Subgroups — C₂×M₃₂×D₃₇ #18007690

Order 4736.

400 conjugacy classes of subgroups.

The subgroup lattice has 176 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	1	yes	maximal, characteristic
2	C₂×#11924476	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11924475	C₂	1	2	yes	maximal, complemented
4	M₃₂×D₃₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₆×D₃₇	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₃₇	C₄	1	1	yes	characteristic
8	C₂²×#64436	C₄	1	1	yes	characteristic
9	C₂×C₈×D₃₇	C₄	1	4	yes	—
10	C₂×C₈×D₃₇	C₄	1	1	yes	characteristic
11	C₂×C₈×D₃₇	C₄	1	1	yes	characteristic
12	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₃₇	C₂²	1	4	yes	—
14	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
15	#11924476	C₂²	1	4	yes	complemented
16	#11924475	C₂²	1	8	yes	complemented
17	C₂×#10596925	C₂²	1	2	yes	complemented
18	C₁₆×D₃₇	C₂²	1	8	yes	complemented
19	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
20	M₃₂×C₃₇	C₂²	1	4	yes	complemented
21	C₂×C₁₆×C₃₇	C₂²	1	2	yes	complemented
22	C₂³×D₃₇	C₈	1	1	yes	characteristic
23	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₃₇	C₈	1	1	yes	characteristic
25	C₂×C₄×D₃₇	C₈	1	1	yes	characteristic
26	C₂×C₄×D₃₇	C₈	1	4	yes	—
27	C₂×C₄×D₃₇	C₈	1	1	yes	characteristic
28	C₂×C₄×D₃₇	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₃₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
33	C₂×C₈×C₃₇	C₂³	1	2	yes	—
34	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
35	C₂×#64436	C₂×C₄	1	1	yes	characteristic
36	C₂×#64436	C₂×C₄	1	4	yes	—
37	C₂×#64436	C₂×C₄	1	1	yes	characteristic
38	C₈×D₃₇	C₂×C₄	1	2	yes	—
39	C₈×D₃₇	C₂×C₄	1	4	yes	—
40	C₈×D₃₇	C₂×C₄	1	4	yes	—
41	C₈×D₃₇	C₂×C₄	1	2	yes	—
42	C₈×D₃₇	C₂³	1	2	yes	—
43	C₈×D₃₇	C₂³	1	2	yes	—
44	C₁₆×C₃₇	C₂³	1	4	yes	complemented
45	#10596925	C₂³	1	4	yes	complemented
46	C₂³×C₃₇	C₂×C₈	1	1	yes	characteristic
47	C₂²×D₃₇	—	2	4	no	—
48	C₂²×D₃₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×D₃₇	C₂×C₈	1	4	yes	—
50	C₂²×D₃₇	C₂×C₈	1	1	yes	characteristic
51	C₂×C₄×C₃₇	C₂×C₈	1	2	yes	—
52	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
53	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×Dic₃₇	C₂×C₈	1	4	yes	—
56	C₂×Dic₃₇	C₂×C₈	1	1	yes	characteristic
57	C₂×Dic₃₇	C₂×C₈	1	1	yes	characteristic
58	C₄×D₃₇	C₂×C₈	1	4	yes	—
59	C₄×D₃₇	C₂×C₈	1	2	yes	—
60	C₄×D₃₇	C₂×C₈	1	2	yes	—
61	C₄×D₃₇	C₂×C₈	1	4	yes	—
62	C₄×D₃₇	C₂²×C₄	1	2	yes	—
63	C₄×D₃₇	C₂²×C₄	1	2	yes	—
64	C₈×C₃₇	C₂²×C₄	1	2	yes	—
65	C₈×C₃₇	C₂²×C₄	1	1	yes	characteristic
66	C₈×C₃₇	C₂⁴	1	1	yes	characteristic
67	#64436	C₂²×C₄	1	2	yes	—
68	#64436	C₂²×C₄	1	2	yes	—
69	C₂²×C₃₇	—	2	2	no	—
70	C₂²×C₃₇	C₂²×C₈	1	2	yes	—
71	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
72	C₂×D₃₇	—	2	2	no	—
73	C₂×D₃₇	—	2	8	no	—
74	C₂×D₃₇	M₃₂	1	4	yes	direct product factor
75	C₂×D₃₇	C₂²×C₈	1	2	yes	—
76	C₂×D₃₇	C₂²×C₈	1	2	yes	—
77	C₄×C₃₇	C₂²×C₈	1	2	yes	—
78	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
79	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
80	Dic₃₇	C₂²×C₈	1	2	yes	—
81	Dic₃₇	C₂²×C₈	1	2	yes	—
82	C₂²×M₃₂	—	37	1	no	maximal, Sylow
83	C₂×C₃₇	—	2	2	no	—
84	C₂×C₃₇	C₂×M₃₂	1	2	yes	complemented
85	C₂×C₃₇	C₂³×C₈	1	1	yes	characteristic
86	D₃₇	—	2	2	no	—
87	D₃₇	C₂×M₃₂	1	4	yes	direct product factor
88	C₂³×C₈	—	37	1	no	—
89	C₂×M₃₂	—	37	1	no	—
90	C₂×M₃₂	—	37	8	no	—
91	C₂×M₃₂	—	37	2	no	—
92	C₂×M₃₂	D₃₇	1	1	yes	characteristic, direct product factor
93	C₂²×C₁₆	—	37	2	no	—
94	C₃₇	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
95	C₂³×C₄	—	37	1	no	—
96	C₂²×C₈	—	37	1	no	—
97	C₂²×C₈	—	37	4	no	—
98	C₂²×C₈	—	37	1	no	—
99	C₂²×C₈	—	37	1	no	—
100	C₂²×C₈	—	37	4	no	—
101	C₂²×C₈	—	37	1	no	—
102	C₂²×C₈	—	37	1	no	—
103	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
104	M₃₂	—	37	4	no	—
105	M₃₂	—	37	8	no	—
106	M₃₂	C₂×D₃₇	1	4	yes	direct product factor
107	C₂×C₁₆	—	37	2	no	—
108	C₂×C₁₆	—	37	8	no	—
109	C₂×C₁₆	C₂×D₃₇	1	2	yes	complemented
110	C₂⁴	—	37	1	no	—
111	C₂²×C₄	—	37	1	no	—
112	C₂²×C₄	—	37	1	no	—
113	C₂²×C₄	—	37	4	no	—
114	C₂²×C₄	—	37	1	no	—
115	C₂²×C₄	—	37	4	no	—
116	C₂²×C₄	—	37	1	no	—
117	C₂²×C₄	—	37	1	no	—
118	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
119	C₂×C₈	—	37	4	no	—
120	C₂×C₈	—	37	1	no	—
121	C₂×C₈	—	37	1	no	—
122	C₂×C₈	—	37	4	no	—
123	C₂×C₈	—	37	4	no	—
124	C₂×C₈	—	37	2	no	—
125	C₂×C₈	—	37	2	no	—
126	C₂×C₈	—	37	2	no	—
127	C₂×C₈	—	37	2	no	—
128	C₂×C₈	C₄×D₃₇	1	1	yes	characteristic
129	C₂×C₈	C₄×D₃₇	1	2	yes	—
130	C₂×C₈	C₂²×D₃₇	1	2	yes	—
131	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
132	C₁₆	—	37	4	no	—
133	C₁₆	C₂²×D₃₇	1	4	yes	complemented
134	C₂³	—	37	1	no	—
135	C₂³	—	37	4	no	—
136	C₂³	—	37	1	no	—
137	C₂³	—	74	4	no	—
138	C₂³	C₈×D₃₇	1	1	yes	characteristic
139	C₂×C₄	—	37	4	no	—
140	C₂×C₄	—	37	1	no	—
141	C₂×C₄	—	37	1	no	—
142	C₂×C₄	—	37	4	no	—
143	C₂×C₄	—	37	2	no	—
144	C₂×C₄	—	37	4	no	—
145	C₂×C₄	—	37	2	no	—
146	C₂×C₄	—	37	2	no	—
147	C₂×C₄	—	37	2	no	—
148	C₂×C₄	C₈×D₃₇	1	2	yes	—
149	C₂×C₄	C₈×D₃₇	1	1	yes	characteristic
150	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
151	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
152	C₈	—	37	2	no	—
153	C₈	—	37	2	no	—
154	C₈	C₂×C₄×D₃₇	1	2	yes	—
155	C₈	C₂×C₄×D₃₇	1	1	yes	characteristic
156	C₈	C₂³×D₃₇	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	—	37	2	no	—
159	C₂²	—	37	4	no	—
160	C₂²	—	37	2	no	—
161	C₂²	—	74	8	no	—
162	C₂²	—	74	2	no	—
163	C₂²	C₂×C₈×D₃₇	1	2	yes	—
164	C₂²	C₂×C₈×D₃₇	1	1	yes	characteristic
165	C₄	—	37	2	no	—
166	C₄	—	37	2	no	—
167	C₄	C₂×C₈×D₃₇	1	2	yes	—
168	C₄	C₂×C₈×D₃₇	1	1	yes	characteristic
169	C₄	C₂²×C₄×D₃₇	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	—	37	4	no	—
172	C₂	—	74	2	no	—
173	C₂	M₃₂×D₃₇	1	2	yes	direct product factor
174	C₂	C₂²×C₈×D₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₃₇
commutator	C₂×C₃₇	C₂³×C₈
Frattini	C₈	C₂³×D₃₇
Fitting	C₂×M₃₂×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×C₈
prime radical	C₂³×D₃₇	C₈
cosocle	C₈×C₃₇	C₂⁴
hypercenter	C₂×M₃₂	D₃₇
hypocenter	C₃₇	C₂²×M₃₂
LCS	G ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#11924476	C₂	semidirect
3. C₂×#11924475	C₂	semidirect
4. M₃₂×D₃₇	C₂	direct (N × H)
5. C₂×C₁₆×D₃₇	C₂	semidirect
6. C₂×M₃₂×C₃₇	C₂	semidirect
15. #11924476	C₂²	semidirect
16. #11924475	C₂²	semidirect
17. C₂×#10596925	C₂²	semidirect
18. C₁₆×D₃₇	C₂²	semidirect
20. M₃₂×C₃₇	C₂²	semidirect
21. C₂×C₁₆×C₃₇	C₂²	semidirect
44. C₁₆×C₃₇	C₂³	semidirect
45. #10596925	C₂³	semidirect
74. C₂×D₃₇	M₃₂	direct (N × H)
84. C₂×C₃₇	C₂×M₃₂	semidirect
87. D₃₇	C₂×M₃₂	direct (N × H)
92. C₂×M₃₂	D₃₇	direct (N × H)
94. C₃₇	C₂²×M₃₂	semidirect
106. M₃₂	C₂×D₃₇	direct (N × H)
109. C₂×C₁₆	C₂×D₃₇	semidirect
133. C₁₆	C₂²×D₃₇	semidirect
173. C₂	M₃₂×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×#11924476	2
C₂×#11924475	2
M₃₂×D₃₇	2
C₂×C₁₆×D₃₇	2
C₂×M₃₂×C₃₇	2
C₂²×M₃₂	37

Subgroups — C2×M32×D37 #18007690

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₃₂×D₃₇ #18007690