Subgroups — C₂²×C₃₇×#504 #22683875

Order 9472.

2118 conjugacy classes of subgroups.

The subgroup lattice has 144 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#131×C₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₇×#504	C₂	1	24	yes	maximal, direct product factor
3	C₂²×M₃₂×C₃₇	C₂	1	3	yes	maximal, complemented
4	C₂³×C₁₆×C₃₇	C₂	1	3	yes	maximal, complemented
5	C₂²×#41×C₃₇	C₄	1	1	yes	characteristic
6	C₂×#131×C₃₇	C₄	1	12	yes	—
7	C₂×#131×C₃₇	C₂²	1	12	yes	—
8	C₂²×M₁₆×C₃₇	C₄	1	3	yes	—
9	C₂³×C₈×C₃₇	C₂²	1	3	yes	—
10	C₃₇×#504	C₂²	1	64	yes	direct product factor
11	C₂×M₃₂×C₃₇	C₂²	1	36	yes	complemented
12	C₂²×C₁₆×C₃₇	C₂²	1	3	yes	—
13	C₂²×C₁₆×C₃₇	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₃₇	C₂²	1	36	yes	complemented
15	C₂×#41×C₃₇	C₈	1	12	yes	—
16	C₂×#41×C₃₇	C₂×C₄	1	12	yes	—
17	C₂²×Q₈×C₃₇	C₈	1	1	yes	characteristic
18	C₂²×D₄×C₃₇	C₈	1	3	yes	—
19	C₂³×C₄×C₃₇	C₂×C₄	1	3	yes	—
20	#131×C₃₇	C₂×C₄	1	48	yes	—
21	#131×C₃₇	C₂³	1	16	yes	—
22	C₂×M₁₆×C₃₇	C₂×C₄	1	36	yes	—
23	C₂²×C₈×C₃₇	C₂×C₄	1	3	yes	—
24	C₂²×C₈×C₃₇	C₂×C₄	1	18	yes	—
25	C₂²×C₈×C₃₇	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₃₇	C₂³	1	18	yes	—
27	M₃₂×C₃₇	C₂³	1	48	yes	complemented
28	C₂×C₁₆×C₃₇	C₂³	1	18	yes	—
29	C₂×C₁₆×C₃₇	C₂³	1	6	yes	—
30	C₂×C₁₆×C₃₇	C₂³	1	48	yes	complemented
31	C₂⁴×C₃₇	C₂×C₈	1	3	yes	—
32	#41×C₃₇	C₂×C₈	1	48	yes	—
33	#41×C₃₇	C₂²×C₄	1	16	yes	—
34	C₂×Q₈×C₃₇	C₂×C₈	1	12	yes	—
35	C₂×D₄×C₃₇	C₂×C₈	1	36	yes	—
36	C₂²×C₄×C₃₇	C₂×C₈	1	3	yes	—
37	C₂²×C₄×C₃₇	C₂×C₈	1	18	yes	—
38	C₂²×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₇	C₂²×C₄	1	18	yes	—
40	M₁₆×C₃₇	C₂²×C₄	1	48	yes	—
41	C₂×C₈×C₃₇	C₂²×C₄	1	3	yes	—
42	C₂×C₈×C₃₇	C₂²×C₄	1	18	yes	—
43	C₂×C₈×C₃₇	C₂²×C₄	1	36	yes	—
44	C₂×C₈×C₃₇	C₂⁴	1	3	yes	—
45	C₂×C₈×C₃₇	C₂⁴	1	12	yes	—
46	C₁₆×C₃₇	C₂⁴	1	4	yes	—
47	C₁₆×C₃₇	C₂⁴	1	12	yes	—
48	C₂³×C₃₇	—	2	12	no	—
49	C₂³×C₃₇	C₂²×C₈	1	18	yes	—
50	C₂³×C₃₇	C₂²×C₈	1	1	yes	characteristic
51	Q₈×C₃₇	C₂²×C₈	1	16	yes	—
52	D₄×C₃₇	C₂²×C₈	1	48	yes	—
53	C₂×C₄×C₃₇	C₂²×C₈	1	3	yes	—
54	C₂×C₄×C₃₇	C₂²×C₈	1	18	yes	—
55	C₂×C₄×C₃₇	C₂²×C₈	1	36	yes	—
56	C₂×C₄×C₃₇	C₂³×C₄	1	3	yes	—
57	C₂×C₄×C₃₇	C₂³×C₄	1	12	yes	—
58	C₈×C₃₇	C₂³×C₄	1	3	yes	—
59	C₈×C₃₇	C₂³×C₄	1	12	yes	—
60	C₈×C₃₇	C₂⁵	1	1	yes	characteristic
61	C₂²×#504	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂²×C₃₇	—	2	36	no	—
63	C₂²×C₃₇	#504	1	4	yes	direct product factor
64	C₂²×C₃₇	C₂³×C₈	1	12	yes	—
65	C₂²×C₃₇	C₂³×C₈	1	3	yes	—
66	C₄×C₃₇	C₂³×C₈	1	3	yes	—
67	C₄×C₃₇	C₂³×C₈	1	12	yes	—
68	C₄×C₃₇	C₂⁴×C₄	1	1	yes	characteristic
69	C₂²×#131	C₂×C₃₇	1	1	yes	characteristic
70	C₂×#504	C₂×C₃₇	1	24	yes	direct product factor
71	C₂²×M₃₂	C₂×C₃₇	1	3	yes	complemented
72	C₂³×C₁₆	C₂×C₃₇	1	3	yes	complemented
73	C₂×C₃₇	—	2	12	no	—
74	C₂×C₃₇	C₂×#504	1	6	yes	direct product factor
75	C₂×C₃₇	C₂⁴×C₈	1	1	yes	characteristic
76	C₂²×#41	C₄×C₃₇	1	1	yes	characteristic
77	C₂×#131	C₄×C₃₇	1	12	yes	—
78	C₂×#131	C₂²×C₃₇	1	12	yes	—
79	C₂²×M₁₆	C₄×C₃₇	1	3	yes	—
80	C₂³×C₈	C₂²×C₃₇	1	3	yes	—
81	#504	C₂²×C₃₇	1	64	yes	direct product factor
82	C₂×M₃₂	C₂²×C₃₇	1	36	yes	complemented
83	C₂²×C₁₆	C₂²×C₃₇	1	1	yes	characteristic
84	C₂²×C₁₆	C₂²×C₃₇	1	3	yes	—
85	C₂²×C₁₆	C₂²×C₃₇	1	36	yes	complemented
86	C₃₇	C₂²×#504	1	1	yes	characteristic, direct product factor, Sylow
87	C₂×#41	C₈×C₃₇	1	12	yes	—
88	C₂×#41	C₂×C₄×C₃₇	1	12	yes	—
89	C₂²×Q₈	C₈×C₃₇	1	1	yes	characteristic
90	C₂²×D₄	C₈×C₃₇	1	3	yes	—
91	C₂³×C₄	C₂×C₄×C₃₇	1	3	yes	—
92	#131	C₂×C₄×C₃₇	1	48	yes	—
93	#131	C₂³×C₃₇	1	16	yes	—
94	C₂×M₁₆	C₂×C₄×C₃₇	1	36	yes	—
95	C₂²×C₈	C₂×C₄×C₃₇	1	3	yes	—
96	C₂²×C₈	C₂×C₄×C₃₇	1	18	yes	—
97	C₂²×C₈	C₂³×C₃₇	1	1	yes	characteristic
98	C₂²×C₈	C₂³×C₃₇	1	18	yes	—
99	M₃₂	C₂³×C₃₇	1	48	yes	complemented
100	C₂×C₁₆	C₂³×C₃₇	1	6	yes	—
101	C₂×C₁₆	C₂³×C₃₇	1	18	yes	—
102	C₂×C₁₆	C₂³×C₃₇	1	48	yes	complemented
103	C₂⁴	C₂×C₈×C₃₇	1	3	yes	—
104	#41	C₂×C₈×C₃₇	1	48	yes	—
105	#41	C₂²×C₄×C₃₇	1	16	yes	—
106	C₂×Q₈	C₂×C₈×C₃₇	1	12	yes	—
107	C₂×D₄	C₂×C₈×C₃₇	1	36	yes	—
108	C₂²×C₄	C₂×C₈×C₃₇	1	3	yes	—
109	C₂²×C₄	C₂×C₈×C₃₇	1	18	yes	—
110	C₂²×C₄	C₂²×C₄×C₃₇	1	1	yes	characteristic
111	C₂²×C₄	C₂²×C₄×C₃₇	1	18	yes	—
112	M₁₆	C₂²×C₄×C₃₇	1	48	yes	—
113	C₂×C₈	C₂²×C₄×C₃₇	1	3	yes	—
114	C₂×C₈	C₂²×C₄×C₃₇	1	18	yes	—
115	C₂×C₈	C₂²×C₄×C₃₇	1	36	yes	—
116	C₂×C₈	C₂⁴×C₃₇	1	3	yes	—
117	C₂×C₈	C₂⁴×C₃₇	1	12	yes	—
118	C₁₆	C₂⁴×C₃₇	1	4	yes	—
119	C₁₆	C₂⁴×C₃₇	1	12	yes	—
120	C₂³	—	2	12	no	—
121	C₂³	C₂²×C₈×C₃₇	1	18	yes	—
122	C₂³	C₂²×C₈×C₃₇	1	1	yes	characteristic
123	Q₈	C₂²×C₈×C₃₇	1	16	yes	—
124	D₄	C₂²×C₈×C₃₇	1	48	yes	—
125	C₂×C₄	C₂²×C₈×C₃₇	1	3	yes	—
126	C₂×C₄	C₂²×C₈×C₃₇	1	18	yes	—
127	C₂×C₄	C₂²×C₈×C₃₇	1	36	yes	—
128	C₂×C₄	C₂³×C₄×C₃₇	1	3	yes	—
129	C₂×C₄	C₂³×C₄×C₃₇	1	12	yes	—
130	C₈	C₂³×C₄×C₃₇	1	3	yes	—
131	C₈	C₂³×C₄×C₃₇	1	12	yes	—
132	C₈	C₂⁵×C₃₇	1	1	yes	characteristic
133	C₂²	—	2	36	no	—
134	C₂²	C₃₇×#504	1	4	yes	direct product factor
135	C₂²	C₂³×C₈×C₃₇	1	12	yes	—
136	C₂²	C₂³×C₈×C₃₇	1	3	yes	—
137	C₄	C₂³×C₈×C₃₇	1	3	yes	—
138	C₄	C₂³×C₈×C₃₇	1	12	yes	—
139	C₄	C₂⁴×C₄×C₃₇	1	1	yes	characteristic
140	C₂	—	2	12	no	—
141	C₂	C₂×C₃₇×#504	1	6	yes	direct product factor
142	C₂	C₂⁴×C₈×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₁₆×C₃₇	C₂²
commutator	C₂	C₂⁴×C₈×C₃₇
Frattini, cosocle	C₈	C₂⁵×C₃₇
socle	C₂³×C₃₇	C₂²×C₈
prime radical	C₂²×#41×C₃₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₇×#504	C₂	direct (N × H)
3. C₂²×M₃₂×C₃₇	C₂	semidirect
4. C₂³×C₁₆×C₃₇	C₂	semidirect
10. C₃₇×#504	C₂²	direct (N × H)
11. C₂×M₃₂×C₃₇	C₂²	semidirect
14. C₂²×C₁₆×C₃₇	C₂²	semidirect
27. M₃₂×C₃₇	C₂³	semidirect
30. C₂×C₁₆×C₃₇	C₂³	semidirect
61. C₂²×#504	C₃₇	direct (N × H)
63. C₂²×C₃₇	#504	direct (N × H)
70. C₂×#504	C₂×C₃₇	direct (N × H)
71. C₂²×M₃₂	C₂×C₃₇	semidirect
72. C₂³×C₁₆	C₂×C₃₇	semidirect
74. C₂×C₃₇	C₂×#504	direct (N × H)
81. #504	C₂²×C₃₇	direct (N × H)
82. C₂×M₃₂	C₂²×C₃₇	semidirect
85. C₂²×C₁₆	C₂²×C₃₇	semidirect
86. C₃₇	C₂²×#504	direct (N × H)
99. M₃₂	C₂³×C₃₇	semidirect
102. C₂×C₁₆	C₂³×C₃₇	semidirect
134. C₂²	C₃₇×#504	direct (N × H)
141. C₂	C₂×C₃₇×#504	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#131×C₃₇	2
C₂×C₃₇×#504	2
C₂²×M₃₂×C₃₇	2
C₂³×C₁₆×C₃₇	2
C₂²×#504	37

Subgroups — C22×C37×#504 #22683875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₇×#504 #22683875