Finite Group Databasebeta

Subgroups — #44533754

Order 31347.

414 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 61 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	#23782924	C₃	1	1	yes	maximal, characteristic, complemented
2	#23782895	C₃	1	2	yes	maximal, complemented
3	#23782886	C₃	1	2	yes	maximal, complemented
4	#23782886	C₃	1	2	yes	maximal, complemented
5	#23782886	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#16812514	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×#16812509	C₃	1	1	yes	maximal, characteristic, complemented
8	#23782806	C₃	1	2	yes	maximal, complemented
9	C₄₃×#6911	C₃	1	1	yes	maximal, characteristic
10	#16812524	—	3	6	no	—
11	#16812524	—	3	3	no	—
12	C₃×#11923958	—	3	3	no	—
13	C₃×#11923958	C₃²	1	2	yes	—
14	C₃×#11923958	C₃²	1	2	yes	—
15	C₃×#11923958	C₃²	1	1	yes	characteristic
16	C₃×#11923958	C₃²	1	1	yes	characteristic
17	C₃×#11923958	C₃²	1	2	yes	—
18	C₃²×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
19	#16812514	—	3	3	no	—
20	#16812514	—	3	6	no	—
21	#16812509	—	3	6	no	—
22	#16812509	—	3	6	no	—
23	#16812509	—	3	3	no	—
24	#16812509	—	3	3	no	—
25	C₃²⋊(C₄₃⋊C₉)	—	3	6	no	—
26	C₃×3₋¹⁺²×C₄₃	C₃²	1	1	yes	characteristic
27	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	2	yes	—
28	C₃×H(3,𝔽₃)×C₄₃	C₃²	1	1	yes	characteristic
29	C₄₃×#732	—	3	3	no	—
30	#11923958	—	3	18	no	—
31	#11923958	—	3	2	no	—
32	#11923958	—	3	2	no	—
33	#11923958	—	3	2	no	—
34	#11923958	—	3	2	no	—
35	#11923958	—	3	2	no	—
36	#11923958	—	3	2	no	—
37	#11923958	—	3	2	no	—
38	#11923958	—	3	2	no	—
39	#11923958	—	3	2	no	—
40	#11923958	—	3	1	no	—
41	#11923958	—	3	2	no	—
42	#11923958	—	3	2	no	—
43	#11923958	H(3,𝔽₃)	1	3	yes	—
44	C₃×C₄₃⋊C₉	—	3	9	no	—
45	C₃×C₄₃⋊C₉	—	3	1	no	—
46	C₃×C₄₃⋊C₉	—	3	2	no	—
47	C₃×C₄₃⋊C₉	—	3	2	no	—
48	C₃×C₄₃⋊C₉	—	3	1	no	—
49	C₃×C₄₃⋊C₉	—	3	2	no	—
50	C₃×C₄₃⋊C₉	—	3	1	no	—
51	C₃×C₄₃⋊C₉	—	3	1	no	—
52	C₃×C₄₃⋊C₉	H(3,𝔽₃)	1	6	yes	—
53	C₃³×C₄₃	—	3	1	no	—
54	C₃³×C₄₃	—	3	2	no	—
55	C₃³×C₄₃	C₃³	1	1	yes	characteristic
56	3₋¹⁺²×C₄₃	—	3	1	no	—
57	3₋¹⁺²×C₄₃	—	3	2	no	—
58	H(3,𝔽₃)×C₄₃	—	3	2	no	—
59	H(3,𝔽₃)×C₄₃	—	3	2	no	—
60	H(3,𝔽₃)×C₄₃	—	3	2	no	—
61	H(3,𝔽₃)×C₄₃	—	3	2	no	—
62	H(3,𝔽₃)×C₄₃	H(3,𝔽₃)	1	3	yes	—
63	C₃×C₉×C₄₃	—	3	1	no	—
64	#10624751	—	43	1	no	maximal, Sylow
65	C₃²×C₄₃	—	3	3	no	—
66	C₃²×C₄₃	—	9	1	no	—
67	C₃²×C₄₃	—	9	6	no	—
68	C₃²×C₄₃	—	9	2	no	—
69	C₃²×C₄₃	—	27	1	no	—
70	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
71	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	2	yes	—
72	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
73	C₉×C₄₃	—	9	1	no	—
74	C₄₃⋊C₉	—	3	3	no	—
75	C₄₃⋊C₉	—	3	6	no	—
76	C₄₃⋊C₉	—	9	1	no	—
77	C₄₃⋊C₉	—	9	2	no	—
78	C₄₃⋊C₉	—	9	2	no	—
79	C₄₃⋊C₉	—	9	1	no	—
80	3₋¹⁺⁴	—	43	1	no	—
81	#6912	—	43	2	no	—
82	#6911	C₄₃⋊C₃	1	1	yes	characteristic
83	#6910	—	43	2	no	—
84	#6910	—	43	2	no	—
85	#6910	—	43	1	no	—
86	C₃×#733	—	43	1	no	—
87	C₃×#731	—	43	1	no	—
88	#6893	—	43	2	no	—
89	C₃×C₄₃	—	3	2	no	—
90	C₃×C₄₃	—	3	1	no	—
91	C₃×C₄₃	—	3	1	no	—
92	C₃×C₄₃	—	9	3	no	—
93	C₃×C₄₃	—	27	2	no	—
94	C₃×C₄₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
95	#737	—	129	6	no	—
96	#737	—	129	3	no	—
97	C₃×3₋¹⁺²	—	43	2	no	—
98	C₃×3₋¹⁺²	—	43	1	no	—
99	C₃×3₋¹⁺²	—	43	2	no	—
100	C₃×3₋¹⁺²	—	43	2	no	—
101	C₃×3₋¹⁺²	—	43	1	no	—
102	C₃×3₋¹⁺²	—	129	3	no	—
103	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
104	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	2	yes	—
105	C₃×H(3,𝔽₃)	C₃×C₄₃⋊C₃	1	1	yes	characteristic
106	C₃²×C₉	—	43	1	no	—
107	#733	—	129	6	no	—
108	#733	—	129	3	no	—
109	#732	—	3	3	no	—
110	#731	—	129	6	no	—
111	#731	—	129	6	no	—
112	#731	—	129	3	no	—
113	#731	—	129	3	no	—
114	C₃²⋊C₉	—	129	6	no	—
115	C₄₃	#10624751	1	1	yes	characteristic, complemented, Sylow
116	C₃³	—	3	1	no	—
117	C₃³	—	3	2	no	—
118	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
119	3₋¹⁺²	—	3	2	no	—
120	3₋¹⁺²	—	3	1	no	—
121	3₋¹⁺²	—	43	3	no	—
122	3₋¹⁺²	—	129	18	no	—
123	3₋¹⁺²	—	129	2	no	—
124	3₋¹⁺²	—	129	2	no	—
125	3₋¹⁺²	—	129	2	no	—
126	3₋¹⁺²	—	129	2	no	—
127	3₋¹⁺²	—	129	2	no	—
128	3₋¹⁺²	—	129	2	no	—
129	3₋¹⁺²	—	129	2	no	—
130	3₋¹⁺²	—	129	2	no	—
131	3₋¹⁺²	—	129	2	no	—
132	3₋¹⁺²	—	129	1	no	—
133	3₋¹⁺²	—	129	2	no	—
134	3₋¹⁺²	—	129	2	no	—
135	H(3,𝔽₃)	—	3	2	no	—
136	H(3,𝔽₃)	—	3	2	no	—
137	H(3,𝔽₃)	—	3	2	no	—
138	H(3,𝔽₃)	—	3	2	no	—
139	H(3,𝔽₃)	C₃²⋊(C₄₃⋊C₃)	1	3	yes	—
140	C₃×C₉	—	3	1	no	—
141	C₃×C₉	—	43	6	no	—
142	C₃×C₉	—	129	9	no	—
143	C₃×C₉	—	129	1	no	—
144	C₃×C₉	—	129	2	no	—
145	C₃×C₉	—	129	2	no	—
146	C₃×C₉	—	129	2	no	—
147	C₃×C₉	—	129	1	no	—
148	C₃×C₉	—	129	1	no	—
149	C₃×C₉	—	129	1	no	—
150	C₃²	—	3	3	no	—
151	C₃²	—	9	1	no	—
152	C₃²	—	9	6	no	—
153	C₃²	—	9	2	no	—
154	C₃²	—	27	1	no	—
155	C₃²	H(3,𝔽₃)×C₄₃⋊C₃	1	2	yes	—
156	C₃²	H(3,𝔽₃)×C₄₃⋊C₃	1	1	yes	characteristic
157	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
158	C₉	—	9	1	no	—
159	C₉	—	129	6	no	—
160	C₉	—	129	3	no	—
161	C₉	—	387	1	no	—
162	C₉	—	387	2	no	—
163	C₉	—	387	2	no	—
164	C₉	—	387	1	no	—
165	C₃	—	3	2	no	—
166	C₃	—	3	1	no	—
167	C₃	—	3	1	no	—
168	C₃	—	9	3	no	—
169	C₃	—	27	2	no	—
170	C₃	(C₃×C₄₃⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	(C₃×C₄₃⋊C₃)⋊H(3,𝔽₃)
commutator, cosocle	C₃³×C₄₃	C₃³
Frattini	C₃³	C₃²×C₄₃⋊C₃
Fitting, prime radical	C₄₃×#6911	C₃
socle	C₃×C₄₃	C₃²⋊H(3,𝔽₃)
hypercenter	#6911	C₄₃⋊C₃
hypocenter	C₄₃	#10624751
LCS	G ▷ C₃³×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6911

Complements

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

N	H ≅ G/N	type
1. #23782924	C₃	semidirect
2. #23782895	C₃	semidirect
3. #23782886	C₃	semidirect
4. #23782886	C₃	semidirect
5. #23782886	C₃	semidirect
6. C₃×#16812514	C₃	semidirect
7. C₃×#16812509	C₃	semidirect
8. #23782806	C₃	semidirect
115. C₄₃	#10624751	semidirect

Maximal subgroups

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

subgroup	index
#23782924	3
#23782895	3
#23782886	3
C₃×#16812514	3
C₃×#16812509	3
#23782806	3
C₄₃×#6911	3
#10624751	43