Subgroups — #18088448

Order 4864.

806 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#10597276	C₂	1	2	yes	maximal
2	C₂×#11926309	C₂	1	2	yes	maximal
3	#14563329	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	4	yes	—
6	C₂×#10597276	—	2	4	no	—
7	C₂×#10597276	—	2	4	no	—
8	C₂×#10597276	C₄	1	8	yes	complemented
9	#11926309	—	2	4	no	—
10	#11926299	—	2	4	no	—
11	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₁₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₉	—	2	4	no	—
15	C₂×C₄×C₈×C₁₉	C₄	1	4	yes	—
16	C₂³×C₄×C₁₉	Q₈	1	2	yes	—
17	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
18	C₂²×C₈×C₁₉	—	2	2	no	—
19	C₂²×C₈×C₁₉	—	2	2	no	—
20	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₉	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₉	D₄	1	2	yes	—
24	C₂²×C₈×C₁₉	D₄	1	2	yes	—
25	C₂²×C₈×C₁₉	D₄	1	4	yes	—
26	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
27	C₂²×C₈×C₁₉	Q₈	1	2	yes	—
28	C₂×C₄²×C₁₉	—	2	2	no	—
29	C₂×C₄²×C₁₉	—	2	2	no	—
30	C₂×C₄²×C₁₉	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₁₉	D₄	1	2	yes	—
32	C₂×#64520	—	2	4	no	—
33	C₂×#64520	—	2	4	no	—
34	C₂×#64520	—	2	4	no	—
35	C₂×#64520	—	2	4	no	—
36	C₂×#64520	—	2	4	no	—
37	C₂×#64520	C₂×C₄	1	8	yes	—
38	C₂²×C₁₉⋊C₈	—	2	4	no	—
39	C₄×C₈×C₁₉	—	2	8	no	—
40	#10597276	—	2	16	no	—
41	C₂⁴×C₁₉	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	—	2	4	no	—
43	C₂²×C₄×C₁₉	—	2	8	no	—
44	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₁₉	C₂²×C₄	1	2	yes	—
46	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₉	#41	1	2	yes	—
54	C₂²×C₄×C₁₉	#41	1	2	yes	—
55	C₂×C₈×C₁₉	—	2	2	no	—
56	C₂×C₈×C₁₉	—	2	2	no	—
57	C₂×C₈×C₁₉	—	2	4	no	—
58	C₂×C₈×C₁₉	—	2	4	no	—
59	C₂×C₈×C₁₉	C₄²	1	4	yes	—
60	C₂×C₈×C₁₉	C₂²⋊C₄	1	8	yes	—
61	C₂×C₈×C₁₉	C₂²⋊C₄	1	4	yes	—
62	C₂×C₈×C₁₉	C₄⋊C₄	1	8	yes	—
63	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
64	C₂×C₈×C₁₉	C₄⋊C₄	1	4	yes	—
65	C₄²×C₁₉	—	2	2	no	—
66	C₄²×C₁₉	—	2	2	no	—
67	C₄²×C₁₉	—	2	4	no	—
68	#64520	—	2	8	no	—
69	#64520	—	2	8	no	—
70	#64520	—	4	4	no	—
71	#64520	—	4	4	no	—
72	C₂×C₁₉⋊C₈	—	2	8	no	—
73	C₂×C₁₉⋊C₈	—	4	4	no	—
74	C₂×C₁₉⋊C₈	—	4	4	no	—
75	#10864	—	19	1	no	maximal, Sylow
76	C₂³×C₁₉	—	2	4	no	—
77	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
79	C₂³×C₁₉	C₂²⋊Q₈	1	2	yes	—
80	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
81	C₂³×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₁₉	—	2	16	no	—
83	C₂×C₄×C₁₉	—	2	4	no	—
84	C₂×C₄×C₁₉	—	2	4	no	—
85	C₂×C₄×C₁₉	#108	1	8	yes	complemented
86	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
93	C₂×C₄×C₁₉	#117	1	1	yes	characteristic
94	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₁₉	C₄×Q₈	1	2	yes	—
97	C₂×C₄×C₁₉	#121	1	2	yes	—
98	C₂×C₄×C₁₉	#121	1	2	yes	—
99	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₁₉	#124	1	2	yes	—
101	C₂×C₄×C₁₉	#125	1	2	yes	—
102	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
103	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
104	C₈×C₁₉	—	2	4	no	—
105	C₈×C₁₉	#95	1	8	yes	—
106	C₁₉⋊C₈	—	4	8	no	—
107	C₂²×#108	—	19	2	no	—
108	C₂²×C₄×C₈	D₁₉	1	1	yes	characteristic
109	C₂×#341	—	19	2	no	—
110	#1728	—	19	2	no	—
111	C₂²×C₁₉	—	2	4	no	—
112	C₂²×C₁₉	—	2	2	no	—
113	C₂²×C₁₉	—	2	2	no	—
114	C₂²×C₁₉	—	2	4	no	—
115	C₂²×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₁₉	#380	1	1	yes	characteristic
117	C₂²×C₁₉	#384	1	1	yes	characteristic
118	C₂²×C₁₉	#389	1	2	yes	—
119	C₂²×C₁₉	#391	1	2	yes	—
120	C₂²×C₁₉	C₂×#108	1	4	yes	—
121	C₄×C₁₉	—	2	8	no	—
122	C₄×C₁₉	C₂×#95	1	1	yes	characteristic
123	C₄×C₁₉	C₄×C₄⋊C₄	1	1	yes	characteristic
124	C₄×C₁₉	#380	1	1	yes	characteristic
125	C₄×C₁₉	#382	1	1	yes	characteristic
126	C₄×C₁₉	#389	1	2	yes	—
127	C₄×C₁₉	#390	1	2	yes	—
128	C₂²×M₁₆	—	19	4	no	—
129	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
130	C₂³×C₈	C₂×D₁₉	1	1	yes	characteristic
131	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
132	C₂×#108	—	19	8	no	—
133	C₂×#108	—	38	4	no	—
134	C₂×#108	—	38	4	no	—
135	C₂×C₄×C₈	—	2	4	no	—
136	C₂×C₄×C₈	Dic₁₉	1	4	yes	—
137	#341	—	38	4	no	—
138	#336	—	38	4	no	—
139	C₂×C₁₉	—	2	2	no	—
140	C₂×C₁₉	—	2	2	no	—
141	C₂×C₁₉	C₄⋊#95	1	1	yes	characteristic
142	C₂×C₁₉	C₄×#108	1	2	yes	—
143	C₂×C₁₉	#1810	1	2	yes	—
144	C₂×C₁₉	#1848	1	2	yes	—
145	C₂³×C₄	Dic₃₈	1	2	yes	—
146	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
147	C₂×M₁₆	—	19	8	no	—
148	C₂×M₁₆	—	38	4	no	—
149	C₂×M₁₆	—	38	4	no	—
150	C₂×M₁₆	—	38	4	no	—
151	C₂×M₁₆	—	38	4	no	—
152	C₂×M₁₆	—	38	4	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	38	4	no	—
156	C₂²×C₈	Dic₃₈	1	2	yes	—
157	C₂²×C₈	Dic₃₈	1	2	yes	—
158	C₂²×C₈	C₄×D₁₉	1	4	yes	—
159	C₂²×C₈	D₇₆	1	2	yes	—
160	C₂²×C₈	D₇₆	1	2	yes	—
161	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
162	C₂²×C₈	C₂×Dic₁₉	1	2	yes	—
163	C₂²×C₈	C₂²⋊D₁₉	1	4	yes	—
164	C₂×C₄²	—	2	2	no	—
165	C₂×C₄²	—	2	2	no	—
166	C₂×C₄²	D₇₆	1	2	yes	—
167	C₂×C₄²	C₂×Dic₁₉	1	2	yes	—
168	#108	—	38	16	no	—
169	C₄×C₈	—	2	8	no	—
170	C₁₉	#10864	1	1	yes	characteristic, complemented, Sylow
171	C₂⁴	C₂×Dic₃₈	1	1	yes	characteristic
172	C₂²×C₄	—	2	4	no	—
173	C₂²×C₄	—	2	8	no	—
174	C₂²×C₄	C₄⋊Dic₁₉	1	4	yes	—
175	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
176	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₄×D₁₉	1	2	yes	—
178	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
181	C₂²×C₄	#64541	1	2	yes	—
182	C₂²×C₄	#64541	1	2	yes	—
183	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
184	C₂²×C₄	C₂×C₂²⋊D₁₉	1	2	yes	—
185	M₁₆	—	38	8	no	—
186	M₁₆	—	38	8	no	—
187	M₁₆	—	76	4	no	—
188	M₁₆	—	76	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	4	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	38	8	no	—
194	C₂×C₈	—	76	4	no	—
195	C₂×C₈	—	76	4	no	—
196	C₂×C₈	C₄×Dic₁₉	1	4	yes	—
197	C₂×C₈	Dic₁₉⋊C₄	1	8	yes	—
198	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
199	C₂×C₈	C₄⋊Dic₁₉	1	4	yes	—
200	C₂×C₈	(C₂×D₁₉)⋊C₄	1	8	yes	—
201	C₂×C₈	C₂²⋊Dic₁₉	1	4	yes	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₂³	—	2	4	no	—
206	C₂³	C₄×Dic₃₈	1	2	yes	—
207	C₂³	#10597315	1	1	yes	characteristic
208	C₂³	#10597315	1	1	yes	characteristic
209	C₂³	#10597370	1	2	yes	—
210	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
211	C₂×C₄	—	2	16	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	#10597276	1	8	yes	complemented
215	C₂×C₄	C₄×Dic₃₈	1	2	yes	—
216	C₂×C₄	#10597315	1	1	yes	characteristic
217	C₂×C₄	#10597316	1	2	yes	—
218	C₂×C₄	C₄×D₇₆	1	2	yes	—
219	C₂×C₄	C₄×D₇₆	1	2	yes	—
220	C₂×C₄	#10597320	1	1	yes	characteristic
221	C₂×C₄	#10597321	1	2	yes	—
222	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
223	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
224	C₂×C₄	#10597370	1	2	yes	—
225	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
226	C₂×C₄	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
227	C₂×C₄	#10597372	1	1	yes	characteristic
228	C₂×C₄	#10597372	1	1	yes	characteristic
229	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
230	C₂×C₄	#10597376	1	2	yes	—
231	C₂×C₄	#10597376	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
233	C₈	—	2	4	no	—
234	C₈	—	76	8	no	—
235	C₈	C₁₉⋊#95	1	8	yes	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	2	no	—
240	C₂²	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
241	C₂²	#11926404	1	1	yes	characteristic
242	C₂²	#11926414	1	1	yes	characteristic
243	C₂²	#11926442	1	2	yes	—
244	C₂²	#11926450	1	2	yes	—
245	C₂²	C₂×#10597276	1	4	yes	—
246	C₄	—	2	8	no	—
247	C₄	C₂×C₁₉⋊#95	1	1	yes	characteristic
248	C₄	C₄×C₄⋊Dic₁₉	1	1	yes	characteristic
249	C₄	#11926404	1	1	yes	characteristic
250	C₄	#11926410	1	1	yes	characteristic
251	C₄	#11926442	1	2	yes	—
252	C₄	#11926446	1	2	yes	—
253	C₂	—	2	2	no	—
254	C₂	—	2	2	no	—
255	C₂	#14561865	1	1	yes	characteristic
256	C₂	C₄×#10597276	1	2	yes	—
257	C₂	#14563673	1	2	yes	—
258	C₂	#14563807	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₂²×C₄×C₈×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₂×Q₈
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	C₂²×C₄×C₈	D₁₉
hypocenter	C₁₉	#10864
LCS	G ▷ C₂×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₂×#10597276	C₄	semidirect
85. C₂×C₄×C₁₉	#108	semidirect
170. C₁₉	#10864	semidirect
214. C₂×C₄	#10597276	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#10597276	2
C₂×#11926309	2
#14563329	2
C₂²×C₄×C₈×C₁₉	2
#10864	19