Finite Group Databasebeta

Subgroups — #28355035

Order 13851.

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

normal subgroups only

This group has 426 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	#17987547	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987514	C₃	1	2	yes	maximal, complemented
3	#17987514	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987509	C₃	1	2	yes	maximal, complemented
5	#17987505	C₃	1	2	yes	maximal, complemented
6	#17987504	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×#732	C₃	1	1	yes	maximal, characteristic, complemented
10	#13095331	—	3	3	no	—
11	#13095331	—	3	6	no	—
12	C₃×#10587025	—	3	3	no	—
13	C₃×#10587025	C₃²	1	2	yes	—
14	C₃×#10587025	C₃²	1	1	yes	characteristic
15	C₃×#10587025	C₃²	1	1	yes	characteristic
16	C₃×#10587025	C₃²	1	2	yes	—
17	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	2	yes	—
18	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
19	#13095319	—	3	3	no	—
20	#13095319	—	3	6	no	—
21	#13095317	—	3	3	no	—
22	#13095316	—	3	6	no	—
23	#13095314	—	3	6	no	—
24	#13095313	—	3	3	no	—
25	#13095303	—	3	6	no	—
26	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	2	yes	—
27	C₃×C₁₉×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
28	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
29	C₁₉×#732	—	3	3	no	—
30	#10624753	—	19	1	no	maximal, Sylow
31	C₃³×C₁₉	—	3	1	no	—
32	C₃³×C₁₉	—	3	2	no	—
33	C₃³×C₁₉	C₃³	1	1	yes	characteristic
34	C₃²×C₁₉⋊C₃	—	3	2	no	—
35	C₃²×C₁₉⋊C₃	—	3	1	no	—
36	C₁₉×H(3,𝔽₃)	—	3	2	no	—
37	C₁₉×H(3,𝔽₃)	—	3	2	no	—
38	C₁₉×H(3,𝔽₃)	—	3	2	no	—
39	C₁₉×H(3,𝔽₃)	—	3	2	no	—
40	C₁₉×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
41	C₃×C₉×C₁₉	—	3	1	no	—
42	C₃×C₉×C₁₉	—	3	2	no	—
43	C₃×C₉×C₁₉	—	3	1	no	—
44	C₃²⋊(C₁₉⋊C₃)	—	3	2	no	—
45	C₃²⋊(C₁₉⋊C₃)	—	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	#10587025	—	3	18	no	—
50	#10587025	—	3	2	no	—
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	1	no	—
59	#10587025	H(3,𝔽₃)	1	3	yes	—
60	C₃×#4294	—	3	9	no	—
61	C₃×#4294	—	3	1	no	—
62	C₃×#4294	—	3	2	no	—
63	C₃×#4294	—	3	1	no	—
64	C₃×#4294	H(3,𝔽₃)	1	6	yes	—
65	3₋¹⁺⁴	—	19	1	no	—
66	#6911	—	19	2	no	—
67	#6911	—	19	1	no	—
68	#6910	—	19	2	no	—
69	#6909	—	19	1	no	—
70	#6909	—	19	2	no	—
71	C₃×#732	C₁₉⋊C₃	1	1	yes	characteristic, complemented
72	C₃×#731	—	19	1	no	—
73	#6893	—	19	2	no	—
74	C₃²×C₁₉	—	3	3	no	—
75	C₃²×C₁₉	—	9	1	no	—
76	C₃²×C₁₉	—	9	2	no	—
77	C₃²×C₁₉	—	27	1	no	—
78	C₃²×C₁₉	—	27	2	no	—
79	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	2	yes	—
80	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
81	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
82	C₃×C₁₉⋊C₃	—	9	6	no	—
83	C₃×C₁₉⋊C₃	—	9	2	no	—
84	C₃×C₁₉⋊C₃	—	9	1	no	—
85	C₃×C₁₉⋊C₃	—	27	1	no	—
86	C₉×C₁₉	—	9	1	no	—
87	#4294	—	3	3	no	—
88	#4294	—	3	6	no	—
89	#4294	—	9	2	no	—
90	#4294	—	9	1	no	—
91	#737	—	57	6	no	—
92	#737	—	57	3	no	—
93	C₃×3₋¹⁺²	—	19	2	no	—
94	C₃×3₋¹⁺²	—	19	1	no	—
95	C₃×3₋¹⁺²	—	19	2	no	—
96	C₃×3₋¹⁺²	—	19	1	no	—
97	C₃×3₋¹⁺²	—	57	3	no	—
98	C₃×H(3,𝔽₃)	—	19	2	no	—
99	C₃×H(3,𝔽₃)	—	19	1	no	—
100	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	2	yes	—
101	C₃×H(3,𝔽₃)	C₃×C₁₉⋊C₃	1	1	yes	characteristic
102	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
103	#732	—	3	3	no	—
104	#732	—	57	6	no	—
105	#732	—	57	3	no	—
106	#731	—	57	3	no	—
107	#731	—	57	6	no	—
108	AΣL(1,𝔽₂₇)	—	57	6	no	—
109	AΣL(1,𝔽₂₇)	—	57	3	no	—
110	C₃²⋊C₉	—	57	6	no	—
111	C₃×C₁₉	—	3	2	no	—
112	C₃×C₁₉	—	3	1	no	—
113	C₃×C₁₉	—	3	1	no	—
114	C₃×C₁₉	—	9	3	no	—
115	C₃×C₁₉	—	27	2	no	—
116	C₃×C₁₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
117	C₁₉⋊C₃	—	27	2	no	—
118	C₁₉⋊C₃	—	27	1	no	—
119	C₃³	—	3	1	no	—
120	C₃³	—	3	2	no	—
121	C₃³	—	57	1	no	—
122	C₃³	—	57	2	no	—
123	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
124	3₋¹⁺²	—	19	3	no	—
125	3₋¹⁺²	—	57	18	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	1	no	—
134	3₋¹⁺²	—	57	2	no	—
135	H(3,𝔽₃)	—	3	2	no	—
136	H(3,𝔽₃)	—	3	2	no	—
137	H(3,𝔽₃)	—	3	2	no	—
138	H(3,𝔽₃)	—	3	2	no	—
139	H(3,𝔽₃)	—	57	2	no	—
140	H(3,𝔽₃)	—	57	1	no	—
141	H(3,𝔽₃)	—	57	2	no	—
142	H(3,𝔽₃)	—	57	2	no	—
143	H(3,𝔽₃)	—	57	2	no	—
144	H(3,𝔽₃)	C₃²⋊(C₁₉⋊C₃)	1	3	yes	—
145	C₃×C₉	—	3	2	no	—
146	C₃×C₉	—	3	1	no	—
147	C₃×C₉	—	3	1	no	—
148	C₃×C₉	—	19	6	no	—
149	C₃×C₉	—	57	9	no	—
150	C₃×C₉	—	57	1	no	—
151	C₃×C₉	—	57	2	no	—
152	C₃×C₉	—	57	1	no	—
153	C₁₉	#10624753	1	1	yes	characteristic, complemented, Sylow
154	C₃²	—	513	1	no	—
155	C₃²	—	3	3	no	—
156	C₃²	—	9	1	no	—
157	C₃²	—	9	2	no	—
158	C₃²	—	27	1	no	—
159	C₃²	—	27	2	no	—
160	C₃²	—	171	6	no	—
161	C₃²	—	171	1	no	—
162	C₃²	—	171	2	no	—
163	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	2	yes	—
164	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
165	C₃²	C₃×C₃²⋊(C₁₉⋊C₃)	1	1	yes	characteristic
166	C₉	—	9	1	no	—
167	C₉	—	57	6	no	—
168	C₉	—	57	3	no	—
169	C₉	—	171	2	no	—
170	C₉	—	171	1	no	—
171	C₃	—	513	1	no	—
172	C₃	—	513	2	no	—
173	C₃	—	3	1	no	—
174	C₃	—	3	1	no	—
175	C₃	—	3	2	no	—
176	C₃	—	9	3	no	—
177	C₃	—	27	2	no	—
178	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×#732	C₃
socle	C₃×C₁₉	C₃²⋊H(3,𝔽₃)
hypercenter	C₃×#732	C₁₉⋊C₃
hypocenter	C₁₉	#10624753
LCS	G ▷ C₃³×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#732

Complements

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

N	H ≅ G/N	type
1. #17987547	C₃	semidirect
2. #17987514	C₃	semidirect
3. #17987514	C₃	semidirect
4. #17987509	C₃	semidirect
5. #17987505	C₃	semidirect
6. #17987504	C₃	semidirect
7. C₃×#13095317	C₃	semidirect
8. #17987431	C₃	semidirect
9. C₃×C₁₉×#732	C₃	semidirect
71. C₃×#732	C₁₉⋊C₃	semidirect
153. C₁₉	#10624753	semidirect

Maximal subgroups

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

subgroup	index
#17987547	3
#17987514	3
#17987509	3
#17987505	3
#17987504	3
C₃×#13095317	3
#17987431	3
C₃×C₁₉×#732	3
#10624753	19