Finite Group Databasebeta

Subgroups — #28355813

Order 13851.

188 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 46 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	#17987501	C₃	1	1	yes	maximal, characteristic, complemented
2	#17987501	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987501	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987404	C₃	1	1	yes	maximal, characteristic, complemented
5	#17987390	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987389	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987388	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987384	C₃	1	1	yes	maximal, characteristic, complemented
9	#17987383	C₃	1	1	yes	maximal, characteristic, complemented
10	#17987382	C₃	1	1	yes	maximal, characteristic, complemented
11	#17987381	C₃	1	1	yes	maximal, characteristic, complemented
12	#17987380	C₃	1	1	yes	maximal, characteristic, complemented
13	C₁₉×#6882	C₃	1	1	yes	maximal, characteristic, complemented
14	#13095333	—	3	1	no	—
15	#13095333	—	3	1	no	—
16	#13095333	—	3	1	no	—
17	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
18	#13095322	C₃²	1	1	yes	characteristic
19	#13095322	C₃²	1	1	yes	characteristic
20	#13095322	C₃²	1	1	yes	characteristic
21	#13095320	C₃²	1	1	yes	characteristic
22	#13095320	C₃²	1	1	yes	characteristic
23	#13095320	C₃²	1	1	yes	characteristic
24	#13095318	—	3	1	no	—
25	#13095318	—	3	1	no	—
26	#13095318	—	3	1	no	—
27	#13095315	—	3	1	no	—
28	#13095315	—	3	1	no	—
29	#13095315	—	3	1	no	—
30	#13095307	C₃²	1	1	yes	characteristic
31	#13095306	C₃²	1	1	yes	characteristic
32	#13095305	—	3	1	no	—
33	#13095303	—	3	1	no	—
34	C₉×#4294	—	3	1	no	—
35	C₁₉×#733	C₃²	1	1	yes	characteristic
36	C₁₉×#733	C₃²	1	1	yes	characteristic
37	C₁₉×#733	C₃²	1	1	yes	characteristic
38	C₁₉×C₉⋊C₉	C₃²	1	1	yes	characteristic
39	#10624849	—	19	1	no	maximal, Sylow
40	C₃²×C₁₉⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
41	C₁₉×3₋¹⁺²	—	3	1	no	—
42	C₁₉×3₋¹⁺²	—	3	1	no	—
43	C₁₉×3₋¹⁺²	—	3	1	no	—
44	C₃×C₉×C₁₉	—	3	1	no	—
45	C₃×C₉×C₁₉	C₃³	1	1	yes	characteristic
46	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
47	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
48	C₃²⋊(C₁₉⋊C₃)	—	3	1	no	—
49	#10587025	—	3	1	no	—
50	#10587025	—	3	1	no	—
51	#10587025	—	3	1	no	—
52	C₃×#4294	—	3	1	no	—
53	C₃×#4294	—	3	1	no	—
54	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
55	C₃×#4294	H(3,𝔽₃)	1	1	yes	characteristic
56	#10587023	—	9	1	no	—
57	#10587023	—	9	1	no	—
58	#10587023	—	9	1	no	—
59	#10587022	—	9	1	no	—
60	#10587022	—	9	1	no	—
61	#10587022	—	9	1	no	—
62	C₉×C₁₉⋊C₃	—	3	1	no	—
63	C₉×C₁₉⋊C₃	—	3	1	no	—
64	C₉×C₁₉⋊C₃	—	3	1	no	—
65	C₉×C₁₉⋊C₃	—	9	1	no	—
66	C₉×C₁₉⋊C₃	—	9	1	no	—
67	C₉×C₁₉⋊C₃	—	9	1	no	—
68	#6908	—	19	1	no	—
69	#6908	—	19	1	no	—
70	#6908	—	19	1	no	—
71	#6887	—	19	1	no	—
72	#6883	—	19	1	no	—
73	#6883	—	19	1	no	—
74	#6883	—	19	1	no	—
75	#6882	—	19	1	no	—
76	#6882	—	19	1	no	—
77	#6882	C₁₉⋊C₃	1	1	yes	characteristic, complemented
78	#6881	—	19	1	no	—
79	#6881	—	19	1	no	—
80	#6881	—	19	1	no	—
81	C₃²×C₁₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
82	C₃×C₁₉⋊C₃	—	3	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	—	9	1	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	C₉×C₁₉	—	9	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	#4294	—	9	1	no	—
98	#4294	#732	1	1	yes	characteristic
99	#4294	#732	1	1	yes	characteristic
100	#4294	#732	1	1	yes	characteristic
101	#737	—	57	1	no	—
102	#737	—	57	1	no	—
103	#737	—	57	1	no	—
104	C₃²×C₉	—	19	1	no	—
105	#733	—	19	1	no	—
106	#733	—	19	1	no	—
107	#733	—	19	1	no	—
108	#733	C₃×C₁₉⋊C₃	1	1	yes	characteristic
109	#733	C₃×C₁₉⋊C₃	1	1	yes	characteristic
110	#733	C₃×C₁₉⋊C₃	1	1	yes	characteristic
111	#732	—	19	1	no	—
112	#732	—	19	1	no	—
113	#732	—	19	1	no	—
114	#731	—	57	1	no	—
115	#731	—	57	1	no	—
116	#731	—	57	1	no	—
117	AΣL(1,𝔽₂₇)	—	57	1	no	—
118	AΣL(1,𝔽₂₇)	—	57	1	no	—
119	AΣL(1,𝔽₂₇)	—	57	1	no	—
120	C₉⋊C₉	—	19	1	no	—
121	C₉⋊C₉	—	19	1	no	—
122	C₉⋊C₉	—	57	1	no	—
123	C₉⋊C₉	C₃×C₁₉⋊C₃	1	1	yes	characteristic
124	C₃²⋊C₉	—	57	1	no	—
125	C₉²	—	57	1	no	—
126	C₃×C₁₉	—	3	1	no	—
127	C₃×C₁₉	C₃×#732	1	1	yes	characteristic
128	C₁₉⋊C₃	—	9	1	no	—
129	C₁₉⋊C₃	—	27	1	no	—
130	C₁₉⋊C₃	—	27	1	no	—
131	C₁₉⋊C₃	—	27	1	no	—
132	C₃³	—	19	1	no	—
133	3₋¹⁺²	—	3	1	no	—
134	3₋¹⁺²	—	3	1	no	—
135	3₋¹⁺²	—	3	1	no	—
136	3₋¹⁺²	—	57	1	no	—
137	3₋¹⁺²	—	57	1	no	—
138	3₋¹⁺²	—	57	1	no	—
139	3₋¹⁺²	—	171	1	no	—
140	3₋¹⁺²	—	171	1	no	—
141	3₋¹⁺²	—	171	1	no	—
142	3₋¹⁺²	—	171	1	no	—
143	3₋¹⁺²	—	171	1	no	—
144	3₋¹⁺²	—	171	1	no	—
145	H(3,𝔽₃)	—	57	1	no	—
146	H(3,𝔽₃)	—	57	1	no	—
147	H(3,𝔽₃)	—	57	1	no	—
148	C₃×C₉	—	3	1	no	—
149	C₃×C₉	—	19	1	no	—
150	C₃×C₉	—	19	1	no	—
151	C₃×C₉	—	57	1	no	—
152	C₃×C₉	—	57	1	no	—
153	C₃×C₉	—	57	1	no	—
154	C₃×C₉	—	57	1	no	—
155	C₃×C₉	—	57	1	no	—
156	C₃×C₉	—	171	1	no	—
157	C₃×C₉	—	171	1	no	—
158	C₃×C₉	—	171	1	no	—
159	C₃×C₉	C₃²×C₁₉⋊C₃	1	1	yes	characteristic
160	C₁₉	#10624849	1	1	yes	characteristic, complemented, Sylow
161	C₃²	—	57	1	no	—
162	C₃²	—	171	1	no	—
163	C₃²	—	171	1	no	—
164	C₃²	—	171	1	no	—
165	C₃²	—	171	1	no	—
166	C₃²	H(3,𝔽₃)×C₁₉⋊C₃	1	1	yes	characteristic
167	C₉	—	3	1	no	—
168	C₉	—	9	1	no	—
169	C₉	—	9	1	no	—
170	C₉	—	9	1	no	—
171	C₉	—	9	1	no	—
172	C₉	—	19	1	no	—
173	C₉	—	19	1	no	—
174	C₉	—	19	1	no	—
175	C₉	—	57	1	no	—
176	C₉	—	171	1	no	—
177	C₉	—	171	1	no	—
178	C₉	—	171	1	no	—
179	C₉	—	171	1	no	—
180	C₉	—	171	1	no	—
181	C₃	—	513	1	no	—
182	C₃	—	513	1	no	—
183	C₃	—	513	1	no	—
184	C₃	—	3	1	no	—
185	C₃	—	171	1	no	—
186	C₃	C₁₉⋊C₃×#732	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₁₉⋊C₃×#732
commutator, cosocle	C₃×C₉×C₁₉	C₃³
Frattini	C₃×C₉	C₃²×C₁₉⋊C₃
Fitting	C₁₉×#6882	C₃
socle	C₃×C₁₉	C₃×#732
hypercenter	#6882	C₁₉⋊C₃
hypocenter	C₁₉	#10624849
LCS	G ▷ C₃×C₉×C₁₉ ▷ C₃²×C₁₉ ▷ C₃×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6882

Complements

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

N	H ≅ G/N	type
1. #17987501	C₃	semidirect
2. #17987501	C₃	semidirect
3. #17987501	C₃	semidirect
4. #17987404	C₃	semidirect
5. #17987390	C₃	semidirect
6. #17987389	C₃	semidirect
7. #17987388	C₃	semidirect
8. #17987384	C₃	semidirect
9. #17987383	C₃	semidirect
10. #17987382	C₃	semidirect
11. #17987381	C₃	semidirect
12. #17987380	C₃	semidirect
13. C₁₉×#6882	C₃	semidirect
77. #6882	C₁₉⋊C₃	semidirect
160. C₁₉	#10624849	semidirect

Maximal subgroups

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

subgroup	index
#17987501	3
#17987404	3
#17987390	3
#17987389	3
#17987388	3
#17987384	3
#17987383	3
#17987382	3
#17987381	3
#17987380	3
C₁₉×#6882	3
#10624849	19