Subgroups — #18072008

Order 4864.

670 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926601	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	#11926601	—	2	4	no	—
5	#11926601	C₄	1	8	yes	—
6	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	4	yes	—
7	C₂×C₂²⋊(C₁₉⋊C₈)	—	2	2	no	—
8	C₂×#10597260	—	2	2	no	—
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	C₂²×C₁₉⋊C₈	—	2	4	no	—
27	C₄×C₈×C₁₉	—	2	4	no	—
28	C₄×C₈×C₁₉	—	2	4	no	—
29	C₂²⋊(C₁₉⋊C₈)	—	2	8	no	—
30	C₄⋊(C₁₉⋊C₈)	—	2	4	no	—
31	C₄⋊(C₁₉⋊C₈)	C₂×C₄	1	8	yes	—
32	#10597260	—	2	8	no	—
33	C₂⁴×C₁₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉	—	2	4	no	—
35	C₂²×C₄×C₁₉	—	2	4	no	—
36	C₂²×C₄×C₁₉	—	2	4	no	—
37	C₂²×C₄×C₁₉	C₄²	1	2	yes	—
38	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
39	C₂²×C₄×C₁₉	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₄×C₁₉	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₉	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₁₉	—	2	16	no	—
48	C₂×C₈×C₁₉	—	2	4	no	—
49	C₂×C₈×C₁₉	—	2	4	no	—
50	C₂×C₈×C₁₉	M₁₆	1	8	yes	—
51	C₄²×C₁₉	C₄²	1	2	yes	—
52	C₄²×C₁₉	C₂²⋊C₄	1	4	yes	—
53	C₄²×C₁₉	C₂²⋊C₄	1	2	yes	—
54	C₄²×C₁₉	C₄⋊C₄	1	4	yes	—
55	C₄²×C₁₉	C₂²×C₄	1	2	yes	—
56	C₄²×C₁₉	C₂×D₄	1	2	yes	—
57	C₂×C₁₉⋊C₈	—	2	8	no	—
58	C₂×C₁₉⋊C₈	—	4	8	no	—
59	#7875	—	19	1	no	maximal, Sylow
60	C₂³×C₁₉	—	2	2	no	—
61	C₂³×C₁₉	—	2	2	no	—
62	C₂³×C₁₉	#95	1	2	yes	—
63	C₂³×C₁₉	#108	1	4	yes	—
64	C₂³×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₄×C₁₉	—	2	8	no	—
66	C₂×C₄×C₁₉	—	2	2	no	—
67	C₂×C₄×C₁₉	—	2	2	no	—
68	C₂×C₄×C₁₉	—	2	2	no	—
69	C₂×C₄×C₁₉	—	2	2	no	—
70	C₂×C₄×C₁₉	—	2	4	no	—
71	C₂×C₄×C₁₉	—	2	4	no	—
72	C₂×C₄×C₁₉	—	2	8	no	—
73	C₂×C₄×C₁₉	#95	1	2	yes	—
74	C₂×C₄×C₁₉	#95	1	2	yes	—
75	C₂×C₄×C₁₉	#95	1	2	yes	—
76	C₂×C₄×C₁₉	C₂×C₄²	1	1	yes	characteristic
77	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	2	yes	—
79	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₁₉	C₂×M₁₆	1	4	yes	—
81	C₂×C₄×C₁₉	#135	1	4	yes	—
82	C₈×C₁₉	—	2	8	no	—
83	C₁₉⋊C₈	—	4	8	no	—
84	C₂×#432	—	19	2	no	—
85	C₂²×C₄×C₈	D₁₉	1	1	yes	characteristic
86	#1277	—	19	4	no	—
87	C₂²×C₁₉	—	2	8	no	—
88	C₂²×C₁₉	—	2	2	no	—
89	C₂²×C₁₉	—	2	2	no	—
90	C₂²×C₁₉	#341	1	4	yes	—
91	C₂²×C₁₉	C₂×#95	1	1	yes	characteristic
92	C₂²×C₁₉	#423	1	2	yes	—
93	C₂²×C₁₉	#427	1	2	yes	—
94	C₂²×C₁₉	C₂×#108	1	2	yes	—
95	C₄×C₁₉	—	2	2	no	—
96	C₄×C₁₉	—	2	2	no	—
97	C₄×C₁₉	—	2	2	no	—
98	C₄×C₁₉	—	2	2	no	—
99	C₄×C₁₉	C₄×M₁₆	1	2	yes	—
100	C₄×C₁₉	#407	1	2	yes	—
101	C₄×C₁₉	#416	1	4	yes	—
102	C₂³×C₈	Dic₁₉	1	2	yes	—
103	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
104	#432	—	19	8	no	—
105	#432	—	38	4	no	—
106	C₂×C₄⋊C₈	—	19	4	no	—
107	C₂×C₂²⋊C₈	—	38	2	no	—
108	C₂×#97	—	38	2	no	—
109	C₂×C₄×C₈	—	2	4	no	—
110	C₂×C₄×C₈	Dic₁₉	1	2	yes	—
111	C₂×C₄×C₈	C₂×D₁₉	1	2	yes	—
112	C₂×C₁₉	—	2	4	no	—
113	C₂×C₁₉	#1568	1	4	yes	—
114	C₂×C₁₉	#1728	1	1	yes	characteristic
115	C₂×C₁₉	#1735	1	1	yes	characteristic
116	C₂×C₁₉	C₂×#341	1	1	yes	characteristic
117	C₂³×C₄	Dic₃₈	1	1	yes	characteristic
118	C₂³×C₄	D₇₆	1	1	yes	characteristic
119	C₂³×C₄	C₂×Dic₁₉	1	1	yes	characteristic
120	C₂²×C₈	—	2	4	no	—
121	C₂²×C₈	—	2	8	no	—
122	C₂²×C₈	—	38	4	no	—
123	C₂²×C₈	C₂×Dic₁₉	1	4	yes	—
124	C₂×C₄²	Dic₃₈	1	1	yes	characteristic
125	C₂×C₄²	C₄×D₁₉	1	4	yes	—
126	C₂×C₄²	D₇₆	1	1	yes	characteristic
127	C₂×C₄²	C₂×Dic₁₉	1	1	yes	characteristic
128	C₂×C₄²	C₂²⋊D₁₉	1	4	yes	—
129	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
130	C₄⋊C₈	—	19	8	no	—
131	C₄⋊C₈	—	38	4	no	—
132	C₂²⋊C₈	—	38	8	no	—
133	#97	—	38	8	no	—
134	C₄×C₈	—	2	4	no	—
135	C₄×C₈	—	2	4	no	—
136	C₁₉	#7875	1	1	yes	characteristic, complemented, Sylow
137	C₂⁴	C₄⋊Dic₁₉	1	1	yes	characteristic
138	C₂²×C₄	—	2	4	no	—
139	C₂²×C₄	—	2	4	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	C₄×Dic₁₉	1	2	yes	—
142	C₂²×C₄	Dic₁₉⋊C₄	1	4	yes	—
143	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
144	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
145	C₂²×C₄	C₄⋊Dic₁₉	1	1	yes	characteristic
146	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
147	C₂²×C₄	C₂²⋊Dic₁₉	1	2	yes	—
148	C₂²×C₄	C₂×Dic₃₈	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
150	C₂²×C₄	C₂²×Dic₁₉	1	1	yes	characteristic
151	C₂×C₈	—	2	16	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	38	8	no	—
155	C₂×C₈	—	76	8	no	—
156	C₂×C₈	#64520	1	8	yes	—
157	C₄²	C₄×Dic₁₉	1	2	yes	—
158	C₄²	Dic₁₉⋊C₄	1	4	yes	—
159	C₄²	(C₂×D₁₉)⋊C₄	1	4	yes	—
160	C₄²	C₂²⋊Dic₁₉	1	2	yes	—
161	C₄²	C₂×C₄×D₁₉	1	2	yes	—
162	C₄²	C₂×C₂²⋊D₁₉	1	2	yes	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	2	no	—
165	C₂³	#10597276	1	4	yes	—
166	C₂³	C₁₉⋊#95	1	2	yes	—
167	C₂³	C₂×C₄⋊Dic₁₉	1	1	yes	characteristic
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	8	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	C₁₉⋊#95	1	2	yes	—
177	C₂×C₄	C₁₉⋊#95	1	2	yes	—
178	C₂×C₄	C₁₉⋊#95	1	2	yes	—
179	C₂×C₄	#10597350	1	4	yes	—
180	C₂×C₄	C₂×#64520	1	4	yes	—
181	C₂×C₄	C₂×C₄×Dic₁₉	1	1	yes	characteristic
182	C₂×C₄	C₂×Dic₁₉⋊C₄	1	2	yes	—
183	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	2	yes	—
184	C₂×C₄	C₂×C₂²⋊Dic₁₉	1	1	yes	characteristic
185	C₈	—	2	8	no	—
186	C₈	—	76	8	no	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	8	no	—
190	C₂²	#11926309	1	4	yes	—
191	C₂²	C₂×C₁₉⋊#95	1	1	yes	characteristic
192	C₂²	#11926568	1	2	yes	—
193	C₂²	#11926582	1	2	yes	—
194	C₂²	C₂×#10597276	1	2	yes	—
195	C₄	—	2	2	no	—
196	C₄	—	2	2	no	—
197	C₄	—	2	2	no	—
198	C₄	—	2	2	no	—
199	C₄	C₄×#64520	1	2	yes	—
200	C₄	#11926508	1	2	yes	—
201	C₄	#11926541	1	4	yes	—
202	C₂	—	2	4	no	—
203	C₂	#14562398	1	4	yes	—
204	C₂	#14563329	1	1	yes	characteristic
205	C₂	#14563352	1	1	yes	characteristic
206	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	#7875
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
136. C₁₉	#7875	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926601	2
#14561470	2
C₂²×C₄×C₈×C₁₉	2
#7875	19