Finite Group Databasebeta

Subgroups — #28355980

Order 13851.

824 conjugacy classes of subgroups; only the 292 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 292 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	C₃³×#4294	C₃	1	1	yes	maximal, characteristic
2	#17987443	C₃	1	3	yes	maximal, complemented
3	#17987442	C₃	1	3	yes	maximal, complemented
4	#17987441	C₃	1	6	yes	maximal, complemented
5	#17987440	C₃	1	6	yes	maximal, complemented
6	#17987435	C₃	1	6	yes	maximal, complemented
7	#17987421	C₃	1	2	yes	maximal
8	#17987415	C₃	1	6	yes	maximal, complemented
9	3₋¹⁺²×#4294	C₃	1	3	yes	maximal, complemented
10	#17987402	C₃	1	3	yes	maximal, complemented
11	C₁₉×#6891	C₃	1	1	yes	maximal, characteristic
12	C₃×#10587025	C₃²	1	6	yes	—
13	C₃²×#4294	—	3	9	no	—
14	C₃²×#4294	C₃²	1	1	yes	characteristic
15	C₃²×#4294	C₃²	1	2	yes	—
16	C₃²×#4294	C₃²	1	2	yes	—
17	C₃²×#4294	C₃²	1	2	yes	—
18	C₃²×#4294	C₃²	1	2	yes	—
19	C₃²×#4294	C₃²	1	1	yes	characteristic
20	C₃²×#4294	C₃²	1	1	yes	characteristic
21	C₃²×#4294	C₃²	1	1	yes	characteristic
22	#13095307	C₃²	1	9	yes	complemented
23	#13095306	C₃²	1	9	yes	complemented
24	#13095305	C₃²	1	18	yes	complemented
25	#13095305	C₃²	1	18	yes	complemented
26	#13095305	C₃²	1	18	yes	complemented
27	C₃²⋊#4294	C₃²	1	6	yes	—
28	C₃²⋊#4294	C₃²	1	6	yes	—
29	C₃²⋊#4294	C₃²	1	6	yes	—
30	C₉×#4294	C₃²	1	9	yes	complemented
31	C₃⁴×C₁₉	C₃²	1	1	yes	characteristic
32	C₃×C₁₉×3₋¹⁺²	C₃²	1	3	yes	—
33	C₁₉×C₃²⋊C₉	C₃²	1	3	yes	—
34	C₁₉×C₃²⋊C₉	C₃²	1	6	yes	—
35	#10624872	—	19	1	no	maximal, Sylow
36	C₃³×C₁₉	—	3	3	no	—
37	C₃³×C₁₉	—	3	3	no	—
38	C₃³×C₁₉	—	3	6	no	—
39	C₃³×C₁₉	C₃³	1	1	yes	characteristic
40	C₃³×C₁₉	C₃³	1	2	yes	—
41	C₃³×C₁₉	C₃³	1	1	yes	characteristic
42	C₁₉×3₋¹⁺²	—	3	9	no	—
43	C₃×C₉×C₁₉	C₃³	1	9	yes	—
44	#10587025	—	3	18	no	—
45	C₃×#4294	—	3	3	no	—
46	C₃×#4294	—	3	3	no	—
47	C₃×#4294	—	3	3	no	—
48	C₃×#4294	—	3	6	no	—
49	C₃×#4294	—	3	6	no	—
50	C₃×#4294	—	3	6	no	—
51	C₃×#4294	—	3	6	no	—
52	C₃×#4294	3₋¹⁺²	1	9	yes	complemented
53	C₃×#4294	C₃³	1	18	yes	—
54	C₃×#4294	C₃³	1	2	yes	—
55	C₃×#4294	C₃³	1	2	yes	—
56	C₃×#4294	C₃³	1	1	yes	characteristic
57	C₃×#4294	C₃³	1	1	yes	characteristic
58	C₃×#4294	C₃³	1	1	yes	characteristic
59	C₃×#4294	C₃³	1	2	yes	—
60	C₃³×C₉	—	19	1	no	—
61	#6895	—	19	3	no	—
62	#6895	—	19	3	no	—
63	#6894	—	19	6	no	—
64	#6894	—	19	6	no	—
65	#6894	—	19	6	no	—
66	#6891	—	19	2	no	—
67	#6891	C₁₉⋊C₃	1	1	yes	characteristic
68	C₉×3₋¹⁺²	—	19	3	no	—
69	C₉×3₋¹⁺²	—	19	6	no	—
70	#6887	—	19	3	no	—
71	C₃²×C₁₉	—	3	27	no	—
72	C₃²×C₁₉	—	3	2	no	—
73	C₃²×C₁₉	—	3	1	no	—
74	C₃²×C₁₉	—	3	1	no	—
75	C₃²×C₁₉	—	3	2	no	—
76	C₃²×C₁₉	—	3	2	no	—
77	C₃²×C₁₉	—	3	2	no	—
78	C₃²×C₁₉	—	3	2	no	—
79	C₃²×C₁₉	C₃×3₋¹⁺²	1	3	yes	—
80	C₃²×C₁₉	#737	1	3	yes	—
81	C₃²×C₁₉	#737	1	6	yes	—
82	C₃²×C₁₉	C₃⁴	1	1	yes	characteristic
83	C₉×C₁₉	—	3	9	no	—
84	#4294	—	3	2	no	—
85	#4294	—	3	2	no	—
86	#4294	—	3	1	no	—
87	#4294	—	3	1	no	—
88	#4294	—	3	18	no	—
89	#4294	C₃×3₋¹⁺²	1	3	yes	—
90	#4294	C₃×3₋¹⁺²	1	6	yes	—
91	C₃⁴	C₃×C₁₉⋊C₃	1	1	yes	characteristic
92	C₃×3₋¹⁺²	—	19	6	no	—
93	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	3	yes	—
94	C₃²×C₉	—	19	1	no	—
95	C₃²×C₉	—	19	1	no	—
96	C₃²×C₉	—	19	2	no	—
97	C₃²×C₉	—	19	2	no	—
98	C₃²×C₉	—	19	2	no	—
99	C₃²×C₉	—	19	2	no	—
100	C₃²×C₉	—	19	1	no	—
101	C₃²×C₉	—	19	1	no	—
102	C₃²×C₉	—	57	9	no	—
103	C₉⋊C₉	—	19	18	no	—
104	C₉⋊C₉	—	19	18	no	—
105	C₉⋊C₉	—	19	18	no	—
106	C₉⋊C₉	—	19	9	no	—
107	C₉⋊C₉	—	19	9	no	—
108	C₃²⋊C₉	—	19	6	no	—
109	C₃²⋊C₉	—	19	6	no	—
110	C₃²⋊C₉	—	19	6	no	—
111	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	3	yes	—
112	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	6	yes	—
113	C₉²	—	19	9	no	—
114	C₃×C₁₉	—	3	3	no	—
115	C₃×C₁₉	—	3	3	no	—
116	C₃×C₁₉	—	3	6	no	—
117	C₃×C₁₉	C₃²×3₋¹⁺²	1	1	yes	characteristic
118	C₃×C₁₉	C₃×#737	1	1	yes	characteristic
119	C₃×C₁₉	C₃×#737	1	2	yes	—
120	C₃³	—	3	3	no	—
121	C₃³	—	3	3	no	—
122	C₃³	—	3	6	no	—
123	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
124	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
125	C₃³	C₃²×C₁₉⋊C₃	1	2	yes	—
126	3₋¹⁺²	—	3	9	no	—
127	3₋¹⁺²	—	57	18	no	—
128	C₃×C₉	—	19	9	no	—
129	C₃×C₉	—	19	1	no	—
130	C₃×C₉	—	19	2	no	—
131	C₃×C₉	—	19	18	no	—
132	C₃×C₉	—	19	2	no	—
133	C₃×C₉	—	19	2	no	—
134	C₃×C₉	—	19	1	no	—
135	C₃×C₉	—	19	1	no	—
136	C₃×C₉	—	57	3	no	—
137	C₃×C₉	—	57	6	no	—
138	C₃×C₉	—	57	6	no	—
139	C₃×C₉	—	57	6	no	—
140	C₃×C₉	—	57	6	no	—
141	C₃×C₉	—	57	3	no	—
142	C₃×C₉	—	57	3	no	—
143	C₃×C₉	C₃²×C₁₉⋊C₃	1	9	yes	—
144	C₁₉	#10624872	1	1	yes	characteristic, complemented, Sylow
145	C₃²	—	3	27	no	—
146	C₃²	—	3	1	no	—
147	C₃²	—	3	2	no	—
148	C₃²	—	3	1	no	—
149	C₃²	—	3	2	no	—
150	C₃²	—	3	2	no	—
151	C₃²	—	3	2	no	—
152	C₃²	—	3	2	no	—
153	C₃²	3₋¹⁺²×C₁₉⋊C₃	1	3	yes	—
154	C₃²	#13095331	1	3	yes	—
155	C₃²	#13095333	1	6	yes	—
156	C₃²	C₃³×C₁₉⋊C₃	1	1	yes	characteristic
157	C₉	—	3	9	no	—
158	C₉	—	19	3	no	—
159	C₉	—	19	6	no	—
160	C₉	—	57	2	no	—
161	C₉	—	57	2	no	—
162	C₉	—	57	1	no	—
163	C₉	—	57	1	no	—
164	C₉	—	57	18	no	—
165	C₃	—	3	3	no	—
166	C₃	—	3	3	no	—
167	C₃	—	3	6	no	—
168	C₃	C₃×3₋¹⁺²×C₁₉⋊C₃	1	1	yes	characteristic
169	C₃	C₃×#13095331	1	1	yes	characteristic
170	C₃	C₃×#13095333	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₁₉⋊C₃
commutator, socle, cosocle	C₃²×C₁₉	C₃⁴
Fitting	C₁₉×#6891	C₃
prime radical	C₃⁴×C₁₉	C₃²
hypercenter	#6891	C₁₉⋊C₃
hypocenter	C₁₉	#10624872
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ #6891

Complements

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

N	H ≅ G/N	type
2. #17987443	C₃	semidirect
3. #17987442	C₃	semidirect
4. #17987441	C₃	semidirect
5. #17987440	C₃	semidirect
6. #17987435	C₃	semidirect
8. #17987415	C₃	semidirect
9. 3₋¹⁺²×#4294	C₃	semidirect
10. #17987402	C₃	semidirect
22. #13095307	C₃²	semidirect
23. #13095306	C₃²	semidirect
24. #13095305	C₃²	semidirect
25. #13095305	C₃²	semidirect
26. #13095305	C₃²	semidirect
30. C₉×#4294	C₃²	semidirect
52. C₃×#4294	3₋¹⁺²	semidirect
144. C₁₉	#10624872	semidirect

Maximal subgroups

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

subgroup	index
C₃³×#4294	3
#17987443	3
#17987442	3
#17987441	3
#17987440	3
#17987435	3
#17987421	3
#17987415	3
3₋¹⁺²×#4294	3
#17987402	3
C₁₉×#6891	3
#10624872	19