Finite Group Databasebeta

Subgroups — #28353554

Order 13851.

332 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 124 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×#13095306	C₃	1	1	yes	maximal, characteristic
2	C₃×#13095305	C₃	1	2	yes	maximal
3	C₃×C₉×#4294	C₃	1	1	yes	maximal, characteristic
4	#17987308	C₃	1	1	yes	maximal, characteristic
5	#17987308	C₃	1	1	yes	maximal, characteristic
6	#17987308	C₃	1	2	yes	maximal
7	#17987308	C₃	1	2	yes	maximal
8	#17987308	C₃	1	2	yes	maximal
9	C₁₉×#6855	C₃	1	1	yes	maximal, characteristic
10	C₃²×#4294	C₃²	1	2	yes	—
11	C₃²×#4294	C₃²	1	2	yes	—
12	C₃²×#4294	C₃²	1	1	yes	characteristic
13	C₃²×#4294	C₃²	1	2	yes	—
14	C₃²×#4294	C₃²	1	2	yes	—
15	#13095306	C₉	1	9	yes	complemented
16	#13095305	—	3	6	no	—
17	C₉×#4294	—	3	3	no	—
18	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
19	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
20	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
21	C₃²×C₉×C₁₉	C₃²	1	1	yes	characteristic
22	#10624559	—	19	1	no	maximal, Sylow
23	C₃³×C₁₉	C₃³	1	1	yes	characteristic
24	C₃×C₉×C₁₉	—	3	1	no	—
25	C₃×C₉×C₁₉	—	3	1	no	—
26	C₃×C₉×C₁₉	—	3	2	no	—
27	C₃×C₉×C₁₉	—	3	2	no	—
28	C₃×C₉×C₁₉	—	3	2	no	—
29	C₃×C₉×C₁₉	—	3	1	no	—
30	C₃×C₉×C₁₉	—	3	1	no	—
31	C₃×C₉×C₁₉	—	3	1	no	—
32	C₃×C₉×C₁₉	—	3	1	no	—
33	C₃×C₉×C₁₉	—	3	2	no	—
34	C₃×C₉×C₁₉	C₃×C₉	1	3	yes	—
35	C₃×C₉×C₁₉	3₋¹⁺²	1	3	yes	—
36	C₃×#4294	—	3	1	no	—
37	C₃×#4294	—	3	2	no	—
38	C₃×#4294	—	3	2	no	—
39	C₃×#4294	—	3	1	no	—
40	C₃×#4294	—	3	1	no	—
41	C₃×#4294	—	3	2	no	—
42	C₃×#4294	—	3	2	no	—
43	C₃×#4294	—	3	2	no	—
44	C₃×#4294	—	3	2	no	—
45	C₃×#4294	—	3	2	no	—
46	C₃×#4294	—	3	2	no	—
47	C₃×#4294	—	3	2	no	—
48	C₃×#4294	—	3	2	no	—
49	C₃×#4294	—	3	2	no	—
50	C₃×#4294	—	3	2	no	—
51	C₃×#4294	—	3	2	no	—
52	C₃×#4294	—	3	2	no	—
53	C₃×#4294	C₃×C₉	1	3	yes	—
54	C₃×#4294	C₃×C₉	1	6	yes	—
55	C₃×#4294	3₋¹⁺²	1	6	yes	—
56	C₃×C₉⋊C₉	—	19	1	no	—
57	C₃×C₉⋊C₉	—	19	2	no	—
58	C₃×C₉²	—	19	1	no	—
59	#6855	—	19	2	no	—
60	#6855	—	19	1	no	—
61	#6855	—	19	2	no	—
62	#6855	—	19	1	no	—
63	#6855	—	19	2	no	—
64	#6855	C₁₉⋊C₃	1	1	yes	characteristic
65	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
66	C₃²×C₁₉	C₃×3₋¹⁺²	1	2	yes	—
67	C₃²×C₁₉	C₃×3₋¹⁺²	1	1	yes	characteristic
68	C₃²×C₁₉	#737	1	1	yes	characteristic
69	C₃²×C₁₉	#737	1	1	yes	characteristic
70	C₃²×C₁₉	#737	1	1	yes	characteristic
71	C₃²×C₁₉	#737	1	2	yes	—
72	C₃²×C₁₉	#737	1	2	yes	—
73	C₃²×C₁₉	#737	1	2	yes	—
74	C₉×C₁₉	—	3	3	no	—
75	C₉×C₁₉	—	3	3	no	—
76	C₉×C₁₉	—	9	1	no	—
77	C₉×C₁₉	—	9	1	no	—
78	#4294	—	3	6	no	—
79	#4294	—	3	3	no	—
80	#4294	—	3	6	no	—
81	#4294	—	9	2	no	—
82	#4294	—	9	2	no	—
83	C₃²×C₉	—	19	1	no	—
84	C₃²×C₉	—	19	2	no	—
85	C₃²×C₉	—	19	2	no	—
86	C₃²×C₉	—	19	2	no	—
87	C₃²×C₉	—	19	2	no	—
88	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
89	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
90	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
91	C₃²×C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
92	C₉⋊C₉	—	19	9	no	—
93	C₉⋊C₉	—	57	6	no	—
94	C₉²	—	57	3	no	—
95	C₃×C₁₉	#6887	1	1	yes	characteristic
96	C₃×C₁₉	C₉×3₋¹⁺²	1	2	yes	—
97	C₃×C₁₉	C₉×3₋¹⁺²	1	1	yes	characteristic
98	C₃×C₁₉	#6894	1	2	yes	—
99	C₃×C₁₉	#6895	1	1	yes	characteristic
100	C₃×C₁₉	#6896	1	1	yes	characteristic
101	C₃×C₁₉	#6897	1	2	yes	—
102	C₃×C₁₉	#6898	1	1	yes	characteristic
103	C₃×C₁₉	#6899	1	2	yes	—
104	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
105	C₃×C₉	—	3	1	no	—
106	C₃×C₉	—	3	1	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	3	2	no	—
109	C₃×C₉	—	3	2	no	—
110	C₃×C₉	—	3	1	no	—
111	C₃×C₉	—	3	1	no	—
112	C₃×C₉	—	3	1	no	—
113	C₃×C₉	—	3	1	no	—
114	C₃×C₉	—	3	2	no	—
115	C₃×C₉	—	19	6	no	—
116	C₃×C₉	—	19	3	no	—
117	C₃×C₉	—	19	6	no	—
118	C₃×C₉	—	57	2	no	—
119	C₃×C₉	—	57	1	no	—
120	C₃×C₉	—	57	2	no	—
121	C₃×C₉	—	57	1	no	—
122	C₃×C₉	—	57	1	no	—
123	C₃×C₉	—	57	2	no	—
124	C₃×C₉	—	57	2	no	—
125	C₃×C₉	—	57	2	no	—
126	C₃×C₉	—	57	2	no	—
127	C₃×C₉	—	57	2	no	—
128	C₃×C₉	—	57	2	no	—
129	C₃×C₉	—	57	2	no	—
130	C₃×C₉	—	57	2	no	—
131	C₃×C₉	—	57	2	no	—
132	C₃×C₉	—	57	2	no	—
133	C₃×C₉	—	57	2	no	—
134	C₃×C₉	—	57	2	no	—
135	C₃×C₉	C₉×C₁₉⋊C₃	1	3	yes	—
136	C₃×C₉	#10587022	1	3	yes	—
137	C₁₉	#10624559	1	1	yes	characteristic, complemented, Sylow
138	C₃²	C₃×C₉×C₁₉⋊C₃	1	1	yes	characteristic
139	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	2	yes	—
140	C₃²	C₃×#10587022	1	1	yes	characteristic
141	C₃²	#13095331	1	1	yes	characteristic
142	C₃²	#13095332	1	1	yes	characteristic
143	C₃²	#13095332	1	1	yes	characteristic
144	C₃²	#13095333	1	2	yes	—
145	C₃²	#13095333	1	2	yes	—
146	C₃²	#13095333	1	2	yes	—
147	C₉	—	3	3	no	—
148	C₉	—	3	3	no	—
149	C₉	—	9	1	no	—
150	C₉	—	9	1	no	—
151	C₉	—	57	6	no	—
152	C₉	—	57	3	no	—
153	C₉	—	57	6	no	—
154	C₉	—	171	2	no	—
155	C₉	—	171	2	no	—
156	C₃	#17987402	1	1	yes	characteristic
157	C₃	#17987410	1	2	yes	—
158	C₃	C₉×#10587022	1	1	yes	characteristic
159	C₃	#17987434	1	2	yes	—
160	C₃	#17987443	1	1	yes	characteristic
161	C₃	#17987446	1	1	yes	characteristic
162	C₃	#17987455	1	2	yes	—
163	C₃	#17987457	1	1	yes	characteristic
164	C₃	#17987465	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₁₉⋊C₃
commutator	C₃²×C₁₉	C₃²×C₉
Fitting	C₁₉×#6855	C₃
socle, prime radical, cosocle	C₃³×C₁₉	C₃³
hypercenter	#6855	C₁₉⋊C₃
hypocenter	C₁₉	#10624559
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃³ ◁ #6855

Complements

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

N	H ≅ G/N	type
15. #13095306	C₉	semidirect
137. C₁₉	#10624559	semidirect

Maximal subgroups

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

subgroup	index
C₃×#13095306	3
C₃×#13095305	3
C₃×C₉×#4294	3
#17987308	3
C₁₉×#6855	3
#10624559	19