Finite Group Databasebeta

Subgroups — #28355839

Order 13851.

188 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 46 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	#17987528	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987528	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987519	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987390	C₃	1	1	yes	maximal, characteristic, complemented
5	#17987390	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987385	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987385	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987385	C₃	1	1	yes	maximal, characteristic, complemented
9	#17987380	C₃	1	1	yes	maximal, characteristic, complemented
10	#17987377	C₃	1	1	yes	maximal, characteristic, complemented
11	#17987370	C₃	1	1	yes	maximal, characteristic, complemented
12	#17987369	C₃	1	1	yes	maximal, characteristic, complemented
13	C₁₉×#6900	C₃	1	1	yes	maximal, characteristic, complemented
14	#13095332	—	3	1	no	—
15	#13095332	—	3	1	no	—
16	C₃×#10587025	—	3	1	no	—
17	#13095322	—	3	1	no	—
18	#13095322	—	3	1	no	—
19	#13095322	C₃²	1	1	yes	characteristic
20	#13095322	C₃²	1	1	yes	characteristic
21	#13095322	C₃²	1	1	yes	characteristic
22	#13095322	C₃²	1	1	yes	characteristic
23	#13095322	C₃²	1	1	yes	characteristic
24	#13095322	C₃²	1	1	yes	characteristic
25	#13095322	C₃²	1	1	yes	characteristic
26	#13095320	C₃²	1	1	yes	characteristic
27	#13095320	C₃²	1	1	yes	characteristic
28	#13095317	—	3	1	no	—
29	#13095317	—	3	1	no	—
30	#13095314	—	3	1	no	—
31	#13095314	—	3	1	no	—
32	C₃×C₁₉×3₋¹⁺²	—	3	1	no	—
33	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
34	C₁₉×C₉⋊C₉	—	3	1	no	—
35	C₁₉×C₉⋊C₉	—	3	1	no	—
36	C₁₉×C₉⋊C₉	C₃²	1	1	yes	characteristic
37	C₁₉×C₉⋊C₉	C₃²	1	1	yes	characteristic
38	C₉²×C₁₉	C₃²	1	1	yes	characteristic
39	#10624852	—	19	1	no	maximal, Sylow
40	C₃³×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
41	C₁₉×3₋¹⁺²	—	9	1	no	—
42	C₁₉×3₋¹⁺²	—	9	3	no	—
43	C₃×C₉×C₁₉	—	3	1	no	—
44	C₃×C₉×C₁₉	—	3	1	no	—
45	C₃×C₉×C₁₉	—	3	1	no	—
46	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
47	C₃×C₉×C₁₉	H(3,𝔽₃)	1	1	yes	characteristic
48	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
49	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
50	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
51	#10587025	—	3	1	no	—
52	#10587025	—	3	1	no	—
53	#10587025	—	3	1	no	—
54	#10587025	—	3	1	no	—
55	#10587025	—	3	1	no	—
56	#10587025	—	3	1	no	—
57	#10587025	—	3	1	no	—
58	#10587025	—	9	2	no	—
59	C₃×#4294	—	9	1	no	—
60	#10587023	—	9	1	no	—
61	#10587023	—	9	1	no	—
62	#10587022	—	9	1	no	—
63	#10587022	—	9	1	no	—
64	C₉×C₁₉⋊C₃	—	9	1	no	—
65	C₉×C₁₉⋊C₃	—	9	1	no	—
66	#6913	—	19	1	no	—
67	#6913	—	19	1	no	—
68	#6912	—	19	1	no	—
69	#6900	C₁₉⋊C₃	1	1	yes	characteristic, complemented
70	#6883	—	19	1	no	—
71	#6883	—	19	1	no	—
72	#6882	—	19	1	no	—
73	#6882	—	19	1	no	—
74	#6882	—	19	1	no	—
75	#6881	—	19	1	no	—
76	#6880	—	19	1	no	—
77	#6878	—	19	1	no	—
78	#6878	—	19	1	no	—
79	C₃²×C₁₉	—	3	1	no	—
80	C₃²×C₁₉	—	9	1	no	—
81	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
82	C₃×C₁₉⋊C₃	—	9	1	no	—
83	C₃×C₁₉⋊C₃	—	9	1	no	—
84	C₉×C₁₉	—	3	3	no	—
85	C₉×C₁₉	—	3	1	no	—
86	C₉×C₁₉	—	3	1	no	—
87	C₉×C₁₉	—	3	1	no	—
88	C₉×C₁₉	—	9	1	no	—
89	C₉×C₁₉	—	9	1	no	—
90	#4294	—	9	1	no	—
91	#4294	—	9	1	no	—
92	#4294	—	9	1	no	—
93	#4294	—	9	1	no	—
94	#4294	—	9	1	no	—
95	#4294	—	9	1	no	—
96	#4294	—	9	1	no	—
97	#737	—	57	1	no	—
98	#737	—	57	1	no	—
99	C₃×3₋¹⁺²	—	3	1	no	—
100	C₃×3₋¹⁺²	—	57	1	no	—
101	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
102	#733	—	19	1	no	—
103	#733	—	19	1	no	—
104	#733	—	19	1	no	—
105	#733	—	19	1	no	—
106	#733	—	19	1	no	—
107	#733	—	19	1	no	—
108	#733	—	19	1	no	—
109	#733	—	57	1	no	—
110	#733	—	57	1	no	—
111	#732	—	19	1	no	—
112	#732	—	19	1	no	—
113	#731	—	57	1	no	—
114	#731	—	57	1	no	—
115	AΣL(1,𝔽₂₇)	—	57	1	no	—
116	AΣL(1,𝔽₂₇)	—	57	1	no	—
117	C₉⋊C₉	—	3	1	no	—
118	C₉⋊C₉	—	3	1	no	—
119	C₉⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
120	C₉⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
121	C₉²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
122	C₃×C₁₉	—	3	1	no	—
123	C₃×C₁₉	—	9	1	no	—
124	C₃×C₁₉	#6911	1	1	yes	characteristic
125	C₁₉⋊C₃	—	27	1	no	—
126	C₁₉⋊C₃	—	27	1	no	—
127	C₃³	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
128	3₋¹⁺²	—	9	1	no	—
129	3₋¹⁺²	—	9	3	no	—
130	3₋¹⁺²	—	57	1	no	—
131	3₋¹⁺²	—	57	1	no	—
132	3₋¹⁺²	—	57	1	no	—
133	3₋¹⁺²	—	57	1	no	—
134	3₋¹⁺²	—	57	1	no	—
135	3₋¹⁺²	—	57	1	no	—
136	3₋¹⁺²	—	57	1	no	—
137	3₋¹⁺²	—	171	1	no	—
138	3₋¹⁺²	—	171	1	no	—
139	3₋¹⁺²	—	171	1	no	—
140	3₋¹⁺²	—	171	1	no	—
141	3₋¹⁺²	—	171	2	no	—
142	H(3,𝔽₃)	—	57	1	no	—
143	H(3,𝔽₃)	—	57	1	no	—
144	C₃×C₉	—	3	1	no	—
145	C₃×C₉	—	3	1	no	—
146	C₃×C₉	—	3	1	no	—
147	C₃×C₉	—	171	1	no	—
148	C₃×C₉	—	171	1	no	—
149	C₃×C₉	—	171	1	no	—
150	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
151	C₃×C₉	C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
152	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
153	C₁₉	#10624852	1	1	yes	characteristic, complemented, Sylow
154	C₃²	—	3	1	no	—
155	C₃²	—	9	1	no	—
156	C₃²	—	171	1	no	—
157	C₃²	—	171	1	no	—
158	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
159	C₉	—	3	3	no	—
160	C₉	—	3	1	no	—
161	C₉	—	3	1	no	—
162	C₉	—	3	1	no	—
163	C₉	—	9	1	no	—
164	C₉	—	9	1	no	—
165	C₉	—	171	1	no	—
166	C₉	—	171	1	no	—
167	C₉	—	171	1	no	—
168	C₉	—	171	1	no	—
169	C₉	—	171	1	no	—
170	C₉	—	171	1	no	—
171	C₉	—	171	1	no	—
172	C₃	—	513	1	no	—
173	C₃	—	513	1	no	—
174	C₃	—	3	1	no	—
175	C₃	—	9	1	no	—
176	C₃	#17987515	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#17987515
commutator, cosocle	C₃×C₉×C₁₉	C₃³
Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6900	C₃
socle	C₃×C₁₉	#6911
hypercenter	#6900	C₁₉⋊C₃
hypocenter	C₁₉	#10624852
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6900

Complements

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

N	H ≅ G/N	type
1. #17987528	C₃	semidirect
2. #17987528	C₃	semidirect
3. #17987519	C₃	semidirect
4. #17987390	C₃	semidirect
5. #17987390	C₃	semidirect
6. #17987385	C₃	semidirect
7. #17987385	C₃	semidirect
8. #17987385	C₃	semidirect
9. #17987380	C₃	semidirect
10. #17987377	C₃	semidirect
11. #17987370	C₃	semidirect
12. #17987369	C₃	semidirect
13. C₁₉×#6900	C₃	semidirect
69. #6900	C₁₉⋊C₃	semidirect
153. C₁₉	#10624852	semidirect

Maximal subgroups

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

subgroup	index
#17987528	3
#17987519	3
#17987390	3
#17987385	3
#17987380	3
#17987377	3
#17987370	3
#17987369	3
C₁₉×#6900	3
#10624852	19