Finite Group Databasebeta

Subgroups — #28354032

Order 13851.

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

normal subgroups only

This group has 204 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	#17987442	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987441	C₃	1	1	yes	maximal, characteristic, complemented
3	3₋¹⁺²×#4294	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987406	C₃	1	1	yes	maximal, characteristic
5	C₁₉⋊#6862	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987323	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987320	C₃	1	1	yes	maximal, characteristic
8	#17987320	C₃	1	1	yes	maximal, characteristic
9	#17987316	C₃	1	1	yes	maximal, characteristic, complemented
10	#17987315	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×C₃²⋊(C₁₉⋊C₃)	C₃²	1	1	yes	characteristic
16	C₃²×#4294	C₃²	1	1	yes	characteristic
17	C₃×C₉×C₁₉⋊C₃	—	3	1	no	—
18	#13095307	—	3	1	no	—
19	#13095307	—	3	1	no	—
20	#13095306	—	3	1	no	—
21	#13095306	—	3	1	no	—
22	#13095305	—	3	1	no	—
23	#13095305	—	3	1	no	—
24	#13095305	—	3	1	no	—
25	#13095305	—	3	1	no	—
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	C₃²⋊#4294	C₃²	1	1	yes	characteristic
31	C₃²⋊#4294	C₃²	1	1	yes	characteristic
32	#13095303	—	3	1	no	—
33	#13095303	—	3	1	no	—
34	C₉×#4294	—	3	1	no	—
35	C₃×C₁₉×3₋¹⁺²	C₃²	1	1	yes	characteristic
36	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
37	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
38	C₁₉×C₃²⋊C₉	C₃²	1	1	yes	characteristic
39	#10624624	—	19	1	no	maximal, Sylow
40	C₃³×C₁₉	C₃³	1	1	yes	characteristic
41	C₃²×C₁₉⋊C₃	—	3	1	no	—
42	C₁₉×3₋¹⁺²	—	3	3	no	—
43	C₃×C₉×C₁₉	—	3	1	no	—
44	C₃×C₉×C₁₉	—	3	1	no	—
45	C₃×C₉×C₁₉	—	3	1	no	—
46	C₃×C₉×C₁₉	—	3	1	no	—
47	C₃²⋊(C₁₉⋊C₃)	—	3	3	no	—
48	#10587025	—	9	1	no	—
49	C₃×#4294	—	3	1	no	—
50	C₃×#4294	—	3	1	no	—
51	C₃×#4294	—	3	1	no	—
52	C₃×#4294	—	3	1	no	—
53	C₃×#4294	—	3	1	no	—
54	C₃×#4294	—	3	1	no	—
55	C₃×#4294	—	3	1	no	—
56	C₃×#4294	—	9	1	no	—
57	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
58	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
59	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
60	C₉×C₁₉⋊C₃	—	9	3	no	—
61	#6895	—	19	1	no	—
62	#6894	—	19	1	no	—
63	C₉×3₋¹⁺²	—	19	1	no	—
64	C₉×H(3,𝔽₃)	—	19	1	no	—
65	#6862	—	19	1	no	—
66	#6861	—	19	1	no	—
67	#6859	—	19	1	no	—
68	#6859	—	19	1	no	—
69	#6858	—	19	1	no	—
70	#6858	—	19	1	no	—
71	#6858	—	19	1	no	—
72	#6858	C₁₉⋊C₃	1	1	yes	characteristic, complemented
73	#6857	—	19	1	no	—
74	C₃²×C₁₉	—	3	1	no	—
75	C₃²×C₁₉	—	3	1	no	—
76	C₃²×C₁₉	—	3	1	no	—
77	C₃²×C₁₉	—	3	1	no	—
78	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
79	C₃×C₁₉⋊C₃	—	3	3	no	—
80	C₃×C₁₉⋊C₃	—	9	1	no	—
81	C₃×C₁₉⋊C₃	—	9	1	no	—
82	C₃×C₁₉⋊C₃	—	9	1	no	—
83	C₉×C₁₉	—	3	3	no	—
84	C₉×C₁₉	—	9	1	no	—
85	C₉×C₁₉	—	9	1	no	—
86	C₉×C₁₉	—	9	1	no	—
87	#4294	—	3	1	no	—
88	#4294	—	3	1	no	—
89	#4294	—	3	1	no	—
90	#4294	—	9	1	no	—
91	#4294	—	9	1	no	—
92	#4294	—	9	1	no	—
93	#4294	—	9	1	no	—
94	#4294	—	9	1	no	—
95	#4294	—	9	1	no	—
96	#4294	—	9	1	no	—
97	C₃×3₋¹⁺²	—	19	1	no	—
98	C₃×3₋¹⁺²	C₃×C₁₉⋊C₃	1	1	yes	characteristic
99	C₃×H(3,𝔽₃)	—	19	1	no	—
100	C₃²×C₉	—	19	1	no	—
101	C₃²×C₉	—	57	1	no	—
102	C₉⋊C₉	—	57	1	no	—
103	C₉⋊C₉	—	57	1	no	—
104	C₉⋊C₉	—	57	1	no	—
105	C₉⋊C₉	—	57	1	no	—
106	C₉⋊C₉	—	57	1	no	—
107	C₉⋊C₉	—	57	1	no	—
108	C₉⋊C₉	—	57	1	no	—
109	C₉⋊C₉	—	57	1	no	—
110	C₃²⋊C₉	—	19	1	no	—
111	C₃²⋊C₉	—	19	1	no	—
112	C₃²⋊C₉	—	19	1	no	—
113	C₃²⋊C₉	—	19	1	no	—
114	C₃²⋊C₉	—	19	1	no	—
115	C₃²⋊C₉	—	19	1	no	—
116	C₃²⋊C₉	—	57	1	no	—
117	C₃²⋊C₉	—	57	1	no	—
118	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
119	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
120	C₃²⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
121	C₉²	—	57	1	no	—
122	C₃×C₁₉	—	9	1	no	—
123	C₃×C₁₉	#6908	1	1	yes	characteristic
124	C₃×C₁₉	#6908	1	1	yes	characteristic
125	C₃×C₁₉	#6910	1	1	yes	characteristic
126	C₃×C₁₉	#6913	1	1	yes	characteristic
127	C₁₉⋊C₃	—	9	3	no	—
128	C₃³	—	57	1	no	—
129	C₃³	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
130	3₋¹⁺²	—	3	3	no	—
131	3₋¹⁺²	—	171	1	no	—
132	H(3,𝔽₃)	—	57	3	no	—
133	C₃×C₉	—	3	1	no	—
134	C₃×C₉	—	3	1	no	—
135	C₃×C₉	—	3	1	no	—
136	C₃×C₉	—	3	1	no	—
137	C₃×C₉	—	19	1	no	—
138	C₃×C₉	—	19	1	no	—
139	C₃×C₉	—	19	1	no	—
140	C₃×C₉	—	57	1	no	—
141	C₃×C₉	—	57	1	no	—
142	C₃×C₉	—	57	1	no	—
143	C₃×C₉	—	57	1	no	—
144	C₃×C₉	—	57	1	no	—
145	C₃×C₉	—	57	1	no	—
146	C₃×C₉	—	57	1	no	—
147	C₃×C₉	—	171	3	no	—
148	C₃×C₉	—	171	1	no	—
149	C₁₉	#10624624	1	1	yes	characteristic, complemented, Sylow
150	C₃²	—	3	1	no	—
151	C₃²	—	3	1	no	—
152	C₃²	—	3	1	no	—
153	C₃²	—	3	1	no	—
154	C₃²	—	57	3	no	—
155	C₃²	—	171	1	no	—
156	C₃²	—	171	1	no	—
157	C₃²	—	171	1	no	—
158	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
159	C₉	—	3	3	no	—
160	C₉	—	9	1	no	—
161	C₉	—	9	1	no	—
162	C₉	—	9	1	no	—
163	C₉	—	57	1	no	—
164	C₉	—	57	1	no	—
165	C₉	—	57	1	no	—
166	C₉	—	171	1	no	—
167	C₉	—	171	1	no	—
168	C₉	—	171	1	no	—
169	C₉	—	171	1	no	—
170	C₉	—	171	1	no	—
171	C₉	—	171	1	no	—
172	C₉	—	171	1	no	—
173	C₃	—	9	1	no	—
174	C₃	—	171	3	no	—
175	C₃	#17987496	1	1	yes	characteristic
176	C₃	#17987501	1	1	yes	characteristic
177	C₃	#17987507	1	1	yes	characteristic
178	C₃	#17987527	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	#10624624
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. #17987442	C₃	semidirect
2. #17987441	C₃	semidirect
3. 3₋¹⁺²×#4294	C₃	semidirect
5. C₁₉⋊#6862	C₃	semidirect
6. #17987323	C₃	semidirect
9. #17987316	C₃	semidirect
10. #17987315	C₃	semidirect
11. #17987314	C₃	semidirect
13. C₁₉×#6858	C₃	semidirect
72. #6858	C₁₉⋊C₃	semidirect
149. C₁₉	#10624624	semidirect

Maximal subgroups

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

subgroup	index
#17987442	3
#17987441	3
3₋¹⁺²×#4294	3
#17987406	3
C₁₉⋊#6862	3
#17987323	3
#17987320	3
#17987316	3
#17987315	3
#17987314	3
#17987311	3
C₁₉×#6858	3
#10624624	19