Finite Group Databasebeta

Subgroups — #44534580

Order 31347.

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	#23782905	C₃	1	1	yes	maximal, characteristic, complemented
2	#23782905	C₃	1	1	yes	maximal, characteristic, complemented
3	#23782896	C₃	1	1	yes	maximal, characteristic, complemented
4	#23782767	C₃	1	1	yes	maximal, characteristic, complemented
5	#23782767	C₃	1	1	yes	maximal, characteristic, complemented
6	#23782762	C₃	1	1	yes	maximal, characteristic, complemented
7	#23782762	C₃	1	1	yes	maximal, characteristic, complemented
8	#23782762	C₃	1	1	yes	maximal, characteristic, complemented
9	#23782757	C₃	1	1	yes	maximal, characteristic, complemented
10	#23782754	C₃	1	1	yes	maximal, characteristic, complemented
11	#23782747	C₃	1	1	yes	maximal, characteristic, complemented
12	#23782746	C₃	1	1	yes	maximal, characteristic, complemented
13	C₄₃×#6900	C₃	1	1	yes	maximal, characteristic, complemented
14	#16812525	—	3	1	no	—
15	#16812525	—	3	1	no	—
16	C₃×#11923958	—	3	1	no	—
17	#16812515	—	3	1	no	—
18	#16812515	—	3	1	no	—
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	#16812515	C₃²	1	1	yes	characteristic
24	#16812515	C₃²	1	1	yes	characteristic
25	#16812515	C₃²	1	1	yes	characteristic
26	#16812513	C₃²	1	1	yes	characteristic
27	#16812513	C₃²	1	1	yes	characteristic
28	#16812510	—	3	1	no	—
29	#16812510	—	3	1	no	—
30	#16812507	—	3	1	no	—
31	#16812507	—	3	1	no	—
32	C₃×3₋¹⁺²×C₄₃	—	3	1	no	—
33	C₃×3₋¹⁺²×C₄₃	C₃²	1	1	yes	characteristic
34	C₄₃×C₉⋊C₉	—	3	1	no	—
35	C₄₃×C₉⋊C₉	—	3	1	no	—
36	C₄₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
37	C₄₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
38	C₉²×C₄₃	C₃²	1	1	yes	characteristic
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	—	3	1	no	—
45	#11923958	—	3	1	no	—
46	#11923958	—	9	2	no	—
47	#11923957	—	9	1	no	—
48	#11923957	—	9	1	no	—
49	#11923956	—	9	1	no	—
50	#11923956	—	9	1	no	—
51	C₃²⋊(C₄₃⋊C₃)	—	3	1	no	—
52	C₃²⋊(C₄₃⋊C₃)	—	3	1	no	—
53	C₃×C₄₃⋊C₉	—	9	1	no	—
54	C₉×C₄₃⋊C₃	—	9	1	no	—
55	C₉×C₄₃⋊C₃	—	9	1	no	—
56	C₃³×C₄₃	H(3,𝔽₃)	1	1	yes	characteristic
57	3₋¹⁺²×C₄₃	—	9	1	no	—
58	3₋¹⁺²×C₄₃	—	9	3	no	—
59	C₃×C₉×C₄₃	—	3	1	no	—
60	C₃×C₉×C₄₃	—	3	1	no	—
61	C₃×C₉×C₄₃	—	3	1	no	—
62	C₃×C₉×C₄₃	H(3,𝔽₃)	1	1	yes	characteristic
63	C₃×C₉×C₄₃	H(3,𝔽₃)	1	1	yes	characteristic
64	C₃×C₉×C₄₃	C₃³	1	1	yes	characteristic
65	#10624852	—	43	1	no	maximal, Sylow
66	C₃²×C₄₃	—	3	1	no	—
67	C₃²×C₄₃	—	9	1	no	—
68	C₃²×C₄₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
69	C₃×C₄₃⋊C₃	—	9	1	no	—
70	C₃×C₄₃⋊C₃	—	9	1	no	—
71	C₉×C₄₃	—	3	3	no	—
72	C₉×C₄₃	—	3	1	no	—
73	C₉×C₄₃	—	3	1	no	—
74	C₉×C₄₃	—	3	1	no	—
75	C₉×C₄₃	—	9	1	no	—
76	C₉×C₄₃	—	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	C₄₃⋊C₉	—	9	1	no	—
83	C₄₃⋊C₉	—	9	1	no	—
84	#6913	—	43	1	no	—
85	#6913	—	43	1	no	—
86	#6912	—	43	1	no	—
87	#6900	C₄₃⋊C₃	1	1	yes	characteristic, complemented
88	#6883	—	43	1	no	—
89	#6883	—	43	1	no	—
90	#6882	—	43	1	no	—
91	#6882	—	43	1	no	—
92	#6882	—	43	1	no	—
93	#6881	—	43	1	no	—
94	#6880	—	43	1	no	—
95	#6878	—	43	1	no	—
96	#6878	—	43	1	no	—
97	C₃×C₄₃	—	3	1	no	—
98	C₃×C₄₃	—	9	1	no	—
99	C₃×C₄₃	#6911	1	1	yes	characteristic
100	C₄₃⋊C₃	—	27	1	no	—
101	C₄₃⋊C₃	—	27	1	no	—
102	#737	—	129	1	no	—
103	#737	—	129	1	no	—
104	C₃×3₋¹⁺²	—	3	1	no	—
105	C₃×3₋¹⁺²	—	129	1	no	—
106	C₃×3₋¹⁺²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
107	#733	—	43	1	no	—
108	#733	—	43	1	no	—
109	#733	—	43	1	no	—
110	#733	—	43	1	no	—
111	#733	—	43	1	no	—
112	#733	—	43	1	no	—
113	#733	—	43	1	no	—
114	#733	—	129	1	no	—
115	#733	—	129	1	no	—
116	#732	—	43	1	no	—
117	#732	—	43	1	no	—
118	#731	—	129	1	no	—
119	#731	—	129	1	no	—
120	AΣL(1,𝔽₂₇)	—	129	1	no	—
121	AΣL(1,𝔽₂₇)	—	129	1	no	—
122	C₉⋊C₉	—	3	1	no	—
123	C₉⋊C₉	—	3	1	no	—
124	C₉⋊C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
125	C₉⋊C₉	C₃×C₄₃⋊C₃	1	1	yes	characteristic
126	C₉²	C₃×C₄₃⋊C₃	1	1	yes	characteristic
127	C₄₃	#10624852	1	1	yes	characteristic, complemented, Sylow
128	C₃³	C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
129	3₋¹⁺²	—	9	1	no	—
130	3₋¹⁺²	—	9	3	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	—	129	1	no	—
137	3₋¹⁺²	—	129	1	no	—
138	3₋¹⁺²	—	387	1	no	—
139	3₋¹⁺²	—	387	1	no	—
140	3₋¹⁺²	—	387	1	no	—
141	3₋¹⁺²	—	387	1	no	—
142	3₋¹⁺²	—	387	2	no	—
143	H(3,𝔽₃)	—	129	1	no	—
144	H(3,𝔽₃)	—	129	1	no	—
145	C₃×C₉	—	3	1	no	—
146	C₃×C₉	—	3	1	no	—
147	C₃×C₉	—	3	1	no	—
148	C₃×C₉	—	387	1	no	—
149	C₃×C₉	—	387	1	no	—
150	C₃×C₉	—	387	1	no	—
151	C₃×C₉	C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
152	C₃×C₉	C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
153	C₃×C₉	C₃²×C₄₃⋊C₃	1	1	yes	characteristic
154	C₃²	—	3	1	no	—
155	C₃²	—	9	1	no	—
156	C₃²	—	387	1	no	—
157	C₃²	—	387	1	no	—
158	C₃²	C₃×C₃²⋊(C₄₃⋊C₃)	1	1	yes	characteristic
159	C₉	—	3	3	no	—
160	C₉	—	3	1	no	—
161	C₉	—	3	1	no	—
162	C₉	—	3	1	no	—
163	C₉	—	9	1	no	—
164	C₉	—	9	1	no	—
165	C₉	—	387	1	no	—
166	C₉	—	387	1	no	—
167	C₉	—	387	1	no	—
168	C₉	—	387	1	no	—
169	C₉	—	387	1	no	—
170	C₉	—	387	1	no	—
171	C₉	—	387	1	no	—
172	C₃	—	3	1	no	—
173	C₃	—	9	1	no	—
174	C₃	—	1161	1	no	—
175	C₃	—	1161	1	no	—
176	C₃	#23782892	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#23782892
commutator, cosocle	C₃×C₉×C₄₃	C₃³
Frattini	C₃×C₉	C₃²×C₄₃⋊C₃
Fitting	C₄₃×#6900	C₃
socle	C₃×C₄₃	#6911
hypercenter	#6900	C₄₃⋊C₃
hypocenter	C₄₃	#10624852
LCS	G ▷ C₃×C₉×C₄₃ ▷ C₃²×C₄₃ ▷ C₃×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×3₋¹⁺² ◁ #6900

Complements

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

N	H ≅ G/N	type
1. #23782905	C₃	semidirect
2. #23782905	C₃	semidirect
3. #23782896	C₃	semidirect
4. #23782767	C₃	semidirect
5. #23782767	C₃	semidirect
6. #23782762	C₃	semidirect
7. #23782762	C₃	semidirect
8. #23782762	C₃	semidirect
9. #23782757	C₃	semidirect
10. #23782754	C₃	semidirect
11. #23782747	C₃	semidirect
12. #23782746	C₃	semidirect
13. C₄₃×#6900	C₃	semidirect
87. #6900	C₄₃⋊C₃	semidirect
127. C₄₃	#10624852	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
#23782767	3
#23782762	3
#23782757	3
#23782754	3
#23782747	3
#23782746	3
C₄₃×#6900	3
#10624852	43