Subgroups — #18091314

Order 4864.

686 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11926410	C₂	1	1	yes	maximal, characteristic
2	C₂×#11926312	C₂	1	4	yes	maximal
3	#14563329	C₂	1	1	yes	maximal, characteristic
4	C₁₉×#1728	C₂	1	1	yes	maximal, characteristic
5	C₂²×#64520	C₂²	1	2	yes	—
6	C₂²×C₄⋊Dic₁₉	C₂²	1	2	yes	—
7	#11926410	—	2	2	no	—
8	#11926410	—	2	2	no	—
9	C₂×C₁₉⋊#95	C₄	1	4	yes	—
10	#11926312	—	2	8	no	—
11	#11926299	—	2	2	no	—
12	C₂²×M₁₆×C₁₉	C₂²	1	2	yes	—
13	C₂²×C₄²×C₁₉	C₂²	1	1	yes	characteristic
14	C₁₉×#336	—	2	2	no	—
15	C₂³×C₄×C₁₉	C₂×C₄	1	2	yes	—
16	C₂³×C₄×C₁₉	C₂³	1	1	yes	characteristic
17	C₂³×Dic₁₉	C₂×C₄	1	4	yes	—
18	C₂×M₁₆×C₁₉	—	2	2	no	—
19	C₂×M₁₆×C₁₉	—	2	4	no	—
20	C₂×M₁₆×C₁₉	—	2	4	no	—
21	C₂×M₁₆×C₁₉	D₄	1	4	yes	—
22	C₂²×C₈×C₁₉	—	2	2	no	—
23	C₂×C₄²×C₁₉	—	2	4	no	—
24	C₂×C₄²×C₁₉	D₄	1	2	yes	—
25	C₂×C₄²×C₁₉	D₄	1	2	yes	—
26	C₂×C₄⋊Dic₁₉	—	2	4	no	—
27	C₂×C₄⋊Dic₁₉	—	2	2	no	—
28	C₂×C₄⋊Dic₁₉	—	2	2	no	—
29	C₂×C₄⋊Dic₁₉	—	2	4	no	—
30	C₂×#64520	—	2	2	no	—
31	C₂×#64520	—	2	4	no	—
32	C₂×#64520	—	2	4	no	—
33	C₂×#64520	Q₈	1	4	yes	—
34	C₂²×C₁₉⋊C₈	—	2	2	no	—
35	C₁₉⋊#95	—	2	8	no	—
36	C₂⁴×C₁₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₉	—	2	4	no	—
38	C₂²×C₄×C₁₉	—	4	4	no	—
39	C₂²×C₄×C₁₉	C₂²⋊C₄	1	4	yes	—
40	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
41	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₄×C₁₉	C₂×Q₈	1	2	yes	—
44	C₂²×C₄×C₁₉	#41	1	2	yes	—
45	C₂²×C₄×C₁₉	#41	1	4	yes	—
46	C₂²×C₄×C₁₉	#41	1	2	yes	—
47	C₂²×Dic₁₉	—	2	4	no	—
48	C₂²×Dic₁₉	—	2	8	no	—
49	C₂²×Dic₁₉	—	4	8	no	—
50	M₁₆×C₁₉	—	2	8	no	—
51	M₁₆×C₁₉	—	4	4	no	—
52	C₂×C₈×C₁₉	—	2	4	no	—
53	C₂×C₈×C₁₉	—	4	2	no	—
54	C₂×C₈×C₁₉	—	4	2	no	—
55	C₄²×C₁₉	—	2	4	no	—
56	C₄²×C₁₉	—	2	4	no	—
57	C₄⋊Dic₁₉	—	4	4	no	—
58	C₄⋊Dic₁₉	—	4	2	no	—
59	C₄⋊Dic₁₉	—	4	2	no	—
60	#64520	—	2	8	no	—
61	#64520	—	4	4	no	—
62	C₂×C₁₉⋊C₈	—	2	4	no	—
63	C₂×C₁₉⋊C₈	—	4	2	no	—
64	C₂×C₁₉⋊C₈	—	4	2	no	—
65	#11355	—	19	1	no	maximal, Sylow
66	C₂³×C₁₉	—	2	4	no	—
67	C₂³×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂³×C₁₉	#117	1	2	yes	—
69	C₂³×C₁₉	C₄×D₄	1	2	yes	—
70	C₂³×C₁₉	C₄×Q₈	1	2	yes	—
71	C₂×C₄×C₁₉	—	2	4	no	—
72	C₂×C₄×C₁₉	—	2	8	no	—
73	C₂×C₄×C₁₉	—	4	8	no	—
74	C₂×C₄×C₁₉	#121	1	2	yes	—
75	C₂×C₄×C₁₉	C₂²⋊Q₈	1	4	yes	—
76	C₂×C₄×C₁₉	C₂²⋊Q₈	1	2	yes	—
77	C₂×C₄×C₁₉	#123	1	2	yes	—
78	C₂×C₄×C₁₉	#123	1	4	yes	—
79	C₂×C₄×C₁₉	#123	1	2	yes	—
80	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
81	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
82	C₂×C₄×C₁₉	#125	1	4	yes	—
83	C₂×C₄×C₁₉	#126	1	4	yes	—
84	C₂×C₄×C₁₉	#127	1	1	yes	characteristic
85	C₂×C₄×C₁₉	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×Dic₁₉	—	4	4	no	—
87	C₂×Dic₁₉	—	4	8	no	—
88	C₂×Dic₁₉	—	8	8	no	—
89	C₈×C₁₉	—	4	4	no	—
90	C₁₉⋊C₈	—	4	4	no	—
91	C₂×#382	—	19	1	no	—
92	C₂×#343	—	19	4	no	—
93	#1728	—	19	1	no	—
94	#1728	D₁₉	1	1	yes	characteristic
95	C₂²×C₁₉	—	2	4	no	—
96	C₂²×C₁₉	—	2	4	no	—
97	C₂²×C₁₉	—	2	4	no	—
98	C₂²×C₁₉	#381	1	1	yes	characteristic
99	C₂²×C₁₉	#387	1	4	yes	—
100	C₂²×C₁₉	#390	1	1	yes	characteristic
101	C₂²×C₁₉	#391	1	1	yes	characteristic
102	C₂²×C₁₉	#413	1	4	yes	—
103	C₄×C₁₉	—	4	4	no	—
104	C₄×C₁₉	#397	1	2	yes	—
105	C₄×C₁₉	#398	1	2	yes	—
106	C₄×C₁₉	#399	1	2	yes	—
107	C₄×C₁₉	#400	1	2	yes	—
108	Dic₁₉	—	8	4	no	—
109	C₂²×M₁₆	—	19	2	no	—
110	C₂²×M₁₆	C₂×D₁₉	1	2	yes	—
111	C₂²×C₄⋊C₄	—	19	2	no	—
112	C₂²×C₄²	C₂×D₁₉	1	1	yes	characteristic
113	#382	—	38	2	no	—
114	#382	—	38	2	no	—
115	C₂×#95	—	19	4	no	—
116	#343	—	38	8	no	—
117	#336	—	2	2	no	—
118	#336	—	38	2	no	—
119	C₂×C₁₉	—	2	4	no	—
120	C₂×C₁₉	#1445	1	1	yes	characteristic
121	C₂×C₁₉	#1822	1	2	yes	—
122	C₂×C₁₉	#1967	1	2	yes	—
123	C₂×C₁₉	#1998	1	2	yes	—
124	C₂³×C₄	—	19	4	no	—
125	C₂³×C₄	C₄×D₁₉	1	2	yes	—
126	C₂³×C₄	C₂²×D₁₉	1	1	yes	characteristic
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	—	2	4	no	—
129	C₂×M₁₆	—	2	4	no	—
130	C₂×M₁₆	—	19	4	no	—
131	C₂×M₁₆	—	38	2	no	—
132	C₂×M₁₆	—	38	4	no	—
133	C₂×M₁₆	—	38	4	no	—
134	C₂×M₁₆	D₇₆	1	4	yes	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	38	2	no	—
137	C₂×C₄⋊C₄	—	38	4	no	—
138	C₂×C₄⋊C₄	—	38	2	no	—
139	C₂×C₄⋊C₄	—	38	4	no	—
140	C₂×C₄⋊C₄	—	38	2	no	—
141	C₂×C₄²	—	2	4	no	—
142	C₂×C₄²	D₇₆	1	2	yes	—
143	C₂×C₄²	C₂²⋊D₁₉	1	2	yes	—
144	#95	—	38	8	no	—
145	C₁₉	#11355	1	1	yes	characteristic, complemented, Sylow
146	C₂⁴	C₂×C₄×D₁₉	1	1	yes	characteristic
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	4	4	no	—
149	C₂²×C₄	—	38	4	no	—
150	C₂²×C₄	—	38	8	no	—
151	C₂²×C₄	—	76	8	no	—
152	C₂²×C₄	(C₂×D₁₉)⋊C₄	1	4	yes	—
153	C₂²×C₄	C₂×D₇₆	1	2	yes	—
154	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
155	C₂²×C₄	#64541	1	2	yes	—
156	C₂²×C₄	#64543	1	4	yes	—
157	C₂²×C₄	#64543	1	2	yes	—
158	C₂²×C₄	Q₈×D₁₉	1	2	yes	—
159	C₂²×C₄	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
160	M₁₆	—	2	8	no	—
161	M₁₆	—	4	4	no	—
162	M₁₆	—	38	8	no	—
163	M₁₆	—	76	4	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	—	38	4	no	—
168	C₂×C₈	—	76	2	no	—
169	C₂×C₈	—	76	2	no	—
170	C₄⋊C₄	—	76	4	no	—
171	C₄⋊C₄	—	76	2	no	—
172	C₄⋊C₄	—	76	2	no	—
173	C₄²	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₂³	—	2	4	no	—
176	C₂³	C₄×D₇₆	1	2	yes	—
177	C₂³	#10597323	1	2	yes	—
178	C₂³	#10597333	1	2	yes	—
179	C₂³	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	8	no	—
182	C₂×C₄	—	4	8	no	—
183	C₂×C₄	—	76	4	no	—
184	C₂×C₄	—	76	8	no	—
185	C₂×C₄	—	152	8	no	—
186	C₂×C₄	#10597320	1	1	yes	characteristic
187	C₂×C₄	#10597321	1	1	yes	characteristic
188	C₂×C₄	#10597325	1	4	yes	—
189	C₂×C₄	#10597332	1	4	yes	—
190	C₂×C₄	#10597332	1	2	yes	—
191	C₂×C₄	#10597335	1	4	yes	—
192	C₂×C₄	#10597343	1	4	yes	—
193	C₂×C₄	#10597343	1	2	yes	—
194	C₂×C₄	#10597376	1	2	yes	—
195	C₂×C₄	#10597381	1	2	yes	—
196	C₂×C₄	#10597382	1	1	yes	characteristic
197	C₂×C₄	#10597390	1	1	yes	characteristic
198	C₈	—	4	4	no	—
199	C₈	—	76	4	no	—
200	C₂²	—	2	4	no	—
201	C₂²	—	2	4	no	—
202	C₂²	—	2	4	no	—
203	C₂²	#11926405	1	1	yes	characteristic
204	C₂²	#11926429	1	4	yes	—
205	C₂²	#11926446	1	1	yes	characteristic
206	C₂²	#11926449	1	1	yes	characteristic
207	C₂²	#11926526	1	4	yes	—
208	C₄	—	4	4	no	—
209	C₄	—	152	4	no	—
210	C₄	#11926469	1	2	yes	—
211	C₄	#11926474	1	2	yes	—
212	C₄	#11926481	1	2	yes	—
213	C₄	#11926484	1	2	yes	—
214	C₂	—	2	4	no	—
215	C₂	#14561902	1	1	yes	characteristic
216	C₂	#14563716	1	2	yes	—
217	C₂	#14564484	1	2	yes	—
218	C₂	#14564627	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×(C₂×D₁₉)⋊C₄
commutator, prime radical	C₂⁴×C₁₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₉
Fitting	C₁₉×#1728	C₂
socle	C₂³×C₁₉	C₂×C₂²⋊C₄
cosocle	C₂³×C₄×C₁₉	C₂³
hypercenter	#1728	D₁₉
hypocenter	C₁₉	#11355
LCS	G ▷ C₂⁴×C₁₉ ▷ C₂×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
145. C₁₉	#11355	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11926410	2
C₂×#11926312	2
#14563329	2
C₁₉×#1728	2
#11355	19