Finite Group Databasebeta

Subgroups — #41006666

Order 26973.

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

normal subgroups only

This group has 456 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	C₃₇⋊H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582647	C₃	1	4	yes	maximal, complemented
3	#22582638	C₃	1	2	yes	maximal, complemented
4	#22582637	C₃	1	2	yes	maximal, complemented
5	C₃×#15688519	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×#15688516	C₃	1	2	yes	maximal, complemented
7	C₃×C₃₇×#732	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#11758608	C₃²	1	4	yes	complemented
9	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
10	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	2	yes	—
11	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
12	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	2	yes	complemented
13	H(3,𝔽₃)×C₃₇⋊C₃	—	3	3	no	—
14	H(3,𝔽₃)×C₃₇⋊C₃	—	3	6	no	—
15	#15688519	—	3	3	no	—
16	#15688518	—	3	12	no	—
17	#15688516	—	3	6	no	—
18	#15688513	—	3	6	no	—
19	#15688512	—	3	6	no	—
20	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
21	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	2	yes	complemented
22	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic, complemented
23	C₃₇×#732	—	3	3	no	—
24	C₃³×C₃₇	—	3	1	no	—
25	C₃³×C₃₇	—	3	2	no	—
26	C₃³×C₃₇	C₃³	1	1	yes	characteristic
27	C₃²×C₃₇⋊C₃	—	3	9	no	—
28	C₃²×C₃₇⋊C₃	—	3	2	no	—
29	C₃²×C₃₇⋊C₃	—	3	1	no	—
30	C₃²×C₃₇⋊C₃	—	3	2	no	—
31	H(3,𝔽₃)×C₃₇	—	3	2	no	—
32	H(3,𝔽₃)×C₃₇	—	3	2	no	—
33	H(3,𝔽₃)×C₃₇	—	3	4	no	—
34	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	3	yes	—
35	C₃×C₉×C₃₇	—	3	1	no	—
36	C₃×C₉×C₃₇	—	3	2	no	—
37	C₃×C₉×C₃₇	—	3	1	no	—
38	C₃²⋊(C₃₇⋊C₃)	—	3	18	no	—
39	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
40	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
41	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
42	C₃²⋊(C₃₇⋊C₃)	—	3	4	no	—
43	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
44	C₃²⋊(C₃₇⋊C₃)	H(3,𝔽₃)	1	3	yes	—
45	C₃²⋊(C₃₇⋊C₃)	H(3,𝔽₃)	1	6	yes	—
46	#11758608	—	3	4	no	—
47	#11758608	—	3	4	no	—
48	#11758608	—	3	4	no	—
49	C₃×#66524	—	3	4	no	—
50	#10624760	—	37	1	no	maximal, Sylow
51	C₃²×C₃₇	—	3	3	no	—
52	C₃²×C₃₇	—	9	1	no	—
53	C₃²×C₃₇	—	9	2	no	—
54	C₃²×C₃₇	—	27	1	no	—
55	C₃²×C₃₇	—	27	2	no	—
56	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
57	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	C₃×C₃₇⋊C₃	—	3	3	no	—
60	C₃×C₃₇⋊C₃	—	3	6	no	—
61	C₃×C₃₇⋊C₃	—	9	2	no	—
62	C₃×C₃₇⋊C₃	—	27	1	no	—
63	C₃×C₃₇⋊C₃	—	27	2	no	—
64	C₃×C₃₇⋊C₃	—	27	9	no	—
65	C₃×C₃₇⋊C₃	—	27	2	no	—
66	C₉×C₃₇	—	9	1	no	—
67	#66524	—	9	4	no	—
68	H(5,𝔽₃)	—	37	1	no	—
69	#6911	—	37	4	no	—
70	#6909	—	37	2	no	—
71	#6909	—	37	2	no	—
72	C₃×#732	—	37	1	no	—
73	C₃×#732	C₃₇⋊C₃	1	1	yes	characteristic, complemented
74	C₃×#731	—	37	2	no	—
75	C₃×C₃₇	—	3	1	no	—
76	C₃×C₃₇	—	3	2	no	—
77	C₃×C₃₇	—	3	1	no	—
78	C₃×C₃₇	—	9	3	no	—
79	C₃×C₃₇	—	27	2	no	—
80	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
81	C₃₇⋊C₃	—	9	3	no	—
82	C₃₇⋊C₃	—	9	6	no	—
83	C₃₇⋊C₃	—	27	2	no	—
84	C₃×3₋¹⁺²	—	37	4	no	—
85	C₃×H(3,𝔽₃)	—	37	2	no	—
86	C₃×H(3,𝔽₃)	—	37	1	no	—
87	C₃×H(3,𝔽₃)	—	37	2	no	—
88	C₃×H(3,𝔽₃)	—	111	6	no	—
89	C₃×H(3,𝔽₃)	—	111	3	no	—
90	C₃×H(3,𝔽₃)	—	111	3	no	—
91	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
92	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	2	yes	complemented
93	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
94	#732	—	3	3	no	—
95	#732	—	111	3	no	—
96	#732	—	111	12	no	—
97	#731	—	111	6	no	—
98	AΣL(1,𝔽₂₇)	—	111	6	no	—
99	AΣL(1,𝔽₂₇)	—	111	6	no	—
100	C₃₇	#10624760	1	1	yes	characteristic, complemented, Sylow
101	C₃³	—	3	1	no	—
102	C₃³	—	3	2	no	—
103	C₃³	—	111	9	no	—
104	C₃³	—	111	2	no	—
105	C₃³	—	111	2	no	—
106	C₃³	—	111	1	no	—
107	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
108	3₋¹⁺²	—	111	4	no	—
109	3₋¹⁺²	—	111	4	no	—
110	3₋¹⁺²	—	111	4	no	—
111	H(3,𝔽₃)	—	3	2	no	—
112	H(3,𝔽₃)	—	3	4	no	—
113	H(3,𝔽₃)	—	3	2	no	—
114	H(3,𝔽₃)	—	37	3	no	—
115	H(3,𝔽₃)	—	37	6	no	—
116	H(3,𝔽₃)	—	111	18	no	—
117	H(3,𝔽₃)	—	111	4	no	—
118	H(3,𝔽₃)	—	111	2	no	—
119	H(3,𝔽₃)	—	111	2	no	—
120	H(3,𝔽₃)	—	111	2	no	—
121	H(3,𝔽₃)	—	111	2	no	—
122	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
123	C₃×C₉	—	3	2	no	—
124	C₃×C₉	—	3	1	no	—
125	C₃×C₉	—	3	1	no	—
126	C₃×C₉	—	111	4	no	—
127	C₃²	—	3	3	no	—
128	C₃²	—	9	1	no	—
129	C₃²	—	9	2	no	—
130	C₃²	—	27	1	no	—
131	C₃²	—	27	2	no	—
132	C₃²	—	333	2	no	—
133	C₃²	—	111	3	no	—
134	C₃²	—	111	6	no	—
135	C₃²	—	999	1	no	—
136	C₃²	—	999	2	no	—
137	C₃²	—	999	2	no	—
138	C₃²	—	999	9	no	—
139	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	2	yes	—
140	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
141	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
142	C₉	—	9	1	no	—
143	C₉	—	333	4	no	—
144	C₃	—	3	1	no	—
145	C₃	—	3	1	no	—
146	C₃	—	3	2	no	—
147	C₃	—	9	3	no	—
148	C₃	—	27	2	no	—
149	C₃	—	333	3	no	—
150	C₃	—	333	6	no	—
151	C₃	—	999	2	no	—
152	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#732	C₃
socle	C₃×C₃₇	C₃²⋊H(3,𝔽₃)
hypercenter	C₃×#732	C₃₇⋊C₃
hypocenter	C₃₇	#10624760
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#732

Complements

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

N	H ≅ G/N	type
1. C₃₇⋊H(5,𝔽₃)	C₃	semidirect
2. #22582647	C₃	semidirect
3. #22582638	C₃	semidirect
4. #22582637	C₃	semidirect
5. C₃×#15688519	C₃	semidirect
6. C₃×#15688516	C₃	semidirect
7. C₃×C₃₇×#732	C₃	semidirect
8. C₃×#11758608	C₃²	semidirect
12. C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	semidirect
21. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
22. C₃²×C₉×C₃₇	C₃²	semidirect
73. C₃×#732	C₃₇⋊C₃	semidirect
92. C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	semidirect
93. C₃²×C₉	C₃×C₃₇⋊C₃	semidirect
100. C₃₇	#10624760	semidirect

Maximal subgroups

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

subgroup	index
C₃₇⋊H(5,𝔽₃)	3
#22582647	3
#22582638	3
#22582637	3
C₃×#15688519	3
C₃×#15688516	3
C₃×C₃₇×#732	3
#10624760	37