Finite Group Databasebeta

Subgroups — #36382186

Order 22599.

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	#21419419	C₃	1	1	yes	maximal, characteristic, complemented
2	#21419410	C₃	1	1	yes	maximal, characteristic, complemented
3	#21419359	C₃	1	1	yes	maximal, characteristic, complemented
4	#21419282	C₃	1	1	yes	maximal, characteristic, complemented
5	#21419279	C₃	1	1	yes	maximal, characteristic, complemented
6	#21419278	C₃	1	1	yes	maximal, characteristic, complemented
7	#21419275	C₃	1	1	yes	maximal, characteristic, complemented
8	#21419274	C₃	1	1	yes	maximal, characteristic, complemented
9	#21419270	C₃	1	1	yes	maximal, characteristic, complemented
10	#21419267	C₃	1	1	yes	maximal, characteristic, complemented
11	#21419263	C₃	1	1	yes	maximal, characteristic, complemented
12	#21419259	C₃	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#6913	C₃	1	1	yes	maximal, characteristic, complemented
14	#14578912	—	3	1	no	—
15	C₃×#11719689	—	3	1	no	—
16	C₃×#11719689	—	3	1	no	—
17	#14578902	—	3	1	no	—
18	#14578902	—	3	1	no	—
19	#14578902	C₃²	1	1	yes	characteristic
20	#14578902	C₃²	1	1	yes	characteristic
21	#14578902	C₃²	1	1	yes	characteristic
22	#14578902	C₃²	1	1	yes	characteristic
23	#14578902	C₃²	1	1	yes	characteristic
24	#14578900	C₃²	1	1	yes	characteristic
25	#14578897	—	3	1	no	—
26	#14578894	—	3	1	no	—
27	#14578887	C₃²	1	1	yes	characteristic
28	#14578886	C₃²	1	1	yes	characteristic
29	(C₃₁⋊C₉)⋊C₉	—	3	1	no	—
30	(C₃₁⋊C₉)⋊C₉	—	3	1	no	—
31	C₉×C₃₁⋊C₉	C₃²	1	1	yes	characteristic
32	C₃₁×#737	—	3	1	no	—
33	C₃×3₋¹⁺²×C₃₁	C₃²	1	1	yes	characteristic
34	C₃₁×#733	C₃²	1	1	yes	characteristic
35	C₃₁×#733	C₃²	1	1	yes	characteristic
36	C₃₁×#732	C₃²	1	1	yes	characteristic
37	C₃₁×#731	—	3	1	no	—
38	C₃₁×AΣL(1,𝔽₂₇)	—	3	1	no	—
39	C₃³×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
40	3₋¹⁺²×C₃₁	—	3	1	no	—
41	3₋¹⁺²×C₃₁	—	3	1	no	—
42	3₋¹⁺²×C₃₁	—	9	1	no	—
43	3₋¹⁺²×C₃₁	—	9	2	no	—
44	H(3,𝔽₃)×C₃₁	—	3	1	no	—
45	C₃×C₉×C₃₁	—	9	1	no	—
46	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
47	C₃×C₉×C₃₁	H(3,𝔽₃)	1	1	yes	characteristic
48	C₃×C₉×C₃₁	C₃³	1	1	yes	characteristic
49	C₃²⋊(C₃₁⋊C₃)	—	3	1	no	—
50	#11719689	—	3	1	no	—
51	#11719689	—	3	1	no	—
52	#11719689	—	3	1	no	—
53	#11719689	—	3	1	no	—
54	#11719689	—	3	1	no	—
55	#11719689	—	9	3	no	—
56	#11719689	—	9	2	no	—
57	C₃×C₃₁⋊C₉	—	3	1	no	—
58	C₃×C₃₁⋊C₉	—	3	1	no	—
59	C₃×C₃₁⋊C₉	—	3	1	no	—
60	C₃×C₃₁⋊C₉	—	9	1	no	—
61	#11719687	—	9	1	no	—
62	#11719686	—	9	1	no	—
63	C₉×C₃₁⋊C₃	—	9	1	no	—
64	#10624852	—	31	1	no	maximal, Sylow
65	C₃²×C₃₁	—	3	1	no	—
66	C₃²×C₃₁	—	9	1	no	—
67	C₃²×C₃₁	—	9	1	no	—
68	C₃²×C₃₁	C₃×H(3,𝔽₃)	1	1	yes	characteristic
69	C₃×C₃₁⋊C₃	—	9	1	no	—
70	C₉×C₃₁	—	3	1	no	—
71	C₉×C₃₁	—	3	1	no	—
72	C₉×C₃₁	—	3	1	no	—
73	C₉×C₃₁	—	9	1	no	—
74	C₉×C₃₁	—	9	1	no	—
75	C₃₁⋊C₉	—	3	3	no	—
76	C₃₁⋊C₉	—	9	1	no	—
77	C₃₁⋊C₉	—	9	1	no	—
78	C₃₁⋊C₉	—	9	1	no	—
79	C₃₁⋊C₉	—	9	1	no	—
80	C₃₁⋊C₉	—	9	1	no	—
81	C₃₁⋊C₉	—	9	1	no	—
82	C₃₁⋊C₉	—	9	1	no	—
83	#6913	—	31	1	no	—
84	#6913	C₃₁⋊C₃	1	1	yes	characteristic, complemented
85	#6912	—	31	1	no	—
86	#6900	—	31	1	no	—
87	#6883	—	31	1	no	—
88	#6883	—	31	1	no	—
89	#6882	—	31	1	no	—
90	#6882	—	31	1	no	—
91	#6882	—	31	1	no	—
92	#6881	—	31	1	no	—
93	#6880	—	31	1	no	—
94	#6878	—	31	1	no	—
95	#6878	—	31	1	no	—
96	C₃×C₃₁	—	3	1	no	—
97	C₃×C₃₁	—	9	1	no	—
98	C₃×C₃₁	—	27	1	no	—
99	C₃×C₃₁	#6911	1	1	yes	characteristic
100	C₃₁⋊C₃	—	27	1	no	—
101	#737	—	3	1	no	—
102	#737	—	93	1	no	—
103	C₃×3₋¹⁺²	—	93	1	no	—
104	C₃×3₋¹⁺²	—	93	1	no	—
105	C₃×3₋¹⁺²	C₃×C₃₁⋊C₃	1	1	yes	characteristic
106	#733	—	31	1	no	—
107	#733	—	31	1	no	—
108	#733	—	31	1	no	—
109	#733	—	31	1	no	—
110	#733	—	31	1	no	—
111	#733	—	93	1	no	—
112	#733	—	93	1	no	—
113	#733	C₃×C₃₁⋊C₃	1	1	yes	characteristic
114	#733	C₃×C₃₁⋊C₃	1	1	yes	characteristic
115	#732	—	31	1	no	—
116	#732	C₃×C₃₁⋊C₃	1	1	yes	characteristic
117	#731	—	3	1	no	—
118	#731	—	93	1	no	—
119	AΣL(1,𝔽₂₇)	—	3	1	no	—
120	AΣL(1,𝔽₂₇)	—	93	1	no	—
121	C₉⋊C₉	—	31	1	no	—
122	C₉⋊C₉	—	31	1	no	—
123	C₉⋊C₉	—	93	1	no	—
124	C₉⋊C₉	—	93	1	no	—
125	C₉²	—	31	1	no	—
126	C₃₁	#10624852	1	1	yes	characteristic, complemented, Sylow
127	C₃³	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
128	3₋¹⁺²	—	3	1	no	—
129	3₋¹⁺²	—	3	1	no	—
130	3₋¹⁺²	—	9	1	no	—
131	3₋¹⁺²	—	9	2	no	—
132	3₋¹⁺²	—	279	1	no	—
133	3₋¹⁺²	—	279	1	no	—
134	3₋¹⁺²	—	279	2	no	—
135	3₋¹⁺²	—	279	3	no	—
136	3₋¹⁺²	—	93	1	no	—
137	3₋¹⁺²	—	93	1	no	—
138	3₋¹⁺²	—	93	1	no	—
139	3₋¹⁺²	—	93	1	no	—
140	3₋¹⁺²	—	93	1	no	—
141	H(3,𝔽₃)	—	3	1	no	—
142	H(3,𝔽₃)	—	93	1	no	—
143	C₃×C₉	—	9	1	no	—
144	C₃×C₉	—	279	1	no	—
145	C₃×C₉	—	279	1	no	—
146	C₃×C₉	—	93	1	no	—
147	C₃×C₉	—	93	1	no	—
148	C₃×C₉	—	93	1	no	—
149	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
150	C₃×C₉	C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
151	C₃×C₉	C₃²×C₃₁⋊C₃	1	1	yes	characteristic
152	C₃²	—	3	1	no	—
153	C₃²	—	9	1	no	—
154	C₃²	—	9	1	no	—
155	C₃²	—	279	1	no	—
156	C₃²	C₃×C₃²⋊(C₃₁⋊C₃)	1	1	yes	characteristic
157	C₉	—	3	1	no	—
158	C₉	—	3	1	no	—
159	C₉	—	3	1	no	—
160	C₉	—	9	1	no	—
161	C₉	—	9	1	no	—
162	C₉	—	279	1	no	—
163	C₉	—	279	1	no	—
164	C₉	—	279	1	no	—
165	C₉	—	279	1	no	—
166	C₉	—	279	1	no	—
167	C₉	—	279	1	no	—
168	C₉	—	279	1	no	—
169	C₉	—	93	3	no	—
170	C₃	—	3	1	no	—
171	C₃	—	9	1	no	—
172	C₃	—	27	1	no	—
173	C₃	—	837	1	no	—
174	C₃	#21419405	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#21419405
commutator, cosocle	C₃×C₉×C₃₁	C₃³
Frattini	C₃×C₉	C₃²×C₃₁⋊C₃
Fitting	C₃₁×#6913	C₃
socle	C₃×C₃₁	#6911
hypercenter	#6913	C₃₁⋊C₃
hypocenter	C₃₁	#10624852
LCS	G ▷ C₃×C₉×C₃₁ ▷ C₃²×C₃₁ ▷ C₃×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6913

Complements

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

N	H ≅ G/N	type
1. #21419419	C₃	semidirect
2. #21419410	C₃	semidirect
3. #21419359	C₃	semidirect
4. #21419282	C₃	semidirect
5. #21419279	C₃	semidirect
6. #21419278	C₃	semidirect
7. #21419275	C₃	semidirect
8. #21419274	C₃	semidirect
9. #21419270	C₃	semidirect
10. #21419267	C₃	semidirect
11. #21419263	C₃	semidirect
12. #21419259	C₃	semidirect
13. C₃₁×#6913	C₃	semidirect
84. #6913	C₃₁⋊C₃	semidirect
126. C₃₁	#10624852	semidirect

Maximal subgroups

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

subgroup	index
#21419419	3
#21419410	3
#21419359	3
#21419282	3
#21419279	3
#21419278	3
#21419275	3
#21419274	3
#21419270	3
#21419267	3
#21419263	3
#21419259	3
C₃₁×#6913	3
#10624852	31