Finite Group Databasebeta

Subgroups — #28354521

Order 13851.

184 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 49 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	#17987443	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987440	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987435	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987427	C₃	1	1	yes	maximal, characteristic
5	C₁₉⋊#6862	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987325	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987320	C₃	1	1	yes	maximal, characteristic
8	#17987319	C₃	1	1	yes	maximal, characteristic
9	#17987317	C₃	1	1	yes	maximal, characteristic, complemented
10	#17987317	C₃	1	1	yes	maximal, characteristic, complemented
11	#17987314	C₃	1	1	yes	maximal, characteristic, complemented
12	#17987311	C₃	1	1	yes	maximal, characteristic
13	C₁₉×#6858	C₃	1	1	yes	maximal, characteristic, complemented
14	C₃×#10587025	C₃²	1	1	yes	characteristic
15	C₃×#10587023	—	3	1	no	—
16	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
17	C₃²×#4294	C₃²	1	1	yes	characteristic
18	#13095307	—	3	1	no	—
19	#13095306	—	3	1	no	—
20	#13095305	—	3	2	no	—
21	#13095305	—	3	1	no	—
22	#13095305	—	3	2	no	—
23	#13095305	—	3	2	no	—
24	C₃²⋊#4294	C₃²	1	1	yes	characteristic
25	C₃²⋊#4294	C₃²	1	1	yes	characteristic
26	C₃²⋊#4294	C₃²	1	1	yes	characteristic
27	C₃²⋊#4294	C₃²	1	1	yes	characteristic
28	C₃²⋊#4294	C₃²	1	1	yes	characteristic
29	C₃²⋊#4294	C₃²	1	1	yes	characteristic
30	#13095303	—	3	2	no	—
31	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
32	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
33	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
34	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
35	#10624672	—	19	1	no	maximal, Sylow
36	C₃³×C₁₉	C₃³	1	1	yes	characteristic
37	C₃²×C₁₉⋊C₃	—	3	1	no	—
38	C₁₉×3₋¹⁺²	—	9	1	no	—
39	C₃×C₉×C₁₉	—	3	1	no	—
40	C₃×C₉×C₁₉	—	3	1	no	—
41	C₃×C₉×C₁₉	—	3	1	no	—
42	C₃×C₉×C₁₉	—	3	1	no	—
43	C₃²⋊(C₁₉⋊C₃)	—	9	1	no	—
44	#10587025	—	9	1	no	—
45	C₃×#4294	—	3	1	no	—
46	C₃×#4294	—	3	1	no	—
47	C₃×#4294	—	3	1	no	—
48	C₃×#4294	—	3	1	no	—
49	C₃×#4294	—	3	1	no	—
50	C₃×#4294	—	3	1	no	—
51	C₃×#4294	—	3	1	no	—
52	C₃×#4294	—	9	1	no	—
53	C₃×#4294	H(3,𝔽₃)	1	2	yes	—
54	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
55	#10587023	—	9	3	no	—
56	#6895	—	19	1	no	—
57	#6894	—	19	1	no	—
58	#6894	—	19	1	no	—
59	C₉⋊H(3,𝔽₃)	—	19	1	no	—
60	#6862	—	19	1	no	—
61	#6861	—	19	1	no	—
62	#6859	—	19	1	no	—
63	#6859	—	19	1	no	—
64	#6858	—	19	1	no	—
65	#6858	—	19	1	no	—
66	#6858	—	19	1	no	—
67	#6858	C₁₉⋊C₃	1	1	yes	characteristic, complemented
68	#6857	—	19	1	no	—
69	C₃²×C₁₉	—	3	1	no	—
70	C₃²×C₁₉	—	3	1	no	—
71	C₃²×C₁₉	—	3	1	no	—
72	C₃²×C₁₉	—	3	1	no	—
73	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
74	C₃×C₁₉⋊C₃	—	9	1	no	—
75	C₃×C₁₉⋊C₃	—	9	1	no	—
76	C₃×C₁₉⋊C₃	—	9	1	no	—
77	C₃×C₁₉⋊C₃	—	9	1	no	—
78	C₉×C₁₉	—	9	1	no	—
79	C₉×C₁₉	—	9	1	no	—
80	C₉×C₁₉	—	9	1	no	—
81	C₉×C₁₉	—	9	1	no	—
82	#4294	—	3	1	no	—
83	#4294	—	3	2	no	—
84	#4294	—	9	1	no	—
85	#4294	—	9	1	no	—
86	#4294	—	9	1	no	—
87	#4294	—	9	1	no	—
88	#4294	—	9	1	no	—
89	#4294	—	9	1	no	—
90	#4294	—	9	1	no	—
91	C₃×3₋¹⁺²	—	19	1	no	—
92	C₃×3₋¹⁺²	—	57	1	no	—
93	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
94	C₃×H(3,𝔽₃)	—	19	1	no	—
95	C₃²×C₉	—	19	1	no	—
96	C₉⋊C₉	—	57	2	no	—
97	C₉⋊C₉	—	57	2	no	—
98	C₉⋊C₉	—	57	1	no	—
99	C₉⋊C₉	—	57	2	no	—
100	C₉⋊C₉	—	57	1	no	—
101	C₉⋊C₉	—	57	1	no	—
102	C₃²⋊C₉	—	19	1	no	—
103	C₃²⋊C₉	—	19	1	no	—
104	C₃²⋊C₉	—	19	1	no	—
105	C₃²⋊C₉	—	19	1	no	—
106	C₃²⋊C₉	—	19	1	no	—
107	C₃²⋊C₉	—	19	1	no	—
108	C₃²⋊C₉	—	57	2	no	—
109	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
110	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
111	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
112	C₃×C₁₉	—	9	1	no	—
113	C₃×C₁₉	#6910	1	1	yes	characteristic
114	C₃×C₁₉	#6913	1	1	yes	characteristic
115	C₃×C₁₉	#6913	1	1	yes	characteristic
116	C₃×C₁₉	#6913	1	1	yes	characteristic
117	C₁₉⋊C₃	—	27	1	no	—
118	C₃³	—	57	1	no	—
119	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
120	3₋¹⁺²	—	9	1	no	—
121	3₋¹⁺²	—	171	3	no	—
122	3₋¹⁺²	—	171	1	no	—
123	H(3,𝔽₃)	—	171	1	no	—
124	C₃×C₉	—	3	1	no	—
125	C₃×C₉	—	3	1	no	—
126	C₃×C₉	—	3	1	no	—
127	C₃×C₉	—	3	1	no	—
128	C₃×C₉	—	19	2	no	—
129	C₃×C₉	—	19	1	no	—
130	C₃×C₉	—	57	1	no	—
131	C₃×C₉	—	57	1	no	—
132	C₃×C₉	—	57	1	no	—
133	C₃×C₉	—	57	1	no	—
134	C₃×C₉	—	57	1	no	—
135	C₃×C₉	—	57	1	no	—
136	C₃×C₉	—	57	1	no	—
137	C₃×C₉	—	171	1	no	—
138	C₁₉	#10624672	1	1	yes	characteristic, complemented, Sylow
139	C₃²	—	3	1	no	—
140	C₃²	—	3	1	no	—
141	C₃²	—	3	1	no	—
142	C₃²	—	3	1	no	—
143	C₃²	—	171	1	no	—
144	C₃²	—	171	1	no	—
145	C₃²	—	171	1	no	—
146	C₃²	—	171	1	no	—
147	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
148	C₉	—	9	1	no	—
149	C₉	—	9	1	no	—
150	C₉	—	9	1	no	—
151	C₉	—	9	1	no	—
152	C₉	—	57	1	no	—
153	C₉	—	57	2	no	—
154	C₉	—	171	1	no	—
155	C₉	—	171	1	no	—
156	C₉	—	171	1	no	—
157	C₉	—	171	1	no	—
158	C₉	—	171	1	no	—
159	C₉	—	171	1	no	—
160	C₉	—	171	1	no	—
161	C₃	—	513	1	no	—
162	C₃	—	9	1	no	—
163	C₃	#17987508	1	1	yes	characteristic
164	C₃	#17987520	1	1	yes	characteristic
165	C₃	#17987526	1	1	yes	characteristic
166	C₃	#17987530	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₁₉⋊C₃
commutator, cosocle	C₃³×C₁₉	C₃³
Frattini	C₃³	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6858	C₃
socle	C₃²×C₁₉	C₃×H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₁₉⋊C₃)	C₃²
hypercenter	#6858	C₁₉⋊C₃
hypocenter	C₁₉	#10624672
LCS	G ▷ C₃³×C₁₉ ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6858

Complements

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

N	H ≅ G/N	type
1. #17987443	C₃	semidirect
2. #17987440	C₃	semidirect
3. #17987435	C₃	semidirect
5. C₁₉⋊#6862	C₃	semidirect
6. #17987325	C₃	semidirect
9. #17987317	C₃	semidirect
10. #17987317	C₃	semidirect
11. #17987314	C₃	semidirect
13. C₁₉×#6858	C₃	semidirect
67. #6858	C₁₉⋊C₃	semidirect
138. C₁₉	#10624672	semidirect

Maximal subgroups

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

subgroup	index
#17987443	3
#17987440	3
#17987435	3
#17987427	3
C₁₉⋊#6862	3
#17987325	3
#17987320	3
#17987319	3
#17987317	3
#17987314	3
#17987311	3
C₁₉×#6858	3
#10624672	19