Subgroups — C₃₇×#10624877 #41004666

Order 26973.

662 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#6899	C₃	1	6	yes	maximal, complemented
2	C₃₇×#6898	C₃	1	3	yes	maximal, complemented
3	C₃₇×#6897	C₃	1	6	yes	maximal, complemented
4	C₃₇×#6896	C₃	1	3	yes	maximal, complemented
5	C₃₇×#6895	C₃	1	2	yes	maximal, complemented
6	C₃₇×#6894	C₃	1	4	yes	maximal, complemented
7	C₃₇×#6893	C₃	1	1	yes	maximal, characteristic
8	C₃₇×C₉⋊H(3,𝔽₃)	C₃	1	2	yes	maximal
9	C₉×3₋¹⁺²×C₃₇	C₃	1	6	yes	maximal, complemented
10	C₉×3₋¹⁺²×C₃₇	C₃	1	2	yes	maximal, complemented
11	C₉×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
12	C₃₇×#6887	C₃	1	3	yes	maximal, complemented
13	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
14	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	—
15	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
16	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
17	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
18	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
19	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
20	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
21	C₃₇×C₉⋊C₉	C₃²	1	12	yes	—
22	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
23	C₃₇×C₉⋊C₉	C₃²	1	12	yes	—
24	C₃₇×C₉⋊C₉	C₃²	1	12	yes	—
25	C₃₇×C₉⋊C₉	C₃²	1	12	yes	—
26	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
27	C₃₇×C₉⋊C₉	C₃²	1	6	yes	—
28	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
29	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
30	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
31	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
32	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
33	C₃₇×C₃²⋊C₉	C₃²	1	4	yes	—
34	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
35	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
36	C₉²×C₃₇	C₃²	1	1	yes	characteristic
37	C₉²×C₃₇	C₃²	1	2	yes	—
38	C₉²×C₃₇	C₃²	1	1	yes	characteristic
39	C₉²×C₃₇	C₃²	1	2	yes	—
40	C₉²×C₃₇	C₃²	1	1	yes	characteristic
41	C₉²×C₃₇	C₃²	1	2	yes	—
42	C₉²×C₃₇	C₃²	1	6	yes	—
43	C₃³×C₃₇	C₃³	1	3	yes	—
44	C₃³×C₃₇	C₃³	1	1	yes	characteristic
45	3₋¹⁺²×C₃₇	—	3	6	no	—
46	3₋¹⁺²×C₃₇	—	3	18	no	—
47	H(3,𝔽₃)×C₃₇	—	3	3	no	—
48	C₃×C₉×C₃₇	—	3	3	no	—
49	C₃×C₉×C₃₇	—	3	3	no	—
50	C₃×C₉×C₃₇	—	3	9	no	—
51	C₃×C₉×C₃₇	—	3	6	no	—
52	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
53	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
54	C₃×C₉×C₃₇	C₃³	1	2	yes	—
55	C₃×C₉×C₃₇	C₃³	1	2	yes	—
56	C₃×C₉×C₃₇	C₃³	1	2	yes	—
57	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
58	C₃×C₉×C₃₇	C₃³	1	12	yes	—
59	C₃×C₉×C₃₇	C₃³	1	6	yes	—
60	C₃×C₉×C₃₇	C₃³	1	6	yes	—
61	C₃×C₉×C₃₇	C₃³	1	2	yes	—
62	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
63	#10624877	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₃²×C₃₇	—	3	1	no	—
65	C₃²×C₃₇	—	3	2	no	—
66	C₃²×C₃₇	—	3	3	no	—
67	C₃²×C₃₇	—	3	3	no	—
68	C₃²×C₃₇	—	3	6	no	—
69	C₃²×C₃₇	#737	1	3	yes	—
70	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
71	C₉×C₃₇	—	3	1	no	—
72	C₉×C₃₇	—	3	2	no	—
73	C₉×C₃₇	—	3	1	no	—
74	C₉×C₃₇	—	3	1	no	—
75	C₉×C₃₇	—	3	2	no	—
76	C₉×C₃₇	—	3	2	no	—
77	C₉×C₃₇	—	3	6	no	—
78	C₉×C₃₇	—	3	6	no	—
79	C₉×C₃₇	—	3	12	no	—
80	C₉×C₃₇	#737	1	3	yes	—
81	C₉×C₃₇	#737	1	6	yes	—
82	#6899	C₃×C₃₇	1	6	yes	complemented
83	#6898	C₃×C₃₇	1	3	yes	complemented
84	#6897	C₃×C₃₇	1	6	yes	complemented
85	#6896	C₃×C₃₇	1	3	yes	complemented
86	#6895	C₃×C₃₇	1	2	yes	complemented
87	#6894	C₃×C₃₇	1	4	yes	complemented
88	#6893	C₃×C₃₇	1	1	yes	characteristic
89	C₉⋊H(3,𝔽₃)	C₃×C₃₇	1	2	yes	—
90	C₉×3₋¹⁺²	C₃×C₃₇	1	6	yes	complemented
91	C₉×3₋¹⁺²	C₃×C₃₇	1	2	yes	complemented
92	C₉×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
93	#6887	C₃×C₃₇	1	3	yes	complemented
94	C₃×C₉²	C₃×C₃₇	1	1	yes	characteristic, complemented
95	C₃×C₃₇	—	3	3	no	—
96	C₃×C₃₇	—	9	3	no	—
97	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
98	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
99	C₃×C₃₇	C₃×#737	1	2	yes	—
100	C₃×3₋¹⁺²	C₃²×C₃₇	1	6	yes	—
101	C₃×3₋¹⁺²	C₃²×C₃₇	1	2	yes	—
102	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
103	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
104	C₃²×C₉	C₃²×C₃₇	1	2	yes	—
105	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
106	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
107	C₉⋊C₉	C₃²×C₃₇	1	12	yes	—
108	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
109	C₉⋊C₉	C₃²×C₃₇	1	12	yes	—
110	C₉⋊C₉	C₃²×C₃₇	1	12	yes	—
111	C₉⋊C₉	C₃²×C₃₇	1	12	yes	—
112	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
113	C₉⋊C₉	C₃²×C₃₇	1	6	yes	—
114	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
115	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
116	C₃²⋊C₉	C₃²×C₃₇	1	2	yes	—
117	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
118	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
119	C₃²⋊C₉	C₃²×C₃₇	1	4	yes	—
120	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
121	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
122	C₉²	C₃²×C₃₇	1	1	yes	characteristic
123	C₉²	C₃²×C₃₇	1	2	yes	—
124	C₉²	C₃²×C₃₇	1	1	yes	characteristic
125	C₉²	C₃²×C₃₇	1	2	yes	—
126	C₉²	C₃²×C₃₇	1	1	yes	characteristic
127	C₉²	C₃²×C₃₇	1	2	yes	—
128	C₉²	C₃²×C₃₇	1	6	yes	—
129	C₃₇	#10624877	1	1	yes	characteristic, direct product factor, Sylow
130	C₃³	C₃³×C₃₇	1	3	yes	—
131	C₃³	C₃³×C₃₇	1	1	yes	characteristic
132	3₋¹⁺²	—	3	6	no	—
133	3₋¹⁺²	—	3	18	no	—
134	H(3,𝔽₃)	—	3	3	no	—
135	C₃×C₉	—	3	3	no	—
136	C₃×C₉	—	3	3	no	—
137	C₃×C₉	—	3	9	no	—
138	C₃×C₉	—	3	6	no	—
139	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
140	C₃×C₉	C₃³×C₃₇	1	2	yes	—
141	C₃×C₉	C₃³×C₃₇	1	2	yes	—
142	C₃×C₉	C₃³×C₃₇	1	2	yes	—
143	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
144	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
145	C₃×C₉	C₃³×C₃₇	1	12	yes	—
146	C₃×C₉	C₃³×C₃₇	1	6	yes	—
147	C₃×C₉	C₃³×C₃₇	1	6	yes	—
148	C₃×C₉	C₃³×C₃₇	1	2	yes	—
149	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
150	C₃²	—	3	1	no	—
151	C₃²	—	3	2	no	—
152	C₃²	—	3	3	no	—
153	C₃²	—	3	3	no	—
154	C₃²	—	3	6	no	—
155	C₃²	C₃₇×#737	1	3	yes	—
156	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
157	C₉	—	3	1	no	—
158	C₉	—	3	2	no	—
159	C₉	—	3	1	no	—
160	C₉	—	3	1	no	—
161	C₉	—	3	2	no	—
162	C₉	—	3	2	no	—
163	C₉	—	3	6	no	—
164	C₉	—	3	6	no	—
165	C₉	—	3	12	no	—
166	C₉	C₃₇×#737	1	3	yes	—
167	C₉	C₃₇×#737	1	6	yes	—
168	C₃	—	3	3	no	—
169	C₃	—	9	3	no	—
170	C₃	C₃×C₃₇×#737	1	1	yes	characteristic
171	C₃	C₃×C₃₇×#737	1	1	yes	characteristic
172	C₃	C₃×C₃₇×#737	1	2	yes	—

	subgroup	G/H
center	C₃×C₉×C₃₇	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
socle	C₃²×C₃₇	C₃⁴
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×#6899	C₃	semidirect
2. C₃₇×#6898	C₃	semidirect
3. C₃₇×#6897	C₃	semidirect
4. C₃₇×#6896	C₃	semidirect
5. C₃₇×#6895	C₃	semidirect
6. C₃₇×#6894	C₃	semidirect
9. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
10. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
12. C₃₇×#6887	C₃	semidirect
13. C₃×C₉²×C₃₇	C₃	semidirect
63. #10624877	C₃₇	direct (N × H)
82. #6899	C₃×C₃₇	semidirect
83. #6898	C₃×C₃₇	semidirect
84. #6897	C₃×C₃₇	semidirect
85. #6896	C₃×C₃₇	semidirect
86. #6895	C₃×C₃₇	semidirect
87. #6894	C₃×C₃₇	semidirect
90. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
91. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
93. #6887	C₃×C₃₇	semidirect
94. C₃×C₉²	C₃×C₃₇	semidirect
129. C₃₇	#10624877	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#6899	3
C₃₇×#6898	3
C₃₇×#6897	3
C₃₇×#6896	3
C₃₇×#6895	3
C₃₇×#6894	3
C₃₇×#6893	3
C₃₇×C₉⋊H(3,𝔽₃)	3
C₉×3₋¹⁺²×C₃₇	3
C₉×H(3,𝔽₃)×C₃₇	3
C₃₇×#6887	3
C₃×C₉²×C₃₇	3
#10624877	37

Subgroups — C37×#10624877 #41004666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624877 #41004666