Finite Group Databasebeta

Subgroups — #28355053

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	C₃×#13095331	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987519	C₃	1	2	yes	maximal, complemented
3	#17987514	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987511	C₃	1	2	yes	maximal, complemented
5	#17987506	C₃	1	2	yes	maximal, complemented
6	#17987504	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×#13095318	C₃	1	2	yes	maximal, complemented
8	C₃×#13095318	C₃	1	1	yes	maximal, characteristic, complemented
9	C₁₉×#6910	C₃	1	1	yes	maximal, characteristic, complemented
10	#13095331	—	3	3	no	—
11	#13095331	—	3	6	no	—
12	C₃×#10587025	—	3	2	no	—
13	C₃×#10587025	C₃²	1	2	yes	—
14	C₃×#10587025	C₃²	1	1	yes	characteristic
15	C₃×#10587025	C₃²	1	2	yes	—
16	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
17	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	2	yes	—
18	C₃²×#4294	—	3	1	no	—
19	C₃²×#4294	C₃²	1	1	yes	characteristic
20	#13095322	—	3	6	no	—
21	#13095319	—	3	3	no	—
22	#13095318	—	3	6	no	—
23	#13095318	—	3	2	no	—
24	#13095318	—	3	1	no	—
25	#13095318	—	3	4	no	—
26	#13095318	—	3	2	no	—
27	#13095315	—	3	6	no	—
28	#13095313	—	3	3	no	—
29	C₃×C₁₉×3₋¹⁺²	C₃²	1	2	yes	—
30	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
31	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
32	C₁₉×#731	—	3	3	no	—
33	#10624756	—	19	1	no	maximal, Sylow
34	C₃³×C₁₉	—	3	1	no	—
35	C₃³×C₁₉	C₃³	1	1	yes	characteristic
36	C₃²×C₁₉⋊C₃	—	3	1	no	—
37	C₃²×C₁₉⋊C₃	—	3	2	no	—
38	C₁₉×3₋¹⁺²	—	3	2	no	—
39	C₁₉×3₋¹⁺²	—	3	2	no	—
40	C₁₉×3₋¹⁺²	—	3	2	no	—
41	C₁₉×3₋¹⁺²	—	3	1	no	—
42	C₁₉×3₋¹⁺²	—	3	2	no	—
43	C₁₉×H(3,𝔽₃)	—	3	2	no	—
44	C₁₉×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
45	C₃×C₉×C₁₉	—	3	1	no	—
46	C₃×C₉×C₁₉	—	3	2	no	—
47	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
48	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
49	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
50	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
51	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
52	#10587025	—	3	18	no	—
53	#10587025	—	3	2	no	—
54	#10587025	—	3	2	no	—
55	#10587025	—	3	2	no	—
56	#10587025	—	3	2	no	—
57	#10587025	—	3	1	no	—
58	#10587025	—	3	2	no	—
59	#10587025	—	3	2	no	—
60	#10587025	H(3,𝔽₃)	1	6	yes	—
61	C₃×#4294	—	3	9	no	—
62	C₃×#4294	—	3	2	no	—
63	C₃×#4294	—	3	2	no	—
64	C₃×#4294	—	3	1	no	—
65	C₃×#4294	—	3	2	no	—
66	C₃×#4294	—	3	1	no	—
67	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
68	C₃×#4294	H(3,𝔽₃)	1	2	yes	—
69	C₃×#737	—	19	1	no	—
70	#6912	—	19	2	no	—
71	#6911	—	19	1	no	—
72	#6910	—	19	2	no	—
73	#6910	C₁₉⋊C₃	1	1	yes	characteristic, complemented
74	#6909	—	19	2	no	—
75	#6909	—	19	1	no	—
76	C₃×#731	—	19	1	no	—
77	C₃×#731	—	19	2	no	—
78	C₃²×C₁₉	—	3	3	no	—
79	C₃²×C₁₉	—	9	1	no	—
80	C₃²×C₁₉	—	9	3	no	—
81	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
82	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	2	yes	—
83	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
84	C₃×C₁₉⋊C₃	—	9	1	no	—
85	C₃×C₁₉⋊C₃	—	9	2	no	—
86	C₃×C₁₉⋊C₃	—	27	1	no	—
87	C₃×C₁₉⋊C₃	—	27	2	no	—
88	C₉×C₁₉	—	9	1	no	—
89	C₉×C₁₉	—	9	2	no	—
90	#4294	—	3	1	no	—
91	#4294	—	3	6	no	—
92	#4294	—	3	2	no	—
93	#4294	—	9	2	no	—
94	#4294	—	9	1	no	—
95	#737	—	57	6	no	—
96	#737	—	57	3	no	—
97	C₃×3₋¹⁺²	—	19	2	no	—
98	C₃×3₋¹⁺²	—	19	1	no	—
99	C₃×3₋¹⁺²	—	19	2	no	—
100	C₃×3₋¹⁺²	—	57	2	no	—
101	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	2	yes	—
102	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
103	C₃×H(3,𝔽₃)	—	19	1	no	—
104	C₃×H(3,𝔽₃)	—	19	2	no	—
105	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
106	C₃²×C₉	—	19	1	no	—
107	C₃²×C₉	—	57	1	no	—
108	#733	—	57	6	no	—
109	#732	—	57	3	no	—
110	#731	—	3	3	no	—
111	#731	—	57	1	no	—
112	#731	—	57	2	no	—
113	#731	—	57	2	no	—
114	#731	—	57	4	no	—
115	#731	—	57	6	no	—
116	AΣL(1,𝔽₂₇)	—	57	6	no	—
117	AΣL(1,𝔽₂₇)	—	57	3	no	—
118	C₃×C₁₉	—	3	2	no	—
119	C₃×C₁₉	—	3	1	no	—
120	C₃×C₁₉	—	3	1	no	—
121	C₃×C₁₉	—	9	3	no	—
122	C₃×C₁₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
123	C₁₉⋊C₃	—	27	1	no	—
124	C₁₉⋊C₃	—	27	2	no	—
125	C₃³	—	3	1	no	—
126	C₃³	—	57	2	no	—
127	C₃³	—	57	1	no	—
128	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
129	3₋¹⁺²	—	3	2	no	—
130	3₋¹⁺²	—	3	2	no	—
131	3₋¹⁺²	—	3	2	no	—
132	3₋¹⁺²	—	3	1	no	—
133	3₋¹⁺²	—	3	2	no	—
134	3₋¹⁺²	—	19	6	no	—
135	3₋¹⁺²	—	57	18	no	—
136	3₋¹⁺²	—	57	2	no	—
137	3₋¹⁺²	—	57	2	no	—
138	3₋¹⁺²	—	57	1	no	—
139	3₋¹⁺²	—	57	2	no	—
140	3₋¹⁺²	—	57	2	no	—
141	3₋¹⁺²	—	57	2	no	—
142	3₋¹⁺²	—	57	2	no	—
143	H(3,𝔽₃)	—	3	2	no	—
144	H(3,𝔽₃)	—	57	2	no	—
145	H(3,𝔽₃)	—	57	2	no	—
146	H(3,𝔽₃)	—	57	1	no	—
147	H(3,𝔽₃)	—	57	2	no	—
148	H(3,𝔽₃)	—	57	2	no	—
149	H(3,𝔽₃)	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
150	C₃×C₉	—	3	2	no	—
151	C₃×C₉	—	3	1	no	—
152	C₃×C₉	—	19	1	no	—
153	C₃×C₉	—	19	2	no	—
154	C₃×C₉	—	57	9	no	—
155	C₃×C₉	—	57	2	no	—
156	C₃×C₉	—	57	2	no	—
157	C₃×C₉	—	57	1	no	—
158	C₃×C₉	—	57	2	no	—
159	C₃×C₉	—	57	1	no	—
160	C₁₉	#10624756	1	1	yes	characteristic, complemented, Sylow
161	C₃²	—	513	2	no	—
162	C₃²	—	513	1	no	—
163	C₃²	—	3	3	no	—
164	C₃²	—	9	1	no	—
165	C₃²	—	9	3	no	—
166	C₃²	—	171	1	no	—
167	C₃²	—	171	2	no	—
168	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	2	yes	—
169	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
170	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
171	C₉	—	9	2	no	—
172	C₉	—	9	1	no	—
173	C₉	—	57	1	no	—
174	C₉	—	57	2	no	—
175	C₉	—	57	6	no	—
176	C₉	—	171	2	no	—
177	C₉	—	171	1	no	—
178	C₃	—	513	2	no	—
179	C₃	—	513	1	no	—
180	C₃	—	3	2	no	—
181	C₃	—	3	1	no	—
182	C₃	—	3	1	no	—
183	C₃	—	9	3	no	—
184	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	C₁₉×#6910	C₃
socle	C₃×C₁₉	C₃²⋊H(3,𝔽₃)
hypercenter	#6910	C₁₉⋊C₃
hypocenter	C₁₉	#10624756
LCS	G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6910

Complements

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

N	H ≅ G/N	type
1. C₃×#13095331	C₃	semidirect
2. #17987519	C₃	semidirect
3. #17987514	C₃	semidirect
4. #17987511	C₃	semidirect
5. #17987506	C₃	semidirect
6. #17987504	C₃	semidirect
7. C₃×#13095318	C₃	semidirect
8. C₃×#13095318	C₃	semidirect
9. C₁₉×#6910	C₃	semidirect
73. #6910	C₁₉⋊C₃	semidirect
160. C₁₉	#10624756	semidirect

Maximal subgroups

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

subgroup	index
C₃×#13095331	3
#17987519	3
#17987514	3
#17987511	3
#17987506	3
#17987504	3
C₃×#13095318	3
C₁₉×#6910	3
#10624756	19