Finite Group Databasebeta

Subgroups — #28355851

Order 13851.

190 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 43 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	#17987529	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987529	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987512	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987466	C₃	1	1	yes	maximal, characteristic, complemented
5	#17987391	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987381	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987381	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987380	C₃	1	1	yes	maximal, characteristic, complemented
9	#17987380	C₃	1	1	yes	maximal, characteristic, complemented
10	#17987373	C₃	1	1	yes	maximal, characteristic, complemented
11	#17987371	C₃	1	1	yes	maximal, characteristic, complemented
12	#17987371	C₃	1	1	yes	maximal, characteristic, complemented
13	C₁₉×#6883	C₃	1	1	yes	maximal, characteristic, complemented
14	#13095333	—	3	1	no	—
15	#13095333	—	3	1	no	—
16	C₃×#10587022	C₃²	1	1	yes	characteristic
17	H(3,𝔽₃)×C₁₉⋊C₃	—	3	1	no	—
18	#13095322	C₃²	1	1	yes	characteristic, complemented
19	#13095322	C₃²	1	1	yes	characteristic, complemented
20	#13095320	C₃²	1	1	yes	characteristic
21	#13095320	C₃²	1	1	yes	characteristic
22	#13095320	C₃²	1	1	yes	characteristic, complemented
23	#13095320	C₃²	1	1	yes	characteristic, complemented
24	#13095319	—	3	1	no	—
25	#13095319	—	3	1	no	—
26	#13095318	—	3	1	no	—
27	#13095318	—	3	1	no	—
28	#13095315	—	3	1	no	—
29	#13095315	—	3	1	no	—
30	#13095307	C₃²	1	1	yes	characteristic, complemented
31	#13095306	—	3	1	no	—
32	#13095305	—	3	1	no	—
33	#13095303	—	3	1	no	—
34	C₉×#4294	C₃²	1	1	yes	characteristic, complemented
35	C₁₉×#733	C₃²	1	1	yes	characteristic, complemented
36	C₁₉×#733	C₃²	1	1	yes	characteristic, complemented
37	C₁₉×#732	C₃²	1	1	yes	characteristic
38	C₁₉×C₉⋊C₉	C₃²	1	1	yes	characteristic, complemented
39	#10624853	—	19	1	no	maximal, Sylow
40	C₃²×C₁₉⋊C₃	—	9	1	no	—
41	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
42	C₁₉×3₋¹⁺²	—	3	1	no	—
43	C₁₉×3₋¹⁺²	—	3	1	no	—
44	C₁₉×H(3,𝔽₃)	—	3	1	no	—
45	C₃×C₉×C₁₉	—	3	1	no	—
46	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
47	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
48	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
49	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
50	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
51	C₃²⋊(C₁₉⋊C₃)	—	9	2	no	—
52	#10587025	—	3	1	no	—
53	#10587025	—	3	1	no	—
54	C₃×#4294	—	3	1	no	—
55	C₃×#4294	—	3	1	no	—
56	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
57	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
58	#10587023	—	9	1	no	—
59	#10587023	—	9	1	no	—
60	#10587022	—	9	1	no	—
61	#10587022	—	9	1	no	—
62	#10587022	—	9	1	no	—
63	C₉×C₁₉⋊C₃	—	9	1	no	—
64	C₉×C₁₉⋊C₃	—	9	1	no	—
65	#6913	—	19	1	no	—
66	#6913	—	19	1	no	—
67	#6911	—	19	1	no	—
68	#6899	—	19	1	no	—
69	#6883	—	19	1	no	—
70	#6883	C₁₉⋊C₃	1	1	yes	characteristic, complemented
71	#6881	—	19	1	no	—
72	#6881	—	19	1	no	—
73	#6881	—	19	1	no	—
74	#6881	—	19	1	no	—
75	#6879	—	19	1	no	—
76	#6878	—	19	1	no	—
77	#6878	—	19	1	no	—
78	C₃²×C₁₉	—	9	1	no	—
79	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
80	C₃×C₁₉⋊C₃	—	3	1	no	—
81	C₃×C₁₉⋊C₃	—	9	1	no	—
82	C₃×C₁₉⋊C₃	—	9	1	no	—
83	C₃×C₁₉⋊C₃	—	9	1	no	—
84	C₃×C₁₉⋊C₃	—	9	1	no	—
85	C₃×C₁₉⋊C₃	—	9	1	no	—
86	C₃×C₁₉⋊C₃	—	27	3	no	—
87	C₉×C₁₉	—	3	1	no	—
88	C₉×C₁₉	—	9	1	no	—
89	C₉×C₁₉	—	9	1	no	—
90	C₉×C₁₉	—	9	1	no	—
91	#4294	—	3	1	no	—
92	#4294	—	3	1	no	—
93	#4294	—	9	1	no	—
94	#4294	—	9	1	no	—
95	#4294	—	9	1	no	—
96	#4294	—	9	1	no	—
97	#737	—	57	1	no	—
98	#737	—	57	1	no	—
99	C₃×3₋¹⁺²	—	19	1	no	—
100	C₃×H(3,𝔽₃)	—	57	1	no	—
101	#733	—	19	1	no	—
102	#733	—	19	1	no	—
103	#733	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
104	#733	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
105	#732	—	19	1	no	—
106	#732	—	19	1	no	—
107	#732	—	19	1	no	—
108	#732	—	19	1	no	—
109	#732	—	57	1	no	—
110	#732	—	57	1	no	—
111	#732	C₃×C₁₉⋊C₃	1	1	yes	characteristic
112	#731	—	57	1	no	—
113	#731	—	57	1	no	—
114	AΣL(1,𝔽₂₇)	—	57	1	no	—
115	AΣL(1,𝔽₂₇)	—	57	1	no	—
116	C₉⋊C₉	—	19	1	no	—
117	C₉⋊C₉	—	57	1	no	—
118	C₉⋊C₉	—	57	1	no	—
119	C₉⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented
120	C₃²⋊C₉	—	57	1	no	—
121	C₉²	—	19	1	no	—
122	C₃×C₁₉	—	3	1	no	—
123	C₃×C₁₉	—	27	1	no	—
124	C₃×C₁₉	#6911	1	1	yes	characteristic
125	C₁₉⋊C₃	—	9	1	no	—
126	C₁₉⋊C₃	—	27	1	no	—
127	C₁₉⋊C₃	—	27	1	no	—
128	C₁₉⋊C₃	—	27	1	no	—
129	C₁₉⋊C₃	—	27	1	no	—
130	C₃³	—	19	1	no	—
131	C₃³	—	171	1	no	—
132	3₋¹⁺²	—	3	1	no	—
133	3₋¹⁺²	—	3	1	no	—
134	3₋¹⁺²	—	57	1	no	—
135	3₋¹⁺²	—	57	1	no	—
136	3₋¹⁺²	—	171	1	no	—
137	3₋¹⁺²	—	171	1	no	—
138	3₋¹⁺²	—	171	1	no	—
139	3₋¹⁺²	—	171	1	no	—
140	3₋¹⁺²	—	171	1	no	—
141	H(3,𝔽₃)	—	3	1	no	—
142	H(3,𝔽₃)	—	57	1	no	—
143	H(3,𝔽₃)	—	57	1	no	—
144	H(3,𝔽₃)	—	57	1	no	—
145	H(3,𝔽₃)	—	57	1	no	—
146	H(3,𝔽₃)	—	171	2	no	—
147	C₃×C₉	—	3	1	no	—
148	C₃×C₉	—	19	1	no	—
149	C₃×C₉	—	19	1	no	—
150	C₃×C₉	—	57	1	no	—
151	C₃×C₉	—	57	1	no	—
152	C₃×C₉	—	171	1	no	—
153	C₃×C₉	—	171	1	no	—
154	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
155	C₁₉	#10624853	1	1	yes	characteristic, complemented, Sylow
156	C₃²	—	513	3	no	—
157	C₃²	—	9	1	no	—
158	C₃²	—	57	1	no	—
159	C₃²	—	171	1	no	—
160	C₃²	—	171	1	no	—
161	C₃²	—	171	1	no	—
162	C₃²	—	171	1	no	—
163	C₃²	—	171	1	no	—
164	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
165	C₉	—	3	1	no	—
166	C₉	—	9	1	no	—
167	C₉	—	9	1	no	—
168	C₉	—	9	1	no	—
169	C₉	—	57	1	no	—
170	C₉	—	57	1	no	—
171	C₉	—	171	1	no	—
172	C₉	—	171	1	no	—
173	C₉	—	171	1	no	—
174	C₉	—	171	1	no	—
175	C₃	—	513	1	no	—
176	C₃	—	513	1	no	—
177	C₃	—	513	1	no	—
178	C₃	—	513	1	no	—
179	C₃	—	3	1	no	—
180	C₃	—	27	1	no	—
181	C₃	—	171	1	no	—
182	C₃	#17987512	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#17987512
commutator, cosocle	C₃×C₉×C₁₉	C₃³
Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6883	C₃
socle	C₃×C₁₉	#6911
hypercenter	#6883	C₁₉⋊C₃
hypocenter	C₁₉	#10624853
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6883

Complements

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

N	H ≅ G/N	type
1. #17987529	C₃	semidirect
2. #17987529	C₃	semidirect
3. #17987512	C₃	semidirect
4. #17987466	C₃	semidirect
5. #17987391	C₃	semidirect
6. #17987381	C₃	semidirect
7. #17987381	C₃	semidirect
8. #17987380	C₃	semidirect
9. #17987380	C₃	semidirect
10. #17987373	C₃	semidirect
11. #17987371	C₃	semidirect
12. #17987371	C₃	semidirect
13. C₁₉×#6883	C₃	semidirect
18. #13095322	C₃²	semidirect
19. #13095322	C₃²	semidirect
22. #13095320	C₃²	semidirect
23. #13095320	C₃²	semidirect
30. #13095307	C₃²	semidirect
34. C₉×#4294	C₃²	semidirect
35. C₁₉×#733	C₃²	semidirect
36. C₁₉×#733	C₃²	semidirect
38. C₁₉×C₉⋊C₉	C₃²	semidirect
70. #6883	C₁₉⋊C₃	semidirect
103. #733	C₃×C₁₉⋊C₃	semidirect
104. #733	C₃×C₁₉⋊C₃	semidirect
119. C₉⋊C₉	C₃×C₁₉⋊C₃	semidirect
155. C₁₉	#10624853	semidirect

Maximal subgroups

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

subgroup	index
#17987529	3
#17987512	3
#17987466	3
#17987391	3
#17987381	3
#17987380	3
#17987373	3
#17987371	3
C₁₉×#6883	3
#10624853	19