Subgroups — #18072005

Order 4864.

758 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926568	C₂	1	2	yes	maximal
2	#14561470	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄×C₈×C₁₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#64520	—	2	2	no	—
5	#11926568	—	2	4	no	—
6	#11926568	C₄	1	8	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
9	C₂³×C₈×C₁₉	C₄	1	2	yes	—
10	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₉	—	2	4	no	—
12	C₂×C₄×C₈×C₁₉	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₁₉	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₉	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₉	D₄	1	1	yes	characteristic
16	C₂³×C₄×C₁₉	Q₈	1	1	yes	characteristic
17	C₂²×C₈×C₁₉	—	2	4	no	—
18	C₂²×C₈×C₁₉	—	2	8	no	—
19	C₂²×C₈×C₁₉	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²×C₁₉	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₁₉	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₉	D₄	1	4	yes	—
24	C₂×C₄²×C₁₉	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
26	C₂×#64520	—	2	8	no	—
27	C₂×#64520	—	4	4	no	—
28	C₂×#64520	—	4	4	no	—
29	C₂²×C₁₉⋊C₈	—	2	4	no	—
30	C₄×C₈×C₁₉	—	2	4	no	—
31	C₄×C₈×C₁₉	—	2	4	no	—
32	C₄⋊(C₁₉⋊C₈)	—	2	8	no	—
33	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
34	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	8	yes	—
35	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₄×C₁₉	—	2	4	no	—
37	C₂²×C₄×C₁₉	—	2	4	no	—
38	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
39	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
40	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	D₈	1	2	yes	—
46	C₂²×C₄×C₁₉	SD₁₆	1	4	yes	—
47	C₂²×C₄×C₁₉	Q₁₆	1	2	yes	—
48	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₉	—	2	16	no	—
52	C₂×C₈×C₁₉	—	2	4	no	—
53	C₂×C₈×C₁₉	—	2	4	no	—
54	C₂×C₈×C₁₉	M₁₆	1	8	yes	complemented
55	C₄²×C₁₉	C₄²	1	2	yes	—
56	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
57	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
58	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
59	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
60	C₄²×C₁₉	C₂×D₄	1	2	yes	—
61	#64520	—	4	16	no	—
62	C₂×C₁₉⋊C₈	—	2	8	no	—
63	C₂×C₁₉⋊C₈	—	4	4	no	—
64	C₂×C₁₉⋊C₈	—	4	4	no	—
65	#7874	—	19	1	no	maximal, Sylow
66	C₂³×C₁₉	—	2	2	no	—
67	C₂³×C₁₉	—	2	2	no	—
68	C₂³×C₁₉	#95	1	2	yes	—
69	C₂³×C₁₉	#106	1	2	yes	—
70	C₂³×C₁₉	#107	1	2	yes	—
71	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₁₉	—	2	2	no	—
73	C₂×C₄×C₁₉	—	2	2	no	—
74	C₂×C₄×C₁₉	—	2	8	no	—
75	C₂×C₄×C₁₉	—	2	4	no	—
76	C₂×C₄×C₁₉	—	2	4	no	—
77	C₂×C₄×C₁₉	#95	1	2	yes	—
78	C₂×C₄×C₁₉	#95	1	2	yes	—
79	C₂×C₄×C₁₉	#95	1	2	yes	—
80	C₂×C₄×C₁₉	#102	1	8	yes	—
81	C₂×C₄×C₁₉	#103	1	8	yes	—
82	C₂×C₄×C₁₉	#106	1	2	yes	—
83	C₂×C₄×C₁₉	#107	1	2	yes	—
84	C₂×C₄×C₁₉	#108	1	4	yes	—
85	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
90	C₂×C₄×C₁₉	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
92	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
93	C₈×C₁₉	—	2	8	no	—
94	C₁₉⋊C₈	—	4	8	no	—
95	C₂×#423	—	19	2	no	—
96	C₂²×C₄×C₈	D₁₉	1	1	yes	characteristic
97	#1277	—	19	4	no	—
98	C₂²×C₁₉	—	2	4	no	—
99	C₂²×C₁₉	—	2	4	no	—
100	C₂²×C₁₉	—	2	2	no	—
101	C₂²×C₁₉	—	2	2	no	—
102	C₂²×C₁₉	#340	1	4	yes	—
103	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
104	C₂²×C₁₉	#423	1	2	yes	—
105	C₂²×C₁₉	C₂×#106	1	1	yes	characteristic
106	C₂²×C₁₉	C₂×#107	1	1	yes	characteristic
107	C₂²×C₁₉	C₂×#108	1	2	yes	—
108	C₄×C₁₉	—	2	2	no	—
109	C₄×C₁₉	—	2	2	no	—
110	C₄×C₁₉	#340	1	4	yes	—
111	C₄×C₁₉	#341	1	4	yes	—
112	C₄×C₁₉	C₄×M₁₆	1	2	yes	—
113	C₄×C₁₉	#407	1	2	yes	—
114	C₄×C₁₉	C₂×#102	1	2	yes	—
115	C₄×C₁₉	C₂×#103	1	2	yes	—
116	C₂²×M₁₆	—	38	2	no	—
117	C₂³×C₈	Dic₁₉	1	2	yes	—
118	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
119	#423	—	19	8	no	—
120	#423	—	38	4	no	—
121	C₂×C₄⋊C₈	—	19	4	no	—
122	C₂×C₄⋊C₈	—	38	2	no	—
123	C₂×C₄×C₈	—	2	4	no	—
124	C₂×C₄×C₈	Dic₁₉	1	2	yes	—
125	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
126	C₂×C₁₉	—	2	4	no	—
127	C₂×C₁₉	#1566	1	2	yes	—
128	C₂×C₁₉	#1567	1	2	yes	—
129	C₂×C₁₉	#1728	1	1	yes	characteristic
130	C₂×C₁₉	C₂×#340	1	1	yes	characteristic
131	C₂×C₁₉	C₂×#341	1	1	yes	characteristic
132	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
133	C₂³×C₄	D₇₆	1	1	yes	characteristic
134	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
135	C₂×M₁₆	—	38	8	no	—
136	C₂×M₁₆	—	76	4	no	—
137	C₂×M₁₆	—	76	4	no	—
138	C₂²×C₈	—	2	4	no	—
139	C₂²×C₈	—	2	8	no	—
140	C₂²×C₈	—	38	4	no	—
141	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
142	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
143	C₂×C₄²	C₄×D₁₉	1	4	yes	—
144	C₂×C₄²	D₇₆	1	1	yes	characteristic
145	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
146	C₂×C₄²	C₂²⋊D₁₉	1	4	yes	—
147	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
148	C₄⋊C₈	—	19	8	no	—
149	C₄⋊C₈	—	38	8	no	—
150	C₄⋊C₈	—	38	4	no	—
151	C₄×C₈	—	2	4	no	—
152	C₄×C₈	—	2	4	no	—
153	C₁₉	#7874	1	1	yes	characteristic, complemented, Sylow
154	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	—	2	4	no	—
157	C₂²×C₄	#64516	1	4	yes	—
158	C₂²×C₄	D₁₅₂	1	2	yes	—
159	C₂²×C₄	Dic₇₆	1	2	yes	—
160	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
161	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
162	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
163	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
164	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
165	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
166	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
167	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
169	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
170	M₁₆	—	76	16	no	—
171	C₂×C₈	—	2	16	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	38	8	no	—
175	C₂×C₈	—	76	4	no	—
176	C₂×C₈	—	76	4	no	—
177	C₂×C₈	#64520	1	8	yes	complemented
178	C₄²	C₄×Dic₁₉	1	2	yes	—
179	C₄²	Dic₁₉⋊C₄	1	4	yes	—
180	C₄²	(C₂×D₁₉)⋊C₄	1	4	yes	—
181	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
182	C₄²	C₂×C₄×D₁₉	1	2	yes	—
183	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	2	no	—
186	C₂³	#10597274	1	2	yes	—
187	C₂³	#10597275	1	2	yes	—
188	C₂³	C₁₉⋊#95	1	2	yes	—
189	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	8	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	#10597273	1	8	yes	—
196	C₂×C₄	#10597274	1	2	yes	—
197	C₂×C₄	#10597275	1	2	yes	—
198	C₂×C₄	#10597276	1	4	yes	—
199	C₂×C₄	#10597278	1	8	yes	—
200	C₂×C₄	C₁₉⋊#95	1	2	yes	—
201	C₂×C₄	C₁₉⋊#95	1	2	yes	—
202	C₂×C₄	C₁₉⋊#95	1	2	yes	—
203	C₂×C₄	C₂×#64516	1	2	yes	—
204	C₂×C₄	C₂×D₁₅₂	1	1	yes	characteristic
205	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
206	C₂×C₄	C₂×#64520	1	4	yes	—
207	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
208	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
209	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
210	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
211	C₈	—	2	8	no	—
212	C₈	—	76	8	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	2	4	no	—
217	C₂²	#11926307	1	4	yes	—
218	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
219	C₂²	#11926568	1	2	yes	—
220	C₂²	C₂×#10597274	1	1	yes	characteristic
221	C₂²	C₂×#10597275	1	1	yes	characteristic
222	C₂²	C₂×#10597276	1	2	yes	—
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	#11926307	1	4	yes	—
226	C₄	#11926309	1	4	yes	—
227	C₄	C₄×#64520	1	2	yes	—
228	C₄	#11926508	1	2	yes	—
229	C₄	C₂×#10597278	1	2	yes	—
230	C₄	C₂×#10597273	1	2	yes	—
231	C₂	—	2	4	no	—
232	C₂	#14562391	1	2	yes	—
233	C₂	#14562394	1	2	yes	—
234	C₂	#14563329	1	1	yes	characteristic
235	C₂	C₂×#11926307	1	1	yes	characteristic
236	C₂	C₂×#11926309	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂×C₄×C₁₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂²×C₄×C₈×C₁₉	C₂
socle	C₂³×C₁₉	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₉	C₄⋊C₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂²×C₄×C₈	D₁₉
hypocenter	C₁₉	#7874
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
2. #14561470	C₂	semidirect
54. C₂×C₈×C₁₉	M₁₆	semidirect
153. C₁₉	#7874	semidirect
177. C₂×C₈	#64520	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926568	2
#14561470	2
C₂²×C₄×C₈×C₁₉	2
#7874	19