Finite Group Databasebeta

Subgroups — #28355018

Order 13851.

414 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17987546	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987518	C₃	1	2	yes	maximal, complemented
3	#17987515	C₃	1	1	yes	maximal, characteristic
4	#17987511	C₃	1	2	yes	maximal, complemented
5	#17987511	C₃	1	2	yes	maximal, complemented
6	#17987510	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×#13095317	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987431	C₃	1	2	yes	maximal, complemented
9	C₃×C₁₉×#733	C₃	1	1	yes	maximal, characteristic, complemented
10	#13095333	—	3	6	no	—
11	#13095332	—	3	3	no	—
12	C₃×#10587025	C₃²	1	2	yes	—
13	C₃×#10587025	C₃²	1	2	yes	—
14	C₃×#10587025	C₃²	1	2	yes	—
15	3₋¹⁺²×C₁₉⋊C₃	—	3	3	no	—
16	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	2	yes	—
17	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
18	#13095321	—	3	6	no	—
19	#13095320	—	3	3	no	—
20	#13095318	—	3	6	no	—
21	#13095318	—	3	6	no	—
22	#13095317	—	3	3	no	—
23	#13095317	—	3	3	no	—
24	#13095303	—	3	6	no	—
25	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
26	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
27	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
28	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
29	C₁₉×#733	—	3	3	no	—
30	#10624751	—	19	1	no	maximal, Sylow
31	C₃³×C₁₉	C₃³	1	1	yes	characteristic
32	C₃²×C₁₉⋊C₃	—	3	1	no	—
33	C₃²×C₁₉⋊C₃	—	3	2	no	—
34	C₁₉×3₋¹⁺²	—	3	2	no	—
35	C₁₉×3₋¹⁺²	—	3	1	no	—
36	C₁₉×3₋¹⁺²	—	3	1	no	—
37	C₁₉×3₋¹⁺²	—	3	2	no	—
38	C₁₉×3₋¹⁺²	—	3	2	no	—
39	C₁₉×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
40	C₃×C₉×C₁₉	—	3	1	no	—
41	C₃×C₉×C₁₉	—	3	1	no	—
42	C₃×C₉×C₁₉	—	3	2	no	—
43	C₃×C₉×C₁₉	—	3	1	no	—
44	C₃×C₉×C₁₉	—	3	1	no	—
45	C₃×C₉×C₁₉	—	3	1	no	—
46	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
47	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
48	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
49	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
50	C₃²⋊(C₁₉⋊C₃)	H(3,𝔽₃)	1	3	yes	—
51	#10587025	—	3	2	no	—
52	#10587025	—	3	2	no	—
53	#10587025	—	3	2	no	—
54	#10587025	—	3	2	no	—
55	#10587025	—	3	2	no	—
56	#10587025	—	3	2	no	—
57	#10587025	—	3	2	no	—
58	#10587025	—	3	2	no	—
59	#10587025	—	3	2	no	—
60	C₃×#4294	—	3	2	no	—
61	C₃×#4294	—	3	2	no	—
62	C₃×#4294	H(3,𝔽₃)	1	6	yes	—
63	#10587023	—	3	9	no	—
64	#10587022	—	3	9	no	—
65	C₉×C₁₉⋊C₃	—	3	9	no	—
66	3₋¹⁺⁴	—	19	1	no	—
67	#6912	—	19	2	no	—
68	#6911	—	19	1	no	—
69	#6910	—	19	1	no	—
70	#6910	—	19	2	no	—
71	#6910	—	19	2	no	—
72	C₃×#733	C₁₉⋊C₃	1	1	yes	characteristic, complemented
73	C₃×#731	—	19	1	no	—
74	#6893	—	19	2	no	—
75	C₃²×C₁₉	—	9	1	no	—
76	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	2	yes	—
77	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
78	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
79	C₃×C₁₉⋊C₃	—	3	3	no	—
80	C₃×C₁₉⋊C₃	—	9	6	no	—
81	C₃×C₁₉⋊C₃	—	9	2	no	—
82	C₃×C₁₉⋊C₃	—	27	1	no	—
83	C₉×C₁₉	—	3	3	no	—
84	C₉×C₁₉	—	9	1	no	—
85	C₉×C₁₉	—	9	1	no	—
86	C₉×C₁₉	—	9	1	no	—
87	#4294	—	3	6	no	—
88	#4294	—	9	2	no	—
89	#4294	—	9	2	no	—
90	#737	—	57	3	no	—
91	#737	—	57	6	no	—
92	C₃×3₋¹⁺²	—	19	2	no	—
93	C₃×3₋¹⁺²	—	19	2	no	—
94	C₃×3₋¹⁺²	—	19	2	no	—
95	C₃×3₋¹⁺²	—	57	3	no	—
96	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
97	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
98	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
99	C₃×H(3,𝔽₃)	—	19	1	no	—
100	C₃×H(3,𝔽₃)	—	19	2	no	—
101	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
102	#733	—	3	3	no	—
103	#733	—	57	6	no	—
104	#732	—	57	3	no	—
105	#731	—	57	3	no	—
106	#731	—	57	3	no	—
107	#731	—	57	6	no	—
108	#731	—	57	6	no	—
109	C₃²⋊C₉	—	57	6	no	—
110	C₃×C₁₉	—	3	2	no	—
111	C₃×C₁₉	—	3	1	no	—
112	C₃×C₁₉	—	3	1	no	—
113	C₃×C₁₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
114	C₁₉⋊C₃	—	9	3	no	—
115	C₁₉⋊C₃	—	27	2	no	—
116	C₃³	—	57	1	no	—
117	C₃³	—	57	2	no	—
118	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
119	3₋¹⁺²	—	3	2	no	—
120	3₋¹⁺²	—	3	2	no	—
121	3₋¹⁺²	—	3	2	no	—
122	3₋¹⁺²	—	3	1	no	—
123	3₋¹⁺²	—	3	1	no	—
124	3₋¹⁺²	—	57	9	no	—
125	3₋¹⁺²	—	57	9	no	—
126	3₋¹⁺²	—	57	2	no	—
127	3₋¹⁺²	—	57	2	no	—
128	3₋¹⁺²	—	57	2	no	—
129	3₋¹⁺²	—	57	2	no	—
130	3₋¹⁺²	—	57	2	no	—
131	3₋¹⁺²	—	57	2	no	—
132	3₋¹⁺²	—	57	2	no	—
133	3₋¹⁺²	—	57	2	no	—
134	3₋¹⁺²	—	57	2	no	—
135	3₋¹⁺²	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
136	H(3,𝔽₃)	—	19	3	no	—
137	H(3,𝔽₃)	—	57	2	no	—
138	H(3,𝔽₃)	—	57	2	no	—
139	H(3,𝔽₃)	—	57	2	no	—
140	H(3,𝔽₃)	—	57	2	no	—
141	C₃×C₉	—	3	1	no	—
142	C₃×C₉	—	3	2	no	—
143	C₃×C₉	—	3	1	no	—
144	C₃×C₉	—	3	1	no	—
145	C₃×C₉	—	3	1	no	—
146	C₃×C₉	—	3	1	no	—
147	C₃×C₉	—	19	6	no	—
148	C₃×C₉	—	57	9	no	—
149	C₃×C₉	—	57	2	no	—
150	C₃×C₉	—	57	2	no	—
151	C₁₉	#10624751	1	1	yes	characteristic, complemented, Sylow
152	C₃²	—	513	1	no	—
153	C₃²	—	9	1	no	—
154	C₃²	—	57	3	no	—
155	C₃²	—	171	6	no	—
156	C₃²	—	171	2	no	—
157	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
158	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	2	yes	—
159	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
160	C₉	—	3	3	no	—
161	C₉	—	9	1	no	—
162	C₉	—	9	1	no	—
163	C₉	—	9	1	no	—
164	C₉	—	57	6	no	—
165	C₉	—	171	2	no	—
166	C₉	—	171	2	no	—
167	C₃	—	513	2	no	—
168	C₃	—	3	1	no	—
169	C₃	—	3	2	no	—
170	C₃	—	3	1	no	—
171	C₃	—	171	3	no	—
172	C₃	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	(C₃×C₁₉⋊C₃)⋊H(3,𝔽₃)
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₃×C₁₉×#733	C₃
socle	C₃×C₁₉	C₃²⋊H(3,𝔽₃)
prime radical	#17987515	C₃
hypercenter	C₃×#733	C₁₉⋊C₃
hypocenter	C₁₉	#10624751
LCS	G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#733

Complements

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

N	H ≅ G/N	type
1. #17987546	C₃	semidirect
2. #17987518	C₃	semidirect
4. #17987511	C₃	semidirect
5. #17987511	C₃	semidirect
6. #17987510	C₃	semidirect
7. C₃×#13095317	C₃	semidirect
8. #17987431	C₃	semidirect
9. C₃×C₁₉×#733	C₃	semidirect
72. C₃×#733	C₁₉⋊C₃	semidirect
151. C₁₉	#10624751	semidirect

Maximal subgroups

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

subgroup	index
#17987546	3
#17987518	3
#17987515	3
#17987511	3
#17987510	3
C₃×#13095317	3
#17987431	3
C₃×C₁₉×#733	3
#10624751	19