊C₈	—	4	2	no	—
79	C₂×C₁₉⋊C₈	—	4	2	no	—
80	C₂×C₁₉⋊C₈	—	8	2	no	—
81	#8606	—	19	1	no	maximal, Sylow
82	C₂³×C₁₉	—	2	1	no	—
83	C₂³×C₁₉	#117	1	1	yes	characteristic
84	Q₈×C₁₉	—	2	2	no	—
85	D₄×C₁₉	—	2	2	no	—
86	D₄×C₁₉	—	4	2	no	—
87	C₂×C₄×C₁₉	—	2	1	no	—
88	C₂×C₄×C₁₉	—	2	1	no	—
89	C₂×C₄×C₁₉	—	2	1	no	—
90	C₂×C₄×C₁₉	—	2	1	no	—
91	C₂×C₄×C₁₉	—	2	1	no	—
92	C₂×C₄×C₁₉	—	4	1	no	—
93	C₂×C₄×C₁₉	—	4	2	no	—
94	C₂×C₄×C₁₉	—	4	2	no	—
95	C₂×C₄×C₁₉	—	4	1	no	—
96	C₂×C₄×C₁₉	—	4	1	no	—
97	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
100	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
101	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₁₉	#131	1	4	yes	—
104	C₈×C₁₉	—	2	2	no	—
105	C₈×C₁₉	—	2	2	no	—
106	C₈×C₁₉	—	4	2	no	—
107	C₁₉⋊C₈	—	8	2	no	—
108	C₁₉⋊C₈	—	8	2	no	—
109	#2922	D₁₉	1	1	yes	characteristic
110	#1564	—	19	1	no	—
111	C₂×#335	—	19	1	no	—
112	#1477	—	19	2	no	—
113	#1283	—	19	2	no	—
114	C₂²×C₁₉	—	2	1	no	—
115	C₂²×C₁₉	—	2	1	no	—
116	C₂²×C₁₉	—	4	1	no	—
117	C₂²×C₁₉	—	4	1	no	—
118	C₂²×C₁₉	—	4	1	no	—
119	C₂²×C₁₉	#380	1	1	yes	characteristic
120	C₂²×C₁₉	#433	1	2	yes	—
121	C₄×C₁₉	—	2	1	no	—
122	C₄×C₁₉	—	2	1	no	—
123	C₄×C₁₉	—	2	1	no	—
124	C₄×C₁₉	—	4	1	no	—
125	C₄×C₁₉	—	4	1	no	—
126	C₄×C₁₉	—	4	1	no	—
127	C₄×C₁₉	#408	1	2	yes	—
128	C₄×C₁₉	#424	1	2	yes	—
129	C₄×C₁₉	#472	1	1	yes	characteristic
130	C₄×C₁₉	#473	1	1	yes	characteristic
131	C₄×#41	C₂×D₁₉	1	1	yes	characteristic
132	#433	—	2	1	no	—
133	#433	—	2	1	no	—
134	#431	—	2	2	no	—
135	#431	—	19	2	no	—
136	#424	Dic₁₉	1	2	yes	—
137	C₂×C₄⋊C₈	—	19	2	no	—
138	#408	Dic₁₉	1	2	yes	—
139	C₄×M₁₆	C₂×D₁₉	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
141	#335	—	38	2	no	—
142	#335	—	38	1	no	—
143	#334	—	38	1	no	—
144	Q₈⋊C₈	—	38	2	no	—
145	D₄⋊C₈	—	38	2	no	—
146	C₂×C₁₉	—	2	2	no	—
147	C₂×C₁₉	—	8	1	no	—
148	C₂×C₁₉	#1798	1	1	yes	characteristic
149	C₂×C₁₉	#1814	1	1	yes	characteristic
150	C₂×C₁₉	#1816	1	1	yes	characteristic
151	C₂×#41	C₂×Dic₁₉	1	1	yes	characteristic
152	C₂×M₁₆	C₂×Dic₁₉	1	2	yes	—
153	C₂²×C₈	—	38	2	no	—
154	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
155	C₄×Q₈	Dic₃₈	1	1	yes	characteristic
156	C₄×Q₈	D₇₆	1	1	yes	characteristic
157	C₄×D₄	—	2	1	no	—
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	Dic₃₈	1	1	yes	characteristic
160	C₄×D₄	D₇₆	1	1	yes	characteristic
161	#117	—	2	1	no	—
162	#117	C₂×Dic₁₉	1	1	yes	characteristic
163	C₂×C₄²	—	2	1	no	—
164	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
165	C₄⋊C₈	—	2	2	no	—
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	38	2	no	—
168	C₄⋊C₈	—	38	2	no	—
169	C₄⋊C₈	—	38	2	no	—
170	C₂²⋊C₈	—	2	2	no	—
171	C₂²⋊C₈	—	2	2	no	—
172	C₂²⋊C₈	—	38	2	no	—
173	#97	—	2	1	no	—
174	C₄×C₈	—	2	1	no	—
175	C₄×C₈	—	38	2	no	—
176	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
177	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
178	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
179	C₁₉	#8606	1	1	yes	characteristic, complemented, Sylow
180	#41	—	2	4	no	—
181	C₂×Q₈	C₄⋊Dic₁₉	1	1	yes	characteristic
182	C₂×D₄	—	2	1	no	—
183	C₂×D₄	C₄⋊Dic₁₉	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	#64541	1	1	yes	characteristic
188	C₂²×C₄	#64541	1	1	yes	characteristic
189	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
190	M₁₆	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	76	2	no	—
194	C₂×C₈	—	76	2	no	—
195	C₂×C₈	—	152	2	no	—
196	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
197	C₂×C₈	C₂²⋊Dic₁₉	1	2	yes	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
201	C₄⋊C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₂²⋊C₄	—	2	1	no	—
205	C₄²	—	2	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
209	C₄²	C₂×D₇₆	1	1	yes	characteristic
210	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
211	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
212	C₂³	—	2	1	no	—
213	C₂³	#10597372	1	1	yes	characteristic
214	Q₈	—	2	2	no	—
215	D₄	—	2	2	no	—
216	D₄	—	4	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	#10597370	1	1	yes	characteristic
228	C₂×C₄	#10597370	1	1	yes	characteristic
229	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
230	C₂×C₄	#10597376	1	1	yes	characteristic
231	C₂×C₄	#10597376	1	1	yes	characteristic
232	C₂×C₄	#10597394	1	4	yes	—
233	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	2	2	no	—
236	C₈	—	4	2	no	—
237	C₈	—	152	2	no	—
238	C₈	—	152	2	no	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	#11926404	1	1	yes	characteristic
245	C₂²	#11926605	1	2	yes	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	#11926511	1	2	yes	—
253	C₄	#11926571	1	2	yes	—
254	C₄	#11926814	1	1	yes	characteristic
255	C₄	#11926819	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	8	1	no	—
258	C₂	#14563619	1	1	yes	characteristic
259	C₂	#14563689	1	1	yes	characteristic
260	C₂	#14563699	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#10597376
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#2922	C₂
socle	C₂²×C₁₉	#380
prime radical	C₂×#41×C₁₉	C₂×C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#2922	D₁₉
hypocenter	C₁₉	#8606
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2922

Complements

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

N	H ≅ G/N	type
1. #14562383	C₂	semidirect
2. C₂×#11926297	C₂	semidirect
4. #14561486	C₂	semidirect
179. C₁₉	#8606	semidirect

Maximal subgroups

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

subgroup	index
#14562383	2
C₂×#11926297	2
#14562022	2
#14561486	2
C₁₉×#2922	2
#8606	19