Finite Group Databasebeta

Subgroups — #72243641

Order 4374.

4072 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 4072 conjugacy classes of subgroups — too many to draw. Only its 219 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	#14555438	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃⋊#92394	—	3	1	no	maximal
3	C₃×#92404	C₃	1	1	yes	maximal, characteristic, complemented
4	#13091656	C₃	1	3	yes	maximal, complemented
5	C₃⋊#92285	—	3	12	no	maximal
6	C₃×3₋¹⁺⁴	D₃	1	1	yes	characteristic, complemented
7	C₃×H(5,𝔽₃)	—	2	1	no	—
8	C₃×H(5,𝔽₃)	C₂×C₃	1	1	yes	characteristic
9	#10624932	—	2	36	no	—
10	#10624932	—	2	3	no	—
11	#10624932	C₂×C₃	1	3	yes	complemented
12	C₃×#6911	—	2	12	no	—
13	C₃×#6911	D₃	1	12	yes	complemented
14	#92404	C₃²	1	1	yes	characteristic, complemented
15	C₃⋊#4208	—	3	3	no	—
16	C₃×C₃⋊#272	—	3	1	no	—
17	C₃×C₃⋊#272	—	3	12	no	—
18	C₃×C₃⋊#272	—	9	9	no	—
19	#92394	—	9	9	no	—
20	C₃⋊D₃×3₋¹⁺²	—	9	4	no	—
21	C₃⋊#4178	—	3	36	no	—
22	#92285	—	9	9	no	—
23	3₋¹⁺⁴	—	3	9	no	—
24	3₋¹⁺⁴	—	6	9	no	—
25	H(5,𝔽₃)	—	2	24	no	—
26	H(5,𝔽₃)	—	2	12	no	—
27	H(5,𝔽₃)	—	2	1	no	—
28	H(5,𝔽₃)	—	2	3	no	—
29	H(5,𝔽₃)	—	6	9	no	—
30	H(5,𝔽₃)	—	6	9	no	—
31	H(5,𝔽₃)	—	6	9	no	—
32	H(5,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
33	C₃×#737	—	2	12	no	—
34	C₃×#737	C₃×D₃	1	3	yes	complemented
35	C₃²×3₋¹⁺²	—	2	4	no	—
36	C₃²×3₋¹⁺²	C₃⋊D₃	1	4	yes	complemented
37	C₃²×H(3,𝔽₃)	—	2	12	no	—
38	C₃²×H(3,𝔽₃)	—	3	9	no	—
39	C₃²×H(3,𝔽₃)	—	6	9	no	—
40	C₃²×H(3,𝔽₃)	C₃×D₃	1	1	yes	characteristic
41	C₃²×H(3,𝔽₃)	C₃×D₃	1	12	yes	—
42	#6911	—	2	72	no	—
43	#6911	—	2	108	no	—
44	#6911	—	2	36	no	—
45	#6911	—	2	36	no	—
46	#6911	—	2	108	no	—
47	#6911	C₃⋊D₃	1	9	yes	complemented
48	C₃×#732	—	2	108	no	—
49	C₃×#732	—	2	36	no	—
50	C₃×#732	C₃×D₃	1	36	yes	complemented
51	C₃²×C₃⋊D₃	—	9	4	no	—
52	C₃²×C₃⋊D₃	—	27	12	no	—
53	C₃⋊#272	—	3	1	no	—
54	C₃⋊#272	—	3	12	no	—
55	C₃⋊#272	—	9	9	no	—
56	#4208	—	9	27	no	—
57	C₃×#272	—	9	9	no	—
58	C₃×#272	—	9	9	no	—
59	C₃×#272	—	27	24	no	—
60	C₉×C₃⋊D₃	—	9	12	no	—
61	D₃×3₋¹⁺²	—	27	12	no	—
62	D₃×3₋¹⁺²	—	27	1	no	—
63	#4178	—	9	27	no	—
64	C₃⁴	—	3	12	no	—
65	C₃⁴	—	6	12	no	—
66	C₃⁴	C₃×C₃⋊D₃	1	4	yes	—
67	#737	—	3	27	no	—
68	#737	—	6	108	no	—
69	C₃×3₋¹⁺²	—	2	12	no	—
70	C₃×3₋¹⁺²	—	2	12	no	—
71	C₃×3₋¹⁺²	—	2	4	no	—
72	C₃×3₋¹⁺²	—	2	12	no	—
73	C₃×3₋¹⁺²	—	3	12	no	—
74	C₃×3₋¹⁺²	—	6	12	no	—
75	C₃×3₋¹⁺²	—	6	24	no	—
76	C₃×3₋¹⁺²	—	6	12	no	—
77	C₃×3₋¹⁺²	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
78	C₃×H(3,𝔽₃)	—	2	4	no	—
79	C₃×H(3,𝔽₃)	—	2	9	no	—
80	C₃×H(3,𝔽₃)	—	2	72	no	—
81	C₃×H(3,𝔽₃)	—	2	36	no	—
82	C₃×H(3,𝔽₃)	—	2	12	no	—
83	C₃×H(3,𝔽₃)	—	2	9	no	—
84	C₃×H(3,𝔽₃)	—	2	36	no	—
85	C₃×H(3,𝔽₃)	—	3	9	no	—
86	C₃×H(3,𝔽₃)	—	3	24	no	—
87	C₃×H(3,𝔽₃)	—	6	72	no	—
88	C₃×H(3,𝔽₃)	—	6	36	no	—
89	C₃×H(3,𝔽₃)	—	6	216	no	—
90	C₃×H(3,𝔽₃)	—	6	24	no	—
91	C₃×H(3,𝔽₃)	—	6	12	no	—
92	C₃×H(3,𝔽₃)	—	6	24	no	—
93	C₃×H(3,𝔽₃)	—	6	12	no	—
94	C₃×H(3,𝔽₃)	—	6	24	no	—
95	C₃×H(3,𝔽₃)	—	6	12	no	—
96	C₃×H(3,𝔽₃)	—	6	24	no	—
97	C₃×H(3,𝔽₃)	—	6	9	no	—
98	C₃×H(3,𝔽₃)	—	6	27	no	—
99	C₃×H(3,𝔽₃)	(C₃⋊D₃)⋊C₃	1	9	yes	—
100	C₃×H(3,𝔽₃)	C₃²×D₃	1	12	yes	—
101	C₃×H(3,𝔽₃)	C₃²×D₃	1	1	yes	characteristic
102	C₃×H(3,𝔽₃)	C₃×C₃⋊D₃	1	9	yes	—
103	C₃²×C₉	—	2	12	no	—
104	C₃²×C₉	C₃×C₃⋊D₃	1	12	yes	complemented
105	#732	—	2	216	no	—
106	#732	—	2	108	no	—
107	#732	—	2	27	no	—
108	#732	C₃×C₃⋊D₃	1	27	yes	complemented
109	C₃×C₃⋊D₃	—	9	4	no	—
110	C₃×C₃⋊D₃	—	27	12	no	—
111	C₃×C₃⋊D₃	—	27	4	no	—
112	C₃×C₃⋊D₃	—	27	36	no	—
113	C₃²×D₃	—	27	12	no	—
114	C₃²×D₃	—	27	1	no	—
115	C₃²×D₃	—	81	9	no	—
116	C₂×3₋¹⁺²	—	81	1	no	—
117	#272	—	9	9	no	—
118	#272	—	9	9	no	—
119	#272	—	27	24	no	—
120	D₃×C₉	—	27	36	no	—
121	D₃×C₉	—	27	3	no	—
122	C₃³	—	2	12	no	—
123	C₃³	—	2	4	no	—
124	C₃³	—	3	12	no	—
125	C₃³	—	3	9	no	—
126	C₃³	—	6	36	no	—
127	C₃³	—	6	9	no	—
128	C₃³	—	6	72	no	—
129	C₃³	—	6	36	no	—
130	C₃³	—	6	12	no	—
131	C₃³	—	27	4	no	—
132	C₃³	—	27	36	no	—
133	C₃³	—	54	36	no	—
134	C₃³	C₃⋊((C₃⋊D₃)⋊C₃)	1	12	yes	—
135	C₃³	C₃²×C₃⋊D₃	1	4	yes	—
136	C₃³	C₃×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
137	3₋¹⁺²	—	3	1	no	—
138	3₋¹⁺²	—	6	1	no	—
139	3₋¹⁺²	—	6	36	no	—
140	3₋¹⁺²	—	6	8	no	—
141	3₋¹⁺²	—	6	36	no	—
142	3₋¹⁺²	—	6	36	no	—
143	3₋¹⁺²	—	6	4	no	—
144	H(3,𝔽₃)	—	2	36	no	—
145	H(3,𝔽₃)	—	2	9	no	—
146	H(3,𝔽₃)	—	2	27	no	—
147	H(3,𝔽₃)	—	3	24	no	—
148	H(3,𝔽₃)	—	6	9	no	—
149	H(3,𝔽₃)	—	6	72	no	—
150	H(3,𝔽₃)	—	6	72	no	—
151	H(3,𝔽₃)	—	6	72	no	—
152	H(3,𝔽₃)	—	6	9	no	—
153	H(3,𝔽₃)	—	6	24	no	—
154	H(3,𝔽₃)	—	6	72	no	—
155	H(3,𝔽₃)	—	6	9	no	—
156	H(3,𝔽₃)	C₃×(C₃⋊D₃)⋊C₃	1	9	yes	—
157	H(3,𝔽₃)	C₃²×C₃⋊D₃	1	9	yes	—
158	C₃×C₉	—	2	12	no	—
159	C₃×C₉	—	3	36	no	—
160	C₃×C₉	—	6	36	no	—
161	C₃×C₉	C₃×C₃⋊(C₃⋊D₃)	1	3	yes	—
162	C₂×C₃²	—	81	1	no	—
163	C₃⋊D₃	—	9	4	no	—
164	C₃⋊D₃	—	27	12	no	—
165	C₃×D₃	—	27	12	no	—
166	C₃×D₃	—	27	1	no	—
167	C₃×D₃	—	81	1	no	—
168	C₃×D₃	—	81	12	no	—
169	C₃×D₃	—	81	9	no	—
170	C₃×D₃	—	81	27	no	—
171	C₂×C₉	—	81	3	no	—
172	C₃²	—	2	12	no	—
173	C₃²	—	2	1	no	—
174	C₃²	—	3	9	no	—
175	C₃²	—	6	27	no	—
176	C₃²	—	6	9	no	—
177	C₃²	—	9	4	no	—
178	C₃²	—	9	12	no	—
179	C₃²	—	9	1	no	—
180	C₃²	—	9	27	no	—
181	C₃²	—	18	4	no	—
182	C₃²	—	18	12	no	—
183	C₃²	—	18	27	no	—
184	C₃²	—	27	12	no	—
185	C₃²	—	54	12	no	—
186	C₃²	—	54	36	no	—
187	C₃²	—	54	36	no	—
188	C₃²	—	54	72	no	—
189	C₃²	C₃×C₃⋊((C₃⋊D₃)⋊C₃)	1	12	yes	—
190	C₃²	C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))	1	1	yes	characteristic
191	C₃²	C₃²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
192	C₉	—	3	3	no	—
193	C₉	—	6	12	no	—
194	C₂×C₃	—	81	1	no	—
195	C₂×C₃	—	243	1	no	—
196	D₃	—	27	1	no	—
197	D₃	—	27	12	no	—
198	D₃	—	81	9	no	—
199	C₃	—	3	1	no	—
200	C₃	—	3	12	no	—
201	C₃	—	3	1	no	—
202	C₃	—	6	1	no	—
203	C₃	—	6	12	no	—
204	C₃	—	9	9	no	—
205	C₃	—	18	9	no	—
206	C₃	—	18	27	no	—
207	C₃	C₃×C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))	1	1	yes	characteristic
208	C₂	—	81	1	no	Sylow

	subgroup	G/H
center, socle	C₃	C₃×C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))
commutator, cosocle, hypocenter	H(5,𝔽₃)	C₂×C₃²
Frattini	C₃²	C₃²×C₃⋊(C₃⋊D₃)
Fitting	#14555438	C₂
hypercenter	C₃²	C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))
LCS	G ▷ H(5,𝔽₃)
UCS	C₁ ◁ C₃ ◁ C₃²

Complements

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

N	H ≅ G/N	type
1. #14555438	C₂	semidirect
3. C₃×#92404	C₃	semidirect
4. #13091656	C₃	semidirect
6. C₃×3₋¹⁺⁴	D₃	semidirect
11. #10624932	C₂×C₃	semidirect
13. C₃×#6911	D₃	semidirect
14. #92404	C₃²	semidirect
34. C₃×#737	C₃×D₃	semidirect
36. C₃²×3₋¹⁺²	C₃⋊D₃	semidirect
47. #6911	C₃⋊D₃	semidirect
50. C₃×#732	C₃×D₃	semidirect
104. C₃²×C₉	C₃×C₃⋊D₃	semidirect
108. #732	C₃×C₃⋊D₃	semidirect

Maximal subgroups

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

subgroup	index
#14555438	2
C₃⋊#92394	3
C₃×#92404	3
#13091656	3
C₃⋊#92285	3