Finite Group Databasebeta

Subgroups — #44534626

Order 31347.

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

normal subgroups only

This group has 178 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	#23782905	C₃	1	1	yes	maximal, characteristic, complemented
2	#23782896	C₃	1	1	yes	maximal, characteristic, complemented
3	#23782838	C₃	1	1	yes	maximal, characteristic, complemented
4	#23782769	C₃	1	1	yes	maximal, characteristic, complemented
5	#23782766	C₃	1	1	yes	maximal, characteristic, complemented
6	#23782763	C₃	1	1	yes	maximal, characteristic, complemented
7	#23782761	C₃	1	1	yes	maximal, characteristic, complemented
8	#23782760	C₃	1	1	yes	maximal, characteristic, complemented
9	#23782755	C₃	1	1	yes	maximal, characteristic, complemented
10	#23782752	C₃	1	1	yes	maximal, characteristic, complemented
11	#23782748	C₃	1	1	yes	maximal, characteristic, complemented
12	#23782745	C₃	1	1	yes	maximal, characteristic, complemented
13	C₄₃×#6913	C₃	1	1	yes	maximal, characteristic, complemented
14	#16812525	—	3	1	no	—
15	C₃×#11923958	—	3	1	no	—
16	#16812515	—	3	1	no	—
17	#16812515	—	3	1	no	—
18	#16812515	C₃²	1	1	yes	characteristic
19	#16812515	C₃²	1	1	yes	characteristic
20	#16812515	C₃²	1	1	yes	characteristic
21	#16812515	C₃²	1	1	yes	characteristic
22	#16812515	C₃²	1	1	yes	characteristic
23	#16812513	C₃²	1	1	yes	characteristic
24	#16812510	—	3	1	no	—
25	#16812507	—	3	1	no	—
26	#16812500	C₃²	1	1	yes	characteristic
27	#16812499	C₃²	1	1	yes	characteristic
28	(C₄₃⋊C₉)⋊C₉	—	3	1	no	—
29	(C₄₃⋊C₉)⋊C₉	—	3	1	no	—
30	C₃²⋊(C₄₃⋊C₉)	—	3	1	no	—
31	C₉×C₄₃⋊C₉	C₃²	1	1	yes	characteristic
32	C₄₃×#737	—	3	1	no	—
33	C₃×3₋¹⁺²×C₄₃	C₃²	1	1	yes	characteristic
34	C₄₃×#733	C₃²	1	1	yes	characteristic
35	C₄₃×#733	C₃²	1	1	yes	characteristic
36	C₄₃×#732	C₃²	1	1	yes	characteristic
37	C₄₃×#731	—	3	1	no	—
38	C₄₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
39	#11923958	—	3	1	no	—
40	#11923958	—	3	1	no	—
41	#11923958	—	3	1	no	—
42	#11923958	—	3	1	no	—
43	#11923958	—	3	1	no	—
44	#11923958	—	9	2	no	—
45	#11923957	—	9	1	no	—
46	#11923956	—	9	1	no	—
47	C₃²⋊(C₄₃⋊C₃)	—	3	1	no	—
48	C₃×C₄₃⋊C₉	—	3	1	no	—
49	C₃×C₄₃⋊C₉	—	3	1	no	—
50	C₃×C₄₃⋊C₉	—	3	1	no	—
51	C₃×C₄₃⋊C₉	—	9	1	no	—
52	C₉×C₄₃⋊C₃	—	9	1	no	—
53	C₃³×C₄₃	H(3,𝔽₃)	1	1	yes	characteristic
54	3₋¹⁺²×C₄₃	—	3	1	no	—
55	3₋¹⁺²×C₄₃	—	3	1	no	—
56	3₋¹⁺²×C₄₃	—	9	1	no	—
57	3₋¹⁺²×C₄₃	—	9	2	no	—
58	H(3,𝔽₃)×C₄₃	—	3	1	no	—
59	C₃×C₉×C₄₃	—	9	1	no	—
60	C₃×C₉×C₄₃	H(3,𝔽₃)	1	1	yes	characteristic
61	C₃×C₉×C₄₃	H(3,𝔽₃)	1	1	yes	characteristic
62	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
63	#10624854	—	43	1	no	maximal, Sylow
64	C₃²×C₄₃	—	3	1	no	—
65	C₃²×C₄₃	—	9	1	no	—
66	C₃²×C₄₃	—	9	1	no	—
67	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
68	C₃×C₄₃⋊C₃	—	9	1	no	—
69	C₉×C₄₃	—	3	1	no	—
70	C₉×C₄₃	—	3	1	no	—
71	C₉×C₄₃	—	3	1	no	—
72	C₉×C₄₃	—	9	1	no	—
73	C₉×C₄₃	—	9	1	no	—
74	C₄₃⋊C₉	—	9	1	no	—
75	C₄₃⋊C₉	—	9	1	no	—
76	C₄₃⋊C₉	—	9	1	no	—
77	C₄₃⋊C₉	—	9	1	no	—
78	C₄₃⋊C₉	—	9	1	no	—
79	C₄₃⋊C₉	—	9	1	no	—
80	C₄₃⋊C₉	—	9	1	no	—
81	C₄₃⋊C₉	—	9	1	no	—
82	#6913	—	43	1	no	—
83	#6913	C₄₃⋊C₃	1	1	yes	characteristic, complemented
84	#6912	—	43	1	no	—
85	#6899	—	43	1	no	—
86	#6883	—	43	1	no	—
87	#6883	—	43	1	no	—
88	#6882	—	43	1	no	—
89	#6882	—	43	1	no	—
90	#6882	—	43	1	no	—
91	#6881	—	43	1	no	—
92	#6880	—	43	1	no	—
93	#6878	—	43	1	no	—
94	#6878	—	43	1	no	—
95	C₃×C₄₃	—	3	1	no	—
96	C₃×C₄₃	—	9	1	no	—
97	C₃×C₄₃	—	27	1	no	—
98	C₃×C₄₃	#6911	1	1	yes	characteristic
99	C₄₃⋊C₃	—	27	1	no	—
100	#737	—	3	1	no	—
101	#737	—	129	1	no	—
102	C₃×3₋¹⁺²	—	129	1	no	—
103	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
104	#733	—	43	1	no	—
105	#733	—	43	1	no	—
106	#733	—	43	1	no	—
107	#733	—	43	1	no	—
108	#733	—	43	1	no	—
109	#733	—	129	1	no	—
110	#733	—	129	1	no	—
111	#733	C₃×C₄₃⋊C₃	1	1	yes	characteristic
112	#733	C₃×C₄₃⋊C₃	1	1	yes	characteristic
113	#732	—	43	1	no	—
114	#732	C₃×C₄₃⋊C₃	1	1	yes	characteristic
115	#731	—	3	1	no	—
116	#731	—	129	1	no	—
117	AΣL(1,𝔽₂₇)	—	3	1	no	—
118	AΣL(1,𝔽₂₇)	—	129	1	no	—
119	C₉⋊C₉	—	43	1	no	—
120	C₉⋊C₉	—	43	1	no	—
121	C₉⋊C₉	—	129	1	no	—
122	C₉⋊C₉	—	129	1	no	—
123	C₃²⋊C₉	—	129	1	no	—
124	C₉²	—	43	1	no	—
125	C₄₃	#10624854	1	1	yes	characteristic, complemented, Sylow
126	C₃³	C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
127	3₋¹⁺²	—	3	1	no	—
128	3₋¹⁺²	—	3	1	no	—
129	3₋¹⁺²	—	9	1	no	—
130	3₋¹⁺²	—	9	2	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	1	no	—
136	3₋¹⁺²	—	387	1	no	—
137	3₋¹⁺²	—	387	1	no	—
138	3₋¹⁺²	—	387	2	no	—
139	H(3,𝔽₃)	—	3	1	no	—
140	H(3,𝔽₃)	—	129	1	no	—
141	C₃×C₉	—	9	1	no	—
142	C₃×C₉	—	129	1	no	—
143	C₃×C₉	—	129	1	no	—
144	C₃×C₉	—	129	1	no	—
145	C₃×C₉	—	387	1	no	—
146	C₃×C₉	—	387	1	no	—
147	C₃×C₉	C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
148	C₃×C₉	C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
149	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
150	C₃²	—	3	1	no	—
151	C₃²	—	9	1	no	—
152	C₃²	—	9	1	no	—
153	C₃²	—	387	1	no	—
154	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
155	C₉	—	3	1	no	—
156	C₉	—	3	1	no	—
157	C₉	—	3	1	no	—
158	C₉	—	9	1	no	—
159	C₉	—	9	1	no	—
160	C₉	—	387	1	no	—
161	C₉	—	387	1	no	—
162	C₉	—	387	1	no	—
163	C₉	—	387	1	no	—
164	C₉	—	387	1	no	—
165	C₉	—	387	1	no	—
166	C₉	—	387	1	no	—
167	C₉	—	387	1	no	—
168	C₃	—	3	1	no	—
169	C₃	—	9	1	no	—
170	C₃	—	27	1	no	—
171	C₃	—	1161	1	no	—
172	C₃	#23782891	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#23782891
commutator, cosocle	C₃×C₉×C₄₃	C₃³
Frattini	C₃×C₉	C₃²×C₄₃⋊C₃
Fitting	C₄₃×#6913	C₃
socle	C₃×C₄₃	#6911
hypercenter	#6913	C₄₃⋊C₃
hypocenter	C₄₃	#10624854
LCS	G ▷ C₃×C₉×C₄₃ ▷ C₃²×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
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. #23782905	C₃	semidirect
2. #23782896	C₃	semidirect
3. #23782838	C₃	semidirect
4. #23782769	C₃	semidirect
5. #23782766	C₃	semidirect
6. #23782763	C₃	semidirect
7. #23782761	C₃	semidirect
8. #23782760	C₃	semidirect
9. #23782755	C₃	semidirect
10. #23782752	C₃	semidirect
11. #23782748	C₃	semidirect
12. #23782745	C₃	semidirect
13. C₄₃×#6913	C₃	semidirect
83. #6913	C₄₃⋊C₃	semidirect
125. C₄₃	#10624854	semidirect

Maximal subgroups

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

subgroup	index
#23782905	3
#23782896	3
#23782838	3
#23782769	3
#23782766	3
#23782763	3
#23782761	3
#23782760	3
#23782755	3
#23782752	3
#23782748	3
#23782745	3
C₄₃×#6913	3
#10624854	43