Subgroups — C₂×C₄×C₈×D₃₇ #18002995

Order 4736.

760 conjugacy classes of subgroups.

The subgroup lattice has 150 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₃₇	C₂	1	2	yes	maximal
2	C₂×C₄²×D₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#64436	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₈×Dic₃₇	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×D₃₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₄×C₈×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₃₇	C₄	1	2	yes	—
8	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
9	C₂²×#64436	C₂²	1	2	yes	—
10	C₂×C₈×D₃₇	C₄	1	16	yes	direct product factor
11	C₂×C₈×D₃₇	C₂²	1	8	yes	—
12	C₂×C₄×Dic₃₇	C₄	1	2	yes	—
13	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
14	C₄²×D₃₇	C₄	1	4	yes	—
15	C₄²×D₃₇	C₂²	1	4	yes	—
16	C₄×#64436	C₂²	1	4	yes	complemented
17	C₈×Dic₃₇	C₂²	1	8	yes	complemented
18	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₃₇	C₂²	1	4	yes	complemented
21	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₃₇	C₈	1	16	yes	direct product factor
27	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₃₇	C₂×C₄	1	8	yes	—
29	C₂×C₄×D₃₇	C₂×C₄	1	8	yes	—
30	C₂×C₄×D₃₇	C₂³	1	2	yes	—
31	C₂×C₈×C₃₇	C₂×C₄	1	8	yes	complemented
32	C₂×C₈×C₃₇	C₂³	1	4	yes	—
33	C₄²×C₃₇	C₂×C₄	1	2	yes	—
34	C₄²×C₃₇	C₂³	1	2	yes	—
35	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
36	C₄×Dic₃₇	C₂×C₄	1	8	yes	—
37	C₄×Dic₃₇	C₂³	1	2	yes	—
38	C₂×#64436	C₂×C₄	1	8	yes	complemented
39	C₂×#64436	C₂³	1	4	yes	—
40	C₈×D₃₇	C₂×C₄	1	32	yes	direct product factor
41	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
42	C₂²×D₃₇	C₄²	1	4	yes	—
43	C₂²×D₃₇	C₂×C₈	1	8	yes	—
44	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₃₇	C₂×C₈	1	8	yes	complemented
46	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₃₇	C₄²	1	4	yes	—
51	C₂×Dic₃₇	C₂×C₈	1	8	yes	complemented
52	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
53	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
54	C₄×D₃₇	C₄²	1	8	yes	—
55	C₄×D₃₇	C₂×C₈	1	32	yes	direct product factor
56	C₄×D₃₇	C₂²×C₄	1	8	yes	—
57	C₈×C₃₇	C₂²×C₄	1	8	yes	complemented
58	#64436	C₂²×C₄	1	8	yes	complemented
59	C₂²×C₃₇	C₂×C₄²	1	2	yes	—
60	C₂²×C₃₇	C₂²×C₈	1	4	yes	—
61	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
62	C₂×D₃₇	C₄×C₈	1	16	yes	direct product factor
63	C₂×D₃₇	C₂×C₄²	1	4	yes	—
64	C₂×D₃₇	C₂²×C₈	1	8	yes	—
65	C₄×C₃₇	C₂×C₄²	1	2	yes	—
66	C₄×C₃₇	C₂²×C₈	1	8	yes	complemented
67	C₄×C₃₇	C₂³×C₄	1	2	yes	—
68	Dic₃₇	C₂×C₄²	1	4	yes	—
69	Dic₃₇	C₂²×C₈	1	8	yes	complemented
70	C₂²×C₄×C₈	—	37	1	no	maximal, Sylow
71	C₂×C₃₇	C₂×C₄×C₈	1	4	yes	complemented
72	C₂×C₃₇	C₂²×C₄²	1	1	yes	characteristic
73	C₂×C₃₇	C₂³×C₈	1	2	yes	—
74	D₃₇	C₂×C₄×C₈	1	8	yes	direct product factor
75	C₂³×C₈	—	37	2	no	—
76	C₂²×C₄²	—	37	1	no	—
77	C₂×C₄×C₈	—	37	2	no	—
78	C₂×C₄×C₈	—	37	1	no	—
79	C₂×C₄×C₈	—	37	8	no	—
80	C₂×C₄×C₈	D₃₇	1	1	yes	characteristic, direct product factor
81	C₃₇	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
82	C₂³×C₄	—	37	2	no	—
83	C₂³×C₄	—	37	1	no	—
84	C₂²×C₈	—	37	2	no	—
85	C₂²×C₈	—	37	16	no	—
86	C₂²×C₈	—	37	8	no	—
87	C₂²×C₈	C₂×D₃₇	1	2	yes	—
88	C₂×C₄²	—	37	2	no	—
89	C₂×C₄²	—	37	1	no	—
90	C₂×C₄²	—	37	4	no	—
91	C₂×C₄²	—	37	4	no	—
92	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
93	C₄×C₈	—	37	8	no	—
94	C₄×C₈	—	37	4	no	—
95	C₄×C₈	C₂×D₃₇	1	4	yes	direct product factor
96	C₂⁴	—	37	1	no	—
97	C₂²×C₄	—	37	1	no	—
98	C₂²×C₄	—	37	2	no	—
99	C₂²×C₄	—	37	16	no	—
100	C₂²×C₄	—	37	2	no	—
101	C₂²×C₄	—	37	8	no	—
102	C₂²×C₄	—	37	8	no	—
103	C₂²×C₄	—	37	2	no	—
104	C₂²×C₄	C₄×D₃₇	1	2	yes	—
105	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
106	C₂×C₈	—	37	8	no	—
107	C₂×C₈	—	37	4	no	—
108	C₂×C₈	—	37	32	no	—
109	C₂×C₈	C₄×D₃₇	1	8	yes	direct product factor
110	C₂×C₈	C₂²×D₃₇	1	4	yes	—
111	C₄²	—	37	2	no	—
112	C₄²	—	37	8	no	—
113	C₄²	—	37	2	no	—
114	C₄²	C₄×D₃₇	1	2	yes	—
115	C₄²	C₂²×D₃₇	1	2	yes	—
116	C₂³	—	37	4	no	—
117	C₂³	—	37	8	no	—
118	C₂³	—	37	2	no	—
119	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
120	C₂×C₄	—	37	4	no	—
121	C₂×C₄	—	37	8	no	—
122	C₂×C₄	—	37	2	no	—
123	C₂×C₄	—	37	4	no	—
124	C₂×C₄	—	37	8	no	—
125	C₂×C₄	—	37	32	no	—
126	C₂×C₄	—	37	8	no	—
127	C₂×C₄	C₈×D₃₇	1	8	yes	direct product factor
128	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
129	C₂×C₄	C₂×C₄×D₃₇	1	4	yes	—
130	C₂×C₄	C₂×C₄×D₃₇	1	4	yes	—
131	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
132	C₈	—	37	8	no	—
133	C₈	C₂×C₄×D₃₇	1	8	yes	direct product factor
134	C₂²	—	37	4	no	—
135	C₂²	—	37	16	no	—
136	C₂²	—	37	8	no	—
137	C₂²	C₄²×D₃₇	1	2	yes	—
138	C₂²	C₂×C₈×D₃₇	1	4	yes	—
139	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
140	C₄	—	37	4	no	—
141	C₄	—	37	8	no	—
142	C₄	C₄²×D₃₇	1	2	yes	—
143	C₄	C₂×C₈×D₃₇	1	8	yes	direct product factor
144	C₄	C₂²×C₄×D₃₇	1	2	yes	—
145	C₂	—	37	8	no	—
146	C₂	C₄×C₈×D₃₇	1	4	yes	direct product factor
147	C₂	C₂×C₄²×D₃₇	1	1	yes	characteristic
148	C₂	C₂²×C₈×D₃₇	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄×C₈	D₃₇
commutator, hypocenter	C₃₇	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂×C₄×C₈×C₃₇	C₂
socle	C₂³×C₃₇	C₂²×C₄
prime radical	C₂³×D₃₇	C₂×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
LCS	G ▷ C₃₇
UCS	C₁ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×#64436	C₂	semidirect
4. C₂×C₈×Dic₃₇	C₂	semidirect
5. C₄×C₈×D₃₇	C₂	direct (N × H)
6. C₂×C₄×C₈×C₃₇	C₂	semidirect
10. C₂×C₈×D₃₇	C₄	direct (N × H)
16. C₄×#64436	C₂²	semidirect
17. C₈×Dic₃₇	C₂²	semidirect
20. C₄×C₈×C₃₇	C₂²	semidirect
26. C₂×C₄×D₃₇	C₈	direct (N × H)
31. C₂×C₈×C₃₇	C₂×C₄	semidirect
38. C₂×#64436	C₂×C₄	semidirect
40. C₈×D₃₇	C₂×C₄	direct (N × H)
45. C₂×C₄×C₃₇	C₂×C₈	semidirect
51. C₂×Dic₃₇	C₂×C₈	semidirect
55. C₄×D₃₇	C₂×C₈	direct (N × H)
57. C₈×C₃₇	C₂²×C₄	semidirect
58. #64436	C₂²×C₄	semidirect
62. C₂×D₃₇	C₄×C₈	direct (N × H)
66. C₄×C₃₇	C₂²×C₈	semidirect
69. Dic₃₇	C₂²×C₈	semidirect
71. C₂×C₃₇	C₂×C₄×C₈	semidirect
74. D₃₇	C₂×C₄×C₈	direct (N × H)
80. C₂×C₄×C₈	D₃₇	direct (N × H)
81. C₃₇	C₂²×C₄×C₈	semidirect
95. C₄×C₈	C₂×D₃₇	direct (N × H)
109. C₂×C₈	C₄×D₃₇	direct (N × H)
127. C₂×C₄	C₈×D₃₇	direct (N × H)
133. C₈	C₂×C₄×D₃₇	direct (N × H)
143. C₄	C₂×C₈×D₃₇	direct (N × H)
146. C₂	C₄×C₈×D₃₇	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×C₈×D₃₇	2
C₂×C₄²×D₃₇	2
C₂×C₄×#64436	2
C₂×C₈×Dic₃₇	2
C₄×C₈×D₃₇	2
C₂×C₄×C₈×C₃₇	2
C₂²×C₄×C₈	37

Subgroups — C2×C4×C8×D37 #18002995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₈×D₃₇ #18002995