Finite Group Databasebeta

Subgroups — #41006980

Order 26973.

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

normal subgroups only

This group has 276 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	#22582568	C₃	1	2	yes	maximal, complemented
2	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃²⋊#66524	C₃	1	1	yes	maximal, characteristic, complemented
4	3₋¹⁺²⋊#66524	C₃	1	2	yes	maximal, complemented
5	3₋¹⁺²⋊#66524	C₃	1	2	yes	maximal, complemented
6	H(3,𝔽₃)⋊#66524	C₃	1	1	yes	maximal, characteristic, complemented
7	H(3,𝔽₃)⋊#66524	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582468	C₃	1	2	yes	maximal, complemented
9	C₃×C₃₇×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic
10	C₃²×#66524	—	3	1	no	—
11	C₃²×#66524	C₃²	1	1	yes	characteristic
12	C₃²×#66524	C₃²	1	1	yes	characteristic
13	C₃²×#66524	C₃²	1	1	yes	characteristic
14	#15688504	—	3	4	no	—
15	#15688504	—	3	2	no	—
16	C₃²⋊#66524	—	3	2	no	—
17	C₃²⋊#66524	—	3	1	no	—
18	C₃²⋊#66524	—	3	2	no	—
19	C₃²⋊#66524	C₃²	1	2	yes	—
20	C₃²⋊#66524	C₃²	1	2	yes	—
21	C₃²⋊#66524	C₃²	1	2	yes	—
22	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
23	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
24	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
25	C₃₇×AΣL(1,𝔽₂₇)	—	3	3	no	—
26	C₃³×C₃₇	—	3	3	no	—
27	C₃³×C₃₇	—	3	1	no	—
28	C₃³×C₃₇	—	3	1	no	—
29	C₃³×C₃₇	—	3	1	no	—
30	C₃³×C₃₇	—	3	2	no	—
31	C₃³×C₃₇	—	9	1	no	—
32	C₃³×C₃₇	—	9	1	no	—
33	C₃³×C₃₇	C₃³	1	1	yes	characteristic
34	3₋¹⁺²×C₃₇	—	3	2	no	—
35	3₋¹⁺²×C₃₇	—	3	4	no	—
36	H(3,𝔽₃)×C₃₇	—	3	2	no	—
37	H(3,𝔽₃)×C₃₇	3₋¹⁺²	1	3	yes	—
38	C₃×C₉×C₃₇	—	3	2	no	—
39	C₃×#66524	—	3	1	no	—
40	C₃×#66524	—	3	1	no	—
41	C₃×#66524	—	3	2	no	—
42	C₃×#66524	—	3	2	no	—
43	C₃×#66524	—	3	2	no	—
44	C₃×#66524	—	9	1	no	—
45	C₃×#66524	—	9	3	no	—
46	C₃×#66524	—	9	1	no	—
47	C₃×#66524	—	9	1	no	—
48	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
49	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
50	#10624797	—	37	1	no	maximal, Sylow
51	C₃²×C₃₇	—	3	1	no	—
52	C₃²×C₃₇	—	3	1	no	—
53	C₃²×C₃₇	—	3	1	no	—
54	C₃²×C₃₇	—	3	3	no	—
55	C₃²×C₃₇	—	3	3	no	—
56	C₃²×C₃₇	—	9	3	no	—
57	C₃²×C₃₇	—	9	6	no	—
58	C₃²×C₃₇	—	9	1	no	—
59	C₃²×C₃₇	—	9	1	no	—
60	C₃²×C₃₇	—	9	1	no	—
61	C₃²×C₃₇	—	9	1	no	—
62	C₃²×C₃₇	—	9	1	no	—
63	C₃²×C₃₇	—	9	1	no	—
64	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
65	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
66	C₃²×C₃₇	#737	1	2	yes	—
67	C₉×C₃₇	—	9	2	no	—
68	#66524	—	3	1	no	—
69	#66524	—	3	2	no	—
70	#66524	—	9	2	no	—
71	#66524	—	9	2	no	—
72	#66524	—	9	1	no	—
73	#66524	—	9	1	no	—
74	#66524	—	9	2	no	—
75	C₃×AΣL(1,𝔽₂₇)	C₃₇⋊C₃	1	1	yes	characteristic
76	#6894	—	37	2	no	—
77	C₉×H(3,𝔽₃)	—	37	1	no	—
78	C₃×C₃²⋊C₉	—	37	1	no	—
79	3₋¹⁺²⋊C₉	—	37	2	no	—
80	3₋¹⁺²⋊C₉	—	37	2	no	—
81	H(3,𝔽₃)⋊C₉	—	37	1	no	—
82	H(3,𝔽₃)⋊C₉	—	37	1	no	—
83	#6866	—	37	2	no	—
84	C₃×C₃₇	—	3	1	no	—
85	C₃×C₃₇	—	3	2	no	—
86	C₃×C₃₇	—	9	3	no	—
87	C₃×C₃₇	—	9	3	no	—
88	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
89	C₃×C₃₇	#6908	1	1	yes	characteristic
90	C₃×C₃₇	#6909	1	1	yes	characteristic
91	C₃×C₃₇	#6913	1	1	yes	characteristic
92	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
93	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
94	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
95	C₃²×C₉	—	37	1	no	—
96	C₃²×C₉	—	37	1	no	—
97	C₃²×C₉	—	37	1	no	—
98	C₃²×C₉	—	111	1	no	—
99	AΣL(1,𝔽₂₇)	—	3	3	no	—
100	C₉⋊C₉	—	111	4	no	—
101	C₉⋊C₉	—	111	2	no	—
102	C₃²⋊C₉	—	37	2	no	—
103	C₃²⋊C₉	—	37	2	no	—
104	C₃²⋊C₉	—	37	2	no	—
105	C₃²⋊C₉	—	111	2	no	—
106	C₃²⋊C₉	—	111	1	no	—
107	C₃²⋊C₉	—	111	2	no	—
108	C₃₇	#10624797	1	1	yes	characteristic, complemented, Sylow
109	C₃³	—	3	3	no	—
110	C₃³	—	3	1	no	—
111	C₃³	—	3	1	no	—
112	C₃³	—	3	1	no	—
113	C₃³	—	3	2	no	—
114	C₃³	—	9	1	no	—
115	C₃³	—	9	1	no	—
116	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
117	3₋¹⁺²	—	3	2	no	—
118	3₋¹⁺²	—	3	4	no	—
119	H(3,𝔽₃)	—	3	2	no	—
120	H(3,𝔽₃)	#11758608	1	3	yes	—
121	C₃×C₉	—	3	2	no	—
122	C₃×C₉	—	37	2	no	—
123	C₃×C₉	—	37	1	no	—
124	C₃×C₉	—	333	3	no	—
125	C₃×C₉	—	333	1	no	—
126	C₃×C₉	—	333	1	no	—
127	C₃×C₉	—	333	1	no	—
128	C₃×C₉	—	111	2	no	—
129	C₃×C₉	—	111	2	no	—
130	C₃×C₉	—	111	2	no	—
131	C₃×C₉	—	111	1	no	—
132	C₃×C₉	—	111	1	no	—
133	C₃²	—	3	1	no	—
134	C₃²	—	3	1	no	—
135	C₃²	—	3	1	no	—
136	C₃²	—	3	3	no	—
137	C₃²	—	3	3	no	—
138	C₃²	—	9	3	no	—
139	C₃²	—	9	6	no	—
140	C₃²	—	9	1	no	—
141	C₃²	—	9	1	no	—
142	C₃²	—	9	1	no	—
143	C₃²	—	9	1	no	—
144	C₃²	—	9	1	no	—
145	C₃²	—	9	1	no	—
146	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
147	C₃²	C₃×#11758608	1	1	yes	characteristic
148	C₃²	#15688530	1	2	yes	—
149	C₉	—	9	2	no	—
150	C₉	—	333	2	no	—
151	C₉	—	333	2	no	—
152	C₉	—	333	1	no	—
153	C₉	—	333	1	no	—
154	C₉	—	333	2	no	—
155	C₉	—	111	1	no	—
156	C₉	—	111	2	no	—
157	C₃	—	3	1	no	—
158	C₃	—	3	2	no	—
159	C₃	—	9	3	no	—
160	C₃	—	9	3	no	—
161	C₃	#22582556	1	1	yes	characteristic
162	C₃	#22582629	1	1	yes	characteristic
163	C₃	#22582635	1	1	yes	characteristic
164	C₃	#22582654	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting, prime radical	C₃×C₃₇×AΣL(1,𝔽₂₇)	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃×AΣL(1,𝔽₂₇)	C₃₇⋊C₃
hypocenter	C₃₇	#10624797
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×AΣL(1,𝔽₂₇)

Complements

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

N	H ≅ G/N	type
1. #22582568	C₃	semidirect
2. H(3,𝔽₃)×#66524	C₃	semidirect
3. C₃×C₃²⋊#66524	C₃	semidirect
4. 3₋¹⁺²⋊#66524	C₃	semidirect
5. 3₋¹⁺²⋊#66524	C₃	semidirect
6. H(3,𝔽₃)⋊#66524	C₃	semidirect
7. H(3,𝔽₃)⋊#66524	C₃	semidirect
8. #22582468	C₃	semidirect
108. C₃₇	#10624797	semidirect

Maximal subgroups

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

subgroup	index
#22582568	3
H(3,𝔽₃)×#66524	3
C₃×C₃²⋊#66524	3
3₋¹⁺²⋊#66524	3
H(3,𝔽₃)⋊#66524	3
#22582468	3
C₃×C₃₇×AΣL(1,𝔽₂₇)	3
#10624797	37