Finite Group Databasebeta

Subgroups — #44533805

Order 31347.

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

normal subgroups only

This group has 402 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	C₃×#16812525	C₃	1	1	yes	maximal, characteristic
2	#23782896	C₃	1	3	yes	maximal, complemented
3	#23782895	C₃	1	3	yes	maximal, complemented
4	#23782888	C₃	1	3	yes	maximal
5	C₃×#16812515	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#16812515	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄₃×#733	C₃	1	1	yes	maximal, characteristic, complemented
8	#16812525	—	3	3	no	—
9	#16812525	—	3	3	no	—
10	#16812525	—	3	3	no	—
11	C₃×#11923958	C₃²	1	3	yes	—
12	C₃×#11923958	C₃²	1	3	yes	—
13	C₃×#11923958	C₃²	1	1	yes	characteristic
14	C₃×#11923958	C₃²	1	1	yes	characteristic
15	C₃×#11923957	—	3	1	no	—
16	C₃×#11923956	—	3	1	no	—
17	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²	1	1	yes	characteristic
18	C₃×C₉×C₄₃⋊C₃	—	3	1	no	—
19	#16812515	—	3	9	no	—
20	#16812515	—	3	1	no	—
21	#16812515	—	3	1	no	—
22	#16812515	—	3	1	no	—
23	#16812515	—	3	1	no	—
24	#16812515	—	3	1	no	—
25	#16812515	—	3	1	no	—
26	#16812514	—	3	9	no	—
27	#16812511	—	3	9	no	—
28	C₃×3₋¹⁺²×C₄₃	C₃²	1	3	yes	—
29	C₃²×C₉×C₄₃	C₃²	1	1	yes	characteristic
30	C₄₃×#733	—	3	1	no	—
31	C₄₃×#733	—	3	1	no	—
32	C₄₃×#733	—	3	1	no	—
33	C₃²×C₄₃⋊C₃	—	3	1	no	—
34	#11923958	—	3	1	no	—
35	#11923958	—	3	1	no	—
36	#11923958	—	3	1	no	—
37	#11923958	—	3	1	no	—
38	#11923958	—	3	3	no	—
39	#11923958	—	3	3	no	—
40	#11923958	—	3	3	no	—
41	#11923958	—	3	3	no	—
42	#11923958	—	3	3	no	—
43	#11923958	—	3	3	no	—
44	#11923958	H(3,𝔽₃)	1	3	yes	—
45	#11923958	H(3,𝔽₃)	1	3	yes	—
46	#11923957	—	3	9	no	—
47	#11923956	—	3	9	no	—
48	C₃²⋊(C₄₃⋊C₃)	—	3	1	no	—
49	C₃²⋊(C₄₃⋊C₃)	—	3	1	no	—
50	C₃²⋊(C₄₃⋊C₃)	H(3,𝔽₃)	1	3	yes	—
51	C₃×C₄₃⋊C₉	—	3	1	no	—
52	C₃×C₄₃⋊C₉	—	3	1	no	—
53	C₃×C₄₃⋊C₉	—	3	3	no	—
54	C₃×C₄₃⋊C₉	—	3	3	no	—
55	C₉×C₄₃⋊C₃	—	3	9	no	—
56	C₃³×C₄₃	C₃³	1	1	yes	characteristic
57	3₋¹⁺²×C₄₃	—	3	3	no	—
58	3₋¹⁺²×C₄₃	—	3	3	no	—
59	3₋¹⁺²×C₄₃	—	3	3	no	—
60	C₃×C₉×C₄₃	—	3	1	no	—
61	C₃×C₉×C₄₃	—	3	1	no	—
62	C₃×C₉×C₄₃	—	3	1	no	—
63	C₃×C₉×C₄₃	—	3	3	no	—
64	C₃×C₉×C₄₃	H(3,𝔽₃)	1	1	yes	characteristic
65	C₃×C₉×C₄₃	H(3,𝔽₃)	1	1	yes	characteristic
66	C₃×C₉×C₄₃	H(3,𝔽₃)	1	1	yes	characteristic
67	#10624758	—	43	1	no	maximal, Sylow
68	C₃²×C₄₃	—	9	1	no	—
69	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
70	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
71	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
72	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
73	C₃×C₄₃⋊C₃	—	3	3	no	—
74	C₃×C₄₃⋊C₃	—	9	3	no	—
75	C₉×C₄₃	—	3	1	no	—
76	C₉×C₄₃	—	3	1	no	—
77	C₉×C₄₃	—	3	1	no	—
78	C₉×C₄₃	—	9	3	no	—
79	C₄₃⋊C₉	—	3	3	no	—
80	C₄₃⋊C₉	—	3	3	no	—
81	C₄₃⋊C₉	—	9	3	no	—
82	C₄₃⋊C₉	—	9	3	no	—
83	C₃×#737	—	43	1	no	—
84	#6912	—	43	3	no	—
85	#6912	—	43	3	no	—
86	#6910	—	43	3	no	—
87	C₃×#733	—	43	1	no	—
88	C₃×#733	—	43	1	no	—
89	C₃×#733	C₄₃⋊C₃	1	1	yes	characteristic, complemented
90	C₃×C₄₃	—	3	1	no	—
91	C₃×C₄₃	—	3	1	no	—
92	C₃×C₄₃	—	3	1	no	—
93	C₃×C₄₃	—	3	1	no	—
94	C₃×C₄₃	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
95	C₄₃⋊C₃	—	9	3	no	—
96	#737	—	129	3	no	—
97	#737	—	129	3	no	—
98	#737	—	129	3	no	—
99	C₃×3₋¹⁺²	—	43	1	no	—
100	C₃×3₋¹⁺²	—	43	1	no	—
101	C₃×3₋¹⁺²	—	43	3	no	—
102	C₃×3₋¹⁺²	—	43	3	no	—
103	C₃×3₋¹⁺²	—	129	1	no	—
104	C₃×3₋¹⁺²	—	129	1	no	—
105	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	3	yes	—
106	C₃×H(3,𝔽₃)	—	43	1	no	—
107	C₃²×C₉	—	129	1	no	—
108	C₃²×C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
109	#733	—	3	1	no	—
110	#733	—	3	1	no	—
111	#733	—	3	1	no	—
112	#733	—	129	9	no	—
113	#733	—	129	9	no	—
114	#733	—	129	1	no	—
115	#733	—	129	1	no	—
116	#733	—	129	1	no	—
117	#733	—	129	1	no	—
118	#733	—	129	1	no	—
119	#733	—	129	1	no	—
120	#731	—	129	9	no	—
121	C₄₃	#10624758	1	1	yes	characteristic, complemented, Sylow
122	C₃³	—	129	1	no	—
123	C₃³	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
124	3₋¹⁺²	—	3	3	no	—
125	3₋¹⁺²	—	3	3	no	—
126	3₋¹⁺²	—	3	3	no	—
127	3₋¹⁺²	—	43	3	no	—
128	3₋¹⁺²	—	43	3	no	—
129	3₋¹⁺²	—	129	9	no	—
130	3₋¹⁺²	—	129	9	no	—
131	3₋¹⁺²	—	129	1	no	—
132	3₋¹⁺²	—	129	1	no	—
133	3₋¹⁺²	—	129	1	no	—
134	3₋¹⁺²	—	129	1	no	—
135	3₋¹⁺²	—	129	3	no	—
136	3₋¹⁺²	—	129	3	no	—
137	3₋¹⁺²	—	129	3	no	—
138	3₋¹⁺²	—	129	3	no	—
139	3₋¹⁺²	—	129	3	no	—
140	3₋¹⁺²	—	129	3	no	—
141	H(3,𝔽₃)	—	43	3	no	—
142	H(3,𝔽₃)	—	129	1	no	—
143	H(3,𝔽₃)	—	129	1	no	—
144	C₃×C₉	—	3	1	no	—
145	C₃×C₉	—	3	1	no	—
146	C₃×C₉	—	3	1	no	—
147	C₃×C₉	—	3	3	no	—
148	C₃×C₉	—	129	9	no	—
149	C₃×C₉	—	129	1	no	—
150	C₃×C₉	—	129	1	no	—
151	C₃×C₉	—	129	3	no	—
152	C₃×C₉	—	129	3	no	—
153	C₃×C₉	C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
154	C₃×C₉	C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
155	C₃×C₉	C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
156	C₃²	—	9	1	no	—
157	C₃²	—	129	3	no	—
158	C₃²	—	387	3	no	—
159	C₃²	H(3,𝔽₃)×C₄₃⋊C₃	1	1	yes	characteristic
160	C₃²	H(3,𝔽₃)×C₄₃⋊C₃	1	1	yes	characteristic
161	C₃²	H(3,𝔽₃)×C₄₃⋊C₃	1	1	yes	characteristic
162	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
163	C₉	—	3	1	no	—
164	C₉	—	3	1	no	—
165	C₉	—	3	1	no	—
166	C₉	—	9	3	no	—
167	C₉	—	129	3	no	—
168	C₉	—	129	3	no	—
169	C₉	—	387	3	no	—
170	C₉	—	387	3	no	—
171	C₃	—	3	1	no	—
172	C₃	—	3	1	no	—
173	C₃	—	3	1	no	—
174	C₃	—	3	1	no	—
175	C₃	—	387	3	no	—
176	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	C₃×C₄₃×#733	C₃
socle	C₃×C₄₃	C₃²⋊H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₄₃⋊C₃)	C₃²
hypercenter	C₃×#733	C₄₃⋊C₃
hypocenter	C₄₃	#10624758
LCS	G ▷ C₃³×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#733

Complements

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

N	H ≅ G/N	type
2. #23782896	C₃	semidirect
3. #23782895	C₃	semidirect
5. C₃×#16812515	C₃	semidirect
6. C₃×#16812515	C₃	semidirect
7. C₃×C₄₃×#733	C₃	semidirect
89. C₃×#733	C₄₃⋊C₃	semidirect
121. C₄₃	#10624758	semidirect

Maximal subgroups

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

subgroup	index
C₃×#16812525	3
#23782896	3
#23782895	3
#23782888	3
C₃×#16812515	3
C₃×C₄₃×#733	3
#10624758	43