Subgroups — C₃₇×#10624899 #41004688

Order 26973.

740 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃₇×#6899	C₃	1	9	yes	maximal, complemented
3	C₃₇×#6896	C₃	1	9	yes	maximal, complemented
4	C₃₇×#6894	C₃	1	6	yes	maximal, complemented
5	C₃₇×#6894	C₃	1	6	yes	maximal, complemented
6	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
7	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
8	C₉×3₋¹⁺²×C₃₇	C₃	1	3	yes	maximal, complemented
9	C₉×3₋¹⁺²×C₃₇	C₃	1	3	yes	maximal, complemented
10	C₃×C₃₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
11	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
13	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
14	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
15	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
16	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
17	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
18	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
19	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
20	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
21	C₃₇×C₉⋊C₉	C₃²	1	18	yes	complemented
22	C₃₇×C₉⋊C₉	C₃²	1	18	yes	complemented
23	C₃₇×C₉⋊C₉	C₃²	1	36	yes	complemented
24	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
26	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
27	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
28	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
29	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
30	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
31	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
32	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
33	C₃₇×C₃²⋊C₉	C₃²	1	3	yes	—
34	C₉²×C₃₇	C₃²	1	9	yes	complemented
35	C₃³×C₃₇	—	3	3	no	—
36	C₃³×C₃₇	—	3	3	no	—
37	C₃³×C₃₇	—	9	1	no	—
38	C₃³×C₃₇	—	9	1	no	—
39	C₃³×C₃₇	C₃³	1	1	yes	characteristic
40	C₃³×C₃₇	C₃³	1	1	yes	characteristic
41	C₃³×C₃₇	C₃³	1	1	yes	characteristic
42	C₃³×C₃₇	C₃³	1	1	yes	characteristic
43	3₋¹⁺²×C₃₇	—	3	6	no	—
44	3₋¹⁺²×C₃₇	—	3	9	no	—
45	3₋¹⁺²×C₃₇	—	3	9	no	—
46	3₋¹⁺²×C₃₇	—	3	9	no	—
47	3₋¹⁺²×C₃₇	—	9	3	no	—
48	3₋¹⁺²×C₃₇	3₋¹⁺²	1	9	yes	complemented
49	C₃×C₉×C₃₇	—	3	9	no	—
50	C₃×C₉×C₃₇	—	9	3	no	—
51	C₃×C₉×C₃₇	C₃³	1	3	yes	—
52	C₃×C₉×C₃₇	C₃³	1	9	yes	—
53	C₃×C₉×C₃₇	C₃³	1	9	yes	—
54	C₃×C₉×C₃₇	C₃³	1	6	yes	—
55	C₃×C₉×C₃₇	C₃³	1	6	yes	—
56	C₃×C₉×C₃₇	C₃³	1	3	yes	—
57	#10624899	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₃²×C₃₇	—	3	1	no	—
59	C₃²×C₃₇	—	3	1	no	—
60	C₃²×C₃₇	—	3	1	no	—
61	C₃²×C₃₇	—	3	1	no	—
62	C₃²×C₃₇	—	3	1	no	—
63	C₃²×C₃₇	—	3	1	no	—
64	C₃²×C₃₇	—	3	1	no	—
65	C₃²×C₃₇	—	3	1	no	—
66	C₃²×C₃₇	—	3	1	no	—
67	C₃²×C₃₇	—	3	1	no	—
68	C₃²×C₃₇	—	3	1	no	—
69	C₃²×C₃₇	—	3	1	no	—
70	C₃²×C₃₇	—	3	1	no	—
71	C₃²×C₃₇	—	3	1	no	—
72	C₃²×C₃₇	—	9	9	no	—
73	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
74	C₃²×C₃₇	#737	1	3	yes	—
75	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
76	C₉×C₃₇	—	3	6	no	—
77	C₉×C₃₇	—	3	9	no	—
78	C₉×C₃₇	—	3	3	no	—
79	C₉×C₃₇	—	3	6	no	—
80	C₉×C₃₇	—	3	9	no	—
81	C₉×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
82	C₃²×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
83	#6899	C₃×C₃₇	1	9	yes	complemented
84	#6896	C₃×C₃₇	1	9	yes	complemented
85	#6894	C₃×C₃₇	1	6	yes	complemented
86	#6894	C₃×C₃₇	1	6	yes	complemented
87	#6891	C₃×C₃₇	1	1	yes	characteristic
88	#6891	C₃×C₃₇	1	1	yes	characteristic
89	C₉×3₋¹⁺²	C₃×C₃₇	1	3	yes	complemented
90	C₉×3₋¹⁺²	C₃×C₃₇	1	3	yes	complemented
91	C₃×C₃²⋊C₉	C₃×C₃₇	1	1	yes	characteristic
92	C₃×C₃₇	—	3	3	no	—
93	C₃×C₃₇	—	3	3	no	—
94	C₃×C₃₇	—	9	1	no	—
95	C₃×C₃₇	—	9	1	no	—
96	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
97	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
98	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
99	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
100	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
101	C₃×3₋¹⁺²	—	3	9	no	—
102	C₃×3₋¹⁺²	C₃²×C₃₇	1	2	yes	—
103	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
104	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
105	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
106	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
107	C₃×3₋¹⁺²	C₃²×C₃₇	1	1	yes	characteristic
108	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
109	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
110	C₉⋊C₉	C₃²×C₃₇	1	18	yes	complemented
111	C₉⋊C₉	C₃²×C₃₇	1	18	yes	complemented
112	C₉⋊C₉	C₃²×C₃₇	1	36	yes	complemented
113	C₃²⋊C₉	C₃²×C₃₇	1	1	yes	characteristic
114	C₃²⋊C₉	C₃²×C₃₇	1	2	yes	—
115	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
116	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
117	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
118	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
119	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
120	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
121	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
122	C₃²⋊C₉	C₃²×C₃₇	1	3	yes	—
123	C₉²	C₃²×C₃₇	1	9	yes	complemented
124	C₃₇	#10624899	1	1	yes	characteristic, direct product factor, Sylow
125	C₃³	—	3	3	no	—
126	C₃³	—	3	3	no	—
127	C₃³	—	9	1	no	—
128	C₃³	—	9	1	no	—
129	C₃³	C₃³×C₃₇	1	1	yes	characteristic
130	C₃³	C₃³×C₃₇	1	1	yes	characteristic
131	C₃³	C₃³×C₃₇	1	1	yes	characteristic
132	C₃³	C₃³×C₃₇	1	1	yes	characteristic
133	3₋¹⁺²	—	3	6	no	—
134	3₋¹⁺²	—	3	9	no	—
135	3₋¹⁺²	—	3	9	no	—
136	3₋¹⁺²	—	3	9	no	—
137	3₋¹⁺²	—	9	3	no	—
138	3₋¹⁺²	3₋¹⁺²×C₃₇	1	9	yes	complemented
139	C₃×C₉	—	3	9	no	—
140	C₃×C₉	—	9	3	no	—
141	C₃×C₉	C₃³×C₃₇	1	3	yes	—
142	C₃×C₉	C₃³×C₃₇	1	9	yes	—
143	C₃×C₉	C₃³×C₃₇	1	9	yes	—
144	C₃×C₉	C₃³×C₃₇	1	6	yes	—
145	C₃×C₉	C₃³×C₃₇	1	6	yes	—
146	C₃×C₉	C₃³×C₃₇	1	3	yes	—
147	C₃²	—	3	1	no	—
148	C₃²	—	3	1	no	—
149	C₃²	—	3	1	no	—
150	C₃²	—	3	1	no	—
151	C₃²	—	3	1	no	—
152	C₃²	—	3	1	no	—
153	C₃²	—	3	1	no	—
154	C₃²	—	3	1	no	—
155	C₃²	—	3	1	no	—
156	C₃²	—	3	1	no	—
157	C₃²	—	3	1	no	—
158	C₃²	—	3	1	no	—
159	C₃²	—	3	1	no	—
160	C₃²	—	3	1	no	—
161	C₃²	—	9	9	no	—
162	C₃²	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
163	C₃²	C₃₇×#737	1	3	yes	—
164	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
165	C₉	—	3	6	no	—
166	C₉	—	3	9	no	—
167	C₉	—	3	3	no	—
168	C₉	—	3	6	no	—
169	C₉	—	3	9	no	—
170	C₉	C₃×3₋¹⁺²×C₃₇	1	9	yes	—
171	C₃	—	3	3	no	—
172	C₃	—	3	3	no	—
173	C₃	—	9	1	no	—
174	C₃	—	9	1	no	—
175	C₃	C₃²×3₋¹⁺²×C₃₇	1	1	yes	characteristic
176	C₃	C₃×C₃₇×#737	1	1	yes	characteristic
177	C₃	C₃₇×3₋¹⁺⁴	1	1	yes	characteristic
178	C₃	C₃₇×3₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#6899	C₃	semidirect
3. C₃₇×#6896	C₃	semidirect
4. C₃₇×#6894	C₃	semidirect
5. C₃₇×#6894	C₃	semidirect
8. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
9. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
21. C₃₇×C₉⋊C₉	C₃²	semidirect
22. C₃₇×C₉⋊C₉	C₃²	semidirect
23. C₃₇×C₉⋊C₉	C₃²	semidirect
34. C₉²×C₃₇	C₃²	semidirect
48. 3₋¹⁺²×C₃₇	3₋¹⁺²	semidirect
57. #10624899	C₃₇	direct (N × H)
83. #6899	C₃×C₃₇	semidirect
84. #6896	C₃×C₃₇	semidirect
85. #6894	C₃×C₃₇	semidirect
86. #6894	C₃×C₃₇	semidirect
89. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
90. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
110. C₉⋊C₉	C₃²×C₃₇	semidirect
111. C₉⋊C₉	C₃²×C₃₇	semidirect
112. C₉⋊C₉	C₃²×C₃₇	semidirect
123. C₉²	C₃²×C₃₇	semidirect
124. C₃₇	#10624899	direct (N × H)
138. 3₋¹⁺²	3₋¹⁺²×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃²×3₋¹⁺²×C₃₇	3
C₃₇×#6899	3
C₃₇×#6896	3
C₃₇×#6894	3
C₃₇×#6891	3
C₉×3₋¹⁺²×C₃₇	3
C₃×C₃₇×C₃²⋊C₉	3
#10624899	37

Subgroups — C37×#10624899 #41004688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624899 #41004688