Subgroups — #18081096

Order 4864.

378 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14564353	C₂	1	2	yes	maximal
2	C₄×#10597274	C₂	1	1	yes	maximal, characteristic
3	C₂×#11926295	C₂	1	1	yes	maximal, characteristic
4	#14561475	C₂	1	2	yes	maximal
5	C₂×C₁₉×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×#10597274	C₄	1	2	yes	—
7	C₂×C₄⋊(C₁₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×Dic₁₉⋊C₈	—	2	2	no	—
9	C₄×C₄⋊Dic₁₉	C₂²	1	2	yes	—
10	#11926307	C₄	1	1	yes	characteristic
11	#11926307	C₄	1	1	yes	characteristic
12	#11926298	—	2	2	no	—
13	#11926295	—	2	2	no	—
14	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
17	C₁₉×#322	—	2	2	no	—
18	C₂²×C₈×C₁₉	—	2	1	no	—
19	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	D₄	1	2	yes	—
22	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊Dic₁₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×Dic₁₉	—	2	2	no	—
26	C₂²×C₁₉⋊C₈	—	2	2	no	—
27	C₄×C₈×C₁₉	—	2	1	no	—
28	C₄×C₈×C₁₉	—	2	1	no	—
29	C₄×C₈×C₁₉	—	2	1	no	—
30	C₄×C₈×C₁₉	D₄	1	2	yes	—
31	C₄×C₈×C₁₉	Q₈	1	2	yes	—
32	C₄×C₈×C₁₉	Q₈	1	2	yes	—
33	C₁₉⋊#95	—	2	2	no	—
34	#10597274	—	2	4	no	—
35	Dic₁₉⋊C₈	—	4	4	no	—
36	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
37	C₄⋊(C₁₉⋊C₈)	—	2	2	no	—
38	C₂²×C₄×C₁₉	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₉	#41	1	1	yes	characteristic
41	C₂²×Dic₁₉	—	2	2	no	—
42	C₂²×Dic₁₉	—	2	2	no	—
43	C₂×C₈×C₁₉	—	2	2	no	—
44	C₂×C₈×C₁₉	—	2	2	no	—
45	C₂×C₈×C₁₉	—	2	2	no	—
46	C₂×C₈×C₁₉	—	2	2	no	—
47	C₂×C₈×C₁₉	—	2	2	no	—
48	C₂×C₈×C₁₉	—	4	1	no	—
49	C₂×C₈×C₁₉	—	4	1	no	—
50	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
51	C₂×C₈×C₁₉	C₄⋊C₄	1	2	yes	—
52	C₂×C₈×C₁₉	SD₁₆	1	4	yes	—
53	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
54	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
55	C₄²×C₁₉	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₁₉	#41	1	1	yes	characteristic
57	C₄⋊Dic₁₉	—	2	2	no	—
58	C₄⋊Dic₁₉	—	2	2	no	—
59	C₄⋊Dic₁₉	—	2	2	no	—
60	C₄⋊Dic₁₉	M₁₆	1	4	yes	—
61	C₄×Dic₁₉	—	4	2	no	—
62	C₄×Dic₁₉	—	4	2	no	—
63	C₂×C₁₉⋊C₈	—	4	2	no	—
64	C₂×C₁₉⋊C₈	—	4	2	no	—
65	C₂×C₁₉⋊C₈	—	4	2	no	—
66	#9597	—	19	1	no	maximal, Sylow
67	C₂³×C₁₉	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₁₉	—	2	1	no	—
69	C₂×C₄×C₁₉	—	2	1	no	—
70	C₂×C₄×C₁₉	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₉	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₁₉	#121	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₂²⋊Q₈	1	1	yes	characteristic
74	C₂×C₄×C₁₉	#125	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₂×M₁₆	1	2	yes	—
77	C₂×C₄×C₁₉	#131	1	2	yes	—
78	C₂×C₄×C₁₉	C₂×SD₁₆	1	2	yes	—
79	C₂×C₄×C₁₉	#135	1	2	yes	—
80	C₂×Dic₁₉	—	2	4	no	—
81	C₂×Dic₁₉	—	4	2	no	—
82	C₂×Dic₁₉	—	4	2	no	—
83	C₂×Dic₁₉	—	4	2	no	—
84	C₂×Dic₁₉	—	4	2	no	—
85	C₂×Dic₁₉	—	4	2	no	—
86	C₈×C₁₉	—	2	2	no	—
87	C₈×C₁₉	—	2	2	no	—
88	C₈×C₁₉	—	2	4	no	—
89	C₈×C₁₉	—	4	2	no	—
90	C₁₉⋊C₈	—	8	2	no	—
91	#1939	—	19	2	no	—
92	C₄×#106	—	19	1	no	—
93	C₂×#334	—	19	1	no	—
94	C₂×#322	D₁₉	1	1	yes	characteristic
95	#1279	—	19	2	no	—
96	C₂²×C₁₉	#389	1	1	yes	characteristic
97	C₂²×C₁₉	#435	1	2	yes	—
98	C₂²×C₁₉	C₄×SD₁₆	1	2	yes	—
99	C₂²×C₁₉	#444	1	1	yes	characteristic
100	C₂²×C₁₉	#444	1	1	yes	characteristic
101	C₄×C₁₉	—	2	1	no	—
102	C₄×C₁₉	—	2	1	no	—
103	C₄×C₁₉	#423	1	1	yes	characteristic
104	C₄×C₁₉	#424	1	1	yes	characteristic
105	C₄×C₁₉	#446	1	2	yes	—
106	C₄×C₁₉	#465	1	2	yes	—
107	C₄×C₁₉	#498	1	1	yes	characteristic
108	C₄×C₁₉	#499	1	1	yes	characteristic
109	Dic₁₉	—	4	4	no	—
110	Dic₁₉	—	8	2	no	—
111	C₂×#106	—	19	2	no	—
112	C₂×C₄⋊C₈	—	19	2	no	—
113	C₂×C₄⋊C₈	—	38	2	no	—
114	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
117	C₄×C₄⋊C₄	—	19	2	no	—
118	#340	—	19	1	no	—
119	#340	—	19	1	no	—
120	#336	—	38	2	no	—
121	#334	—	38	2	no	—
122	#322	—	2	2	no	—
123	C₂×C₁₉	#1582	1	2	yes	—
124	C₂×C₁₉	#1583	1	2	yes	—
125	C₂×C₁₉	#1940	1	1	yes	characteristic
126	C₂×C₁₉	#1941	1	1	yes	characteristic
127	C₂×C₁₉	#1946	1	1	yes	characteristic
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	38	2	no	—
130	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
131	C₂²×C₈	C₄×D₁₉	1	1	yes	characteristic
132	C₂²×C₈	D₇₆	1	2	yes	—
133	C₂×C₄⋊C₄	—	19	2	no	—
134	C₂×C₄⋊C₄	—	19	2	no	—
135	C₂×C₄²	—	38	2	no	—
136	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
137	#106	—	38	4	no	—
138	C₄⋊C₈	—	38	2	no	—
139	C₄⋊C₈	—	38	2	no	—
140	C₄⋊C₈	—	76	4	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	Dic₃₈	1	2	yes	—
145	C₄×C₈	Dic₃₈	1	2	yes	—
146	C₄×C₈	C₂²⋊D₁₉	1	2	yes	—
147	#95	—	38	2	no	—
148	C₁₉	#9597	1	1	yes	characteristic, complemented, Sylow
149	C₂²×C₄	—	38	2	no	—
150	C₂²×C₄	—	38	2	no	—
151	C₂²×C₄	C₂×C₄×D₁₉	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
153	C₂²×C₄	#64541	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	76	2	no	—
162	C₂×C₈	—	76	2	no	—
163	C₂×C₈	—	76	2	no	—
164	C₂×C₈	#64516	1	4	yes	—
165	C₂×C₈	Dic₁₉⋊C₄	1	2	yes	—
166	C₂×C₈	Dic₁₉⋊C₄	1	2	yes	—
167	C₄⋊C₄	—	19	4	no	—
168	C₄⋊C₄	—	38	2	no	—
169	C₄⋊C₄	—	38	2	no	—
170	C₄⋊C₄	—	38	2	no	—
171	C₄²	—	76	2	no	—
172	C₄²	—	76	2	no	—
173	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
174	C₄²	C₂×Dic₃₈	1	1	yes	characteristic
175	C₄²	#64541	1	1	yes	characteristic
176	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
177	C₂³	C₄×D₇₆	1	1	yes	characteristic
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	38	4	no	—
181	C₂×C₄	—	76	2	no	—
182	C₂×C₄	—	76	2	no	—
183	C₂×C₄	—	76	2	no	—
184	C₂×C₄	—	76	2	no	—
185	C₂×C₄	—	76	2	no	—
186	C₂×C₄	C₄×Dic₃₈	1	1	yes	characteristic
187	C₂×C₄	#10597315	1	1	yes	characteristic
188	C₂×C₄	#10597316	1	1	yes	characteristic
189	C₂×C₄	C₂×#64516	1	2	yes	—
190	C₂×C₄	#10597350	1	2	yes	—
191	C₂×C₄	M₁₆×D₁₉	1	2	yes	—
192	C₂×C₄	#10597353	1	2	yes	—
193	C₂×C₄	C₂×Dic₁₉⋊C₄	1	1	yes	characteristic
194	C₂×C₄	#10597370	1	1	yes	characteristic
195	C₂×C₄	#10597376	1	1	yes	characteristic
196	C₈	—	2	2	no	—
197	C₈	—	2	2	no	—
198	C₈	—	2	4	no	—
199	C₈	—	4	2	no	—
200	C₈	—	152	2	no	—
201	C₂²	#11926442	1	1	yes	characteristic
202	C₂²	#11926617	1	2	yes	—
203	C₂²	C₄×#64516	1	2	yes	—
204	C₂²	#11926667	1	1	yes	characteristic
205	C₂²	#11926667	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	76	4	no	—
209	C₄	—	152	2	no	—
210	C₄	#11926567	1	1	yes	characteristic
211	C₄	#11926572	1	1	yes	characteristic
212	C₄	#11926672	1	2	yes	—
213	C₄	#11926773	1	2	yes	—
214	C₄	#11926942	1	1	yes	characteristic
215	C₄	#11926946	1	1	yes	characteristic
216	C₂	#14562473	1	2	yes	—
217	C₂	#14562480	1	2	yes	—
218	C₂	#14564358	1	1	yes	characteristic
219	C₂	#14564363	1	1	yes	characteristic
220	C₂	#14564388	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₇₆
commutator	C₂²×C₄×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₂×C₁₉×#322	C₂
socle, prime radical	C₂³×C₁₉	C₄×D₄
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	C₂×#322	D₁₉
hypocenter	C₁₉	#9597
LCS	G ▷ C₂²×C₄×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
148. C₁₉	#9597	semidirect

Maximal subgroups

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

subgroup	index
#14564353	2
C₄×#10597274	2
C₂×#11926295	2
#14561475	2
C₂×C₁₉×#322	2
#9597	19