Subgroups — #18081913

Order 4864.

394 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14574149	C₂	1	1	yes	maximal, characteristic
2	#14563799	C₂	1	1	yes	maximal, characteristic
3	#14562200	C₂	1	4	yes	maximal
4	C₁₉×#1450	C₂	1	1	yes	maximal, characteristic
5	#11927101	C₂²	1	2	yes	—
6	#11926949	C₄	1	2	yes	complemented
7	#11926946	—	2	1	no	—
8	#11926946	—	2	1	no	—
9	#11926942	C₄	1	2	yes	complemented
10	#11926582	—	2	1	no	—
11	#11926582	—	2	1	no	—
12	C₂×#10597275	—	2	1	no	—
13	C₂×#10597274	—	2	1	no	—
14	#11926410	C₂²	1	2	yes	—
15	#11926282	—	2	8	no	—
16	C₁₉×#431	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₁₉	C₂²	1	1	yes	characteristic
18	C₁₉×#322	—	2	1	no	—
19	C₈²×C₁₉	—	2	1	no	—
20	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₉	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₉	C₂³	1	1	yes	characteristic
23	#10597372	—	2	2	no	—
24	C₂×C₄⋊Dic₁₉	—	2	2	no	—
25	#10597370	—	2	4	no	—
26	#10597316	—	2	2	no	—
27	#10597315	C₂×C₄	1	4	yes	—
28	C₄×Dic₃₈	—	2	4	no	—
29	C₁₉×C₄⋊C₈	—	2	2	no	—
30	C₁₉×C₂²⋊C₈	D₄	1	4	yes	—
31	C₄×C₈×C₁₉	—	2	2	no	—
32	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₁₉	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
35	C₄×C₈×C₁₉	D₄	1	1	yes	characteristic
36	C₁₉⋊#95	—	2	4	no	—
37	#10597275	—	2	2	no	—
38	#10597275	—	2	2	no	—
39	#10597275	—	4	2	no	—
40	#10597274	—	2	2	no	—
41	#10597274	—	2	2	no	—
42	#10597274	—	4	2	no	—
43	C₂²×C₄×C₁₉	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₁₉	C₂×D₄	1	1	yes	characteristic
45	C₂²×Dic₁₉	—	4	2	no	—
46	C₂×Dic₃₈	—	2	4	no	—
47	C₂×C₈×C₁₉	—	2	4	no	—
48	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₁₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
53	C₂×C₈×C₁₉	SD₁₆	1	2	yes	—
54	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
55	C₂×C₈×C₁₉	Q₁₆	1	2	yes	—
56	C₄²×C₁₉	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₉	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₁₉	#41	1	2	yes	—
59	C₂²⋊Dic₁₉	—	4	2	no	—
60	C₄⋊Dic₁₉	—	2	4	no	—
61	C₄⋊Dic₁₉	—	2	4	no	—
62	C₄⋊Dic₁₉	—	4	2	no	—
63	C₄⋊Dic₁₉	—	4	2	no	—
64	Dic₁₉⋊C₄	—	4	4	no	—
65	C₄×Dic₁₉	—	4	2	no	—
66	#9755	—	19	1	no	maximal, Sylow
67	C₂³×C₁₉	#127	1	1	yes	characteristic
68	C₂×C₄×C₁₉	—	4	2	no	—
69	C₂×C₄×C₁₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₁₉	C₄×D₄	1	2	yes	—
71	C₂×C₄×C₁₉	#121	1	2	yes	—
72	C₂×C₄×C₁₉	#124	1	1	yes	characteristic
73	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₁₉	C₂×SD₁₆	1	1	yes	characteristic
75	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
76	C₂×C₄×C₁₉	C₂×Q₁₆	1	1	yes	characteristic
77	C₂×Dic₁₉	—	4	4	no	—
78	C₂×Dic₁₉	—	8	4	no	—
79	C₂×Dic₁₉	—	8	2	no	—
80	Dic₃₈	—	8	4	no	—
81	C₈×C₁₉	—	4	4	no	—
82	C₈×C₁₉	#103	1	2	yes	—
83	C₈×C₁₉	#103	1	2	yes	—
84	C₈×C₁₉	#103	1	2	yes	—
85	C₈×C₁₉	#103	1	2	yes	—
86	#3160	—	19	1	no	—
87	#1846	—	19	1	no	—
88	#1518	—	19	4	no	—
89	#1450	D₁₉	1	1	yes	characteristic
90	C₂²×C₁₉	—	2	2	no	—
91	C₂²×C₁₉	#390	1	1	yes	characteristic
92	C₂²×C₁₉	#492	1	1	yes	characteristic
93	C₂²×C₁₉	#494	1	1	yes	characteristic
94	C₄×C₁₉	—	2	2	no	—
95	C₄×C₁₉	—	2	2	no	—
96	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
97	C₄×C₁₉	C₂×#103	1	1	yes	characteristic
98	C₄×C₁₉	#487	1	1	yes	characteristic
99	C₄×C₁₉	#487	1	1	yes	characteristic
100	Dic₁₉	—	8	4	no	—
101	Dic₁₉	—	16	2	no	—
102	#533	—	19	2	no	—
103	#500	—	19	2	no	—
104	#499	—	38	1	no	—
105	#499	—	38	1	no	—
106	#498	—	19	2	no	—
107	#431	C₂×D₁₉	1	2	yes	—
108	#427	—	38	1	no	—
109	#427	—	38	1	no	—
110	C₂×#107	—	38	1	no	—
111	C₂×#106	—	38	1	no	—
112	C₂×C₄×C₈	C₂×D₁₉	1	1	yes	characteristic
113	#382	—	19	2	no	—
114	#328	—	38	8	no	—
115	#322	—	2	1	no	—
116	C₈²	—	2	1	no	—
117	C₂×C₁₉	—	2	1	no	—
118	C₂×C₁₉	—	2	1	no	—
119	C₂×C₁₉	#1956	1	1	yes	characteristic
120	C₂×C₁₉	#1976	1	2	yes	—
121	C₂²×C₈	D₇₆	1	1	yes	characteristic
122	C₂²×C₈	D₇₆	1	1	yes	characteristic
123	C₄⋊Q₈	—	19	4	no	—
124	#125	—	38	2	no	—
125	C₂²⋊Q₈	—	38	4	no	—
126	C₄×Q₈	—	38	4	no	—
127	#117	—	38	2	no	—
128	C₂×C₄⋊C₄	—	38	2	no	—
129	C₂×C₄²	C₂²×D₁₉	1	1	yes	characteristic
130	#107	—	38	2	no	—
131	#107	—	38	2	no	—
132	#107	—	76	2	no	—
133	#106	—	38	2	no	—
134	#106	—	38	2	no	—
135	#106	—	76	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₂²⋊C₈	D₇₆	1	4	yes	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
140	C₄×C₈	C₄×D₁₉	1	1	yes	characteristic
141	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
142	C₄×C₈	C₂²⋊D₁₉	1	1	yes	characteristic
143	#95	—	38	4	no	—
144	C₁₉	#9755	1	1	yes	characteristic, complemented, Sylow
145	C₂×Q₈	—	38	4	no	—
146	C₂²×C₄	—	76	2	no	—
147	C₂²×C₄	C₂×D₇₆	1	2	yes	—
148	C₂²×C₄	C₂×D₇₆	1	1	yes	characteristic
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	#64516	1	2	yes	—
151	C₂×C₈	#64516	1	2	yes	—
152	C₂×C₈	Dic₇₆	1	2	yes	—
153	C₂×C₈	Dic₇₆	1	2	yes	—
154	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
155	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
156	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
157	C₂×C₈	(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
158	C₄⋊C₄	—	38	4	no	—
159	C₄⋊C₄	—	38	4	no	—
160	C₄⋊C₄	—	76	4	no	—
161	C₄⋊C₄	—	76	2	no	—
162	C₄⋊C₄	—	76	2	no	—
163	C₂²⋊C₄	—	76	2	no	—
164	C₄²	—	76	2	no	—
165	C₄²	C₂×C₄×D₁₉	1	1	yes	characteristic
166	C₄²	#64541	1	2	yes	—
167	C₄²	C₂×C₂²⋊D₁₉	1	1	yes	characteristic
168	C₂³	#10597320	1	1	yes	characteristic
169	Q₈	—	152	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	76	4	no	—
172	C₂×C₄	—	152	4	no	—
173	C₂×C₄	—	152	2	no	—
174	C₂×C₄	C₄×D₇₆	1	2	yes	—
175	C₂×C₄	#10597321	1	1	yes	characteristic
176	C₂×C₄	C₂×#64516	1	1	yes	characteristic
177	C₂×C₄	C₂×#64516	1	1	yes	characteristic
178	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
179	C₂×C₄	C₂×Dic₇₆	1	1	yes	characteristic
180	C₂×C₄	C₂×(C₂×D₁₉)⋊C₄	1	1	yes	characteristic
181	C₂×C₄	#10597376	1	2	yes	—
182	C₈	—	4	4	no	—
183	C₈	#10597273	1	2	yes	—
184	C₈	#10597273	1	2	yes	—
185	C₈	#10597273	1	2	yes	—
186	C₈	#10597273	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	#11926446	1	1	yes	characteristic
189	C₂²	#11926917	1	1	yes	characteristic
190	C₂²	#11926924	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	304	2	no	—
194	C₄	—	152	4	no	—
195	C₄	C₂×#10597273	1	1	yes	characteristic
196	C₄	C₂×#10597273	1	1	yes	characteristic
197	C₄	#11926895	1	1	yes	characteristic
198	C₄	#11926895	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#14564437	1	1	yes	characteristic
202	C₂	#14564533	1	2	yes	—

	subgroup	G/H
center	C₂²	#11926446
commutator	C₄²×C₁₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₁₉
Fitting	C₁₉×#1450	C₂
socle	C₂²×C₁₉	#390
prime radical	C₂³×C₁₉	#127
cosocle	C₂×C₄²×C₁₉	C₂³
hypercenter	#1450	D₁₉
hypocenter	C₁₉	#9755
LCS	G ▷ C₄²×C₁₉ ▷ C₂²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

Complements

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

N	H ≅ G/N	type
6. #11926949	C₄	semidirect
9. #11926942	C₄	semidirect
144. C₁₉	#9755	semidirect

Maximal subgroups

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

subgroup	index
#14574149	2
#14563799	2
#14562200	2
C₁₉×#1450	2
#9755	19