Finite Group Databasebeta

Subgroups — #28357017

Order 13851.

196 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₃×(C₁₉⋊C₃)⋊C₂₇	C₃	1	1	yes	maximal, characteristic
2	#17987573	C₃	1	1	yes	maximal, characteristic, complemented
3	#17987573	C₃	1	1	yes	maximal, characteristic, complemented
4	#17987572	C₃	1	1	yes	maximal, characteristic, complemented
5	#17987572	C₃	1	1	yes	maximal, characteristic, complemented
6	#17987571	C₃	1	1	yes	maximal, characteristic, complemented
7	#17987570	C₃	1	1	yes	maximal, characteristic, complemented
8	#17987557	C₃	1	1	yes	maximal, characteristic
9	#17987557	C₃	1	1	yes	maximal, characteristic
10	#17987555	C₃	1	1	yes	maximal, characteristic, complemented
11	#17987555	C₃	1	1	yes	maximal, characteristic, complemented
12	#17987555	C₃	1	1	yes	maximal, characteristic, complemented
13	#17987409	C₃	1	1	yes	maximal, characteristic
14	(C₁₉⋊C₃)⋊C₂₇	—	3	3	no	—
15	C₃×#10587020	C₃²	1	1	yes	characteristic
16	C₃×#10587020	C₃²	1	1	yes	characteristic
17	C₃×#10587020	C₃²	1	1	yes	characteristic
18	C₃×#10587020	C₃²	1	1	yes	characteristic
19	C₃×#10587020	C₃²	1	1	yes	characteristic
20	C₃×#10587020	C₃²	1	1	yes	characteristic
21	C₃×#10587020	C₃²	1	1	yes	characteristic
22	C₃×#10587020	C₃²	1	1	yes	characteristic
23	C₃×#10587020	C₃²	1	1	yes	characteristic
24	C₃×#10587023	C₉	1	1	yes	characteristic
25	C₃×C₉×C₁₉⋊C₃	C₃²	1	1	yes	characteristic
26	#13095306	C₉	1	1	yes	characteristic
27	#13095305	C₉	1	1	yes	characteristic
28	#13095305	C₉	1	1	yes	characteristic
29	#13095305	C₉	1	1	yes	characteristic
30	#13095305	C₉	1	1	yes	characteristic
31	#13095303	C₉	1	1	yes	characteristic
32	#13095303	C₉	1	1	yes	characteristic
33	C₉×#4294	C₃²	1	1	yes	characteristic
34	C₉×#4294	C₃²	1	1	yes	characteristic
35	C₉×#4294	C₃²	1	1	yes	characteristic
36	C₁₉×C₉⋊C₉	C₉	1	1	yes	characteristic
37	#10624727	—	19	1	no	maximal, Sylow
38	C₃²×C₁₉⋊C₃	C₃×C₉	1	1	yes	characteristic
39	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
40	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
41	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
42	C₃×C₉×C₁₉	C₃×C₉	1	1	yes	characteristic
43	C₃×#4294	C₃×C₉	1	1	yes	characteristic
44	C₃×#4294	C₃×C₉	1	1	yes	characteristic
45	C₃×#4294	C₃×C₉	1	1	yes	characteristic
46	C₃×#4294	C₃×C₉	1	1	yes	characteristic
47	C₃×#4294	C₃×C₉	1	1	yes	characteristic
48	C₃×#4294	C₃×C₉	1	1	yes	characteristic
49	C₃×#4294	C₃×C₉	1	1	yes	characteristic
50	C₃×#4294	C₃³	1	1	yes	characteristic
51	#10587023	—	3	3	no	—
52	C₉×C₁₉⋊C₃	—	3	1	no	—
53	C₉×C₁₉⋊C₃	—	3	1	no	—
54	C₉×C₁₉⋊C₃	—	3	1	no	—
55	#10587020	—	3	1	no	—
56	#10587020	—	3	1	no	—
57	#10587020	—	3	1	no	—
58	#10587020	—	3	1	no	—
59	#10587020	—	3	1	no	—
60	#10587020	—	3	1	no	—
61	#10587020	—	3	1	no	—
62	#10587020	—	3	1	no	—
63	#10587020	3₋¹⁺²	1	3	yes	complemented
64	C₃²×C₂₇	—	19	1	no	—
65	C₉×3₋¹⁺²	—	19	1	no	—
66	C₉⋊C₂₇	—	19	1	no	—
67	C₉⋊C₂₇	—	19	1	no	—
68	C₉⋊C₂₇	—	19	1	no	—
69	C₉⋊C₂₇	—	19	1	no	—
70	C₉⋊C₂₇	—	19	1	no	—
71	C₉⋊C₂₇	—	19	1	no	—
72	C₃²⋊C₂₇	—	19	1	no	—
73	C₃²⋊C₂₇	—	19	1	no	—
74	#6864	—	19	1	no	—
75	#6864	—	19	1	no	—
76	#6864	—	19	1	no	—
77	C₃²×C₁₉	C₃²×C₉	1	1	yes	characteristic
78	C₃×C₁₉⋊C₃	—	3	1	no	—
79	C₃×C₁₉⋊C₃	—	3	1	no	—
80	C₃×C₁₉⋊C₃	—	3	1	no	—
81	C₃×C₁₉⋊C₃	C₂₇⋊C₃	1	3	yes	—
82	C₉×C₁₉	—	3	1	no	—
83	C₉×C₁₉	—	3	1	no	—
84	C₉×C₁₉	—	3	1	no	—
85	C₉×C₁₉	—	3	1	no	—
86	#4294	—	3	1	no	—
87	#4294	—	3	1	no	—
88	#4294	—	3	1	no	—
89	#4294	—	3	1	no	—
90	#4294	—	3	1	no	—
91	#4294	—	3	1	no	—
92	#4294	—	3	1	no	—
93	#4294	C₃×3₋¹⁺²	1	1	yes	characteristic
94	#4294	#737	1	1	yes	characteristic
95	#4294	#737	1	1	yes	characteristic
96	C₃×3₋¹⁺²	—	19	1	no	—
97	C₃²×C₉	—	19	1	no	—
98	C₃×C₂₇	—	19	1	no	—
99	C₃×C₂₇	—	19	1	no	—
100	C₃×C₂₇	—	19	1	no	—
101	C₃×C₂₇	—	19	1	no	—
102	C₃×C₂₇	—	19	1	no	—
103	C₃×C₂₇	—	19	1	no	—
104	C₃×C₂₇	—	19	1	no	—
105	C₃×C₂₇	—	19	1	no	—
106	C₃×C₂₇	—	19	1	no	—
107	C₃×C₂₇	—	57	3	no	—
108	C₉⋊C₉	—	19	1	no	—
109	C₉⋊C₉	—	19	1	no	—
110	C₉⋊C₉	—	19	1	no	—
111	C₉⋊C₉	—	19	1	no	—
112	C₉⋊C₉	—	19	1	no	—
113	C₉⋊C₉	#4296	1	1	yes	characteristic
114	C₃²⋊C₉	—	19	1	no	—
115	C₃²⋊C₉	—	19	1	no	—
116	C₉²	—	19	1	no	—
117	C₉²	—	19	1	no	—
118	C₉²	—	19	1	no	—
119	C₃×C₁₉	C₉×3₋¹⁺²	1	1	yes	characteristic
120	C₃×C₁₉	C₃×C₂₇⋊C₃	1	1	yes	characteristic
121	C₃×C₁₉	#6903	1	1	yes	characteristic
122	C₃×C₁₉	#6903	1	1	yes	characteristic
123	C₁₉⋊C₃	—	3	3	no	—
124	C₃³	—	19	1	no	—
125	3₋¹⁺²	—	57	3	no	—
126	C₃×C₉	—	19	1	no	—
127	C₃×C₉	—	19	1	no	—
128	C₃×C₉	—	19	1	no	—
129	C₃×C₉	—	19	1	no	—
130	C₃×C₉	—	19	1	no	—
131	C₃×C₉	—	19	1	no	—
132	C₃×C₉	—	19	1	no	—
133	C₃×C₉	—	19	1	no	—
134	C₃×C₉	—	57	1	no	—
135	C₃×C₉	—	57	1	no	—
136	C₃×C₉	—	57	1	no	—
137	C₃×C₉	C₃×#4296	1	1	yes	characteristic
138	C₃×C₉	C₃×#4296	1	1	yes	characteristic
139	C₃×C₉	C₃×#4296	1	1	yes	characteristic
140	C₃×C₉	C₃×#4296	1	1	yes	characteristic
141	C₂₇	—	19	3	no	—
142	C₂₇	—	57	1	no	—
143	C₂₇	—	57	1	no	—
144	C₂₇	—	57	1	no	—
145	C₂₇	—	57	1	no	—
146	C₂₇	—	57	1	no	—
147	C₂₇	—	57	1	no	—
148	C₂₇	—	57	1	no	—
149	C₂₇	—	57	1	no	—
150	C₁₉	#10624727	1	1	yes	characteristic, complemented, Sylow
151	C₃²	—	19	3	no	—
152	C₃²	—	57	1	no	—
153	C₃²	—	57	1	no	—
154	C₃²	—	57	1	no	—
155	C₃²	C₃²×#4296	1	1	yes	characteristic
156	C₉	—	3	1	no	—
157	C₉	—	3	1	no	—
158	C₉	—	3	1	no	—
159	C₉	—	3	1	no	—
160	C₉	—	19	1	no	—
161	C₉	—	19	1	no	—
162	C₉	—	19	1	no	—
163	C₉	—	57	1	no	—
164	C₉	—	57	1	no	—
165	C₉	—	57	1	no	—
166	C₉	—	57	1	no	—
167	C₉	—	57	1	no	—
168	C₉	—	57	1	no	—
169	C₉	—	57	1	no	—
170	C₃	—	57	3	no	—
171	C₃	3₋¹⁺²×#4296	1	1	yes	characteristic
172	C₃	(C₁₉⋊C₃)⋊(C₂₇⋊C₃)	1	1	yes	characteristic
173	C₃	#17987599	1	1	yes	characteristic
174	C₃	#17987599	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×#4296
commutator, socle	C₃²×C₁₉	C₃²×C₉
Fitting	C₁₉×C₉⋊C₉	C₉
prime radical	C₃²×C₁₉⋊C₃	C₃×C₉
cosocle	C₃×#4294	C₃³
hypercenter	C₉⋊C₉	#4296
hypocenter	C₁₉	#10624727
LCS	G ▷ C₃²×C₁₉ ▷ C₁₉
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

Complements

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

N	H ≅ G/N	type
2. #17987573	C₃	semidirect
3. #17987573	C₃	semidirect
4. #17987572	C₃	semidirect
5. #17987572	C₃	semidirect
6. #17987571	C₃	semidirect
7. #17987570	C₃	semidirect
10. #17987555	C₃	semidirect
11. #17987555	C₃	semidirect
12. #17987555	C₃	semidirect
63. #10587020	3₋¹⁺²	semidirect
150. C₁₉	#10624727	semidirect

Maximal subgroups

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

subgroup	index
C₃×(C₁₉⋊C₃)⋊C₂₇	3
#17987573	3
#17987572	3
#17987571	3
#17987570	3
#17987557	3
#17987555	3
#17987409	3
#10624727	19