Finite Group Databasebeta

Subgroups — #28356901

Order 13851.

362 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17987589	C₃	1	3	yes	maximal, complemented
2	#17987560	C₃	1	3	yes	maximal, complemented
3	#17987559	C₃	1	3	yes	maximal, complemented
4	#17987558	C₃	1	3	yes	maximal
5	C₃×3₋¹⁺²×C₁₉⋊C₃	C₃	1	1	yes	maximal, characteristic
6	#13095339	—	3	3	no	—
7	#13095338	—	3	3	no	—
8	C₃²⋊#4296	C₃²	1	9	yes	—
9	C₉×#4296	—	3	3	no	—
10	C₃³×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
11	C₃×#10587025	C₉	1	1	yes	characteristic, complemented
12	C₃×#10587025	C₉	1	1	yes	characteristic, complemented
13	C₃×#10587022	C₉	1	1	yes	characteristic, complemented
14	C₃×#10587022	C₉	1	1	yes	characteristic, complemented
15	C₃×#10587022	C₉	1	1	yes	characteristic, complemented
16	C₃×#10587023	C₉	1	1	yes	characteristic, complemented
17	C₃×#10587023	C₉	1	1	yes	characteristic, complemented
18	C₃×#10587023	C₉	1	1	yes	characteristic, complemented
19	3₋¹⁺²×C₁₉⋊C₃	—	3	3	no	—
20	3₋¹⁺²×C₁₉⋊C₃	—	3	6	no	—
21	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
22	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
23	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
24	C₃×C₁₉×3₋¹⁺²	C₉	1	1	yes	characteristic, complemented
25	#10624692	—	19	1	no	maximal, Sylow
26	C₃³×C₁₉	C₃×C₉	1	1	yes	characteristic
27	C₃²×C₁₉⋊C₃	—	3	1	no	—
28	C₃²×C₁₉⋊C₃	—	3	2	no	—
29	C₃²×C₁₉⋊C₃	—	9	3	no	—
30	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
31	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
32	C₃²×C₁₉⋊C₃	C₃³	1	1	yes	characteristic
33	C₃×#4296	—	3	9	no	—
34	C₁₉×3₋¹⁺²	—	3	3	no	—
35	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
36	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
37	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
38	#10587025	—	3	3	no	—
39	#10587025	—	3	3	no	—
40	C₃×#4294	C₃×C₉	1	1	yes	characteristic
41	C₃×#4294	C₃×C₉	1	1	yes	characteristic
42	C₃×#4294	C₃×C₉	1	1	yes	characteristic
43	C₃×#4294	C₃×C₉	1	1	yes	characteristic
44	C₃×#4294	C₃×C₉	1	1	yes	characteristic
45	C₃×#4294	C₃×C₉	1	1	yes	characteristic
46	#10587023	—	3	3	no	—
47	#10587023	—	3	3	no	—
48	#10587023	—	3	3	no	—
49	#10587022	—	3	3	no	—
50	#10587022	—	3	3	no	—
51	#10587022	—	3	3	no	—
52	C₉×C₁₉⋊C₃	—	3	1	no	—
53	C₉×C₁₉⋊C₃	—	3	1	no	—
54	C₉×C₁₉⋊C₃	—	3	2	no	—
55	C₉×C₁₉⋊C₃	—	3	2	no	—
56	C₉×C₁₉⋊C₃	H(3,𝔽₃)	1	3	yes	—
57	C₉×C₁₉⋊C₃	3₋¹⁺²	1	6	yes	—
58	C₃²×3₋¹⁺²	—	19	1	no	—
59	#6887	—	19	3	no	—
60	#6868	—	19	3	no	—
61	#6867	—	19	3	no	—
62	#6866	—	19	3	no	—
63	C₃²×C₁₉	—	3	1	no	—
64	C₃²×C₁₉	—	9	1	no	—
65	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
66	C₃×C₁₉⋊C₃	—	3	3	no	—
67	C₃×C₁₉⋊C₃	—	3	1	no	—
68	C₃×C₁₉⋊C₃	—	3	1	no	—
69	C₃×C₁₉⋊C₃	—	9	6	no	—
70	C₃×C₁₉⋊C₃	—	9	3	no	—
71	C₃×C₁₉⋊C₃	—	9	1	no	—
72	C₃×C₁₉⋊C₃	—	9	1	no	—
73	C₃×C₁₉⋊C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
74	C₃×C₁₉⋊C₃	C₃×3₋¹⁺²	1	2	yes	—
75	#4296	—	9	9	no	—
76	C₉×C₁₉	—	3	1	no	—
77	C₉×C₁₉	—	3	1	no	—
78	C₉×C₁₉	C₃²⋊C₉	1	3	yes	—
79	#4294	—	3	1	no	—
80	#4294	—	3	1	no	—
81	#4294	—	3	1	no	—
82	#4294	—	3	1	no	—
83	#4294	C₃²⋊C₉	1	3	yes	—
84	#4294	C₃²⋊C₉	1	3	yes	—
85	C₃⁴	—	19	1	no	—
86	C₃×3₋¹⁺²	—	19	1	no	—
87	C₃×3₋¹⁺²	—	19	1	no	—
88	C₃×3₋¹⁺²	—	19	1	no	—
89	C₃×3₋¹⁺²	—	19	1	no	—
90	C₃×3₋¹⁺²	—	19	1	no	—
91	C₃×3₋¹⁺²	—	19	1	no	—
92	C₃×3₋¹⁺²	—	19	1	no	—
93	C₃×3₋¹⁺²	—	19	1	no	—
94	C₃×3₋¹⁺²	—	57	3	no	—
95	C₃×3₋¹⁺²	—	57	6	no	—
96	C₃×3₋¹⁺²	#4296	1	1	yes	characteristic, complemented
97	C₃²×C₉	—	19	1	no	—
98	C₃²×C₉	—	19	1	no	—
99	C₃²×C₉	—	19	1	no	—
100	C₉⋊C₉	—	57	3	no	—
101	C₉⋊C₉	—	57	3	no	—
102	C₃²⋊C₉	—	19	9	no	—
103	C₉²	—	57	3	no	—
104	C₃×C₁₉	—	3	1	no	—
105	C₃×C₁₉	—	9	1	no	—
106	C₃×C₁₉	C₃×C₃²⋊C₉	1	1	yes	characteristic
107	C₁₉⋊C₃	—	3	2	no	—
108	C₁₉⋊C₃	—	9	1	no	—
109	C₁₉⋊C₃	—	9	1	no	—
110	C₁₉⋊C₃	#6913	1	3	yes	—
111	C₃³	—	19	1	no	—
112	C₃³	—	19	1	no	—
113	C₃³	—	19	1	no	—
114	C₃³	—	57	1	no	—
115	C₃³	—	57	2	no	—
116	C₃³	—	171	3	no	—
117	C₃³	C₃×#4296	1	1	yes	characteristic
118	3₋¹⁺²	—	3	3	no	—
119	3₋¹⁺²	—	57	3	no	—
120	3₋¹⁺²	—	57	3	no	—
121	3₋¹⁺²	—	57	3	no	—
122	3₋¹⁺²	—	57	3	no	—
123	3₋¹⁺²	—	57	3	no	—
124	3₋¹⁺²	—	57	3	no	—
125	3₋¹⁺²	—	57	3	no	—
126	3₋¹⁺²	—	57	3	no	—
127	C₃×C₉	—	19	1	no	—
128	C₃×C₉	—	19	1	no	—
129	C₃×C₉	—	19	3	no	—
130	C₃×C₉	—	19	6	no	—
131	C₃×C₉	—	19	1	no	—
132	C₃×C₉	—	19	1	no	—
133	C₃×C₉	—	19	1	no	—
134	C₃×C₉	—	19	1	no	—
135	C₃×C₉	—	57	1	no	—
136	C₃×C₉	—	57	1	no	—
137	C₃×C₉	—	57	2	no	—
138	C₃×C₉	—	57	2	no	—
139	C₃×C₉	—	57	9	no	—
140	C₃×C₉	C₃×#4296	1	1	yes	characteristic
141	C₃×C₉	C₃×#4296	1	1	yes	characteristic
142	C₃×C₉	C₃×#4296	1	1	yes	characteristic
143	C₁₉	#10624692	1	1	yes	characteristic, complemented, Sylow
144	C₃²	—	3	1	no	—
145	C₃²	—	9	1	no	—
146	C₃²	—	19	1	no	—
147	C₃²	—	19	2	no	—
148	C₃²	—	57	3	no	—
149	C₃²	—	57	1	no	—
150	C₃²	—	57	1	no	—
151	C₃²	—	171	1	no	—
152	C₃²	—	171	1	no	—
153	C₃²	—	171	3	no	—
154	C₃²	—	171	6	no	—
155	C₃²	C₃²×#4296	1	1	yes	characteristic
156	C₉	—	3	1	no	—
157	C₉	—	3	1	no	—
158	C₉	—	19	3	no	—
159	C₉	—	19	3	no	—
160	C₉	—	57	1	no	—
161	C₉	—	57	1	no	—
162	C₉	—	57	1	no	—
163	C₉	—	57	1	no	—
164	C₉	—	171	9	no	—
165	C₉	C₃²⋊#4296	1	3	yes	—
166	C₃	—	3	1	no	—
167	C₃	—	9	1	no	—
168	C₃	—	19	3	no	—
169	C₃	—	57	2	no	—
170	C₃	—	171	1	no	—
171	C₃	—	171	1	no	—
172	C₃	C₃×C₃²⋊#4296	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊#4296
commutator	C₃²×C₁₉	C₃²×C₉
Frattini	C₃²	C₃²×#4296
Fitting	C₃×C₁₉×3₋¹⁺²	C₉
socle	C₃×C₁₉	C₃×C₃²⋊C₉
prime radical	C₃³×C₁₉⋊C₃	C₃²
cosocle	C₃²×C₁₉⋊C₃	C₃³
hypercenter	C₃×3₋¹⁺²	#4296
hypocenter	C₁₉	#10624692
LCS	G ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. #17987589	C₃	semidirect
2. #17987560	C₃	semidirect
3. #17987559	C₃	semidirect
11. C₃×#10587025	C₉	semidirect
12. C₃×#10587025	C₉	semidirect
13. C₃×#10587022	C₉	semidirect
14. C₃×#10587022	C₉	semidirect
15. C₃×#10587022	C₉	semidirect
16. C₃×#10587023	C₉	semidirect
17. C₃×#10587023	C₉	semidirect
18. C₃×#10587023	C₉	semidirect
24. C₃×C₁₉×3₋¹⁺²	C₉	semidirect
96. C₃×3₋¹⁺²	#4296	semidirect
143. C₁₉	#10624692	semidirect

Maximal subgroups

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

subgroup	index
#17987589	3
#17987560	3
#17987559	3
#17987558	3
C₃×3₋¹⁺²×C₁₉⋊C₃	3
#10624692	19